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

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the onedimensional 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.


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 −  −  +  ()  3 = 0, ( > 0) for () a positive periodic coefficient.The study of such equation was motivated by previous works on some models of Bose-Einstein condensates considered in [2][3][4].
A more classical form of Nonlinear Schrödinger Equation (NLSE from now on) which has been studied by many authors is given by where ℏ denotes the Planck constant,  is the imaginary unit,  is a positive constant, ]() is the potential, and  > 1.
The search of stationary waves, namely, solutions of the form (, ) = exp(ℏ −1 )(), where  ∈ R and () is a real valued function, leads to the study of This latter equation, which is usually written as or equivalently (by a standard rescaling) as has motivated a great deal of research from different points of view (see, for instance, [5][6][7][8][9][10][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 hypothesis  > − inf ] has been assumed.Setting () fl  + ](), this is equivalent to consider the equation
Looking at (6) in one-dimension, we can interpret it as a slowly varying perturbation of the planar Hamiltonian system Figure 1: Energy level lines of system (7) in the phase-plane (,   ),  ≥ 0, for  = 1, and  = 3.A darker color represents a lower value of the energy of the orbits.
which 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][15][16][17][18]).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 where the dielectric function (, ) takes into account the presence of layers with different refractive indexes.A possible choice of (, ) for three layers (one "internal" and two "external") is given by For some other typical forms of (, ), see [20] and the references therein.Also this class of equations has been widely investigated in the last decades [20][21][22][23].
In the present paper we restrict ourselves to the onedimensional 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 −  +  ()  =  ()  () , 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, () for the linear part of the equation and () for the nonlinear part) is useful in view of dealing with the more general Schrödinger equation (previously considered in [32]).The nonlinear term () in ( 14) is assumed to be a continuously 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 ()).The main hypothesis on the coefficients () and (), which are supposed to be nonnegative, is that they are close in the  1 -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][38][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   +  − () 3 = 0 and   + ( − ()) −  3 = 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][43][44][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 of topological 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 on a bounded domain Ω ⊂ R  ,  ≥ 2 with interior and/or boundary peaks [53][54][55][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 R  .
For simplicity in the exposition, we will focus our presentation mainly to the study of positive solutions.We shall explain how to obtain sign changing solutions with prescribed nodal properties with some illustrative remarks at the end of the article.

Setting of the Problem and Main Results
We consider the second order nonlinear equation (of Schrödinger type) where  : R → R is a continuously differentiable function of the form with ℎ : R → R satisfying As a consequence of ( * ) it follows that (0) = 0, () > 0 for  > 0 and lim ( * * ) Since we are looking for positive solutions, the actual behavior of () for  < 0 will not affect our result.For simplicity, we suppose that  is odd (that is, ℎ is even).We assume such a symmetry condition also in order to cover the classical example ℎ() = || −1 with  > 1.All the results of the present paper could be proved for a locally Lipschitz continuous function  satisfying ( * * ) and with ℎ() = ()/ strictly increasing on ]0, +∞) (and strictly decreasing on (−∞, 0[).
We prefer to consider the smooth case for simplicity in the presentation.
For the potential () we suppose that  : R → R + 0 fl ]0, +∞) is a -periodic stepwise function (for some  > 0) of the form with  1 ̸ =  2 .Writing (17) as the equivalent first order system in the phase-plane, we can describe the presence of a piecewise constant -periodic coefficient as follows: the trajectories are governed by the autonomous system in the time interval [0,  1 [.At the time  =  1 we have a switching to system which, in turns, rules the motions for a time interval of length All this switching behavior is then repeated in a -periodic fashion.
