pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Integrability of Two-Dimensional Flows

AbstractThis paper deals with the notion of integrability of flows or vector fields on two-dimensional manifolds. We consider the following two key points about first integrals: (1) They must be defined on the whole domain of definition of the flow or vector field, or defined on the complement of some special orbits of the system; (2) How are they computed? We prove that every local flow ϕ on a two-dimensional manifold M always has a continuous first integral on each component of M\Σ where Σ is the set of all separatrices of ϕ. We consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two-dimensional flow or vector field. Finally, we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus, of a non-resonant hyperbolic node, and of a Siegel hyperbolic saddle
The Inverted Pendulum: A Singularity Theory Approach

AbstractThe inverted pendulum with small parametric forcing is considered as an example of a wider class of parametrically forced Hamiltonian systems. The qualitative dynamics of the Poincaré map corresponding to the central periodic solution is studied via an approximating integrable normal form. At bifurcation points we construct local universal models in the appropriate symmetry context, using equivariant singularity theory. In this context, structural stability can be proved under generic conditions
Some Aspects of the Dynamic of V=H−H

AbstractWe consider the evolution of a surface Γ(t) according to the equation V=H−H, where V is the normal velocity of Γ(t), H is the sum of the two principal curvatures and H is the average of H on Γ(t). We study the case where Γ(t) intersects orthogonally a fixed surface Σ and discuss some aspects of the dynamics of Γ(t) under the assumption that the volume of the region enclosed between Γ(t) and Σ is small. We show that, in this case, if Γ(0) is near a hemisphere, Γ(t) keeps its almost hemispherical shape and slides on Σ crawling approximately along orbits of the tangential gradient ∇HΣ of the sum HΣ of the two principal curvatures of Σ. We also show that, if p∈Σ is a nondegenerate zero of ∇HΣ and a>0 is sufficiently small, then there is a surface of constant mean curvature which is near a hemisphere of radius a with center near p and intersects Σ orthogonally
Existence of a Global Attractor for Semilinear Dissipative Wave Equations on RN

AbstractWe consider the semilinear hyperbolic problem utt+δut−φ(x)Δu+λf(u)=η(x), x∈RN, t>0, with the initial conditions u(x, 0)=u0(x) and ut(x, 0)=u1(x) in the case where N⩾3 and (φ(x))−1:=g(x) lies in LN/2(RN). The energy space X0=D1, 2(RN)×L2g(RN) is introduced, to overcome the difficulties related with the noncompactness of operators which arise in unbounded domains. We derive various estimates to show local existence of solutions and existence of a global attractor in X0. The compactness of the embedding D1, 2(RN)⊂L2g(RN) is widely applied
Projective Differential Geometrical Structure of the Painlevé Equations

AbstractThe necessary and sufficient conditions that an equation of the form y″=f(x, y, y′) can be reduced to one of the Painlevé equations under a general point transformation are obtained. A procedure to check these conditions is found. The theory of invariants plays a leading role in this investigation. The reduction of all six Painlevé equations to the form y″=f(x, y) is obtained. The structure of equivalence classes is investigated for all the Painlevé equations. Following Cartan the space of the normal projective connection which is uniquely associated with any class of equivalent equations is considered. The specific structure of the spaces under investigation allows us to immerse them into RP3. Each immersion generates a triple of two-dimensional manifolds in RP3. The surfaces corresponding to all the Painlevé equations are presented
Riemann Problems for a Class of Coupled Hyperbolic Systems of Conservation Laws

AbstractWe study a class of coupled hyperbolic systems of conservation laws which contain the one-dimensional zero-pressure gas dynamics as a prototypical example. The Riemann problems are solved constructively. The Riemann solutions exactly include two kinds: delta-shock wave solutions and vacuum solutions. Under the generalized Rankine–Hugoniot relation and entropy condition, all of the existence, uniqueness, and stability of solutions to viscous perturbations are proved. Two typical examples are presented finally
Reversible Equivariant Linear Systems

