Strange attractors and wandering domains near a homoclinic cycle to a bifocus

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Abstract

In this paper, we explore the three-dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits. Under an additional hypothesis about the coexistence of two heteroclinically related periodic points (one without dominated splitting into one-dimensional sub-bundles), the heteroclinic tangencies can be slightly modified in order to satisfy Tatjer's conditions for a generalized tangency of codimension two. This configuration may be seen as the organizing center, by which one can obtain Bogdanov-Takens bifurcations and therefore, strange attractors, infinitely many sinks and non-trivial contracting wandering domains.

Introduction

The homoclinic cycle to a bifocus provides one of the main examples of the occurrence of chaotic dynamics in four-dimensional vector fields. Examples of dynamical systems from applications where these homoclinic cycles play a basic role can be found in [12], [13]. Results in [2] show that homoclinic orbits to bifoci arise generically in unfoldings of four-dimensional nilpotent singularities of codimension four. Family (1.2) in [2] has been widely studied in the literature because of its relevance in many physical settings as, for instance, the study of travelling waves in the Korteweg-de Vries model.

The striking complexity of the dynamics near this type of homoclinic cycles has been discovered and investigated by Shilnikov [46], [47], who claimed the existence of a countable set of periodic solutions of saddle type. It was shown that, for any NN and for any local transverse section to the homoclinic cycle, there exists a compact invariant hyperbolic set on which the Poincaré map is topologically conjugate to the Bernoulli shift on N symbols. A sketch of the proof has been presented in [53]. In the works [10], [30], the formation and bifurcations of periodic solutions were studied. Motivated by [1], [43], the authors of [20] describe the hyperbolic suspended horseshoes that are contained in any small neighbourhood of a double homoclinic cycle to a bifocus and showed that switching and suspended horseshoes are strongly connected.

The spiralling geometry of the non-wandering set near the homoclinic cycle associated to the bifocus has been partially described in [10], where the authors studied generic unfoldings of the cycle. Breaking the cycle, and using appropriate first return maps, the authors visualized the structure of the spiralling invariant set which exists near the cycle. In the reversible setting, the authors of [17], [41] proved the existence of a family of non-trivial (non-hyperbolic) closed trajectories and subsidiary connections near this type of cycle. See also the works by Lerman's team [24], [32] who studied cycles to a bifocus in the hamiltonian context. In the general context, the complete understanding of the structure of this spiralling set is a hard task.

An important open question related to the homoclinic cycle to a bifocus is what type of dynamics typically occurs. In this paper, we study the dynamics near a homoclinic cycle to a bifocus at which the vector field has negative divergence, so that the flow near the equilibrium contracts volume. We are particularly interested in the occurrence of strange attractors and non-trivial wandering domains. We show that these phenomena occur for small C1-perturbations of the vector field which, in principle, no longer have the original homoclinic cycle.

Many aspects contribute to the richness and complexity of a dynamical system. One of them is the existence of strange attractors. According to [19], [51]:

Definition 1.1

A (Hénon-type) strange attractor of a three-dimensional dissipative diffeomorphism R, defined in a compact and riemannian manifold, is a compact invariant set Λ with the following properties:

  • Λ equals the closure of the unstable manifold of a hyperbolic periodic point;

  • the basin of attraction of Λ contains an open set (and thus has positive Lebesgue measure);

  • there is a dense orbit in Λ with a positive Lyapounov exponent (exponential growth of the derivative along its orbit);

  • Λ is not hyperbolic.

A vector field possesses a (Hénon-type) strange attractor if the first return map to a cross section does. In [52], there is another definition of strange atractor contemplating two expanding directions.

The rigorous proof of the strange character of an invariant set is a great challenge and the proof of the existence of such attractors in the unfolding of the homoclinic tangency is a very involving task. Mora and Viana [35] proved the emergence (and persistence) of strange attractors in the process of creation or destruction of planar Smale horseshoes that appear through a bifurcation of a tangential homoclinic (sectionally dissipative) point.