Recall that, given a first order differential system   = (, ), its Poincaré map, on a time interval [ 0 ,  1 ], is the function which maps any initial point  0 to ( 1 ;  0 ,  0 ), where () fl (;  0 ,  0 ), is the solution of the differential system satisfying the initial condition ( 0 ) =  0 .In our setting, it is straightforward to check that the Poincaré map on [0, ] for system (), that we denote by Φ, can be decomposed as where Φ  is the Poincaré map associated with the autonomous system (  ) along the time interval [0,   ].Notice that, due to the autonomous nature of the subsystem, the map Φ 2 coincides with the Poincaré map of ( 2 ) on [ 1 , ].The assumptions on () 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 of chaotic dynamics have been proposed by various authors.For the reader convenience, we recall now the concept of chaos 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.Let Φ :  Φ (⊆ R 2 ) → R 2 be a homeomorphism and let D ⊆  Φ be a nonempty set.Assume also that  ≥ 2 is an integer.We say that Φ induces chaotic dynamics on  symbols in the set D if there exist  nonempty pairwise disjoint compact sets such that, for each two-sided sequence of  symbols there exists a corresponding sequence (  ) ∈Z ∈ D Z with and, whenever (  ) ∈Z is a −periodic sequence (that is,  + =   , ∀ ∈ Z) for some  ≥ 1, there exists a −periodic sequence (  ) ∈Z ∈ D Z satisfying (24).
Note that, as a particular consequence of this definition, we have that for each  ∈ {0, . . .,  − 1} there is at least one fixed point of Φ in K  .Since Φ is a homeomorphism from onto 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  on  symbols according to the commutative diagram where  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 ) of any periodic sequence in Σ  contains a periodic point of Φ (see [58] for the details).As usual, in Σ  , the set of two-sided sequence of  symbols, we take its standard metric [18] for which Σ  turns out to be a compact set with the product topology.
We observe that Definition 1 is related to the concept of chaos 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][62][63][64][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).
The constants  * 1 and  * 2 can be explicitly determined in terms of  and some Abelian integrals depending by  1 ,  2 and () as in formula (41).The set D is explicitly exhibited in the course of the proof.Indeed, we have D fl A with A defined in (54).
The proof is based on a topological technique, named stretching 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 of topological horseshoes of Kennedy and Yorke [50] as well as to the concept of covering 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 selfcontained, we recall the main notation and the results which are needed for the proof of Theorem 2.
By path  we mean a continuous mapping  : [ 0 ,  1 ] → R 2 and we set  fl ([ 0 ,  1 ]).Without loss of generality we will usually take [ 0 ,  1 ] = [0, 1].By a sub-path  of  we mean the restriction of  to a compact subinterval of its domain.An arc is the homeomorphic image of the compact interval [0, 1].We define an oriented rectangle in R 2 as a pair where R ⊆ R 2 is homeomorphic to the unit square [0, 1] 2 (we usually refer to R as a topological rectangle) and is the disjoint union of two disjoint compact arcs R −  , R −  ⊆ R (which are called the components or sides of R − ).We also denote by R + the closure of R \ (R −  ∪ R −  ) which is the union of two compact arcs R +  and R +  .The subscripts , , ,  stand, conventionally, for left, right, up, and down.
Suppose that Φ :  Φ (⊆ R 2 ) → R 2 is a planar homeomorphism of  Φ onto its image.Let M fl (M, M − ) and N fl (N, N − ) be oriented rectangles.Definition 3. Let H ⊆ M ∩  Φ be a compact set.We say that (H, Φ) stretches M to N along the paths and write if for every path  : and, moreover, Φ((  )) and Φ((  )) belong to different components of N − .In the special case in which H = M, we simply write The 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).
) be oriented rectangles.Suppose that the following conditions are satisfied: Then the map Φ fl Φ 2 ∘ Φ 1 induces chaotic dynamics on  symbols in the set A dual version of Theorem 4 holds if we interchange the hypotheses on Φ 1 and Φ 2 , namely, if we suppose that (ii) there exist  ≥ 2 pairwise disjoint compact sets H 0 , . . ., The corresponding conclusion has to be modified accordingly.
The application of Theorem 4 to Theorem 2 is possible thanks to a linked twist maps geometry which appears from the phase-plane analysis of the systems ( 1 ) and ( 2 ).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][71][72][73][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 (Φ 1 , Φ 2 ) acts on them as a linked twist map.