AbstractIn this paper we classify the structure of linear reversible systems (vector fields) on Rn that are equivariant with respect to a linear representation of a compact Lie group H. We assume the time-reversal symmetry R also acts linearly and is such that the group G that is generated by H and R is again a compact Lie group. The main tool for the classification is the representation theory of compact Lie groups. The results are applied to some generic eigenvalue movements of linear reversible equivariant systems
Isochronicity for Several Classes of Hamiltonian Systems

AbstractIn this paper we study isochronous centers of analytic Hamiltonian systems giving special attention to the polynomial case. We first revisit the potential systems and we show the connection between isochronicity and involutions. We then study a more general system, namely the ones associated to Hamiltonians of the form H(x, y)=A(x)+B(x)y+C(x)y2. As an application we classify the cubic polynomial Hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial Hamiltonian isochronous centers
Lq-Decay Rates of Solutions for Benjamin–Bona–Mahony–Burgers Equations

AbstractThis paper studies the asymptotic behavior of solutions for the Benjamin–Bona–Mahony–Burgers equations ut−uxxt−αuxx+βux+upux=0, x∈R, t⩾0, with the initial data u|t=0=u0(x)→0 as x→±∞. Under the restrictions ∫∞−∞u0(x)dx=0 and ∫x−∞u0(y)dy∈W2p+1, 1, we obtain more results on the energy decay rates of the solutions in the forms that if p⩾1, then ‖∂jxu(t)‖L2=O(1)t−(2j+3)/4 for j=0, 1, …, 2p−1, and ‖∂jxu(t)‖Lq=O(1)t−((j+2)q−1)/(2q) for 2<q⩽∞ and j=0, 1, …, 2p−2; furthermore, if p⩾2, then ‖∂jxu(t)‖Lq=O(1)t−((j+2)q−1)/(2q) for 1⩽q<2, j=0, 1, …, 2p−3, and ‖∂jxut(t)‖Lq=O(1)t−((j+4)q−1)/(2q) for 2⩽q⩽∞, j=0, 1, …, 2p−3, which are optimal. The proof is dependent on the Fourier transform method, the energy method and the point wise method of the Green function
Shape Derivative in the Wave Equation with Dirichlet Boundary Conditions

AbstractThe aim of this paper is to give a full analysis of the the shape differentiability for the solution to the second order hyperbolic equation with Dirichlet boundary conditions. The implicit function theorem does not work to solve the problem of weak regularity of the data; nevertheless by a more technical approach we prove an analogous result. We will first prove the theorem under strong regularity of the right hand side, then using the hidden regularity we will prove the shape derivative continues to exist under weak condition of regularity. We end up with a second order shape derivative for this problem
Asymptotic Behavior of Parabolic Equations Arising from One-Dimensional Null-Recurrent Diffusions

AbstractThis work is concerned with the asymptotic behavior of homogeneous and nonhomogeneous parabolic equations arising from one-dimensional null-recurrent diffusion processes. First, we review the concepts of regularity, recurrence, and positive recurrence of Markov processes and recall the connections of these concepts with properties of solutions of the corresponding differential equations. Next, we examine the rate of convergence of the solutions of both homogeneous and nonhomogeneous parabolic equations when the initial function and the forcing function are integrable with respect to the invariant measure. Weaker and verifiable conditions compared with the existing work in the literature are obtained. Then the corresponding problems when the initial and forcing functions are not integrable with respect to the invariant measure are dealt with. Convergence under suitable scaling for the solutions of the parabolic equations is proved, and the explicit limit is obtained
Initial Boundary Value Problem for the System of Compressible Adiabatic Flow Through Porous Media

