diff --git "a/UtE2T4oBgHgl3EQftgij/content/tmp_files/load_file.txt" "b/UtE2T4oBgHgl3EQftgij/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/UtE2T4oBgHgl3EQftgij/content/tmp_files/load_file.txt" @@ -0,0 +1,747 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf,len=746 +page_content='Truncation of contact defects in reaction-diffusion systems Milen Ivanov∗, Bj¨orn Sandstede† January 11, 2023 Abstract Contact defects are time-periodic patterns in one space dimension that resemble spatially homoge- neous oscillations with an embedded defect in their core region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Keywords: spatial dynamics, nonlinear waves, reaction-diffusion systems, defects, Lin’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 1 Introduction Solutions of reaction-diffusion systems exhibit a wide variety of patterns, which makes them ubiquitous in modeling chemical, biological and ecological models [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For example, Turing patterns are potential mechanism for the emergence of stripes and spots on animal coats [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In chemistry, spontaneous pattern generation occurs in experiments of the Belousov–Zhabotinsky reaction [25] and in numerical simulations of model systems, in which both rigidly-rotating spiral waves and spiral waves exhibiting one or more line defects have been observed (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Our motivation comes from the line defects visible in the two center panels of Figure 1, which are caused by the destabilization of a rigidly-rotating spiral wave through a period-doubling bifurcation [22]: across the line defect, the phase of the spatio-temporal oscillations jumps by half a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Over long time scales, these line defects attract and annihilate each other in pairs unless only one line defect is left: Figure 1c illustrates this behavior through the two pairs of co-located line defects that are about to merge and disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' It is difficult to analyse these interaction properties between adjacent line defects in the full planar case, and we instead consider a simpler scenario in one space dimension that is more manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' This scenario consists of a one-dimensional system that admits a point defect that mediates between two spatially homogeneous oscillations, whose phases jump by half their period across the defect: see Figure 1d for an illustration of the resulting space-time plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' A slightly different way to think about this scenario is to restrict the planar pattern with the line defect to the small red rectangle shown in Figure 1b: the resulting image resembles Figure 1d, and its time dynamics is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In the one-dimensional case shown in Figure 1d, we could now concatenate several defects and attempt to understand their interaction properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' As a first step, we need to prove that we can actually truncate such ∗Institute of Mathematics and Informatics, Bulgarian Academy of Sciences †Division of Applied Mathematics, Brown University 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='04071v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='DS] 10 Jan 2023 (a) (b) (c) (d) Figure 1: Panel (a) shows a snapshot of a rigidly-rotating planar spiral wave, while panels (b) and (c) contain snapshots of spiral waves that exhibit line defects (images taken from [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Panel (d) shows a space-time plot (space horizontal and time vertical) of a one-dimensional defect (shown in cyan) that mediates between two spatially homogeneous oscillations that are out of phase by half the period, thus representing one-dimensional versions of the line defects shown in panels (b) and (c): note the resemblance of this pattern with the pattern inside the red box shown in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' a defect sitting at, say, x = 0 from the entire line to a large bounded interval (−L, L) supplemented by Neumann boundary conditions: once we know this, we can use reversibility or symmetry to create multiple copies by reflecting the truncated defect across x = L or x − L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' It is problem of establishing the existence of truncated defects on large intervals (−L, L) that we will focus on in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' A different motivation for the same problem comes from validating numerical computations that are also conducted on bounded intervals rather than on the whole line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1 Discussion of Defects In this section, we will review the necessary definitions and results from the theory of one-dimensional defect patterns [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Consider the reaction-diffusion system ut = Duxx + f(u), with x ∈ R, t ∈ R+, u(x, t) ∈ Rd, f ∈ C∞(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Rd), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) where D is a constant, positive-definite diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Informally, defects are time-periodic solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) that converge to spatio-temporally periodic structures as x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' More formally, assume that uwt(kx − ωt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' k) is a family of solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) whose profiles are periodic in the first argument and parame- terized by the wave number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' These solutions are called wave trains, and ω is referred to as their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Typically, ω is uniquely determined by k via the so-called nonlinear dispersion relation ω = ωnl(k), and uwt = uwt(kx − ωnl(k)t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' k) is therefore a one-parameter family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Amongst the four types of generic defects, namely sources, sinks, transmission defects, and contact defects discussed in [20], we focus here on contact defects, which are typically symmetric under reflections in x and resemble spatially homogeneous oscillations uwt(−ω(0)t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 0) as x → ±∞: they therefore reflect the pattern shown in Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' It will be useful to define ωnl(0) =: ωd and use the rescaled time variable τ := ωdt, so that the spatially homogeneous oscillations uwt(−τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 0) are 2π-periodic in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' With this notation, we can define contact defects more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' A function ud(x, τ) is called a contact defect with frequency ωd if it is 2π−periodic in τ, satisfies the reaction-diffusion system ωduτ = Duxx + f(u), where x ∈ R, τ ∈ R+, u(x, τ) ∈ Rd, f ∈ C∞(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Rd), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2) and, for some phase correcting functions θ±(x) with θ′ ±(x) → 0 as x → ±∞, obeys ud(x, τ) − uwt(−τ − θ±(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 0) → 0 as x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2 CThe convergence is assumed to be uniform in τ as x → ±∞ for the functions and their first derivatives with respect to x, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' It is worth noting that the phase-correcting functions θ±(x) will necessarily diverge logarithmically as x → ±∞, see [21, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1], which will pose difficulties later on as the phase of the defect does not converge to that of a single limiting wave train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Contact defects were shown to exist in the complex cubic-quintic Ginzburg–Landau equa- tion [20], and Smoller [23, Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='17] provided another existence result of contact defects as contact discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Our goal is to prove that contact defects persist under domain truncation to a sufficiently large interval [−L, L] with suitable boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2 Main Results Before stating our persistence result, we reformulate the existence problem in terms of a spatial dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We will state our hypotheses for the spatial dynamics problem rather than for the original reaction- diffusion system to keep the discussion concise and make it easier to connect the hypotheses more directly with the proofs in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Since our focus is on time-periodic solutions, we proceed as in [20] and rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) as a first-order system � ux vx � = � v −D−1(−ωuτ + f(u)) � =: