A Reinjected Cuspidal Horseshoe

Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finite iterations. In this work we construct a model that possesses an attracting set that contains a cuspidal horseshoe with positive entropy. This model is obtained by reinjecting the points that escape the horseshoe and can be realized in a 3-dimensional vector field.


Introduction
Smale's horseshoe [19] was introduced in late 1960's as an example of a chaotic and hyperbolic dynamical system which is topologically transitive, and contains a countable set of periodic orbits, see [17] for more details.It has been shown that such model is observed (up to a topological conjugacy) when the stable manifold and the unstable manifold of a hyperbolic fixed point of a twodimensional diffeomorphism intersect transversally [16].
The dynamics of the horseshoe is chaotic on the topological product of two Cantor sets, and its complement is dense in a neighborhood so that most nearby points escape after finite iterates.In this article we propose a model that contains similar features but also richer dynamics.This model is essentially obtained by 'reinjecting' the points that escape a neighborhood of the horseshoe.We show that this model can be realized by considering the unfolding of a 3-dimensional vector field that possesses two homoclinic orbits to the same singularity, one being degenerate and the other nondegenerate.

The model
In the (x, y)-plane consider the domain S = S + ∪ S − where S + = (0, 1] × [−1, 1], and The image of S under Φ is described in Figure 1: the restriction of Φ on S + is like a horseshoe except that one boundary is collapsed onto a cusp.The points that escape S + are reinjected into S + via the action of Φ on S − .More precisely the model satisfies the following properties: (i) Φ maps S + and S − diffeomorphically onto their respective images and (v) there exists P + and P − in S + such that lim where the limits are taken with respect to the Hausdorf distance, Define the following set: We further assume (vii) the restriction of Φ on Ω + Z is hyperbolic.The dynamics of such a model is very rich since it contains the set Ω + Z .This set is often called a cuspidal horseshoe, which was introduced almost at the same time in [4] and [10], see also [20].More precisely the authors show that a suspended cuspidal horseshoe can be observed in the unfolding of a degenerate homoclinic orbit in R 3 .For typical values of the parameters, the corresponding dynamics generalize that of Smale's horseshoe: it is conjugate with a partial shift on two symbols i.e., there exists a set B ⊂ {0, 1} Z that is invariant under the shift σ : {0, 1} → {0, 1} and a homeomorphism ϕ : However, in general, the corresponding dynamics is not a subshift of finite type, see for instance [10] for more details.The unfolding of a degenerate homoclinic orbit can lead to dynamics with high complexity, including for instance, the geometric Lorenz attractor [3,18] or a suspended Hénon-like attractor [14].
The goal of this paper is to show that the dynamics of the set is realizable in the generic unfolding of a double homoclinic orbit in 3-dimensional space.In the next section we describe how we retrieve that dynamics and state the main theorem of the article.Section 4 is devoted to the sketch of the proof of Theorem 1.We finish the article with a brief discussion of the dynamics on Ω Z .