Technical Estimates and Proofs
As already observed in Section 2, the motion associated with system () is given by a switching in a -periodic fashion between the orbits of the two autonomous systems ( 1 ) and ( 2 ).Such systems have the same qualitative structure and differ only for the value of the -coefficient.For this reason, we first perform a phase-plane analysis of the planar system for  > 0 a given parameter.System ( 36) is a conservative one with associated energy where As a consequence of ( * ), there is a unique  0 = ℎ −1 () > 0 solution of the equation ℎ() = .The corresponding equilibrium point  fl ( 0 , 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 line Γ 0 at energy zero contained in the halfplane  ≥ 0. We denote by (, 0) the intersection point of the homoclinic orbit with the (positive) -axis.Notice that  is the unique (positive) solution of the equation 2()/ 2 = .As a consequence of ( * ), both  0 and , thought as functions of the parameter , are strictly monotone increasing.Observe also that, for every constant  with the level line is a closed curve which is a (positive) periodic orbit of (36).
The period () of Γ  can be computed by the quadrature formula where () and () are the solutions of the equation F  () = , with 0 < () <  0 < () < .Moreover, we have that lim Without further assumptions on () (or, equivalently, on ℎ()) we cannot guarantee the monotonicity of the timemapping function   → ().Sufficient conditions ensuring that () is strictly increasing can be found in literature.For instance, according to [75], the convexity of the auxiliary function guarantees that () is increasing.
Moreover, we find that , 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 () fl (F   ()) 4    () on the open interval ]0, [.After performing the required computations and using the change of variable  =  0 , one can see that the sign of () for 0 <  <  is the same of the expression for 0 <  < (( + 1)/2) 1/(−1) .For instance, if  = 3, it is easy to check that the above expression is strictly positive for  ̸ = 1 (which corresponds to  ̸ =  0 ) and therefore the time-mapping function () is strictly increasing on ] 0 , 0[.The case  = 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 arbitrary  > 1.The proof in this case is a more complicated task (see [76,77]).
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 () for a periodic potential () defined as in (19), we need to make a comparison between the phase-portraits associated with the autonomous systems ( 1 ) and ( 2 ).Keeping the notation just introduced, we set F  fl F   and denote by   the associated energy, for  = 1, 2. Accordingly, we indicate by   = ( 0 , 0) and (  , 0) the corresponding equilibrium points and the intersection points of the homoclinic orbits with the positive -axis.Moreover,   () denotes the fundamental  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,  1 ] and oscillate in the phase-plane around the point (− 01 , 0), and then, in the time interval [ 1 , ] oscillate a certain number of times around the origin.More in detail, given any positive integer , we can produce solutions () of ( 17) which have precisely 2, 4, . . ., 2 simple zeros in the interval ] 1 , [, provided that  2 =  −  1 is sufficiently large.A lower estimate for  2 can be easily determined by the knowledge of the period of the closed trajectory Γ  2 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 () =  3 and  1 = 2,  2 = 1.For graphical reasons a slightly different and -scaling has been used.period of the closed orbits defined in (41) for the potential functions F  .
Just to fix a case of study, we suppose that The case when  1 <  2 can be treated in a similar manner.
Observe that from (47) it follows that the homoclinic orbit Γ 0 2 of system ( 2 ) is contained in the part of the right half-plane bounded by the homoclinic orbit Γ 0 1 of system ( 1 ).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 Γ   94).We start with system ( 1 ) by considering initial points  0 = ( 0 , 0) with  01 <  0 ≤  1 (recall that  1 = ( 01 , 0) is the positive center of ( 1 ), while ( 1 , 0) is the intersection point of the homoclinic orbit Γ 0 1 with the positive -axis).We parameterize such initial points as an arc ().If  1 is sufficiently large (with a lower bound which can be easily estimated from the equation) we find that Φ 1 (()) is a spiral-like curve winding a certain number of times around  1 and with an end point on Γ 0 1 in the fourth quadrant near the origin.If we fix a (small) positive constant   and denote by C fl {(, ) : 0 ≤  2 (, ) ≤   } the region between the homoclinic trajectories of ( 2 ) and the level line Γ  2 fl Γ   2 , we observe that the points of Φ 1 (()) ∩ Γ 0 2 remain on Γ 0 2 under the action of Φ 2 , while the points of Φ 1 (()) ∩ Γ  2 run around the origin along the periodic orbit Γ  2 and will perform a certain number of revolutions if  2 is sufficiently large.More precisely, if we denote by   the period of Γ  2 and suppose that  2 /  > , we can find solutions of (94) having precisely 2zeros in the interval ] 1 ,  1 +  2 [, for every  = 1, . . ., .As in the preceding figures, we have considered () =  3 and  1 = 2,  2 = 1.For graphical reasons a slightly different and -scaling has been used.