In the unfolding of a non-contracting Shilnikov cycle associated to a saddle-focus in R3 (details in [45]), horseshoes appear and disappear by means of generic homoclinic bifurcations, leading to persistent non hyperbolic strange attractors like those described in [35]. These tangencies give rise to suspended Hénon-like strange attractors. Without breaking the cycle, Homburg [19] proved the coexistence of strange attractors and attracting 2-periodic solutions near a homoclinic cycle to a saddle-focus in R3, when moving the saddle-value (see also [37]). He also proved that, despite the existence of strange attractors, a large proportion of points near the homoclinic cycle lies outside the basin of the attractor. Under a particular configuration of the spectrum of the vector field at the saddle (equal to 1), Pumariño and Rodríguez [40] proved that infinitely many of these strange attractors can coexist in non generic families of vector fields with a Shilnikov cycle, for a positive Lebesgue measure set of parameters.

The homoclinic cycle to a bifocus in R4 seems to be the scenario for more complicated dynamics than those inherent to the saddle-focus in R3, where the existence of such strange attractors has been proved. As far as we know, no result has been established relating the existence of bifocal homoclinic bifurcations with the existence of (persistent) strange attractors. In Theorem A, we prove that the first return map to a given cross section of the cycle associated to a bifocus can be C1-approximated by another map exhibiting strange attractors.

A wandering domain for a diffeomorphism is a non-empty connected open set whose forward orbit is a sequence of pairwise disjoint open sets. More precisely:

Definition 1.2

A non-trivial wandering domain (or just wandering domain) for a given map R on a Riemannian manifold M is a non-empty connected open set DM which satisfies the following conditions:

  • Ri(D)Rj(D)= for every i,j0 (ij)

  • the union of the ω-limit sets of points in D for R, denoted by ω(D,R), is not equal to a single periodic orbit.

A wandering domain D is called contracting if the diameter of Rn(D) converges to zero as n+.

In the early 20th century, the authors of [4], [9] constructed examples of C1 diffeomorphisms on a circle which have contracting wandering domains where the union of the ω–limit sets of points is a Cantor set. See also [34] and references therein. Similar behaviours in different contexts may be found in [5], [26], [27], [36]. The existence of non-trivial wandering domains in nonhyperbolic dynamics has been studied by Colli and Vargas [7], through a countable number of perturbations on the gaps of an affine thick horseshoe with persistent tangencies. The conjecture about the existence of contracting wandering sets near Newhouse regions was recently proved in [22] for diffeomorphisms, and partially in [29] for flows, when the authors were exploring persistent historic behaviour realised by a set with positive Lebesgue measure.

In the context of diffeomorphisms of the circle, if sufficient differentiability exists, Denjoy [9] proved that non-trivial wandering domains could not exist. The absence of wandering domains is the key for the classification of one-dimensional unimodal and multimodal maps, in real analytic category, a subject which has been discussed in [8], [33], [50]. For rational maps on the Riemannian sphere, we address the reader to [34]. Very recently, Kiriki et al. [23] presented a sufficient condition for three-dimensional diffeomorphisms having heterodimensional cycles (and thus non-transverse equidimensional cycles C1-close) which can be C1-approximated by diffeomorphisms with non-trivial contracting wandering domains and strange attractors.

A natural question arises: is there a configuration for a flow having a first return map with equidimensional cycles similar to those described in §3 of [23]? In other words, could we describe a general configuration (for a flow) giving a criterion for the existence of non-trivial wandering domains? Theorem B gives a partial answer to this question.

The goal of this paper is to show that a homoclinic cycle associated to a bifocus may be considered as a criterion for four-dimensional flows to be C1-approximated by other flows exhibiting strange attractors and contracting non-trivial wandering domains.

