# Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

Authors: Chiara Zanini; Fabio Zanolin
Journal: Complexity
 (2018)

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## Abstract

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation-ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated.


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## Body

## 1. Introduction

In a recent paper [1] we have proved the existence of chaotic dynamics associated with a class of second order nonlinear ODEs of Schrödinger type of the form(1)-u′′-μu+gxu3=0,μ>0 for g(x) a positive periodic coefficient. The study of such equation was motivated by previous works on some models of Bose-Einstein condensates considered in [2–4].A more classical form of Nonlinear Schrödinger Equation (NLSE from now on) which has been studied by many authors is given by(2)iħ∂ψ∂t=-ħ22mΔψ+νxψ-ψp-1ψ,x∈RN,where ħ denotes the Planck constant, i is the imaginary unit, m is a positive constant, ν(x) is the potential, and p>1. The search ofstationary waves, namely, solutions of the form ψ(t,x)=exp(iλħ-1t)u(x), where λ∈R and u(x) is a real valued function, leads to the study of (3)-ħ22mΔu+νx+λu=up-1u.This latter equation, which is usually written as (4)-ε2Δu+λu+νxu=up-1u,ε>0or equivalently (by a standard rescaling) as (5)-Δu+λu+νεxu=up-1u,ε>0,has motivated a great deal of research from different points of view (see, for instance, [5–11] just to quote a few classical contributions among a very large and constantly increasing literature on the subject). The case of a periodic potential has been considered as well (see [12, 13]).In various articles, the hypothesisλ>-inf⁡ν has been assumed. Setting V(ξ)≔λ+ν(ξ), this is equivalent to consider the equation(6)Δu-Vεxu+up-1u=0,ε>0,with V(·) a positive weight function.Looking at (6) in one-dimension, we can interpret it as a slowly varying perturbation of the planar Hamiltonian system(7)u′=y,y′=Vu-up-1uwhich presents a hyperbolic equilibrium point (the origin which is a saddle point) with a homoclinic orbit enclosing a region that contains another equilibrium point which is a (local) center (see Figure 1). Similar phase-portraits are common in many different situations and it is known that their perturbations can produce chaotic-like dynamics (see, for instance, [14–18]).Figure 1
Energy level lines of system (7) in the phase-plane (u,u′), u≥0, for V=1, and p=3. A darker color represents a lower value of the energy of the orbits.Analogous equations appear in some mathematical models of nonlinear optics derived from Maxwell’s equations [19]. For instance, in [20] the study of the propagation of electromagnetic waves in layered media leads to the scalar equation(8)u′′-λu+ϵx,12u2u=0,λ>0,where the dielectric function ϵ(x,s) takes into account the presence of layers with different refractive indexes. A possible choice of ϵ(x,s) for three layers (one “internal” and two “external”) is given by (9)ϵx,s=ϵ1x,s,x≤dϵ2s,x>d. For some other typical forms of ϵ(x,s), see [20] and the references therein. Also this class of equations has been widely investigated in the last decades [20–23].Finally, we briefly mention another area of research where similar equations arise, that is in the context of wave propagation or stationary solutions for bistable reaction diffusion equations in excitable media (see, for instance, [24–27] and the references therein). In the above quoted papers, a typical one-dimensional model equation takes the form (10)ut=uxx+gu,x, where the function g(u,x) is defined piecewise as follows: (11)gu,x=fu,-∞<x≤x1,x2<x≤x3,…,x2n<x<+∞g0u,x,otherwise(see [26, 27] for different choices of g0(s,x)). The kinetic term is usually taken to be a Nagumo type cubic function f(s)=s(1-s)(s-a) (with 0<a<1/2) or a McKean’s piecewise linear reaction term [28]. Other possible variants concern the equation (12)ut=Dxuxx+gu,x, with D(x) a stepwise function [24]. The case of periodicity in the x-variable has been also considered. With this respect we mention a paper of Keener [29], dealing with the equation (13)ut=Duxx+1+g′xLkfu-kau,where g(x) is the so-called 1-periodic “sawtooth function”. Analogous investigations have been addressed also in [30, 31].In the present paper we restrict ourselves to the one-dimensional case and we study a second order nonlinear equation which is related to the models considered above. More in detail, we deal with a class of equations of the form(14)-u′′+Vtu=qtfu,where, for notational convenience, we consider the independent variable (which usually refers in the above quoted models as a space variable) as a time variable. Such a convention is also motivated by the dynamical systems approach which is followed in the present paper. The choice of introducing two weight functions (that is, V(t) for the linear part of the equation and q(t) for the nonlinear part) is useful in view of dealing with the more general Schrödinger equation (15)-ε2Δu+λu+νxu=Kxup-1u(previously considered in [32]).The nonlinear termf(s) in (14) is assumed to be acontinuously differentiable function and is chosen in order to include, as a particular case, the polynomial nonlinearities which usually appear in the context of the NLSEs (see Section 2 for the precise assumptions on f(s)). The main hypothesis on the coefficients V(t) and q(t), which are supposed to be nonnegative, is that they are close in the L1-norm to stepwise functions. Such a particular choice for the shape of the coefficients is mainly motivated by mathematical convenience, as it permits to develop the proofs in a simpler and more transparent way and thus to avoid more complicated technicalities. However, it is interesting to observe that second order equations or, more generally, first order planar systems with piecewise constant coefficients naturally appear in several applications, such as the theory of switching control [33], electric or mechanical systems [34, 35], periodically forced Nagumo equations [36], and biological models subject to seasonal dynamics [37–39], as well as in the context of NLSEs arising in nonlinear optics [23] and in mathematical modelling of structures like crystals or switches in optical fibers [40]. In this connection and as already observed at the beginning of the Introduction, stepwise periodic coefficients have been recently considered also in some (Gross-Pitaevskii) equations describing the phenomenon of Bose-Einstein condensation where the existence of complex dynamics for u′′+μu-g(t)u3=0 and u′′+(ω-U(t))u-u3=0 was proved in [1] and [41], respectively. Periodically forced second order nonlinear equations with stepwise coefficients are widely analyzed also in connection with Littlewood’s example of unbounded solutions to Duffing equations and its generalizations [42–45]. Finally, we observe that variants of these equations with a stepwise weight function have been considered with respect to the search of multiple “large” solutions, namely, solutions presenting a blow-up phenomenon at the boundary of a given interval (see [46, 47]).The main part of the paper is devoted to the study of the periodic boundary value problem associated with (14). In doing so, we prove the presence of infinitely many subharmonic solutions and also the existence of solutions with certain oscillatory properties which can reproduce any prescribed sequence of coin-tossing type [48] (see Definition 1 in Section 2). Such chaotic-like solutions are obtained by an application of the theory oftopological horseshoes [49, 50], in a variant developed in [51, 52]. We will also discuss how the arguments of the proofs can be modified in order to deal with the Neumann and the Dirichlet boundary value problems. In this latter context, the recent years have witnessed a growing interest toward the search of existence and multiplicity results of solutions of (16)-ε2Δu+u=up,u>0on a bounded domain Ω⊂RN,N≥2 with interior and/or boundary peaks [53–56]. With this respect, we stress the fact that our multiplicity results appear to be of completely different nature; they are typically one-dimensional, even if they could be applied to PDEs on thin annular domains of RN.For simplicity in the exposition, we will focus our presentation mainly to the study ofpositive solutions. We shall explain how to obtain sign changing solutions with prescribed nodal properties with some illustrative remarks at the end of the article.