G(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ω), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) with frequency ω near ωd, where the right-hand side is defined on the dense subspace Y := H1(S1)×H1/2(S1) of X := H1/2(S1) × L2(S1), and S1 := R/2πZ denotes the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In other words, we are exchanging the evolution in time for evolution in the space variable x, hence the term ”spatial dynamics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' This method was pioneered by Kirchg¨assner [11, 12] and Mielke [17], see also [3, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' While the initial-value problem for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) is ill-posed, many approaches from dynamical-systems theory, including invariant-manifold theory, continue to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) is posed on Sobolev spaces on S1, so there is a translation operator Sα : u(τ) → u(τ + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The corresponding translation operator on X will be denoted by Tα = Sα × Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Given B ⊂ X, we will denote by Γ(B) = {Tαp| α ∈ S1, p ∈ B} the union of the group orbits of the elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We will use the notation ud(x, τ) = (ud, ∂xud), and similar for uwt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The wave train uwt(−τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 0), together with its τ-translates, satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) when ω = ωd, so uwt is an equilibrium of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3), and thus Γ(uwt) is a circle of equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By definition, the contact defect ud(x, τ) converges to Γ(uwt) as x → ±∞, and it is therefore a homoclinic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The circle of equilibria has center, stable, and unstable manifolds by [20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1], and we use these to state our assumption that a contact defect exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume that ud(x, ·) ∈ W cs(Γ(uwt)) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) for ω = ωd = ωnl(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We assume that ud(x, ·) ̸∈ W ss(Γ(uwt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Our next hypothesis will be on the derivative Gp(uwt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ωd)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' One can readily check that (∂τuwt, 0) is an eigenvector and (0, ∂τuwt) a generalized eigenvector of the eigenvalue zero of Gp(uwt, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ωd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The eigenvector is generated by the Tα symmetry by the circle group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We assume that there are no other eigenvalues, counted with multiplicity, on the imaginary axis so that W c(Γ(uwt)) has dimension two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 1We will use the notation Gp to denote the derivative of G(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ω) with respect to (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3 Hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We assume that zero is an eigenvalue of algebraic multiplicity two of Gp(uwt, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ωd) and that all other elements of the spectrum are bounded away from the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Besides τ-symmetry, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) has symmetry with respect to its evolution variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Recall that a reverser of a dynamical system [16] is a linear bounded involution such that v(x) := Ru(−x) is a solution whenever u(x) is a solution: alternatively, we can require that the reverser anti-commutes with the right-hand side of the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) has two reversers, namely the operators R0 : (u, v)(τ) → (u, −v)(τ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4) Rπ : (u, v)(τ) → (u, −v)(τ + π) = R0Tπ(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hypothesis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume that the defect ud is reversible, so that ud(0) ∈ Fix R where R is either R0 or Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Hypothesis 1, we have ud(0) ∈ W cs(Γ(uwt)), and Hypothesis 3 implies that ud(0) ∈ W cu(Γ(uwt)), so that W cs and W cu intersect at the contact defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Tα invariance, the intersection of these manifolds contains all time translates of the contact defect, and, generically, we do not expect it to contain anything else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hypothesis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume that W cs(Γ(uwt)) and W cu(Γ(uwt)) intersect transversely at ud(0), that is, the sum of their tangent spaces at each point p ∈ Γ(uwt) is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Our notation for transversality will be W cs(Γ(uwt)) ⋔ W cu(Γ(uwt)) at ud(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' So far, our assumptions have been statements for the case ω = ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' When we change ω, the circle of equilibria will disappear, and we will assume this is due to a non-degenerate saddle-node bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hypothesis 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We assume that the circle of equilibria undergoes a non-degenerate saddle-node bifurcation as we vary ω ≈ ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [3, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='15)] show that, as we vary ω = ωd + ω∗, ω∗ ≈ 0, the reduced vector field on the two-dimensional center manifold W c(Γ(uwt)) is of the form α′(x) = y, y′(x) = − 2ω∗ λ′′ lin(0) + ω′′ nl(0) λ′′ lin(0)y2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=', where α represents the coordinate given by time translation, y is orthogonal to α, λlin is the linear dispersion relation, and ωnl is the nonlinear dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In particular, Hypothesis 5 holds when λ′′ lin(0), ω′′ nl(0) are both nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The following theorem is our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' (Existence and uniqueness of truncated contact defects) Assume that Hypotheses 1-5 hold, then there exist positive constants �L, C and a function ϵ∗ : [�L, ∞) → (0, ∞) so that the following is true for each L ≥ �L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' First, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) with ω = ωd + ϵ2 ∗(L) has an R-reversible solution uL(x) = (uL, u′ L) : [−L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' L] → X that is uniformly at most C/L2 away from Γ(ud(x)) and satisfies the boundary conditions uL(±L) ∈ Fix R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Furthermore, if uL and ˜uL are two such solutions, then there exists an α ∈ S1 such that TαuL(x) = ˜uL(x) for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Finally, the function ϵ∗(L) is C2 and satisfies the estimates ϵ∗(L) = 2 πL + O � 1 L2 � , ϵ′ ∗(L) = −2 πL2 + O � 1 L3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5) 4 We note that if R = R0, then the truncated contact defects uL(x, τ) extend to smooth 2L-periodic solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3), since R0-reversibility of uL(x, τ) together with uL(±L, τ) ∈ Fix R0 implies uL(L, τ) = uL(−L, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' This is not true if the contact defect ud(x, τ) is Rπ-reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4, we will need the following auxiliary result on passage times through non- degenerate saddle-node bifurcations, which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Consider the system y′(x) = ϵ2 + y2 + g(y, ϵ2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) with parameter ω = ϵ2 ≥ 0, where g is Cr for some r ≥ 4 in both arguments, and g(0, 0) = gy(0, 0) = gyy(0, 0) = gω(0, 0) = gyω(0, 0) = 0, then the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' There exist positive constants ϵ0, δ0 and a function T = T(ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) defined for ϵ ∈ (0, ϵ0] and δ ∈ [δ0/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2δ0] such that the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) with y(0) = −δ satisfies y(T(ϵ, δ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' There exists an L0 > 0 and a unique function ϵ∗(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) : (L0, ∞) × [δ0/2, δ0] → (0, ϵ0), such that whenever L ≥ L0, L = T(ϵ∗(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) for all L ≥ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For each fixed β ∈ [0, 1), the function ϵ∗ is C1+β in both arguments, and there is a C1,β function Q(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) such that ϵ∗(L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) = 2 πL + Q(L−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) = 2 πL + O(L−β−1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='7) dϵ∗ dL (L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) = −2 πL2 − Qz(L−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) L2 = −2 πL2 + O(L−β−2), where the constant in the big-O term may blow up as β → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' If, in addition, g(−y, ω) = g(y, ω), then Q(ϵ, δ) ∈ Cr, and the above estimates hold with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Analogous statements hold for the problem y(0) = 0, y(T(ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ)) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3 Related work Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4 can be viewed as a result on the existence of periodic orbits with large periods near a given homoclinic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Homoclinic bifurcations have been studied for many decades, and we refer to the survey [8] for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Most results are for the case where the underlying equilibria are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Homoclinic bifurcations for nonhyperbolic equilibria have been considered for generic fold bifurcations, and we refer to [8, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='10] for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The case where homoclinic orbits approach a circle of equilibria with a two- dimensional center manifold was investigated first in the finite-dimensional case, and in fact for arbitrary Galerkin approximations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3), by the first author in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The proof in [10] relies on the persistence of normally invariant manifolds for well-posed dynamical systems [7, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Since similar results are not known for the infinite-dimensional ill-posed spatial dynamics problem considered here, we instead utilize Lin’s method [15] to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5 provides expansions of the travel from y = −δ to y = 0 (and similarly from y = δ backwards in time to y = 0) in the unfolding of a non-degenerate saddle-node bifurcation at y = 0: our result shows that the travel times typically contain logarithmic terms log ϵ are therefore not differentiable in ϵ regardless 5 of how smooth the right-hand side is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In contrast, Fontich and Sardanyes [4] considered the travel time from y = −δ to y = δ for the unfolding of a possibly degenerate saddle-node bifurcations: for analytic vector fields, they used the residue theorem to prove that the resulting travel times are analytic in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' These two results are reconciled by noting that the logarithmic terms in the travel times from y = −δ to y = 0 and from y = 0 to y = δ cancel, yielding a smooth expression for the travel times from y = −δ to y = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We also note that we cannot assume analyticity since our results are needed for the vector field on a center manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Finally, we remark that Kuehn [13] showed that travel times may exhibit many different scaling laws when the right-hand side depends only continuously on ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In §2 we discuss the dynamics on the center manifold and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4 is proved in §3 using Lin’s method, and we will provide additional estimates on the truncated contact defect uL in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We end with a brief discussion in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2 Dynamics on the Center Manifold Our goal in this section is to analyze the equations of the slow dynamics of a local equivariant center manifold near the circle of equilibria Γ(uwt) using equivariant local coordinates (α, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Henceforth, we will frequently use α as a coordinate of a two-dimensional center manifold, that corresponds to the drift along the group action Tα, and we will denote the coordinate, perpendicular to α, by y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [3, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='15)] show that, as we vary ω = ωd + ω∗, ω∗ ≈ 0, the dynamics of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) on the two-dimensional center manifold is of the form α′(x) = y, y′(x) = − 2ω∗ λ′′ lin(0) + ω′′ nl(0) λ′′ lin(0)y2 + O(|y|3 + |yω∗| + ω2 ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In our situation, both reversers R = R0, Rπ act on the reversible local center manifold by R(α, y) = (α, −y), and the right-hand side of the y equation is therefore even in y for all sufficiently small ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hence, up to rescaling by constant factors, we may assume the dynamics on the center manifold is α′(x) = y, y′(x) = ω∗ + y2 + g(y, ω∗), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) where α ∈ S1, y ∈ [−2δ0, 2δ0] and ω∗ ∈ [−ϵ2 0, ϵ2 0] (here δ0, ϵ0 are sufficiently small positive constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We know g(−y, ω∗) = g(y, ω∗), so in particular g(y, ω∗) = O(y4 + ω2 ∗) and g contains no y, ω∗y, y3, ω∗y3 terms in its Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In order to choose the value of ω∗ in terms of the parameter L, we are going to need to need to study travel time in saddle-node bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We answer these questions in the next section, and we remark its results may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1 Passage Time Near Saddle Node Bifurcations The previous section shows that we need to study the dynamics of the saddle-node bifurcation y′(x) = ω∗ +y2 +g(y, ω∗), where g(y, ω∗) = O(y3 +ω2 ∗) is a C4 function and g has other properties to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Whenever there are no equilibria (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ω∗ = ϵ2 > 0), we want to answer the following questions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Given ω∗, δ, where δ ≫ |ω∗| > 0 are sufficiently small, in what time does the solution of a saddle-node bifurcation travel between y = 0 and y = δ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Given a sufficiently large travel time L > 1 and a sufficiently small δ > 0, can we find an ω∗, such that the solution of a saddle-node bifurcation travels between y = 0 and y = δ in time L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We answer the first question in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1 and use it to answer the second question in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5: The key result we need to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5 is Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1 below, where we compute the time of flight from 0 to δ in terms of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Consider the non-degenerate saddle-node bifurcation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) in the regime with no equilibria (ϵ > 0) and with the same assumptions on g as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5, then there exist numbers ϵ0, δ0 > 0 such that the following holds for all ϵ ∈ (0, ϵ0], δ ∈ [δ0/2, 2δ0]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' There is an unique function T+(ϵ, δ), such that, the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) with the initial condition y(0) = 0, satisfies y(T+(ϵ, δ)) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We call this function T+ the travel time between 0 and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' There exist functions η(ϵ) ∈ Cr([0, ϵ0]), ζ(ϵ, δ) ∈ Cr([0, ϵ0]×[δ0/2, 2δ0]), such that η(ϵ) = O(ϵ), ζ(ϵ, δ) = O(ϵ) and ϵT+(ϵ, δ) = η(ϵ) log ϵ + π 2 + ζ(ϵ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In particular, ϵT+(ϵ, δ) is continuous up to ϵ = 0, uniformly in δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' If, in addition, we assume g(−y, ϵ2) = g(y, ϵ2) for all y, ϵ, then the function η(ϵ) is identically zero, and then ϵT+(ϵ, δ) ∈ Cr([0, ϵ0] × [δ0/2, 2δ0]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The idea of the proof is to construct an appropriate normal form for a saddle-node bifurcation and then to use partial fractions to compute the travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We start with the computation of the normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By [9, Theorem 5 and Corollary 1], each saddle-node bifurcation has the normal form ˜y′(x) = (˜ω∗ + ˜y2)(1 + b(˜ω∗)˜y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2) where b = b(˜ω∗) is a Ck function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We are interested in the case, where there is no ˜y term, so we will do the substitution ˜y = z + ζ ˜ω∗b(˜ω∗), where ζ will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' This yields z′ = ˜ω∗ + ˜ω∗b(z + ζ ˜ω∗b) + (z + ζ ˜ω∗b)2 + (z + ζ ˜ω∗b)3b = ˜ω∗ + ζ ˜ω2 ∗b2 + ζ2˜ω2 ∗b2 + ζ3˜ω3 ∗b4 + z(˜ω∗b + 2ζ ˜ω∗b + 3ζ2˜ω2 ∗b3) + z2(1 + 3ζ ˜ω∗b2) + z3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, we will pick ζ ≈ −1/2, so that the z coefficient is zero, or 3ζ2˜ω∗b2 + 2ζ + 1 = 0, ζ = −1/2 − 3˜ω∗b2/8 + O(˜ω2 ∗b4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By the Inverse Function Theorem, we can rename the ˜ω∗ + ζ ˜ω2 ∗b2 + ζ2˜ω2 ∗b2 + ζ3˜ω3 ∗b4 as ω∗, 3ζ ˜ω∗b(˜ω∗) as a(ω∗) and b(˜ω∗) as b(ω∗), so that the saddle-node bifurcation equation takes the normal form z′ = ω∗ + z2(1 + a(ω∗)) + z3b(ω∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) Per our computation, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) is a normal form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6), so in fact there is a change of variables y = Ψ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ω∗), such that Ψ′(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ω∗) = 1, which converts (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Also, let ˜δ be such that Ψ(˜δ, ω∗) = δ (of course, ˜δ depends smoothly on ω∗, but we suppress this in our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=') The travel time of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) from 0 to δ will be the same as the travel time of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) from 0 to ˜δ, namely T+(ϵ, δ) = � ˜δ 0 1 dz/dxdz = � ˜δ 0 1 ϵ2 + z2(1 + a(ϵ2)) + z3b(ϵ2)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 7 The idea of the proof is to analyze the above integral via partial fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' One obstacle to this approach is the fact that ϵ2 is small and b might be zero, which obstructs the partial fraction decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' To remedy this issue, we multiply the integral by ϵ and then substitute z = ϵ/u: ϵT+(ϵ, δ) = � ∞ ϵ/˜δ ϵ ϵ2 + ϵ2 u2 (1 + a) ϵ3 u3 b 1 u2 du = � ∞ 1 u u3 + u(1 + a) + ϵbdu + � 1 ϵ/˜δ u u3 + u(1 + a) + ϵbdu =: I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By the Dominated Convergence Theorem ϵT+(ϵ, δ)|ϵ=0 = π/2 (here we use the assumption that a(0) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Again by the Dominated Convergence Theorem, the integral I1 is as smooth in ϵ as the functions a(ϵ2), b(ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For I2 we use partial fractions: let u1,2,3 be the roots of u3 + u(1 + a) + ϵb, where u1 = −ϵb + O(ϵ2b2), u2,3 = ±i + O(ϵb) and u2 = ¯u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Then, there exist complex numbers A1, A2, A3, such that u u3 + u(1 + a) + ϵb = A1 u − u1 + A2 u − u2 + A3 u − u3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We can find Aj, j = 1, 2, 3 by multiplying the above equation by u − uj and then substituting u = uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' This yields A1 = u1 (u1 − u2)(u1 − u3) = u1 d du(u3 + u(1 + a) + ϵb)|u=u1 = u1 3u2 1 + 1 + a, and similar for A2, A3, so that Aj = uj 3u2 j + 1 + a(ϵ2), j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4) We analyze the sum of A2/(u − u2) and A3/u − u3, where we use u2 = ¯u3, A2 = ¯A3: A2 u − u2 + A3 u − u3 = A2(u − u3) + A3(u − u2) (u − u2)(u − u3) = 2uℜ(A2) − 2ℜ(A2u3) (u − 2ℜu2u + (ℜu2)2 + (ℑu2)2 = 1 ℑu2 u−ℜu2 ℑu2 B(ϵ) + C(ϵ) ( u−ℜu2 ℑu2 )2 + 1 , where B, C are Cr functions of ϵ, which can be computed explicitly from u2, u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, integrating from ϵ/˜δ to 1 yields � 1 ϵ/˜δ A2 u − u2 + A3 u − u3 du = � 1 ϵ/˜δ 1 ℑu2 u−ℜu2 ℑu2 B(ϵ) + C(ϵ) ( u−ℜu2 ℑu2 )2 + 1 du = 1 2B(ϵ) log ��u − ℜu2 ℑu2 �2 + 1 � + C(ϵ) arctan �u − ℜu2 ℑu2 � ����� 1 ϵ/˜δ , which are Cr−smooth in ϵ, δ up to ϵ = 0 (note that ℑu2 = 1 + O(ϵ), so the denominators do not blow up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, the smoothness properties of I2 are determined by � A1/(u − u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In the case when b(ϵ2) ≡ 0, u1(ϵ) = 0, so A1(ϵ) = 0 and this term vanishes: this proves the third part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' If b(ϵ2) ̸≡ 0, � 1 ϵ/˜δ A1 u − u1 du = u1 log(1 + u1) 3u2 1 + 1 + a(ϵ2) − u1 log(ϵ(1/˜δ − u1/ϵ)) 3u2 1 + 1 + a(ϵ2) = −u1 log ϵ + R(ϵ, ˜δ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5) where R is a Cr function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, we proved that η(ϵ) = −u1(ϵ) and this finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Under the same assumptions as Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1, the travel time from −δ to 0 is given by ϵT−(ϵ, δ) = −η(ϵ) log ϵ + π 2 + ζ−(ϵ, δ), where η(ϵ) is the same as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1 and ζ− satisfies the same smoothness assumptions as ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The travel time T+(ϵ, δ) + T−(ϵ, δ) from −δ to δ satisfies the following: ϵ[T+(ϵ, δ) + T−(ϵ, δ)] = π + ζ(ϵ, δ) + ζ−(ϵ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In particular the right-hand side is smooth in ϵ as ϵ → 0, even without the extra assumption about g being even in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The first part follows from the proof of the lemma, with the substitution z → −z, τ = −t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' This will have the net effect of reversing the sign of b, while keeping a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, for the partial fraction decomposition, we would be looking for the roots of u3 + u(1 + a) − ϵb, and the first root u− 1 will be −u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5) we would see +u1 log ϵ instead of −u1 log ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The second part of this corollary follows directly when we add the two travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Now we can prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' we solve for ϵ as a function of the total travel time T+(ϵ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The above corollary shows L = T+(ϵ, δ) = 1 ϵ �π 2 + η(ϵ) log ϵ + ζ(ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) , where η(ϵ) = O(ϵ), ζ(ϵ, δ) = O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Taking d/dϵ shows there is an ϵ0, such that for ϵ ∈ (0, ϵ0], the right-hand side is decreasing in ϵ, hence bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We expect ϵ ≈ π/2L, so we solve for π/2L: π 2L = ϵ 1 + π 2 η(ϵ)ϵ log ϵ + π 2 ϵζ(ϵ, δ) =: ϵ 1 + W(ϵ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We note that for all β ∈ [0, 1) W(ϵ, δ) is C1+β in both arguments, as η(ϵ)ϵ log ϵ ∈ C1+α([0, ϵ0]), and W would be Cr in both arguments if the η(ϵ)ϵ log ϵ term did not exist (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' for g even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' With z := π/(2L), we have z = ϵ 1 + W(ϵ, δ), so by the Implicit Function Theorem there exist constants ϵ0, κ1, κ2, such that, if ϵ ∈ (−2ϵ0, 2ϵ0), z ∈ (κ1, κ2), the equation has an unique solution ϵ∗(z, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By implicit differentiation ∂zϵ∗(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' δ0) = 1 ∂ϵ ϵ 1+W (ϵ,δ) = 1 1 + O(ϵβ ∗) = 1 + O(ϵβ ∗) = 1 + O(zβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In the specific case η(ϵ) = 0, we would obtain ϵ′ ∗(z) = 1 + O(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Integrating in z and recalling gives us ϵ∗ = z + O(zη+1), or ϵ∗(z) = z + O(z2) in the case η(ϵ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Substituting z = π/2L and defining Q to be the remaining term finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' It is worth commenting on the size of the solutions for large x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Below we derive some estimates for the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Fix β ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' When ϵ = 0, the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) has an asymptotic expansion y(x) = −1/x + O(x1+β) as x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Furthermore, if gyyy(0, 0) = 0, the expansion is y(x) = −1/x + O(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 9 (uwt,0) (0,∂τuwt) ud(x) ∂τud(x) (uwt,0) ∂xud(x) ud(0) (∂τuwt,0) Fix R Figure 2: The contact defect ud(x), in blue, converging to the circle of equilibria Γ((uwt, 0)) for x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For the sake of clarity, the analogous behavior as x → −∞ is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We can assume y(0) < 0, so that y(x) → 0 as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The proof in the other case is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume that δ is so small that when |y| ≤ δ, |g(y, 0)| ≤ C|y|3δ ≤ |y|2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Then y′ ∈ [|y2|/2, 3|y|2/2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' since solutions of y′ = y2/2 and y′ = 3y2/2 are both O(1/x) as x → ∞ we see that the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) is O(1/x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Now use this estimate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) to get y′ = y2(1 + O(x−1)), and the remainder would be O(x−2) if