AbstractWe study the initial value problem for the system of compressible adiabatic flow through porous media in the one space dimension with fixed boundary condition. Under the restriction on the oscillations in the initial data, we establish the global existence and large time behavior for the classical solutions via the combination of characteristic analysis and energy estimate methods. The time-asymptotic states for the solution are found and the exponential convergence to the asymptotic states is proved. It is also shown that this problem can be approximated very well time-asymptotically by an initial boundary problem for the related diffusion equations obtained from the original hyperbolic system by Darcy's law, provided that the initial total mass is the same. The difference between these two solutions tends to zero exponentially fast as time goes to infinity in the sense of H1. The diffusive phenomena caused by damping mechanism with boundary effects are thus observed
Small-Amplitude Limit Cycles in Liénard Systems via Multiplicity

AbstractA recent paper of Christopher and Lloyd reduces the computation of the order of degeneracy of a weak focus for a polynomial Liénard system x=y−F(x), y=−g(x) to the computation of the multiplicity of a polynomial map. In this paper, we first take advantage of that approach to obtain new lower and upper bounds for the maximum order of degeneracy of the origin in terms of the degrees of F′ and g. Later on, we implement an algorithm to compute this maximum order for concrete values of these degrees. As far as we know we enlarge the set of degrees for which this maximum order was known. Finally we extend Christopher and Lloyd's result to analytic degenerate (or not) Liénard equations
Traveling Waves in Time Almost Periodic Structures Governed by Bistable Nonlinearities I. Stability and Uniqueness

AbstractThe current series of two papers is concerned with traveling wave solutions of time almost periodic nonlinear evolution equations. In the first of the series, we introduce a definition of almost periodic traveling wave solutions and establish the stability and uniqueness of such solutions to time almost periodic bistable nonlinear equations with one dimensional continuous space variable. We also investigate the stability and uniqueness of such solutions to time almost periodic bistable equations with one dimensional discrete space variable. In the second of the series, we will study the existence of almost periodic traveling wave solutions to time almost periodic bistable equations with continuous as well as discrete space variables
Traveling Waves in Time Almost Periodic Structures Governed by Bistable Nonlinearities II. Existence

AbstractThe current series of two papers is concerned with traveling wave solutions of time almost periodic nonlinear evolution equations. In the first of the series, we introduced the definition of almost periodic traveling wave solutions and established the stability and uniqueness of such solutions for time almost periodic bistable equations in one space dimension. In the second of the series, we prove the existence of such solutions for time almost periodic bistable equations with continuous space variables. We also consider the existence of traveling wave solutions and occurrence of propagation failure and spatial chaos in bistable equations with discrete space variables
Relaxation Limit for Piecewise Smooth Solutions to Systems of Conservation Laws

AbstractIn this paper we study the asymptotic equivalence of a general system of 1-D conservation laws and the corresponding relaxation model proposed by S. Jin and Z. Xin (1995, Comm. Pure Appl. Math.48, 235–277) in the limit of small relaxation rate. It is shown that if the relaxation system satisfies the subcharacteristic condition and the solution of the hyperbolic conservation laws is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions of the relaxation systems that converge to the solution of the original conservation laws (“equilibrium” system) at a rate of order ε as the rate of relaxation ε goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the nonlinear stability theory for viscous shock profiles
On the Symbiotic Lotka–Volterra Model with Diffusion and Transport Effects

AbstractIn this work we analyze the existence, stability, and multiplicity of coexistence states for a symbiotic Lotka–Volterra model with general diffusivities and transport effects. Global bifurcation theory, blowing up arguments for a priori bounds, singular perturbation results, singularity theory, and fixed point index in cones are among the techniques used to get our results and to explain the drastic change of behavior exhibited by the dynamics of the model between the cases of weak and strong mutualism between the species. Our methodology works out to treat much more general classes of symbiotic models
Exponential Decay Rates for a Full von Karman System of Dynamic Thermoelasticity