of system ( 1 ) which intersects in two distinct points the homoclinic trajectory Γ 0 2 \ {(0, 0)} of system ( 2 ).From an analytic point of view, this corresponds to solving the system for a suitable  ∈] 01 , 0[, with  01 fl F 1 ( 01 ).It is clear that this system has a pair of solutions (  , ±  ) with   ,   > 0 if and only if The latter condition holds if and only if 0 <   <  2 .We conclude that the desired geometry can be produced if and only if we choose an energy level  for  1 (, ) such that From now on, we suppose to have fixed a constant  satisfying (50).Let us call  1 such a constant and denote by ( 1 , 0) the intersection of the closed orbit Γ  1 1 with the positive -axis which is closer to the origin.Next, we choose  2 with 0 <  2 <  1 and consider the level line of system ( 1 ) passing through ( 2 , 0).This is the closed orbit Γ  2 1 , for  2 fl  1 ( 2 , 0).For  2 we further require that We notice that it is always possible to find an open interval ]0,  * [⊆]0,  1 [ such that for each  2 ∈]0,  * [ the condition (51) holds.This follows from the fact that  1 () → +∞ as  → 0 − .In the special case in which the time-mapping is strictly monotone increasing, one can take an arbitrary  2 ∈]0,  1 [.Observe that, by construction, we also have As a last step, we fix a constant  such that By this latter choice, the corresponding (periodic) trajectory Γ  2 of system ( 2 ) intersects both Γ  1 1 and Γ  2 1 in the region {(, ) :  > 0,  2 (, ) < 0} bounded by the homoclinic orbit Γ 0 2 .Figure 2 illustrates the geometric construction performed above.
Next we define and its specular image B with respect to the -axis, namely, For these regions we select an orientation as follows: (see Figure 3).Now we describe the behavior of the points in the regions A and B under the action of the Poincaré maps Φ 1 and Φ 2 , respectively.
For each point  ∈ A the solution (; 0, ) of () is indeed a solution of the autonomous conservative system ( 1 ) and therefore,  1 ((; 0, )) =  1 (), for all  ∈ [0, 1 ].The orbit Γ  1 for  fl  1 () is closed trajectory surrounding the equilibrium point  1 = ( 01 , 0) of ( 1 ).Consistently with the previous notation, the period of the points in Γ  1 is denoted by  1 ().Since all the points of Γ  1 (for  1 ≤  ≤  2 ) move in the clockwise sense along the orbit under the action of the dynamical system associated with ( 1 ), it will be convenient to introduce a system of polar coordinates with center at  1 and take the clockwise orientation as a positive orientation for the angles starting from the positive half-line {(, 0) :  >  01 }.
In this manner, we can associate an angular coordinate (; ) to any solution (; 0, ) for any initial point  ∈ A and  ∈ [0,  1 ].
After these preliminary observations, we are now in position to prove the validity of the first condition of Theorem 4 provided that  1 is large enough.