gyyy(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We can solve this by separation of variables to obtain y(x) = 1 −x + O(log x) + O(1), where the O(log x) term would not be present if fyyy(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We can add 1/x to this equation and obtain y(x) + 1 x = O(log x) + O(1) x(−x + O(log x) + O(1)), so the right hand-side is O(x−(1+β)) for all β ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' When gyyy(0, 0) = 0 there would be no logarithmic term, so we would just obtain O(x−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3 Existence of Truncated Contact Defects The main goal of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4, namely that we can truncate a contact defect to a large, bounded interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The main geometric configuration in the case ω = ωd is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The proof formalizes the idea that, when we perturb ω, the circle of equilibria Γ(uwt) will disappear, but the invariant torus will persist and consist of the time translates of the truncated defects uL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' At the end of the section, we will explain in what sense the truncated contact defect is close to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1 Exponential Trichotomies We first discuss exponential trichotomies of the linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) about the contact defect, which we will use to construct the truncated contact defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Trichotomies allow us the decompose the underlying space into three complementary subspaces that contain, respectively, initial conditions of solutions that decay exponentially in forward time or backward time, or that grow only mildly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We note that Hypothesis 2 shows that the linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) will have a two-dimensional center space, so we cannot expect that exponential dichotomies exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The following theorem stating the existence of trichotomies was proved in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume Hypotheses 1-3, then the linearization � ux vx � = � v −D−1(−ωduτ + f ′(ud(x)))u � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) about ud(x) at ω = ωd has an exponential trichotomy on R, that is, there exist strongly continuous families {Φs(ξ, ζ)}ξ,ζ∈J,ξ≥ζ, {Φc(ξ, ζ)}ξ,ζ∈J,ξ≥ζ, {Φu(ξ, ζ)}ξ,ζ∈J,ξ≤ζ of operators in L(X) with the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Φj(ξ, σ)Φj(σ, ζ) = Φj(ξ, ζ) for j = s, c, u and Φs(ξ, ξ) + Φc(ξ, ξ) + Φu(ξ, ξ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' There exist constants C, κ > 0, such that ∥Φs(ξ, ζ)∥ + ∥Φu(ζ, ξ)∥ ≤ C exp(−κ|ξ − ζ|) for all ξ, ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Given η ∈ (0, κ), there exists a constant C(η), such that ∥Φc(ξ, ζ)∥ ≤ C(η) exp(η|ξ − ζ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Φs(ξ, ζ)u0 and Φc(ξ, ζ)u0 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) for ξ > ζ and Φu(ξ, ζ) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) for ξ < ζ whenever u0 ∈ Y, ξ, η ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We need reversibility (Hypothesis 3) to ensure the exponential trichotomies are defined on R, otherwise we would only have exponential trichotomies on R±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The key feature of exponential trichotomies is roughness (see [2, §4] and [6, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='10]), that is, sensitivity to perturbations of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Loosely speaking, exponential trichotomies persist when we perturb the solution ud we linearize about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2 Description of the center and center-stable manifolds In this section we describe the center-stable and center-unstable manifolds near uwt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' It is shown in [20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1] that in a neighborhood of the wave train Γuwt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) exhibits center, center- stable manifolds W cs(ω∗), W c(ω∗), smooth in the parameter ω∗, and equivariant with respect to translation in τ (Tα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Hypothesis 2, the center space has dimension two, and is spanned by ∂τuwt, ∂xuwt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Find a sufficiently small δ0 > 0, such that we can parameterize W c by local coordinates α ∈ S1, y ∈ (−2δ0, 2δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In addition, W cs is parameterized by W c and strong-stable fibers Fss(p, bs, ω∗), p ∈ W c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We can find δ1 > 0, such that this fibration is valid for |b| < δ1, uniformly in p = (α, z) ∈ S1 × (−2δ0, 2δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' When ω∗ = 0, the contact defect ud(x) ∈ W cs by Hypothesis 1, so we can find L0 ≫ 1, such that ud(L0) belongs to a fiber of the point p = (0, −δ0) ∈ W c(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The fibration in the case ω∗ = 0 can be seen on Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3 Flow on the center manifold In this section we will describe the flow on W c(ω∗), given by the equivariant coordinates (α, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By [21], the dynamics are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The normal form for the saddle-node bifurcation in y was computed in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3), which we restate here: there are a coordinate transformation y = Ψ(z), with Ψ′(0) = 1, and functions a(ω∗), b(ω∗), such that the y equation transforms to z′ = ω∗ + z2(1 + a(ω∗)) + z3b(ω∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' However, by reversibility, the y equation is symmetric with respect to the transformation y → −y, so it must be the case that b(ω∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, we obtain z′ = ω∗ + z2(1 + a(ω∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2) We will now outline some estimates on the travel time of solutions of the equations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Namely, we look for solutions, which satisfy y(−L) = −δ0, y(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Let ω∗ = ϵ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The following estimates hold for the solution of our saddle-node bifurcation equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) with y(0) = 0, y(−L) = −δ1: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' If |x| < 1/3, y(x) = ϵ2x + O(ϵ4x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' If |x| < 1, y(−L + x) = −δ0 + δ2 0(1 + o(δ0))x + O(ϵ|x| + δ3 0x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We will use the normal form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2), y = Ψ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For the first inequality, let X = arg min{|z(x)| = ϵ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Then z(X) = ϵ2 ≤ � x 0 ϵ2 + 2ϵ4dy ≤ 3ϵ2x, so X ≥ 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Taylor’s theorem with integral remainder, z(x) = z(0) + xz′(0) + 1 2x2 � 1 0 z′′(sx)ds = ϵ2x + O(ϵ4x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, by y(x) = Ψ(z(x)) with Ψ′(0) = 1, y(x) = ϵ2x + O(ϵ4x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The second inequality can be proven in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Define ˜δ by z(−L) = −˜δ and do a Taylor expansion z(−L + x) = z(−L) + z′(−L)x + x2 2 � 1 0 z′′(sx)ds = −˜δ + (ϵ2 + ˜δ2(1 + a(ϵ2)))x + x2 2 O(2zz′) = −˜δ + ˜δ2x + O(ϵ2|x| + x2˜δ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By definition of a normal form transformation, Ψ(z) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' and in particular Ψ(−˜δ) = −δ0, so y(−L + x) = Ψ(z(−L + x)) = Ψ(−˜δ + ˜δ2x + O(ϵ2|x| + x2˜δ3)) = Ψ(−˜δ + ˜δ2x) + O(ϵ2|x| + x2˜δ3) = −δ0 + Ψ′(−˜δ)˜δ2x + O(˜δ4x2) + O(ϵ2|x| + x2˜δ3) = −δ0 + δ2 0(1 + o(δ0))x + O(ϵ2|x| + x2˜δ3), where in the last line we used −δ0 = Ψ(−˜δ) = −˜δ(1 + o(˜δ)) = −˜δ(1 + o(δ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 12 uwt WC WCS(uwt,0) defect ud ud(L0) ��� Fss(p,bs,0) Fss(p,0,0) (a) defect ud Pushforward FixL0,ε R FixR uwt FixR WCS(uwt,ε²) Truncated defect uL uL(0) uL(L) (b) Figure 3: Panel (a) shows the fibration of the center-stable manifold and the defect when ϵ = 0 (the time symmetry is factored out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Panel (b) displays the intersection of the pushforward FixL0,ϵ R and the center- stable manifold W cs(uwt, ϵ2 0), and the corresponding solution uL (the time symmetry is factored out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4 Geometry near Fix R Assume that uc(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) ∈ W c(ϵ2) and ∂xuc(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Then, Ruc x(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) ⊕ Ran(P u(uwt)) ⊕ Fix R = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Indeed, this holds for ϵ = 0, and the mapping Lϵ : Ruc x(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) ⊕ Ran(P u(uwt)) ⊕ Fix R → Y is bijective and bounded when ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, it is bijective and bounded uniformly for ϵ near 0, and, by the Open Mapping Theorem, its inverse is uniformly bounded in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5 Pushforward of Fix R The goal of this section is to compute the pushforward FixL0,ϵ R of Fix R along the defect ud from x = 0 to x = L0 for each L0 ≫ 1 and each ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Let u(x) = ud(x) + v(x), so vx = fu(ud(x))v + O(|v|2 + ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Let Φc,s,u d (x, y) be an exponential trichotomy of the linearized about ud equation, and, additionally, let R(Ran Φu d(0, 0)) = Ran Φs d(0, 0) (this is possible because the contact defect is reversible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For each L0 ≫ 1, ϵ ≪ 1, there exists a constant C1 > 0, such that the pushforward FixL0,ϵ R of Fix R exists and is parameterized by FixL0,ϵ R = {ud(L0) + au + O(e−ηL0|au| + |au|2 + ϵ2) : au ∈ Ran Φu d(L0, L0) with |au|, ϵ ≤ C1 L0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The idea of the proof is to use variation of parameters and the Banach Fixed Point theorem to construct the pushforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We start with deriving the fixed-point equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) with v = u − ud: vx = G(ud + v, ω∗) − G(ud, 0) = Gu(ud, 0)v + (G(ud + v, ω∗) − Gu(ud, 0)v − G(ud, 0)) =: Gu(ud, 0)v + H(v, ω∗), where H(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ω∗) := G(ud + v, ω∗) − Gu(ud, 0)v − G(ud, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Taylor’s theorem2, H(v, ϵ2) = O(|v|2 + ϵ2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) Hv(v, ϵ2) = O(|v| + ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 2It will be convenient to write ω∗ = ϵ2 13 Apply variation of parameters: v(x) = Φs d(x, 0)as + Φu d(x, L0)au + � x 0 Φcs d (x, y)H(v(y), ϵ2)dy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4) + � x L0 Φu d(x, y)H(v(y), ϵ2)dy, 0 ≤ x ≤ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) and the estimates for exponential trichotomies, the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4) is bounded by C0(|as| + |au| + L0(∥v∥2 + ϵ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' With C1, C2 small, we will choose |as|, |au|, ϵ ≤ C1/L0 and ∥v∥∞ ≤ C2/L0, so that ∥RHS∥ ≤ C0(2C1 + C2 2 + C2 1) 1 L0 ≤ C2 L0 and, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3): ∥DvRHS∥ ≤ 2C0L0∥v∥∞ ≤ 2C0C2 ≤ 1 2 whenever C1, C2 are chosen small, depending on C0, but not L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, by the Banach Fixed Point Theorem, there is an unique solution v in the ball of radius C2/L0 for |as|, |au|, ϵ ≤ C1/L0, and ∥v∥∞ ≤ C0(|as| + |au| + L0ϵ2) ≤ C2 L0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We impose the condition v(0) ∈ Fix R, so we can solve uniquely for as = O(e−κL|au| + |au|2 + ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hence, there exists an unique v(L0), subject to v(0) ∈ Fix R, and it is given by v(L0) = Φs d(L0, 0)as + au + � L0 0 Φcs d (x, y)H(v(y), ϵ2)dy = au + O(e−κL0|au| + |au|2 + ϵ2) Therefore, the following holds: FixL0,ϵ R = {ud(L0) + au + O(e−κL0|au| + |au|2 + ϵ2) : au ∈ Ran Φu d(L0, L0), with |au|, ϵ ≤ C1 L0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The pushforward and the center-stable manifoldd are displayed on Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The goal of the proof is to show that they intersect, and to adjust the parameter ϵ to ensure the resulting orbit travels from Fix R to Fix R in time L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In the above discussion we chose not to add a component in the ∂τ direction, so it would not be incorrect to say the above result is on the pushforward of Fix R/∂τud(0)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6 Description of the center-stable manifold As noted in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2, the defect ud is in the center-stable manifold and W cs is fibered over W c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In this section we will introduce notation for this fibration and we will express ud in said coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We parameterize the strong-stable fibers in W cs with base points p ∈ W c as Fss(p, bs, ϵ) = p + bs + O(δ0|bs|), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5) where bs ∈ Ran P s(uwt) and Fss(p, 0, ϵ) = p ∈ W c(ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In other words, p is the base point of the fibration and bs parameterizes the fiber Fss(p, bs, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 14 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For all ϵ ≥ 0, ϵ ≪ 1, and all L0 ≫ 1, there is a base point pd(L0, ϵ) ∈ W c and bs d(L0, ϵ) ∈ Ran P uuwt, so that FixL0,ϵ R ∩ W cs(ϵ2) = Fss(pd(L0, ϵ), bs d(L0, ϵ), ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The intersection FixL0,ϵ R ∩ W cs(ϵ2) is the point ud(L0)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The intersection is transverse when ϵ = 0: in this case, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3 yields FixL0,0 R = {ud(L0) + au + O(e−ηL0|au| + |au|2) : au ∈ Ran Φu d(L0, L0) with |au| ≤ C1 L0 }, so the tangent space Tud(L0) FixL0,0 R is Ran Φu d(L0, L0) + O(e−ηL0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The tangent space Tud(L0)W cs is Ran Φcs d (L0, L0) by [20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1], so indeed we have transversality when ϵ = 0 and L0 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Both FixL0,ϵ R and W cs(ϵ2) are C1 in ϵ, so transversality persists when we perturb ϵ > 0 by the stability theorem for transversality [5, §6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume ϵ > 0, ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' There is an unique number L(ϵ) and unique uc(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) ∈ W c, such that uc(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) = 0 and uc(−L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) = pd(L0, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Furthermore, ϵL(ϵ) ∈ C1 with ϵL(ϵ) = π/2 + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In the notation we omit the dependence of uc and L on L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5, there is always a base point pd(L0, ϵ) ∈ W c of ud(L0, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By the results in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3, the dynamics on the center manifold is determined by the y equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' when ϵ > 0, there is a finite travel time of pd(L0, ϵ) to Fix R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' to {y = 0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1 shows that, as long as L0 ≫ 1 is fixed, the travel time of pd(L0, ϵ) to Fix R is L(ϵ), such that ϵL(ϵ) = π/2 + O(ϵ) is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='7 Transversality of the pushforward We observe the following lemma holds: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' FixL0,ϵ R is transverse to Ruc x(−L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) � Ran P s(uwt) near ud(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The proof follows from the transversality outlined in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='8 Solving near the center-stable manifold The goal of this section is to apply variation of parameters to solve for orbits u near the center-stable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We start with some notation on fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We will use the coordinates pd(L0, ϵ), bs d(L0, ϵ) from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5 to define Fss R (bs, ϵ) := Fss(pd(L0, ϵ), bs + bs d(L0, ϵ), ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We will define uR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) to be the solution such that uR(−L(ϵ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) = Fss R (bs, ϵ), where L(ϵ) is given from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In particular, for ϵ > 0, uR(−L(ϵ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, 0) satisfies uR(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, 0) ∈ Fix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The variable bs will account for changes within the stable fiber (to be used later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We will look for solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3) of the type u(x) = uR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) + v(x), 3Note that, had we added the ∂τ direction in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='5, the intersection would have been a curve, instead of a point (but transversality would still hold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 15 so that vx = u(x)x − uR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) = G(uR + v, ϵ2) − G(uR, ϵ2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='e vx = Gu(uR(x, ϵ, bs), ϵ2)v + G(uR + v, ϵ2) − G(uR, ϵ2) − Gu(uR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs), ϵ2)v (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6) =: Gu(uR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs), ϵ2)v + HR(v, ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Taylor’s theorem, HR(v, ϵ2) = O(|v|2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='7) ∂vHR(v, ϵ2) = O(|v|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, we will use the fixed-point equation v(x) = Φu ϵ2,bs(x, 0)au 0 + � x −L+l0 Φcs ϵ2,bs(x, y)HR(v(y), ϵ2)dy + � x l1 Φu ϵ2,bs(x, y)HR(v(y), ϵ2)dy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='8) for x ∈ [−L + l0, l1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' A couple of remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' First, Φu ϵ2,bs, Φcs ϵ2,bs come from the exponential trichotomies when linearizing about uR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs), hence they depend on ϵ, bs, but by roughness of exponential dichotomies and trichotomies, the dependence is smooth and the bounds on exponential trichotomies can be chosen independently of ϵ, bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Second, the reason why the equation for vx has no component like Φcs(x, 0)acs 0 is that here we aim to account for the unstable direction only, and we will use bs to account for the stable direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The parameters l0, l1 are considered to be small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' they need not be zero, because we will need them to match in the ∂xuR direction near FixL0,ϵ R and near uwt respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' To apply the Contraction Mapping Theorem to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='8) we will introduce the exponentially weighed norm ∥v∥η := supx∈[−L+l0,l1] eη|x||v(x)|, where η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' As long as η < κ (κ is the exponent in the definition of exponential trichotomies), the following inequalities hold for the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='8): ∥RHS∥η ≤ C|au 0| + eη|x| �� x −L+l0 eϵ(x−y)e−2η|y|dy + � x l0 e−κ(y−x)e−2η|y|dy � C∥v∥2 η ≤ C|au 0| + eη|x| � e−2η|x| + e−2η(L−l0)eϵ(L−l0) + e−κ|x| + e−2η|x|� C∥v∥2 η ≤ C(|au 0| + ∥v∥2 η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Similarly, we can show that∥∂vRHS∥η ≤ 1/2, so Banach’s Fixed Point Theorem shows there is a unique solution v(x) = v∗(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, η, l0, l1, L, bs) and |v(x)| ≤ Ce−η|x||au 0| for all small au 0 ∈ Ran Φu R(−l1, −l1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In particular, we can estimate v(−L + l0) ≤ Ce−ηL|au 0|, where C is independent of l0, l1, so uR(−L + l0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) + v(−L + l0) = uR(−L + l0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' bs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) + O(e−ηL|au 0|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='9) Furthermore, our solution at l1 is uR(l1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) + v(l1) = uR(l1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) + au 0 + O(|au 0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='10) In the next section, we will need (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='10) to do the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='9 Matching Matching at x = l1: by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='10) we have uR(l1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) + au 0 + O(e−ηL + |au 0|2) ∈ Fix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Our matching at x = −L + l0 looks like this: uR(−L + l0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) + v(−L + l0) ∈ FixL0,ϵ R, and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='9), the condition at x = −L + l0 is φl0(Fss R (bs, ϵ)) + O(e−ηx|au 0|) = ud(L0) + au + O(e−ηL0|au| + |au|2 + ϵ2) , where φl0 denotes the local flow on the center-stable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Taylor’s theorem, this yields φl0(Fss R (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ)) + bs + O(δ0|bs|) + O(e−ηL|au 0|) = Fss R (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) + au + O(e−ηL0|au| + |au|2 + ϵ2) We will write the two matching conditions together an explain why they can be solved: uR(l1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ, bs) + au 0 + O(e−ηL + |au 0|2) ∈ Fix R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='11) φl0(Fss R (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ)) − Fss R (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' ϵ) + bs − au = O(δ0|bs|) + O(e−ηL|au 0|) + O(e−ηL0|au| + |au|2 + ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The left-hand side of the first equation is a perturbation of a linear isomorphism (l1, au 0) → Fix R by §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4: varying l1 corresponds to motion in the ∂xuR(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 0, 0) direction and varying au 0 allows one to traverse the remainder of Fix R, namely Fix R/(∂xuR(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 0, 0)R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The left-hand side of the second equation is a perturbation of a boundedly invertible linear isomorphism as well (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='7: varying l0 takes care of the motion in ∂x direction, and bs, au span the stable and unstable direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='11) is of the type Az = G(z, ϵ), G(z) = O(|z|2 + |ϵ|), where z = (l0, l1, au, au 0, bs) (here we are using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='2 to solve for l0, l1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The linear operator A is boundedly invertible by the arguments above, so such equations can be solved by the Banach Fixed Point Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Therefore, l0, l1, au, au 0, bs are all parameterized by ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Finally, the solution, which we constructed, travels from Fix R to Fix R in time L0 +L(ϵ)+l0(ϵ)+l1(ϵ)), hence, if we want that travel time to be a fixed constant L, we can use the Implicit Function Theorem to solve for ϵ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' To obtain the nonzero ∂ϵ derivative, one can check that ∂ϵlj(ϵ) = O(ϵ), j = 0, 1, and then use Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' This finishes the existence part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Finally, the uniqueness follows from the uniqueness in Banach’s Fixed Point Theorem and by Tα symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 4 Estimates on Truncated Contact Defects In this section, we estimate the distance between the truncated defect uL(x) to the original defect ud(x) on (−L, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' To do so, we can use the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We remark that Γ(ud) ∪ Γ(uwt), Γ(uL) are two invariant tori, which we proved are O(ϵ2) away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' These tori inherit the local coordinates (α, y) and (αL, yL) from the center manifold, and we can extend these to global coordinates on the tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume Hypotheses 1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Assume ud(x), uL(x) have local coordinates as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 17 The following estimates hold: |ω∗(L)| = |ϵ2 ∗(L)| = O(L−2), max |x|≤L |y(x) − yL(x)| = O(L−1), max |x|≤L |α(x) − αL(x)| = O(log L), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) max |x|≤L |α′(x) − α′ L(x)| = O(L−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The first estimate in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='4, where ϵ∗(L) = O(1/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' For any given L0, max |x|≤L0 |y(x) − yL(x)| + |α(x) − αL(x)| = O(ϵ2), so we need to compute the maxima only over the interval [L0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='3 we know y(x) = O(1/x), so for x ∈ [L0, L], y(x) = O(1/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Over the same interval |yL(x)| < |y(x)| = O(1/L), so the second inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1) α′ = y, α′ L = y′ L, so the fourth inequality is identical to the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' When we integrate it, we obtain max|x|≤L |α(x) − αL(x)| = O(log L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 5 Conclusion and Future Work The present work answers in the affirmative the question of existence and uniqueness of truncated contact defects in reaction-diffusion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' A forthcoming paper will address the issue of spectral stability of the constructed solutions under the assumption that the contact defect on the whole line is spectrally stable: it turns out that R0-reversible truncated contact defects are spectrally stable when periodic boundary con- ditions are used,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' while reversible truncated