AbstractThe full von Karman system accounting for in plane acceleration and thermal effects is considered. The main results of the paper are: (i) the wellposedness of regular and weak (finite energy) solutions, (ii) the uniform decay rates obtained for the energy function in the presence of mechanical damping affecting only the solenoidal part of the velocity field representing the horizontal displacements of the plate. The obtained decay rates are uniform with respect to the parameter γ which represents the momenta of inertia and whose presence distinguishes the “parabolic-like” from the hyperbolic character of the dynamics
An Inverse Problem in Elastodynamics: Uniqueness of the Wave Speeds in the Interior

AbstractWe consider the unique determination of internal properties of a nonhomogeneous, isotropic elastic object from measurements made at the surface. The 3-dimensional object is modelled by solutions of the linear hyperbolic system of equations for elastodynamics, whose (leading) coefficients correspond to the internal properties of the object (its density and elasticity). We model surface measurements by the Dirichlet-to-Neumann map on a finite time interval. In a previous paper the author has shown that the density and elastic properties of the surface of the object are uniquely determined by the Dirichlet-to-Neumann map. Here we apply that result to conclude that certain properties of the interior of the object (the wave speeds) are also determined. We then observe that the elastodynamic polarization is determined outside the object by the Dirichlet-to-Neumann map. We conclude, in the case that the wave speeds are constant, that the polarization data do not determine the density in the interior. This problem and techniques used in its study are closely related to those in, for example, seismology and medical imaging. The techniques used here, though (from geometric optics, integral geometry, and microlocal analysis) lead to the solution of this fully three-dimensional problem
Equilibria with Many Nuclei for the Cahn–Hilliard Equation

AbstractLet f be a bistable nonlinearity such as u−u3. We consider multi-peaked stationary solutions to the Cahn–Hilliard equation ut=−Δ(ε2Δu+f(u)) in Ω, ∂u/∂n=∂Δu/∂n=0 on ∂Ω, with the average value of u in the metastable region. By “multi-peaked” we mean states which, as ε→0, tend to a constant value everywhere except for a finite number of points, which we call nuclei, in Ω, where the states tend to a different constant value. For any N we find such solutions with N peaks located at certain geometrically identified points. The proof is based on a dynamical systems viewpoint where the stationary solutions being sought are equilibrium points on a finite-dimensional invariant manifold of multi-peaked states. In addition to the existence of these solutions we also discuss their strong instability, justifying the name nuclei for the points of concentration
The Fucik Spectrum of General Sturm–Liouville Problems

AbstractConsider the boundary value problem−(pu′)′+qu=αu+−βu−,in(0, π),c00u(0)+c01u′(0)=0,c10u(π)+c11u′(π)=0, where u±=max{±u, 0}. The set of points (α, β)∈R2 for which this problem has a non-trivial solution is called the Fucik spectrum. When p≡1, q≡0, and either Dirichlet or periodic boundary conditions are imposed, the Fucik spectrum is known explicitly and consists of a countable collection of curves, with certain geometric properties. In this paper we show that similar properties hold for the general problem above, and also for a further generalization of the Fucik spectrum. We also discuss some spectral type properties of a positively homogeneous, “half-linear” problem and use these results to consider the solvability of a nonlinear problem with jumping nonlinearities
Lagrange Stability for Duffing-Type Equations

AbstractIn this paper the Duffing-type equation (d2x/dt2)+x2n+1+p2n(t)x2n+…+p1(t)x+p0(t)=0 is studied where the pj's are 1-periodic. It is shown that all solutions of this equation are bounded, provided that the pj(t) (0⩽j⩽n) are of bounded variation in [0, 1] and that the derivatives of pj(t) (n⩽j⩽2n) are Lipschitzian. It is also shown that there exist pj's being discontinuous everywhere such that all solutions of the equation are bounded. This implies that the continuity of pj's is not necessary for the boundedness of solutions of the equation
Evolution Semigroups, Translation Algebras, and Exponential Dichotomy of Cocycles