Let us fix an integer  ≥ 2 and set Given  * 1 as above, we also fix  1 >  * 1 and define ⌉ . ( The position (61) and the choice of  1 imply that and hence  1 −  2 > .Now we consider the motion associated with First of all, we note that On the other hand, since 1 , we know that Similarly, since 1 , we know that We thus conclude that the range of the angular function {( we obtain  nonempty and pairwise disjoint compact subsets H 0 , . . ., H −1 of A. Let  : [0, 1] → A be a (continuous) path such that (0) ∈ A −  and (1) ∈ A −  and consider the path [0, 1] ∋   → Φ 1 (()).Passing to the polar coordinates we have that and, moreover, Hence, for every  = 0, . . .,  − 1 there exists an interval and, moreover, Using the fact that  1 ≤ Complexity and, moreover, We also know that According to Definition 3 we conclude that (ℋ j , Φ 1 ) :    ℬ, ＠ＩＬ j = 0, . . ., m − 1 (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 time  1 we switch to equation ( 2 ).As previously observed, due to the autonomous nature of the system, the study of the solutions in the time interval [ 1 , ] is equivalent to the study for 0 ≤  ≤  2 .Now we set and fix  2 >  * 2 .Arguing in a similar manner as before, we introduce a system of polar coordinates with center at the point  2 = ( 02 , 0) and take the clockwise orientation as a positive orientation for the angles starting from the positive half-line {(, 0) :  >  02 }.In this manner, we can associate an angular coordinate (; ) to any solution ( 2 ) with initial point  ∈ B and  ∈ [0,  2 ].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   (; ) ⋛  (0, ) + 2 if and only if  ⋛  2 () (76) (of course, the above relation holds provided that  ∈ [0,  2 ] and  2 >  2 () for  ∈ [, 0]).
Hence, there exists an interval Using the fact that  ≤  2 (Φ 2 (())) ≤ 0 for every  ∈ [0, 1] (by the invariance of the region bounded by Γ  2 and Γ 0 2 with respect to system ( 2 )), we conclude that the set {Φ 2 (()) :  ∈ [  ,   ]} crosses the region A. Hence, by the continuity of the map [  ,   ] ∋   → Φ 2 (()), we can find a subinterval and, moreover, According to Definition 3 we conclude that and thus the second condition in Theorem 4 is fulfilled for K fl B (see Figure 5 for a graphical description of this last step in the proof).Then, Theorem 4 applies and the proof of Theorem 2 is complete (with respect to the "chaotic part"), providing the existence of complex dynamics on  symbols for the Poincaré map Φ (of system ()) on the compact set A, according to Definition 1.
Regarding the fact that all the solutions we find via Theorem 4 are always positive in the -variable, we need only to observe that, in the first step of the proof concerning the property we have also found that () > 0 for all  ∈ [0,  1 ] when ((0), (0)) ∈ H  .Moreover, by construction, it follows that () > 0 for all  ∈ [0, 2 ] when ((0), (0)) ∈ B. This concludes the proof.
Remark 6.From the proof of Theorem 2 (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, ], a "small" perturbation of the coefficients in the  1 -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. Let  : R → R and  : R → R + 0 be as in Theorem 2, with  1 ̸ =  2 .Given  ≥ 2 and  * 1 and  * 2 according to Theorem 2, let us fix  1 >  * 1 and  2 >  * 2 and let  fl  1 +  2 .Then, there is  = (,  1 ,  2 ) > 0 such that equation has infinitely many periodic (subharmonic) solutions as well as solutions presenting a complex dynamics, with () > 0, provided that (⋅) : R → R is a (measurable) -periodic function satisfying In Theorem 7 the solutions are considered in the Carathéodory sense [78] (when () is only measurable).On the other hand, we can also take an arbitrarily smooth function () which approximates the step function (), 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

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 for V : R → R + 0 a periodic stepwise function of fixed period T > 0, of the form with V 1 ̸ = V 2 and 0 < T 1 < T. As before, we also set T 2 fl T−T 1 .Writing (87) as an equivalent first order system in the phase-plane, we obtain the following result.
The constant  *  can be explicitly determined in terms of  and T 1 , T 2 .
Proof.As in the proof of Theorem 2, we suppose (the treatment of the other situation is completely similar and thus is omitted).The change of variables  fl  and () fl (), () fl V(), transforms system (89) to the equivalent first order system for () fl V() a stepwise periodic function of period  fl T/.By (88) and setting  1 fl T 1 / and  2 fl T 2 /, it follows that () = Notice that if we denote by Φ the Poincaré map associated with system (91) on [0, ], then it follows that Ψ  = Φ (one can easily check this fact, because ((), ()) = ((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 ⊆ R + 0 × R and two constants  * 1 and  * 2 such that the chaotic dynamics for Φ (according to Definition 1) is ensured provided that  1 >  * 1 and  2 >  * 2 .The lower estimates on  1 and  2 transfer to an upper bound for , so that if then the chaotic dynamics for Ψ  on the same set D is guaranteed.In particular, recalling the definition of  * 1 in (61) and  * 2 in (75), we derive a precise estimate to  *  .For instance, if we take  1 = 2,  2 = 1 and () =  3 (like in the example of Figure 2) then we know that  * 2 ≥ min{T 1 /50, T 2 /10} and therefore, if we choose  < min{T 1 /50, T 2 /10}, the conclusion of Theorem 8 holds.