A suspension
Let X γ be a smooth family of vector fields on R 3 , γ ∈ V, where V ⊂ R 3 is a neighborhood of the origin.Assume that for all γ ∈ V, X γ admits a hyperbolic singularity of saddle type.By the implicit function theorem, after a parameter dependent translation, one can assume that the singularity is fixed at the origin.We also assume that the eigenvalues {−α(γ), −β(γ), λ(γ)} of dX γ (0) satisfy After a time rescaling, one can assume that λ(γ) ≡ 1 and X γ takes the following form where α(γ) > 1 and R stands for the nonlinear terms.This implies that X γ admits a 2-dimensional local stable manifold, denoted by W s loc and 1-dimensional local unstable manifold denoted by W u loc .We extend W s loc by the backward iteration of the flow and obtain the global stable manifold denoted by W s .The local strong stable manifold W ss loc ⊂ W s loc has its tangent space at the origin spanned by the eigenspace associated to −α(γ).These manifolds are invariant, smooth, and unique.Furthermore, there exists a local invariant manifold which is tangent at the origin to the eigenspace associated with the eigenvalues 1 and −β.This manifold is denoted by W s,u loc is called an extended unstable manifold.Such an invariant set contains W u loc , is not unique, and is C k where k is the integer part of α/β.However its tangent space along the local unstable manifold W u loc is unique.See [8] for more details.In this setting we have and we can choose W u,s loc ⊂ {y = 0}.
We further assume that X 0 admits two homoclinic orbits to the origin, Γ 1 is said to be an inclination-flip homoclinic orbit if the global stable manifold W s intersects any extended unstable manifold along Γ 1 in a non-transversal manner.At the same time we assume that Γ 2 is a nondegenerate homoclinic orbit, i.e., the global stable manifold W s intersects any extended unstable manifold along Γ 2 transversally, and Observe that the latter assumptions amounts to assuming that Γ 2 is not of orbit-flip type.See [3,4,10,12,14,20] for more details and discussions.
In the 3-dimensional context, a double homoclinic loop forms either an 'eight' shape or a 'butterfly'.In this article we assume that Γ 1 ∪ Γ 2 forms a butterfly: W ss loc splits W s loc into two connected components, say and we assume that Γ 1 (t) ∪ Γ 2 (t) ⊂ W s + for t large enough.Let S be a 2-dimensional section transverse to the local stable manifold, and let Σ + , Σ − be 2-dimensional sections, each transverse to a branch of the unstable manifold, as indicated in Figure 3.
After some rescaling, we choose those sections to be where δ 1 > 0 and δ 2 > 0 are sufficiently small in such a way that the corresponding Poincaré transition maps In this setting, X 0 admits Γ 1 as inclination flip homoclinic orbit if and only if ε 1 (0) = 0 = µ(0), and Γ 2 is a nondegenerate homoclinic orbit if and only if ε 2 (0) = 0 but b(0) = 0. We also assume that, for the unperturbed system, W s and W u,s loc have a tangency of quadratic contact, which further implies that We finally assume that the map γ → (ε 1 (γ), ε 2 (γ), µ(γ)) is a diffeomorphism near 0. ( From now on we identify γ with (ε 1 , ε 2 , µ).
Lemma 1 Let X γ be a family of vector fields as above.Let Σ be any section transverse to Γ 1 .Then the Poincaré return map F associated to X 0 is semiconjugate with a C 1 diffeomorphism G, Γ 1 being in one-to-one correspondence with a fixed point Q of G. Morever, the eigenvalues of dG(Q) are 0 and c M = 0.This latter eigenvalue is called the Melnikov exponent.We are now in position to state the main theorem of this article.
Theorem 1 Let X γ , be a family of smooth vector fields as above.We further assume that the Melnikov exponent satisfies 1 < c M < 4. Then there exists an open set V * ⊂ V such that for all γ ∈ V * , X γ possesses a suspended reinjected cuspidal horsehoe.

Sketch of proof
To study the dynamics of the system, our strategy is to estimate the Poincaré return map on the section S. We take δ 1 sufficiently small in such a way that this map is well defined.Observe that the flow starting from a point on the stable manifold accumulates to the origin, therefore the stable manifold acts as a separatrix and the Poincaré return map Φ γ is defined on S + ∪ S − where We state the following lemma.

Lemma 2
The transition maps take the form The proof of this lemma follows from results in [2] and in [15].By Lemma 2, we are able to compute the Poincaré return map onto S. We state the following proposition.
Proposition 1 The Poincaré return map can be expressed as Note that the Poincaré return map depends on y only through the higher-order terms.In what follows we assume that c(0) > 0, b(0) < 0 (see Figure 2) but the other cases can be treated similarly.Let ε 0 > 0 be small.We define the following blow-up in the parameter space and the following rescaling in the phase space where h 1 , h 2 , g 1 and g 2 are smooth functions and where From this proposition one can prove that the Poincaré return map satisfies properties (i) and (ii).Now in order to show the remaining items (iii) to (vii), we have to show that there exists a system of coordinates where some vertical lines (i.e.lines of the form {b} × [−1, 1]) are mapped onto vertical lines.It is sufficient to show that this holds for some leaves of a foliation.The existence of such a foliation follows using the invariance under dF of a cone-field.More precisely we state the following lemma.

Lemma 3 Let P ∈ Ω +
Z and V , W ∈ T P S + where Let L {n} = dF n (P ) for n ∈ Z.  5).In this new system of coordinates the Poincaré return map F is 'close' to f .
For all ε 0 sufficiently small, there exists τ > 0 such that if From this lemma we deduce the hyperbolicity of Ω + Z , item (vii).Furthermore following the techniques developed by Moser, [13] we construct a C 1 foliation where each leaf F u0 is a vertical graph i.e., for each u 0 ∈ [−1/2, 1], there exists a C 1 function g with |g (v)| ≤ τ and Observe that F b3 coincides with the stable manifold of a fixed point.Since such an invariant C 1 foliation exists, there exists a C 1 change of coordinates that trivializes the foliation so that each leaf F a maps to {a} × [−1, 1].In particular after a linear rescaling one can set b 3 = 1 and ignore leaves of the form F b4 where b 4 > 1. Items (iii) to (vi) follow.