## 2. Setting of the Problem and Main Results

We consider the second order nonlinear equation (of Schrödinger type)(17)-u′′+Vtu=fuwhere f:R→R is a continuously differentiable function of the form(18)fs=shswith h:R→R satisfying ∗h0=0andh′s>0fors>0,withh+∞=+∞. As a consequence of ∗ it follows that f(0)=0, f(s)>0 for s>0 and ∗∗lims→0+fss=0,lims→+∞fss=+∞.Since we are looking forpositive solutions, the actual behavior of f(s) for s<0 will not affect our result. For simplicity, we suppose that f isodd (that is, h iseven). We assume such a symmetry condition also in order to cover the classical example h(s)=sp-1 with p>1. All the results of the present paper could be proved for a locally Lipschitz continuous function f satisfying ∗∗ and with h(s)=f(s)/s strictly increasing on ]0,+∞) (and strictly decreasing on (-∞,0[). We prefer to consider the smooth case for simplicity in the presentation.For the potentialV(t) we suppose that V:R→R0+≔]0,+∞) is a T-periodic stepwise function (for some T>0) of the form(19)Vt≔V1fort∈0,T1V2fort∈T1,Twith V1≠V2. Writing (17) as the equivalent first order system Sx′=yy′=Vtx-fxin the phase-plane, we can describe the presence of a piecewise constant T-periodic coefficient as follows: the trajectories are governed by the autonomous system S1x′=yy′=V1x-fx in the time interval [0,T1[. At the time t=T1 we have a switching to system S2x′=yy′=V2x-fx which, in turns, rules the motions for a time interval of length (20)T2≔T-T1. All this switching behavior is then repeated in a T-periodic fashion.Recall that, given a first order differential systemz′=Z(t,z), its Poincaré map, on a time interval [t0,t1], is the function which maps any initial point w0 to ζ(t1;t0,w0), where ζ(t)≔ζ(t;t0,w0), is the solution of the differential system satisfying the initial condition z(t0)=w0. In our setting, it is straightforward to check that the Poincaré map on [0,T] for system S, that we denote by Φ, can be decomposed as(21)Φ=Φ2∘Φ1,where Φi is the Poincaré map associated with the autonomous system (Si) along the time interval [0,Ti]. Notice that, due to the autonomous nature of the subsystem, the map Φ2 coincides with the Poincaré map of S2 on [T1,T]. The assumptions on f(s) guarantee the global existence of the solutions for all the Cauchy problems and therefore Φ is a global homeomorphism of the plane.Our goal is to prove the existence of periodic solutions (harmonic and subharmonic) for (17). Following a classical procedure [57], this will be achieved by looking for the fixed points of Φ and its iterates. In our approach we apply some recent results on planar maps which provide not only the existence of fixed points and periodic points, but also the fact that the associated discrete dynamical system is “chaotic”. In the literature one can find several different methods which guarantee the presence of chaos for planar maps or, more generally, for homeomorphisms (or diffeomorphisms) in finite dimensional spaces. Moreover, different definitions ofchaotic dynamics have been proposed by various authors. For the reader convenience, we recall now the concept ofchaos that we are going to consider. Although the main definitions and the abstract setting can be presented in the framework of metric spaces, we confine ourselves to the case of homeomorphisms of the plane, which is the situation encountered by dealing with the Poincaré map associated with a planar system.Definition 1.
LetΦ:DΦ(⊆R2)→R2 be a homeomorphism and let D⊆DΦ be a nonempty set. Assume also that m≥2 is an integer. We say that Φinduces chaotic dynamics onmsymbols in the setD if there exist m nonempty pairwise disjoint compact sets (22)K0,K1,…,Km-1⊆Dsuch that, for each two-sided sequence of m symbols (23)sii∈Z∈Σm≔0,…,m-1Z,there exists a corresponding sequence (wi)i∈Z∈DZ with(24)wi∈Ksiandwi+1=Φwi,∀i∈Zand, whenever (si)i∈Z is a k-periodic sequence (that is, si+k=si,∀i∈Z) for some k≥1, there exists a k-periodic sequence (wi)i∈Z∈DZ satisfying (24).Note that, as a particular consequence of this definition, we have that for eachi∈{0,…,m-1} there is at least one fixed point of Φ in Ki. Since Φ is a homeomorphism from (25)K≔⋃i=0m-1Ki⊆Donto its image, it follows also that there exists a nonempty compact set Λ⊆K which is invariant for Φ (i.e., Φ(Λ)=Λ) and such that Φ|Λ is semiconjugate to the two-sided Bernoulli shift σ on m symbols (26)σ:Σm→Σm,σsii∈Z=si+1i∈Z,according to the commutative diagram (27)where g is a continuous and surjective function. Moreover, as a consequence of Definition 1 we can take Λ such that it contains as a dense subset the periodic points of Φ and such that the counterimage (by the semiconjugacy g) of any periodic sequence in Σm contains a periodic point of Φ (see [58] for the details). As usual, in Σm, the set of two-sided sequence of m symbols, we take its standard metric [18] for which Σm turns out to be a compact set with the product topology.We observe that Definition1 is related to the concept ofchaos in the sense of coin-tossing [48] and it also implies the presence of chaotic dynamics according to Block and Coppel [59, 60], as well as a positive topological entropy for the map Φ|Λ. Similar examples of complex dynamics for the Poincaré map associated with differential systems have been discussed, e.g., in [61–65], using different methods. See also [1, 31, 66] for recent contributions in this direction.Now we are in position to state our main result for (17).Theorem 2.
Letf:R→R be a C1-function of the form (18), with h satisfying ∗. Let V:R→R0+ be a T-periodic stepwise function as in (19), such that V1≠V2. Then, there exist a compact set D⊂R0+×R and, for every integer m≥2, two positive constants T1∗ and T2∗ such that, if T1>T1∗ and T2>T2∗, the Poincaré map Φ for system S on [0,T] (with T=T1+T2) induces chaotic dynamics on m symbols in the set D. Moreover, for the corresponding solutions (x(t),y(t)) of S we have x(t)>0 for every t∈R.The constantsT1∗ and T2∗ can be explicitly determined in terms of m and some Abelian integrals depending by V1, V2 and f(x) as in formula (41). The set D is explicitly exhibited in the course of the proof. Indeed, we have D≔A with A defined in (54).The proof is based on a topological technique, namedstretching along the paths (SAP), which is a variant of the classical Smale’s horseshoe geometry (see [67]). Our approach is closely related to the theory oftopological horseshoes of Kennedy and Yorke [50] as well as to the concept ofcovering relations introduced by Zgliczyński in [68]. The general theory concerning the “SAP method” has been already exposed in some previous papers (see, for instance, [58] and the references therein). In order to make our paper self-contained, we recall the main notation and the results which are needed for the proof of Theorem 2.Bypathγ we mean a continuous mapping γ:[t0,t1]→R2 and we set γ¯≔γ([t0,t1]). Without loss of generality we will usually take [t0,t1]=[0,1]. By asub-pathσ of γ we mean the restriction of γ to a compact subinterval of its domain. Anarc is the homeomorphic image of the compact interval [0,1]. We define anoriented rectangle in R2 as a pair (28)R^≔R,R-,where R⊆R2 is homeomorphic to the unit square [0,1]2 (we usually refer to R as atopological rectangle) and (29)R-≔Rl-∪Rr- is the disjoint union of two disjoint compact arcs Rl-,Rr-⊆∂R (which are called the components or sides of R-). We also denote by R+ the closure of ∂R∖(Rl-∪Rr-) which is the union of two compact arcs Rd+ and Ru+. The subscripts l,r,u,d stand, conventionally, forleft, right, up, and down.Suppose thatΦ:DΦ(⊆R2)→R2 is a planar homeomorphism of DΦ onto its image. Let M^≔(M,M-) and N^≔(N,N-) be oriented rectangles.Definition 3.
LetH⊆M∩DΦ be a compact set. We say that (H,Φ)stretchesM^toN^along the paths and write(30)if for every path γ:[a,b]→M such that γ(a)∈Ml- and γ(b)∈Mr- (or γ(a)∈Mr- and γ(b)∈Ml-), there exists a subinterval t′,t′′⊆[a,b] such that (31)γt∈H,Φγt∈N,∀t∈t′,t′′and, moreover, Φ(γ(t′)) and Φ(γ(t′′)) belong to different components of N-. In the special case in which H=M, we simply writeThe next result, taken from [69, Theorem 2.1], provides the existence of periodic points and chaotic-like dynamics according to Definition 1, when Φ admits a splitting as in (21).Theorem 4.
LetΦ1:DΦ1(⊆R2)→R2 and Φ2:DΦ2(⊆R2)→R2 be continuous maps and let A^≔(A,A-), B^≔(B,B-) be oriented rectangles. Suppose that the following conditions are satisfied: (i)
there existm≥2 pairwise disjoint compact sets H0,…,Hm-1⊆A∩DΦ1 such that (32)(ii)
there is a compact setK⊆B∩DΦ2 such that
Then the mapΦ≔Φ2∘Φ1 induces chaotic dynamics on m symbols in the set (33)H∗≔⋃j=0,…,m-1Hj′,forHj′≔Hj∩Φ1-1K.Moreover, for each sequence of m symbols s=(sn)n∈{0,…,m-1}N, there exists a compact connected set Cs⊆Hs0′ with Cs∩Ad+≠∅ and Cs∩Au+≠∅, such that, for every w∈Cs, there exists a sequence (yn)n with y0=w and (34)yn∈Hsn′,Φyn=yn+1,∀n≥0.A dual version of Theorem4 holds if we interchange the hypotheses on Φ1 and Φ2, namely, if we suppose that(i)
there is a compact setK⊆A∩DΦ1 such that(ii)
there existm≥2 pairwise disjoint compact sets H0,…,Hm-1⊆B∩DΦ2 such that (35) The corresponding conclusion has to be modified accordingly.The application of Theorem4 to Theorem 2 is possible thanks to alinked twist maps geometry which appears from the phase-plane analysis of the systems S1 and S2. The theory of “linked twist maps” regards the case in which a map can be expressed as a composition of two twist maps acting on two annuli crossing each other (see [70–74] for an introduction of the topic and for interesting applications to chaotic mixing). The main argument in the proof of Theorem 2 relies on the construction of two annular regions which cross each other in a suitable manner (see Figures 2 and 3) and such that (Φ1,Φ2) acts on them as a linked twist map.Figure 2
The present figure describes our geometric construction. The points on thex-axis marked with a black circle (from right to the left) represent (a1,0) and (a2,0) on the orbits Γ1α1 and Γ1α2. For this figure we have considered f(x)=x3 and V1=2, V2=1. For graphical reasons a slightly different x- and y-scaling has been used.Figure 3
The present figure describes the oriented regionsA^ and B^ (in lighter and darker colors, respectively). As in Figure 2 we have considered f(x)=x3 and V1=2, V2=1.