contact defects are always spectrally unstable under Neumann boundary conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' regardless of which of the two reversers R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='π is present,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' since the eigenvalue corre- sponding to the approximate eigenfunction ∂xuL becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' These results will, in particular, explain why these defect pairwise attract each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' We believe that nonlinear stability of contact defects and their truncation are difficult to establish due to the logarithmically diverging phase correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Already in the case of source defects (whose spectrum appears to be ”nicer” than the spectrum of contact defects [20, Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='1]), the proof of nonlinear stability is highly nontrivial [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Acknowledgments Ivanov was partially supported by the NSF under grant DMS-1714429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede was partially supported by the NSF under grant DMS-1714429 and DMS-2106566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Ivanov was partially supported by the Program of Enhancing Research Capacity in the Mathematical Sciences of the Bulgarian Ministry of Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Beck, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Nguyen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Zumbrun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Nonlinear stability of source defects in the complex ginzburg–landau equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Nonlinearity, 27(4):739, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Coppel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Dichotomies in stability theory, volume 629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Doelman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Scheel, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Schneider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The dynamics of modulated wave trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 18 [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Fontich and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sardanyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' General scaling law in the saddle–node bifurcation: a complex phase space study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Journal of Physics A: Mathematical and Theoretical, 41(1):015102, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Guillemin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Pollack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Differential topology, volume 370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Henry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Geometric theory of semilinear parabolic equations, volume 840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hirsch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Pugh, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Shub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Invariant manifolds, volume 583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Homburg and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Homoclinic and heteroclinic bifurcations in vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Broer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Takens, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Hasselblatt, editors, Handbook of Dynamical Systems III, pages 379–524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Elsevier, Amsterdam, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Ilyashenko and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Yakovenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Finitely smooth normal forms of local families of diffeomorphisms and vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Russian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Surveys, 46(1):1–43, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Ivanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Truncation of contact defects in reaction-diffusion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' PhD thesis, Brown University, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' URL https://repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='edu/studio/item/bdr:gut5c42r/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Kirchg¨assner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Nonlinearly resonant surface waves and homoclinic bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In Advances in applied mechanics, volume 26, pages 135–181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Elsevier, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Kirchg¨assner and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Scheurle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Bifurcation of non-periodic solutions of some semilinear equations in unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In Surveys Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Works Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 6 (Applications of Nonlinear Analysis in the Physical Sciences), pages 41–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Pitman Boston, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Kuehn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Journal of Physics A: Mathematical and Theoretical, 42(4):045101, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Kuehn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Multiple time scale dynamics, volume 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [15] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Using melnikov’s method to solve silnikov’s problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Edinburgh A, volume 116, pages 295–325, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Meiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Differential dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' SIAM, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Mielke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' A spatial center manifold approach to steady state bifurcations from spatially periodic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' In Dynamics of Nonlinear Waves in Dissipative Systems, volume 352 of Pitman Research Notes in Mathematics, chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Longman, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Murray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Mathematical biology: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' An introduction, volume 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Springer Science & Business Media, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Scheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' On the structure of spectra of modulated travelling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Mathematische Nachrichten, 232(1):39–93, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [20] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Scheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Defects in oscillatory media: toward a classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' SIAM Journal on Applied Dynamical Systems, 3(1):1–68, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Scheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Evans function and blow-up methods in critical eigenvalue problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Discrete and Continuous Dynamical Systems, 10(2004), 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Sandstede and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Scheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Period-doubling of spiral waves and defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' SIAM Journal on Applied Dynamical Systems, 6(2):494–547, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 19 [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Smoller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Shock waves and reaction—diffusion equations, volume 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Springer Science & Business Media, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Turing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' The chemical basis of morphogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Philosophical Transactions of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Series B, Biological Sciences, 237(641):37–72, 1952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Yoneyama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Fujii, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Maeda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Wavelength-doubled spiral fragments in photosensitive mono- layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' Journal of the American Chemical Society, 117(31):8188–8191, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE2T4oBgHgl3EQftgij/content/2301.04071v1.pdf'}