Dynamics of the model
From the definition of the model reinjected cuspidal horseshoe, S is forwardinvariant with Φ(S) ⊂ int (S), and Ω Z = ∩ n≥0 Φ n (S) is a (non-compact) "attracting set".This implies that the suspension of Φ in a neighborhood of a double homoclinic loop in a 3-dimensional flow possesses an attractor A such that A ∩ S = Ω Z .
As described previously, Ω Z contains the cuspidal horseshoe Ω + Z , which typically is a hyperbolic, chaotic set for a large set of parameters, see [4,10,20] for more details.A natural question is how much richer is the dynamics on Ω Z ?
We leave a detailed investigation of the dynamics of the model reinjected cuspidal horseshoe for a future work.However, we briefly describe the methods we expect to employ.Since Ω Z is not a hyperbolic set, obtaining an analytical characterization in terms of symbolic dynamics, for example, is difficult.We plan to compute rigorous symbolic dynamics for the model utilizing computational topological tools that have been developed in recent years that exploit the Conley index theory [6,5,11,1].
The first step is to combinatorialize the dynamics by dividing the phase space into a set of small rectangles (boxes) B. A rigorous covering of the image of each box by a subset of boxes in B using interval arithmetic produces a multivalued map F : B ⇒ B which outer approximates the dynamics of Φ. Figure 4 concerns the map .75, M = 3.47098748700613, and ε = 1/40.These parameters were chosen so that the re-injected cusp (image of the left side of the domain) is exaggerated in size for better visualization.
In both plots the green region is a rigorous box covering B of the (chain) recurrent set of the map, i.e. all periodic and chaotic orbits in Ω Z must be contained in this region.The bottom plot superimposes a numerically simulated orbit starting from initial condition x = y = 0.1 where the first 1000 points are ignored and the next 30, 000 points are plotted.Note that there is good agreement between the numerically simulated orbit and the rigorous outer approximation.The simulated orbit is qualitatively the same if the initial conditions are chosen randomly in the rectangle.In the top plot, the blue region is a rigorous covering of the (chain) recurrent set of the map restricted to the right side.Since Ω + Z is not an attractor, we cannot readily simulate the dynamics here.However, this region is clearly a much smaller portion of the phase space than the green region, which suggests significantly more structure in the dynamics on Ω Z versus Ω + Z .The multivalued map F can be interpreted as a directed graph, and using efficient graph algorithms, we can extract boxes that form cycles of various periods.Following the algorithms described in [5], it is possible to then construct a rigorous symbolic dynamics for Φ, and lower bounds on the topological entropy, using the Conley index.These set-based techniques can also be used to approximate invariant measures using the methods outlined in [7].
The results of these computations will be the subject of future work.However, we can provide some preliminary information through a rigorous comparison of periodic orbits using the Conley index.At the resolution indicated in the figures, we were able to isolate and verify the existence of 21 distinct periodic orbits: 2 fixed points, 1 period-2 orbit, 2 period-3 orbits, 3 period-4 orbits, 4 period-5 orbits, 4 period-6 orbits, 3 period-7 orbits, and 2 period-8 orbits.Of those periodic orbits that are found, only 3 of them lie completely in the right side: the 2 fixed points and the period-2 orbit.Thus we are able to see a significant difference in the structure of the dynamics of the full system versus the system restricted to the right hand side.

Figure 1 :
Figure 1: Description of the map Φ.

Figure 2 :
Figure 2: Phase portrait of X 0 .The top figure emphasizes Γ 1 and the quadratic tangency between W ss and W u,s loc .The bottom figure is deduced from that on the top after a rotation about the z-axis and emphasizes Γ 2 .

Figure 3 :
Figure3: Graph of the function f (u) in equation(5).In this new system of coordinates the Poincaré return map F is 'close' to f .

Figure 4 :
Figure 4: The sets Ω Z and Ω + Z for a model map, see Section 5.The red circles indicate the cusps.