## 3. Technical Estimates and Proofs

As already observed in Section2, the motion associated with system S is given by a switching in a T-periodic fashion between the orbits of the two autonomous systems S1 and S2. Such systems have the same qualitative structure and differ only for the value of the V-coefficient. For this reason, we first perform a phase-plane analysis of the planar system(36)x′=yy′=Vx-fxfor V>0 a given parameter. System (36) is a conservative one with associated energy (37)Ex,y≔12y2+FVx, where (38)FVx≔-12Vx2+FxwithFx≔∫0xfsds.As a consequence of ∗, there is a unique x0=h-1(V)>0 solution of the equation h(x)=V. The corresponding equilibrium point P≔(x0,0) is a center surrounded by a trajectory which represents the homoclinic solution at zero.The origin and the homoclinic trajectory determine the part of the level lineΓ0 at energy zero contained in the half-plane x≥0. We denote by (x¯,0) the intersection point of the homoclinic orbit with the (positive) x-axis. Notice that x¯ is the unique (positive) solution of the equation 2F(x)/x2=V. As a consequence of ∗, both x0 and x¯, thought as functions of the parameter V, are strictly monotone increasing. Observe also that, for every constant c with (39)0>c>c0≔FVx0, the level line (40)Γc≔x,y∈R+×R:Ex,y=c, is a closed curve which is a (positive) periodic orbit of (36). The period τ(c) of Γc can be computed by the quadrature formula(41)τc=2∫αcβcdξ2c-FVξ,where α(c) and β(c) are the solutions of the equation FV(x)=c, with 0<α(c)<x0<β(c)<x¯. Moreover, we have that (42)limc→c0+τc=2πFV′′x0=2πx0h′x0,limc→0-τc=+∞. Without further assumptions on f(x) (or, equivalently, on h(x)) we cannot guarantee the monotonicity of the time-mapping function c↦τ(c). Sufficient conditions ensuring that τ(c) is strictly increasing can be found in literature. For instance, according to [75], the convexity of the auxiliary function (43)ϕVx≔FVx-FVx0FV′x2guarantees that τ(c) is increasing.Example 5.
Consider the typical nonlinear termf(x)=xp-1x, with p>1. In this case, condition ∗ holds for h(x)=xp-1 and (44)FVx=-12Vx2+1p+1xp+1,forx≥0.Moreover, we find that (45)x0=V1/p-1,x¯=p+1V21/p-1,FV′′x0=x0h′x0=p-1V.In order to prove the monotonicity of the time-map, via the Chicone theorem in [75] we have to study the sign of auxiliary function N(x)≔(FV′(x))4ϕV′′(x) on the open interval ]0,x¯[. After performing the required computations and using the change of variable x=sx0, one can see that the sign of N(x) for 0<x<x¯ is the same of the expression (46)3+pp-7sp-1+p2p+1s2p-2-pp-2sp+1-2p2-3p+3s2p+ps3p-1,for 0<s<((p+1)/2)1/p-1.
For instance, ifp=3, it is easy to check that the above expression is strictly positive for s≠1 (which corresponds to x≠x0) and therefore the time-mapping function τ(c) is strictly increasing on ]c0,0[. The case p=3 is the model situation that we have chosen in all our illustrative examples of Figures 1–8.
The monotonicity of the period map still holds for an arbitraryp>1. The proof in this case is a more complicated task (see [76, 77]).Figure 4
The present figure describes how a pathγ(s) crossing A from Al- to Ar- is stretched by Φ1 to a path Φ1γs which crosses twice B from Bl- to Br-. As in Figure 2 we have considered f(x)=x3 and V1=2, V2=1. For this example we have taken T1=50.Figure 5
The present figure describes how a pathγ(s) crossing B from Bl- to Br- is stretched by Φ2 to a path Φ2γs which crosses (once) A from Al- to Ar-. As in Figure 2 we have considered f(x)=x3 and V1=2, V2=1. For this example we have taken T2=10.Figure 6
Phase-portrait of (36) for f(x)=x3 and V=2. For graphical reasons a slightly different x- and y-scaling has been used.Figure 7
The present figure shows some possible rectangular regions which can be considered for the application of Theorem4. The sets A and B are the same as in Figure 3 and (as we have already proved in Section 3) they can be used to provide a complex dynamics on positive solutions. If we choose, for instance, the sets A′ and B′ we can prove the presence of a complex dynamics generated by solutions which are negative on the time interval [0,T1] and oscillate in the phase-plane around the point (-x01,0), and then, in the time interval [T1,T] oscillate a certain number of times around the origin. More in detail, given any positive integer M, we can produce solutions u(t) of (17) which have precisely 2,4,…,2M simple zeros in the interval ]T1,T[, provided that T2=T-T1 is sufficiently large. A lower estimate for T2 can be easily determined by the knowledge of the period of the closed trajectory Γ2′ which bounds “externally” A′ and B′. Similar remarks can be made by selecting other pairs of topological rectangles among those put in evidence with a color. As in the preceding figures, we have considered f(x)=x3 and V1=2, V2=1. For graphical reasons a slightly different x- and y-scaling has been used.Figure 8
The present figure suggests a possible argument to prove multiplicity of sign changing solutions to the Neumann problem (94). We start with system S1 by considering initial points w0=(u0,0) with x01<u0≤x¯1 (recall that P1=(x01,0) is the positive center of S1, while (x¯1,0) is the intersection point of the homoclinic orbit Γ10 with the positive x-axis). We parameterize such initial points as an arc σ(s). If T1 is sufficiently large (with a lower bound which can be easily estimated from the equation) we find that Φ1(σ(s)) is a spiral-like curve winding a certain number of times around P1 and with an end point on Γ10 in the fourth quadrant near the origin. If we fix a (small) positive constant c′′ and denote by C≔{(x,y):0≤E2(x,y)≤c′′} the region between the homoclinic trajectories of S2 and the level line Γ2′′≔Γ2c′′, we observe that the points of Φ1(σ(s))∩Γ20 remain on Γ20 under the action of Φ2, while the points of