We observe that, similarly to Remark 6, the stability of the result with respect to small perturbations of the coefficients is guaranteed, too.

Boundary Value Problems on Finite Intervals: Positive Solutions
We start this section by briefly describing how the method applied in the proof of Theorem (see (17)) and look for multiplicity results, where the number of the solutions increases as the time-interval length grows.
As in Section 2 we suppose that  : [0, ] → R + 0 is a stepwise function of the form (19) for  1 >  2 (the case in which  1 <  2 can be treated in a similar manner).The assumptions on () are the same as in Section 2.
Following the argument of the proof of Theorem 2 we consider the Poincaré map Φ associated with the planar system () as well as its components Φ  associated with systems (  ).The difference with respect to the proof of Theorem 2, consists into the fact that this time we look for initial points  0 fl ( 0 , 0), with  0 > 0 such that the second component of Φ( 0 ) fl (; 0,  0 ) vanishes.In other words, we apply a shooting method, looking for a solution which departs at time  = 0 from a point on the positive -axis and hits again the (positive) -axis at the time  = , with () > 0 for all  ∈ [0, ].
We repeat step by step (keeping the same notation) the geometrical construction in the proof of Theorem 2. In particular, as before, we choose the closed orbits Γ of system ( 1 ) and Γ  2 of system ( 2 ) in order to produce the regions A and B in the phase-plane.Recall also that  1 ( 2 ) >  1 ( 1 ) (see (51)).We have already indicated by (  , 0) the intersection point of Γ   1 with the positive -axis which is closer to the origin.We introduce now also the second intersection point of Γ   1 with the positive -axis, which will be denoted by (  , 0).Clearly we have (compared also with Figure 2).We produce the solutions of (94) by shooting from initial points  0 = ( 0 , 0) with 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 [ 2 ,  1 ] × {0} of the phase-plane and apply the Poincaré map Φ 1 for a time  1 sufficiently long so that the image of such segment crosses at least  times the set B. Then we switch to the Poincaré map Φ 2 and apply it for a time  2 sufficiently long so that the above arcs crossing B will be transformed (by Φ 2 ) to some curves winding around the point  2 and crossing at least  times the -axis.Putting all together these facts we conclude that there are at least  ×  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 of ( 1 ) in polar coordinates with respect to the center  1 = ( 01 , 0), using a clockwise orientation for the angular coordinate.In this case, instead of taking  ∈ A, we have  ∈ [ 2 ,  1 ] × {0} and therefore (0, ) = −.Accordingly, we replace the path  : [0,1] → A with the map which parameterizes the segment [ 2 ,  1 ] × {0}, namely we take As a consequence, for we have that  (,  ()) ⋛ ( − 1)  if and only if  ⋛   1 ( ()) 2 . (99) For an integer  ≥ 2, we define  * 1 as in ( 61) and fix  1 >  * 1 .Then, repeating the same argument of the proof of Theorem 2, we define the integers  1 and  2 and obtain and (which, in this case, is an obvious choice).As a consequence we find  pairwise disjoint intervals [   ,    ] ⊆ [ 2 ,  1 ] for  = 0, . . .,  − 1 such that and, moreover, After the time  1 we switch to equation ( 2 ).As remarked before, the study of the solutions in the time interval [ 1 , ] is equivalent to the study for 0 ≤  ≤  2 , where, for the moment,  2 is not yet fixed.We introduce another system of polar coordinates with center at the point  2 = ( 02 , 0) and take the clockwise orientation as a positive orientation for the angles starting from the positive half-line {(, 0) :  >  02 }.In this manner, we can associate an angular coordinate (; ) to any solution ( 2 ) with initial point  ∈ B and  ∈ [0,  2 ].In order to have the condition   () = 0 satisfied, we look for solutions (in the phase-plane) such that ( 2 ) = 0.In terms of these new angular coordinates, this corresponds to the condition ( 2 , ) =  for some positive integer .