The main results are stated and discussed in §3, after collecting relevant notions in §2. Normal form techniques are used in §4 to construct local and return maps. Section 5 deals with the geometrical structures which allow to get an understanding of the dynamics. After reviving the proof of the existence of hyperbolic horseshoes whose suspension accumulates on the cycle (see §6), owing the results of [6], [23], [37], [49], in §7 we C1-approximate the first return map to the cycle by another diffeomorphism exhibiting a Tatjer tangency. This codimension-two bifurcation leads to Bogdanov-Takens bifurcations and subsidiary homoclinic connections associated to a sectionally dissipative saddle. In §8, we prove Theorem A, Theorem B. The itinerary of their proofs is summarised in Appendix A (Table 1).

The last step of the proof of Theorem B is similar to [23]. For the sake of completeness, we revisit the proof, addressing the reader to the original paper where the proof has been done. Throughout this paper, we have endeavoured to make a self contained exposition bringing together all topics related to the proofs. We have stated short results and we have drawn illustrative figures to make the paper easily readable.

Section snippets

Preliminaries

For k5 and A a compact and boundaryless subset of R4, we consider a Ck vector field f:AR4 defining a differential equation:x˙=f(x) and denote by φ(t,x), with tR, the associated flow. In this section, we introduce some essential topics that will be used in the sequel.

Description of the problem

The object of our study is the dynamics around a homoclinic cycle Γ to a bifocus defined on R4 for which we give a rigorous description here. Let X5(R4) the Banach space of C5 vector fields on R4 endowed with the C5-Whitney topology. Our object of study is a one-parameter family of C5 vector fields fλ:R4R4 with a flow given by the unique solution x(t)=φ(t,x)R4 ofx˙=fλ(x)x(0)=x0R4λR satisfying the following hypotheses for λ=0:

    (P1)

    The point O=(0,0,0,0) is an equilibrium point.

    (P2)

    The

Return maps

Using local coordinates near the bifocus we will provide a construction of local and global transition maps. In the end, a return map around the homoclinic cycle will be defined.

Notions related with spiralling behaviour

In this section, we introduce the notions of segment, spiral, helix, spiralling sheet and scroll. These definitions are adapted from [17], [20].

Definition 5.1

A segment s in ΣOin is a regular curve s:[0,1]ΣOin parametrized by t that meets Wlocs(O) transversely and only at a point s(0) and such that writing s(t)=(ϕsin(t),ruin(t),ϕuin(t)), then:

  • the components are monotonic functions of t and

  • ϕsin(t) and ϕuin(t) are bounded.

Similarly, we define a segment in ΣOout.

Definition 5.2

Let aR, D be a disc centered at pR2. A spiral

Three-dimensional whiskered horseshoes revisited

The existence of a homoclinic cycle Γ is considered as a mechanism to create three-dimensional chaos in the spirit of Shilnikov [45], [46] and Lerman [32]. In this section, we recall the main steps of the construction of the invariant horseshoe given in [20], adapted to our purposes. We address the reader to [53] for more details in the definitions.

Generalized homoclinic tangency

The goal of this section is to prove that, in the C1-topology, the map R01 may be approximated by a Cr diffeomorphism with a homoclinic tangency satisfying [T2]–[T4] of Definition 2.6, for r5. Once this is proved, by definition, the map R0 may be approximated by a Cr diffeomorphism with a Tatjer homoclinic tangency satisfying [T1]–[T4] – see page 257 of [49].

Remark 7.1

We suggest the reader to think on the geometry of R01 as it was the first return map for a cycle to a bifocus at which the vector field

Proof of the main results

Using the previous sections, it is easy to check that the map R51 satisfies Lemma 7.12 for the dissipative periodic point PM. Therefore, the map R51 may be seen as the organizing center by which we can obtain strange attractors and non-trivial contracting wandering domains. In Lemma 7.12, the parameter a is responsible for splitting the manifolds Ws(PM) and Wu(PM) and b is the parameter unfolding the degeneracy related to condition [T3].

Acknowledgements

The author thanks the helpful and valuable comments from Pablo Barrientos, Artem Raibekas, Shin Kiriki and Teruhiko Soma about the main results of this paper. A. Rodrigues was partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. The author also acknowledges financial support from Program INVESTIGADOR FCT (IF/00107/2015). Part of this work has been written

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