Φ1(σ(s))∩Γ2′′ run around the origin along the periodic orbit Γ2′′ and will perform a certain number of revolutions if T2 is sufficiently large. More precisely, if we denote by τ′′ the period of Γ2′′ and suppose that T2/τ′′>m, we can find solutions of (94) having precisely 2j-zeros in the interval ]T1,T1+T2[, for every j=1,…,m. As in the preceding figures, we have considered f(x)=x3 and V1=2, V2=1. For graphical reasons a slightly different x- and y-scaling has been used.Until now we have considered some general properties of the solutions of system (36). As a next step, in order to investigate the dynamics associate to system S for a T-periodic potential V(t) defined as in (19), we need to make a comparison between the phase–portraits associated with the autonomous systems S1 and S2. Keeping the notation just introduced, we set Fi≔FVi and denote by Ei the associated energy, for i=1,2. Accordingly, we indicate by Pi=(x0i,0) and (x¯i,0) the corresponding equilibrium points and the intersection points of the homoclinic orbits with the positive x-axis. Moreover, τi(c) denotes the fundamental period of the closed orbits defined in (41) for the potential functions Fi.Just to fix a case of study, we suppose that(47)V1>V2.The case when V1<V2 can be treated in a similar manner. Observe that from (47) it follows that the homoclinic orbit Γ20 of system S2 is contained in the part of the right half-plane bounded by the homoclinic orbit Γ10 of system S1.Now we are in position to introduce the rectangular regions (topological rectangles)A and B in order to apply Theorem 4. As a first step, we chose a closed trajectory Γ1c of system S1 which intersects in two distinct points the homoclinic trajectory Γ20∖{(0,0)} of system S2. From an analytic point of view, this corresponds to solving the system (48)E2x,y=12y2-12V2x2+Fx=0E1x,y=12y2-12V1x2+Fx=c, for a suitable c∈]c01,0[, with c01≔F1(x01). It is clear that this system has a pair of solutions (ξc,±ηc) with ξc,ηc>0 if and only if (49)ξc≔2cV1-V2,F2ξc<0. The latter condition holds if and only if 0<ξc<x¯2. We conclude that the desired geometry can be produced if and only if we choose an energy level c for E1(x,y) such that(50)0>c>c∗≔F1x¯2≥c01.From now on, we suppose to have fixed a constant c satisfying (50). Let us call α1 such a constant and denote by (a1,0) the intersection of the closed orbit Γ1α1 with the positive x-axis which is closer to the origin. Next, we choose a2 with 0<a2<a1 and consider the level line of system S1 passing through (a2,0). This is the closed orbit Γ1α2, for α2≔E1(a2,0). For α2 we further require that(51)τ1α2>τ1α1.We notice that it is always possible to find an open interval ]0,a∗[⊆]0,a1[ such that for each a2∈]0,a∗[ the condition (51) holds. This follows from the fact that τ1(c)→+∞ as c→0-. In the special case in which the time-mapping is strictly monotone increasing, one can take an arbitrary a2∈]0,a1[. Observe that, by construction, we also have(52)c01≤c∗<α1<α2<0.As a last step, we fix a constant β such that(53)0>β>maxE2a1,0,E2a2,0≥c02≔F2x02.By this latter choice, the corresponding (periodic) trajectory Γ2β of system S2 intersects both Γ1α1 and Γ1α2 in the region {(x,y):x>0,E2(x,y)<0} bounded by the homoclinic orbit Γ20. Figure 2 illustrates the geometric construction performed above.Next we define(54)A≔x,y:x>0,y>0,α1≤E1x,y≤α2,β≤E2x,y≤0and its specular image B with respect to the x-axis, namely,(55)B≔x,y:x>0,y<0,α1≤E1x,y≤α2,β≤E2x,y≤0.For these regions we select an orientation as follows: (56)A-≔Al-∪Ar-withAl-≔A∩Γ1α2,Ar-≔A∩Γ1α1,B-≔Bl-∪Br-withBl-≔B∩Γ20,Br-≔B∩Γ2β (see Figure 3).Now we describe the behavior of the points in the regionsA and B under the action of the Poincaré maps Φ1 and Φ2, respectively.Suppose that(57)0≤t≤T1. For each point z∈A the solution ζ(t;0,z) of S is indeed a solution of the autonomous conservative system S1 and therefore, E1(ζ(t;0,z))=E1(z), for all t∈[0,T1]. The orbit Γ1c for c≔E1(z) is closed trajectory surrounding the equilibrium point P1=(x01,0) of S1. Consistently with the previous notation, the period of the points in Γ1c is denoted by τ1(c). Since all the points of Γ1c (for α1≤c≤α2) move in the clockwise sense along the orbit under the action of the dynamical system associated with S1, it will be convenient to introduce a system of polar coordinates with center at P1 and take the clockwise orientation as a positive orientation for the angles starting from the positive half-line {(x,0):x>x01}. In this manner, we can associate an angular coordinate θ(t;z) to any solution ζ(t;0,z) for any initial point z∈A and t∈[0,T1].In such a situation, we find that(58)ddtθt;z=ytx′t-xt-x01y′txt-x012+yt2forxt,yt=ζt;0,z.Since xt,yt is a solution of S1 and, moreover, α1≤E1xt,yt≤α2 with x(t)>0, for all t∈[0,T1], we have that d/dtθ(t;z)>0 for all t∈[0,T1]. In fact, by (18) and ∗, we obtain (59)yx′-x-x01y′=y2-x-x01V1x-fx=y2+xx-x01hx-hx01>0,∀x>0. We have thus proved that the angle is a strictly increasing function of the time variable. Hence, for any positive integer k we conclude that(60)θt;z⋛θ0,z+2kπifandonlyift⋛kτ1c(of course, the above relation holds provided that t∈[0,T1] and T1>kτ1(c)).After these preliminary observations, we are now in position to prove the validity of the first condition of Theorem4 provided that T1 is large enough.Let us fix an integerm≥2 and set(61)T1∗≔m+2τ1α1τ1α2τ1α2-τ1α1.GivenT1∗ as above, we also fix T1>T1∗ and define (62)k1≔T1τ1α1,k2≔T1τ1α2. The position (61) and the choice of T1 imply that (63)T1τ1α1-T1τ1α2>m+2 and hence k1-k2>m.Now we consider the motion associated withS1 for t∈[0,T1].First of all, we note that(64)θ0,z∈-π,0,∀z∈A. On the other hand, since Ar-⊆Γ1α1, we know that (65)θT1,z≥θ0,z+2k1π>2k1-1π,∀z∈Ar-. Similarly, since Al-⊆Γ1α2, we know that (66)θT1,z≤θ0,z+2k2π<2k2π,∀z∈Al-.We thus conclude that the range of the angular function {θ(T1,z):z∈A} covers the interval [2k2π,(2k1-1)π]⊇[2k2π,(2k2+1)π+2(m-1)π]. This interval contains m closed disjoint intervals of the form [2k2π+2jπ,(2k2+1)π+2jπ], for j=0,…,m-1. Hence, if for each nonnegative integer j, we define (67)Hj≔z∈A:θT1,z∈2k2π+2jπ,2k2+1π+2jπ, we obtain m nonempty and pairwise disjoint compact subsets H0,…,Hm-1 of A.Letγ:[0,1]→A be a (continuous) path such that γ(0)∈Al- and γ(1)∈Ar- and consider the path [0,1]∋s↦Φ1(γ(s)). Passing to the polar coordinates we have that (68)θ0,γs∈-π,0,∀s∈0,1 and, moreover, (69)θT1,γ0<2k2π,θT1,γ1>2k1-1π>2k2+1π+2m-1π.Hence, for every j=0,…,m-1 there exists an interval [sj′,sj′′]⊆[0,1] such that γ(s)∈Hj for all s∈[sj′,sj′′], and, moreover, (70)θT1,γsj′=2k2π+2jπ,θT1,γsj′′=2k2+1π+2jπ.Using the fact that α1≤E1(Φ1(γ(s)))≤α2 for every s∈[0,1] (by the invariance of the annular region bounded by Γ1α1 and Γ1α2 with respect to system S1), we conclude that the set {Φ1(γ(s)):s∈[sj′,sj′′]} crosses the region B. Hence, by the continuity of the map [sj′,sj′′]∋s↦Φ1(γ(s)), we can find a subinterval [tj′,tj′′]⊆[sj′,sj′′] such that (71)Φ1γs∈B,∀s∈tj′,tj′′ and, moreover, (72)Φ1γtj′∈Bl-,Φ1γtj′′∈Br-. We also know that (73)γs∈Hj,∀s∈tj′,tj′′. According to Definition 3 we conclude that (74) and thus the first condition in Theorem 4 is fulfilled (see Figure 4 for a graphical description of this step in the proof).After the timeT1 we switch to equation S2. As previously observed, due to the autonomous nature of the system, the study of the solutions in the time interval [T1,T] is equivalent to the study for 0≤t≤T2. Now we set(75)T2∗≔τ2βand fix T2>T2∗.Arguing in a similar manner as before, we introduce a system of polar coordinates with center at the pointP2=(x02,0) and take the clockwise orientation as a positive orientation for the angles starting from the positive half-line {(x,0):x>x02}. In this manner, we can associate an angular coordinate ϑ(t;z) to any solution S2 with initial point z∈B and t∈[0,T2]. Repeating the same argument as above, one can see that the angle is a strictly increasing function of the time variable and, moreover, for any positive integer k(76)ϑt;z⋛ϑ0,z+2kπifandonlyift⋛kτ2c (of course, the above relation holds provided that t∈[0,T2] and T2>kτ2(c) for c∈[β,0]).Letγ:[0,1]→B be a (continuous) path such that γ(0)∈Bl- and γ(1)∈Br- and consider the path [0,1]∋s↦Φ2γs. Passing to the polar coordinates we have that (77)ϑ0,γs∈0,π,∀s∈0,1and, moreover, (78)ϑT2,γ0<π,ϑT2,γ1>2π.Indeed, the point γ(0)∈Bl- lies on the homoclinic trajectory Γ20 and therefore Φ2(γ(0)) remains in the fourth quadrant. On the other hand, the point γ(1)∈Br- lies on the periodic orbit Γ2β and, since T2>τ2(β) (which is the period of Γ2β), it follows that γ(1) makes at least one turn around the center P2 during the time interval [0,T2].Hence, there exists an interval[s′,s′′]⊆[0,1] such that (79)ϑT2,γs′=π,ϑT2,γs′′=2π. Using the fact that β≤E2Φ2γs≤0 for every s∈[0,1] (by the invariance of the region bounded by Γ2β and Γ20 with respect to system S2), we conclude that the set Φ2γs:s∈s′,s′′ crosses the region A. Hence, by the continuity of the map s′,s′′∋s↦Φ2γs, we can find a subinterval t′,t′′⊆s′,s′′ such that (80)Φ2γs∈A,∀s∈t′,t′′ and, moreover, (81)Φ2γt′∈Al-,Φ2γt′′∈Ar-. According to Definition 3 we conclude that (82) and thus the second condition in Theorem 4 is fulfilled for K≔B (see Figure 5 for a graphical description of this last step in the proof).Then, Theorem4 applies and the proof of Theorem 2 is complete (with respect to the “chaotic part”), providing the existence of complex dynamics on m symbols for the Poincaré map Φ (of system S) on the compact set A, according to Definition 1.Regarding the fact that all the solutions we find via Theorem4 are always positive in the x-variable, we need only to observe that, in the first step of the proof concerning the property (83) we have also found that x(t)>0 for all t∈[0,T1] when (x(0),y(0))∈Hi. Moreover, by construction, it follows that x(t)>0 for all t∈[0,T2] when (x(0),y(0))∈B. This concludes the proof.Remark 6.
From the proof of Theorem2 (and the geometric construction involving A and B) it follows that our result is stable with respect to small perturbations of the coefficients. Indeed, from the fundamental theory of ODEs we know that, for any fixed time interval [0,T], a “small” perturbation of the coefficients in the L1-norm on [0,T] produces a “small” perturbation on the Poincaré map (in the sense of the continuous dependence of the solutions from the data). With this respect, the following result holds.Theorem 7.
Letf:R→R and V:R→R0+ be as in Theorem 2, with V1≠V2. Given m≥2 and T1∗ and T2∗ according to Theorem 2, let us fix T1>T1∗ and T2>T2∗ and let T≔T1+T2. Then, there is ε=ε(m,T1,T2)>0 such that equation(84)-u′′+atu=fu,has infinitely many periodic (subharmonic) solutions as well as solutions presenting a complex dynamics, with u(t)>0, provided that a(·):R→R is a (measurable) T-periodic function satisfying(85)∫0Tat-Vtdt<ε.In Theorem7 the solutions are considered in the Carathéodory sense [78] (when a(t) is only measurable). On the other hand, we can also take an arbitrarily smooth function a(t) which approximates the step function V(t), provided that (85) is satisfied. The stability of Theorem 2 with respect to small perturbations of the Poincaré map is not confined to the coefficient of the nonlinearity. For instance, we can obtain the same result for a perturbed equation of the form (86)-u′′+cu′+atu=qtfu+et,u,u′,provided that c, ∫0Tat-Vtdtand |e(⋯)| are sufficiently small and q(·):R→R is a T-periodic function with ∫0Tqt-1dt small, too. Finally, we stress that T1∗ and T2∗ can be easily computed and are not necessarily “large” (see, for instance, the examples considered in Figures 4-5).