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 orbits with respect to the -axis, we find that, for any positive integer  and for any  ∈ B, such that ( 2 ;    ) = .The corresponding solution () has precisely -zeros of the derivative in the interval ]0,  2 ].
The solutions that we have produced are only those obtained by shooting from [ 2 ,  1 ] × {0}, achieving the set B at the time  1 and coming back at the -axis at the time .With obvious changes in the argument, we could produce solutions which are in A at the time  1 .Moreover, we could also start from the segment [ 1 ,  2 ] × {0} and reach the region B (or, respectively, A) at the time  1 .Therefore, with the same technique, we can produce four different classes of × solutions.Finally, we can consider (by suitably modifying the same approach) also the case  2 >  1 > 0.
A variant of Theorem 9 for problem (93) with  sufficiently small can be also obtained via the same change of variables as in the proof of Theorem 8.
with  =  1 +  2 +  3 and  1 ,  2 ,  3 positive constants with In fact, under these assumptions, we can find solutions of (108) which are positive on ]0, [ and oscillate a certain (prescribed) number of times in the phase-plane around the point  2 = ( 02 , 0), during the time interval [ 1 ,  1 +  2 ].The number of such solutions can be larger than a preassigned number , provided that  2 >  * 2 for  * 2 a sufficiently large time (depending on ).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.

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

Figure 2 :
Figure 2: The present figure describes our geometric construction.The points on the -axis marked with a black circle (from right to the left) represent ( 1 , 0) and ( 2 , 0) on the orbits Γ  1 1 and Γ  2 1 .For this figure we have considered () =  3 and  1 = 2,  2 = 1.For graphical reasons a slightly different and -scaling has been used.

Figure 4 :Figure 5 :
Figure 4: The present figure describes how a path () crossing A from A −  to A −  is stretched by Φ 1 to a path Φ 1 (()) which crosses twice B from B −  to B −  .As in Figure 2 we have considered () =  3 and  1 = 2,  2 = 1.For this example we have taken  1 = 50.

Figure 7 :
Figure 7: The present figure shows some possible rectangular regions which can be considered for the application of Theorem 4.The sets A and B are the same as in Figure3and (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,  1 ] and oscillate in the phase-plane around the point (− 01 , 0), and then, in the time interval [ 1 , ] oscillate a certain number of times around the origin.More in detail, given any positive integer , we can produce solutions () of (17) which have precisely 2, 4, . . ., 2 simple zeros in the interval ] 1 , [, provided that  2 =  −  1 is sufficiently large.A lower estimate for  2 can be easily determined by the knowledge of the period of the closed trajectory Γ  2 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 () =  3 and  1 = 2,  2 = 1.For graphical reasons a slightly different and -scaling has been used.

Figure 8 :
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 ( 1 ) by considering initial points  0 = ( 0 , 0) with  01 <  0 ≤  1 (recall that  1 = ( 01 , 0) is the positive center of ( 1 ), while ( 1 , 0) is the intersection point of the homoclinic orbit Γ 0 1 with the positive -axis).We parameterize such initial points as an arc ().If  1 is sufficiently large (with a lower bound which can be easily estimated from the equation) we find that Φ 1 (()) is a spiral-like curve winding a certain number of times around  1 and with an end point on Γ 0 1 in the fourth quadrant near the origin.If we fix a (small) positive constant   and denote by C fl {(, ) : 0 ≤  2 (, ) ≤   } the region between the homoclinic trajectories
2 can be adapted to obtain multiplicity results for the Neumann boundary value problem − 2   + V ()  =  ()   (0) =   () = 0 (93) (see (87)).Our aim is to find multiple positive solutions for (93), where the number of the solutions becomes arbitrarily Complexity large as  → 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 −  +  ()  =  ()   (0) =   () = 0