AbstractWe study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space Θ acting on a Banach space X. Our main tool is the associated evolution semigroup on C0(Θ; X). We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mapping/annular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c0(Z; X). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, we extend to our situation the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded, continuous, mild solutions to the inhomogeneous equation
Singularly Perturbed Boundary Value Problems for Systems of Tichonov's Type in Case of Exchange of Stabilities

AbstractWe consider a system of ordinary differential equations consisting of a singularly perturbed scalar differential equation of second order and a scalar differential equation of first or second order and study a Neumann–Cauchy or a Neumann–Dirichlet problem. We assume that the degenerate equation has two intersecting solutions such that the standard theory for systems of Tichonov's type cannot be applied. We introduce the notation of a degenerate stable solution. By means of the technique of ordered lower and upper solutions we prove the existence of a solution of our problems near the degenerate stable solution for sufficiently small ε and determine its asymptotic behavior in ε
On the Rates of Asymptotic Regularity for Some Unbounded Trajectories

AbstractLet T be a nonexpansive self-mapping of C where C is a nonempty closed convex subset of a Banach space E. We define Tλ for 0<λ<1 by Tλ=λT+(1−λ)I, where I is the identity operator on C, and denote xn=Tnλx0 where x0∈C. Then the related initial value problem is du/dt=−(I−T)u(t) with u(0)=x0∈C. The facts that ‖xn−Txn‖=O(1/n) as n→∞ and ‖u′(t)‖=O(1/t) as t→∞ are known when C is bounded. In this paper we look for a rate of asymptotic regularity for ‖u′(t)‖ if ‖u(t)‖=O(tα) where 0⩽α⩽1. We prove ‖u′(t)‖=O(t−β) as t→∞, where α+2β=1 and obtain an estimate on ‖u′(t)‖ with the universal constant Cα depending only on α
A Decomposition Theorem for Bounded Solutions and the Existence of Periodic Solutions of Periodic Differential Equations

AbstractWe prove a decomposition theorem for bounded uniformly continuous mild solutions to τ-periodic evolution equations of the form dx/dt=A(t)x+f(t) (*) with (in general, unbounded) τ-periodic A(·), τ-periodic f (·), and compact monodromy operator. By this theorem, every bounded uniformly continuous mild solution to (*) is a sum of a τ-periodic solution to (*) and a quasi periodic solution to its homogeneous equation. An analog of this for bounded solutions has been proved for abstract functional differential equations dx/dt=Ax+F(t)xt+f(t) with finite delay, where A generates a compact semigroup. As an immediate consequence, the existence of such a solution implies the existence of a τ-periodic solution to the inhomogeneous equation as well as a formula for its Fourier coefficients. This, even for the classical case of equations, improves considerably the previous results on the subject
Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

AbstractWe study reversible, SO(2)-invariant vector fields in R4 depending on a real parameter ε which possess for ε=0 a primary family of homoclinic orbits TαH0, α∈S1. Under a transversality condition with respect to ε the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values ε(n)k→0 for k→∞. The existence of cascades of 2l3m-pulse solutions follows by showing their transversality and then using induction. The method relies on the construction of an SO(2)-equivariant Poincaré map which, after factorization, is a composition of two involutions: A logarithmic twist map and a smooth global map. Reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem. As an application we treat the steady complex Ginzburg–Landau equation for which a primary homoclinic solution is known explicitly
Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems

AbstractIn this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type. In particular we obtain a complete theory for the 2×2 systems of genuinely nonlinear hyperbolic balance laws in 1≳D with a strong dissipative term. A different method, which combines the div–curl lemma with accretive operators, is then applied to study the limiting profiles in the case of nonhomogeneous isentropic gas dynamics. We also investigate relaxation results for some 2≳D cases, which include the Cattaneo model for nonlinear heat conduction and the compressible Euler flow. Moreover, convergence result is also obtained for general semilinear systems in 1≳D