## 4. Related Results

In the previous sections we have discussed the presence of chaotic-like dynamics (including the existence of infinitely many subharmonic solutions) for (17), by assuming that the period is large enough. From this point of view, our results can be interpreted in line with analogous theorem on Hamiltonian systems with slowly varying coefficients (see [14, 16]). On the other hand, via a simple change of variable, we can apply our results to a typical Schrödinger equation of the form(87)-ε2u′′+Vtu=fu,for V:R→R0+ a periodic stepwise function of fixed period T>0, of the form(88)Vt≔V1fort∈0,T1V2fort∈T1,Twith V1≠V2 and 0<T1<T. As before, we also set T2≔T-T1. Writing (87) as an equivalent first order system(89)u′=ε-1vv′=ε-1Vtu-fuin the phase-plane, we obtain the following result.Theorem 8.
Letf:R→R be a C1-function of the form (18), with h satisfying ∗. Let V:R→R0+ be a T-periodic stepwise function as in (88). Then, there exist a compact set D⊂R0+×R and, for every integer m≥2, a constant εm∗>0 such that, for every ε∈]0,εm∗[ the Poincaré map Ψε associated with (89) on [0,T] induces chaotic dynamics on m symbols in the set D. Moreover, for the corresponding solutions (u(t),v(t)) of (89) we have u(t)>0 for every t∈R.The constantεm∗ can be explicitly determined in terms of m and T1,T2.Proof.
As in the proof of Theorem2, we suppose (90)0<V2<V1(the treatment of the other situation is completely similar and thus is omitted).
The change of variablest≔εs and x(s)≔u(εs), y(s)≔v(εs), transforms system (89) to the equivalent first order system(91)x′=yy′=Vsx-fxfor V(s)≔V(εs) a stepwise periodic function of period T≔T/ε. By (88) and setting T1≔T1/ε and T2≔T2/ε, it follows that V(s)=V1≔V1 on [0,T1[ and V(s)=V2≔V2 on [T1,T[ with T=T1+T2. Notice that if we denote by Φ the Poincaré map associated with system (91) on [0,T], then it follows that Ψε=Φ (one can easily check this fact, because (x(T),y(T))=(u(T),v(T))).
Now, for (91) we can apply Theorem 2. In particular, through the proof of that result in Section 3, we find a compact region D⊆R0+×R and two constants T1∗ and T2∗ such that the chaotic dynamics for Φ (according to Definition 1) is ensured provided that T1>T1∗ and T2>T2∗. The lower estimates on T1 and T2 transfer to an upper bound for ε, so that if (92)0<ε<εm∗≔minT1T1∗,T2T2∗,then the chaotic dynamics for Ψε on the same set D is guaranteed. In particular, recalling the definition of T1∗ in (61) and T2∗ in (75), we derive a precise estimate to εm∗. For instance, if we take V1=2, V2=1 and f(x)=x3 (like in the example of Figure 2) then we know that ε2∗≥min{T1/50,T2/10} and therefore, if we choose ε<min{T1/50,T2/10}, the conclusion of Theorem 8 holds.We observe that, similarly to Remark6, the stability of the result with respect to small perturbations of the coefficients is guaranteed, too.


## 5. Boundary Value Problems on Finite Intervals: Positive Solutions

We start this section by briefly describing how the method applied in the proof of Theorem2 can be adapted to obtain multiplicity results for the Neumann boundary value problem(93)-ε2u′′+Vtu=fuu′0=u′T=0(see (87)). Our aim is to find multiple positive solutions for (93), where the number of the solutions becomes arbitrarily large as ε→0+. Actually, such kind of result can be obtained by a variant of Theorem 2, via a change of variables as in the proof of Theorem 8. With this respect, we study the equivalent problem(94)-u′′+Vtu=fuu′0=u′T=0(see (17)) and look for multiplicity results, where the number of the solutions increases as the time-interval length grows.As in Section2 we suppose that V:[0,T]→R0+ is a stepwise function of the form (19) for V1>V2 (the case in which V1<V2 can be treated in a similar manner). The assumptions on f(x) are the same as in Section 2.Following the argument of the proof of Theorem2 we consider the Poincaré map Φ associated with the planar system S as well as its components Φi associated with systems (Si). The difference with respect to the proof of Theorem 2, consists into the fact that this time we look for initial points w0≔(u0,0), with u0>0 such that the second component of Φ(w0)≔ζ(T;0,w0) vanishes. In other words, we apply ashooting method, looking for a solution which departs at time t=0 from a point on the positive x-axis and hits again the (positive) x-axis at the time t=T, with x(t)>0 for all t∈[0,T].We repeat step by step (keeping the same notation) the geometrical construction in the proof of Theorem2. In particular, as before, we choose the closed orbits Γ1α1 and Γ1α2 of system S1 and Γ2β of system S2 in order to produce the regions A and B in the phase-plane. Recall also that τ1(α2)>τ1(α1) (see (51)). We have already indicated by (ai,0) the intersection point of Γ1αi with the positive x-axis which is closer to the origin. We introduce now also the second intersection point of Γ1αi with the positive x-axis, which will be denoted by (bi,0). Clearly we have (95)0<a2<a1<x01<b1<b2 (compared also with Figure 2).We produce the solutions of (94) by shooting from initial points w0=(u0,0) with (96)u0∈a2,a1oru0∈b1,b2.Just to fix one case for our discussion, let us assume that the former of the above alternative occurs. More precisely, we shall develop the following argument that we first briefly describe in an heuristic manner for the reader’s convenience. We start from an initial point in the segment [a2,a1]×{0} of the phase-plane and apply the Poincaré map Φ1 for a time T1 sufficiently long so that the image of such segment crosses at least m times the set B. Then we switch to the Poincaré map Φ2 and apply it for a time T2 sufficiently long so that the above arcs crossing B will be transformed (by Φ2) to some curves winding around the point P2 and crossing at least k times the x-axis. Putting all together these facts we conclude that there are at least m×k solutions to (94). We present now the technical justification, by slightly modifying the argument of the proof of Theorem 2.As before, we represent the solutions ofS1 in polar coordinates with respect to the center P1=(x01,0), using a clockwise orientation for the angular coordinate. In this case, instead of taking z∈A, we have z∈[a2,a1]×{0} and therefore θ(0,z)=-π. Accordingly, we replace the path γ:[0,1]→A with the map which parameterizes the segment [a2,a1]×{0}, namely we take (97)γs≔s,0,fors∈a2,a1. As a consequence, for (98)cs≔E1γs,s∈a2,a1, we have that (99)θt,γs⋛k-1πifandonlyift⋛kτ1cs2. For an integer m≥2, we define T1∗ as in (61) and fix T1>T1∗. Then, repeating the same argument of the proof of Theorem 2, we define the integers k1 and k2 and obtain (100)θT1,γa1=θT1,a1,0≥2k1-1π and (101)θT1,γa2=θT1,a2,0<2k2π (which, in this case, is an obvious choice). As a consequence we find m pairwise disjoint intervals [tj′,tj′′]⊆[a2,a1] for j=0,…,m-1 such that (102)Φ1s,0∈B,∀,s∈tj′,tj′′ and, moreover, (103)Φ1tj′,0∈Bl-,Φ1tj′′,0∈Br-. After the time T1 we switch to equation S2. As remarked before, the study of the solutions in the time interval [T1,T] is equivalent to the study for 0≤t≤T2, where, for the moment, T2 is not yet fixed.We introduce another system of polar coordinates with center at the pointP2=(x02,0) and take the clockwise orientation as a positive orientation for the angles starting from the positive half-line {(x,0):x>x02}. In this manner, we can associate an angular coordinate ϑ(t;z) to any solution S2 with initial point z∈B and t∈[0,T2]. In order to have the condition u′(T)=0 satisfied, we look for solutions (in the phase-plane) such that y(T2)=0. In terms of these new angular coordinates, this corresponds to the condition ϑ(T2,z)=kπ for some positive integer k.We have already proved that the angle is a strictly increasing function of the time variable. Moreover, as a consequence of (60) and the symmetry of the orbits with respect to the x-axis, we find that, for any positive integer k and for any z∈B, (104)ϑt;z>kπwhenevert>kτ2c2 (of course, the above relation holds provided that t∈[0,T2] and T2>kτ2(c)/2 for c∈[β,0]).Now we are ready to introduce the constantT2∗ which will be defined as follows. Let us fix a positive integer k and set(105)T2∗=T2∗k≔kτ2β2and fix T2>T2∗.The pointΦ1((tj′,0))∈Bl- lies on the level line Γ20 of the homoclinic trajectory and therefore, ϑ(T2;Φ1((tj′,0)))<π. On the other hand, Φ1((tj′′,0))∈Br- lies on the level line Γ2β of the closed trajectory of period τ2(β) and therefore we have ϑ(T2;Φ1((tj′′,0)))>kπ. Hence, we can conclude that the arc {Φ1((s,0)):s∈[tj′,tj′′]}, which connect in B the opposite sides Bl- and Br-, is mapped by Φ2 to an arc which crosses at least k times the x-axis. In other words, for each integer i∈{1,…,k} there exists one point (106)zji∈Φ1s,0:s∈tj′,tj′′ such that ϑ(T2;zji)=iπ. The corresponding solution u(t) has precisely i-zeros of the derivative in the interval ]0,T2].Finally, recalling that them×k points zji∈B are images through Φ1 of corresponding points in the interval [a2,a1]×{0}, allows us to conclude with the following claim.Theorem 9.
Letf:R→R be a C1-function of the form (18), with h satisfying ∗. Let V:[0,T]→R0+ be a stepwise function as in (19), and such that V1>V2>0. Then, for every pair of integers (m,k) with m≥2 and k≥1, there exist two positive constants T1∗ and T2∗ such that, if T1>T1∗ and T2>T2∗, the problem (94) has at least m×k positive solutions on [0,T] (with T=T1+T2). More precisely, for each i=1,…,k there are at least m solutions of (94) with u′(T1)<0 and such that u′(t) has precisely i-zeros in [T1,T].The solutions that we have produced are only those obtained by shooting from[a2,a1]×{0}, achieving the set B at the time T1 and coming back at the x-axis at the time T. With obvious changes in the argument, we could produce solutions which are in A at the time T1. Moreover, we could also start from the segment [b1,b2]×{0} and reach the region B (or, respectively, A) at the time T1. Therefore, with the same technique, we can producefour different classes of m×k solutions. Finally, we can consider (by suitably modifying the same approach) also the case V2>V1>0.A variant of Theorem9 for problem (93) with ε sufficiently small can be also obtained via the same change of variables as in the proof of Theorem 8.In order to obtainpositive solutions for the Dirichlet (two-point) boundary value problem(107)-ε2u′′+Vtu=fuu0=uT=0or (after a suitable rescaling)(108)-u′′+Vtu=fuu0=uT=0a natural implementation of the shooting approach considered for the periodic and the Neumann problem consists into shooting from the positive y-axis of the phase-plane. More precisely, we consider the initial condition (109)x0,y0=w0=0,r,withr>0 for system S and look for a suitable value of the parameter r such that (110)Φw0≔xT;0,w0,yT;0,w0 satisfies (111)xT;0,w0=0,withxt>0∀t∈0,T. In such a situation, for a weight function with only two steps, we do not obtain multiplicity results (in general). However, one could easily provide a mechanism for multiple solutions by taking a weight function V(t) with three steps defined as follows:(112)Vt≔V1fort∈0,T1V2fort∈T1,T1+T2V3fort∈T1+T2,Twith T=T1+T2+T3 and V1,V2,V3 positive constants with (113)V2>maxV1,V3.In fact, under these assumptions, we can find solutions of (108) which are positive on ]0,T[ and oscillate a certain (prescribed) number of times in the phase-plane around the point P2=(x02,0), during the time interval [T1,T1+T2]. The number of such solutions can be larger than a preassigned number m, provided that T2>T2∗ for T2∗ a sufficiently large time (depending on m). An analogous conclusion can be derived for problem (107).As a consequence of the technique we can also prove the stability of the multiplicity results with respect to small perturbations of the coefficients.


## 6. Final Remarks

### 6.1. Sign Changing Solutions

In the previous sections we have focused our attention only to the search of positive solutions. We stress the fact that the same approach works well in order to producenodal solutions, that is solutions with a prescribed number of zeros in a given interval. For the sake of simplicity in the exposition and consistently with the classical case f(u)=up-1u, we confine ourselves to the case of anodd nonlinear term in the equation. If f(s)=sh(s) is not odd, we can still adapt the same argument with minor efforts.The assumption (18), with h(s) aneven function satisfying ∗, implies that the phase–portrait of (36) is similar to that of Figure 1 with a mirror symmetry with respect to the y-axis (see Figure 6).Now we consider again (17) with a periodic stepwise weight function V(t) as in (19), with V1≠V2. Actually, in order to enter in the same situation analyzed in Section 3, we assume(114)V1>V2>0.As previously observed, the dynamics associated with system S can be described as that of a couple of switching systems of the form S1 and S2. This time we are interested also in sign changing solutions and therefore we exploit the properties of the trajectories in the negative half-plane, too. If we overlap the energy level lines of the two autonomous systems we have the possibility of applying our abstract result (namely, Theorem 4) to different choices of rectangular regions A and B which are determined through the intersections of the level lines of the two systems. An illustrative example of possible choices of topological rectangles is given in Figure 7 where we have put in evidence some sets which are suitable for the application of Theorem 4.As a consequence, we can find different rectangular regions where chaotic dynamics occur. The corresponding solutions can have different qualitative behaviors on the intervals]jT,jT+T1[ and/or ]jT+T1,(j+1)T[, depending on the regions where we apply Theorem 4. For instance, we can find solutions with prescribed nodal properties on the intervals ]jT,jT+T1[ and ]jT+T1,(j+1)T[, as well as solutions which are positive (or negative) on ]jT,jT+T1[ and have a prescribed number of zeros on ]jT+T1,(j+1)T[.For the search of sign changing solutions to the Neumann problems (93) or (94), we suppose (114) and apply a shooting method by selecting initial points on the x-axis of the phase-plane. A possible argument is illustrated in Figure 8.Finally, for the Dirichlet problem (108), where again we suppose that V(t) is step function like in (19), we can prove the existence of multiple sign changing solutions by shooting from the y-axis in the phase-plane and suitably adapting the argument described above for the Neumann problem.


### 6.2. Remarks on the Weight Functions

Throughout the paper we have focused our analysis to an equation of the form (17), namely -u′′+V(t)u=f(u). We point out that the same technique works for the case of equation(115)-u′′+Vu=qtfu,with q(·) a positive stepwise function. This follows from the elementary observation that the equation -y′′+ay=bf(y), with constant coefficients a,b>0, is equivalent to -u′′+(a/b)u=f(u), for u(t)≔y(b-1/2t). Hence, if we take q(t) as a step function on an interval [0,Tq] (with q(t) possibly periodic), we find that (115) turns out to be equivalent to (17). For the Dirichlet problem, existence, multiplicity and stability of positive solutions to equations of the form (115) have been obtained in [79] in the more general PDEs case for q(t) a sign changing weight function. In our setting, we suppose q(t)>0 for every t.Finally, we observe that a simple adaptation of our approach (as described in Sections2 and 3 can be applied to (14) provided that both V(t) and q(t) are close to stepwise functions.


## 6.1. Sign Changing Solutions

In the previous sections we have focused our attention only to the search of positive solutions. We stress the fact that the same approach works well in order to producenodal solutions, that is solutions with a prescribed number of zeros in a given interval. For the sake of simplicity in the exposition and consistently with the classical case f(u)=up-1u, we confine ourselves to the case of anodd nonlinear term in the equation. If f(s)=sh(s) is not odd, we can still adapt the same argument with minor efforts.The assumption (18), with h(s) aneven function satisfying ∗, implies that the phase–portrait of (36) is similar to that of Figure 1 with a mirror symmetry with respect to the y-axis (see Figure 6).Now we consider again (17) with a periodic stepwise weight function V(t) as in (19), with V1≠V2. Actually, in order to enter in the same situation analyzed in Section 3, we assume(114)V1>V2>0.As previously observed, the dynamics associated with system S can be described as that of a couple of switching systems of the form S1 and S2. This time we are interested also in sign changing solutions and therefore we exploit the properties of the trajectories in the negative half-plane, too. If we overlap the energy level lines of the two autonomous systems we have the possibility of applying our abstract result (namely, Theorem 4) to different choices of rectangular regions A and B which are determined through the intersections of the level lines of the two systems. An illustrative example of possible choices of topological rectangles is given in Figure 7 where we have put in evidence some sets which are suitable for the application of Theorem 4.As a consequence, we can find different rectangular regions where chaotic dynamics occur. The corresponding solutions can have different qualitative behaviors on the intervals]jT,jT+T1[ and/or ]jT+T1,(j+1)T[, depending on the regions where we apply Theorem 4. For instance, we can find solutions with prescribed nodal properties on the intervals ]jT,jT+T1[ and ]jT+T1,(j+1)T[, as well as solutions which are positive (or negative) on ]jT,jT+T1[ and have a prescribed number of zeros on ]jT+T1,(j+1)T[.For the search of sign changing solutions to the Neumann problems (93) or (94), we suppose (114) and apply a shooting method by selecting initial points on the x-axis of the phase-plane. A possible argument is illustrated in Figure 8.Finally, for the Dirichlet problem (108), where again we suppose that V(t) is step function like in (19), we can prove the existence of multiple sign changing solutions by shooting from the y-axis in the phase-plane and suitably adapting the argument described above for the Neumann problem.


## 6.2. Remarks on the Weight Functions

Throughout the paper we have focused our analysis to an equation of the form (17), namely -u′′+V(t)u=f(u). We point out that the same technique works for the case of equation(115)-u′′+Vu=qtfu,with q(·) a positive stepwise function. This follows from the elementary observation that the equation -y′′+ay=bf(y), with constant coefficients a,b>0, is equivalent to -u′′+(a/b)u=f(u), for u(t)≔y(b-1/2t). Hence, if we take q(t) as a step function on an interval [0,Tq] (with q(t) possibly periodic), we find that (115) turns out to be equivalent to (17). For the Dirichlet problem, existence, multiplicity and stability of positive solutions to equations of the form (115) have been obtained in [79] in the more general PDEs case for q(t) a sign changing weight function. In our setting, we suppose q(t)>0 for every t.Finally, we observe that a simple adaptation of our approach (as described in Sections2 and 3 can be applied to (14) provided that both V(t) and q(t) are close to stepwise functions.



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*Source: 2101482-2018-12-02.xml*