Abstract
This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits an attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the network and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov–Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the “classical” Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory.
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Notes
Also called Belyakov transitions.
A \(\mathcal {F}_{\mu }\)-fixed point O is dissipative if O is hyperbolic and \(|\det \mathcal {F}_{\mu }(O)| <1\).
The open set is defined in the phase space; the set \(\tilde{\mathcal {V}}\) is defined in the space of parameters.
The invariant circles, in the first return map, do not envelop the phase cylinder.
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Acknowledgements
The author is grateful to Isabel Labouriau for the fruitful discussions during the research work performed in [37]. Special thanks to Andrey Shilnikov for pointing out the paper [52] on bifurcations analysis of a low-order atmospheric circulation model. The author is indebted to the two reviewers for the constructive comments, corrections and suggestions which helped to improve the readability of this manuscript.
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AR was partially supported by CMUP (UID/MAT/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. AR also acknowledges financial support from Program INVESTIGADOR FCT (IF/00107/2015)
Appendix A. Glossary
Appendix A. Glossary
We record a miscellaneous collection of terms and terminology that are used throughout the text. For \(\varepsilon >0\) small, consider the 3-parameter family of \(C^3\)–smooth autonomous differential equations
Since \({\mathbb {S}}^3\) is a compact set without boundary, the local solutions of (A.1) could be extended to \({\mathbb {R}}\). Denote by \(\varphi _{(A, \lambda , \omega )}(t,x)\), \(t \in {\mathbb {R}}\), the associated flow.
1.1 Symmetry
Given a compact Lie group \(\mathcal {G}\) of endomorphisms of \({\mathbb {R}}^4\), we will consider 3-parameter families of vector fields \((f_{(A, \lambda , \omega )})\) under the equivariance assumption
for all \(x \in {\mathbb {S}}^3\), \(\gamma \in \mathcal {G}\) and \((A, \lambda , \omega )\in [0, \varepsilon ]^2\times {\mathbb {R}}^+.\) For an isotropy subgroup \(\widetilde{\mathcal {G}}< \mathcal {G}\), we write \({\text {Fix}}(\widetilde{\mathcal {G}})\) for the vector subspace of points that are fixed by the elements of \(\widetilde{\mathcal {G}}\). Note that, for \(\mathcal {G}-\)equivariant differential equations, the subspace \({\text {Fix}}(\widetilde{\mathcal {G}})\) is flow-invariant.
1.2 Attracting Set
A subset \(\Omega \) of \( {\mathbb {S}}^3\) for which there exists a neighborhood \(U \subset {\mathbb {S}}^3\) satisfying \(\varphi _{(A, \lambda , \omega )}(t,U)\subset U\) for all \(t\ge 0\) and
is called an attracting set by the flow of (A.1). This set is not necessarily connected. Its basin of attraction, denoted by \(\mathbf{B} (\Omega )\), is the set of points in \( {\mathbb {S}}^3\) whose orbits have \(\omega -\)limit in \(\Omega \). We say that \(\Omega \) is asymptotically stable (or \(\Omega \) is a global attractor) if \(\mathbf{B} (\Omega )= {\mathbb {S}}^3\). An attracting set is said to be quasi-stochastic if it encloses periodic solutions with different Morse indices (dimension of the unstable manifold), structurally unstable cycles, sinks and saddle-type invariant sets (cf. [24]).
1.3 Heteroclinic Structures
Suppose that \(O_1\) and \(O_2\) are two hyperbolic equilibria of (A.1) with different Morse indices (dimension of the unstable manifold). There is a heteroclinic cycle associated to \(O_1\) and \(O_2\) if
For \( i\ne j \in \{1,2\}\), the non-empty intersection of \(W^{u}(O_i)\) with \(W^{s}(O_j)\) is called a heteroclinic connection between \(O_i\) and \(O_j\), and will be denoted by \([O_i \rightarrow O_j]\). Although heteroclinic cycles involving equilibria are not a generic property within differential equations, they may be structurally stable within families of vector fields which are equivariant under the action of a compact Lie group \(\mathcal {G}\subset \mathbb {O}(4)\), due to the existence of flow-invariant subspaces [26].
A heteroclinic cycle between two hyperbolic saddle-foci of different Morse indices, where one of the connections is transverse while the other is structurally unstable, is called a Bykov cycle. We address the reader to [30] for an overview of heteroclinic bifurcations and substantial information on the dynamics near different types of structures.
1.4 Historic Behaviour
We say that the solution of (A.1), \(\varphi _{(A, \lambda , \omega )}(t,x)\) with \(x \in {\mathbb {S}}^3\), has historic behaviour if there is a continuous function \({H}:{\mathbb {S}}^3\rightarrow {\mathbb {R}}\) such that the time average \(\displaystyle \frac{1}{T}\int _{0}^{T} {H} (\varphi _{(A, \lambda , \omega )}(t,x)) dt\, \, \) fails to converge.
1.5 Strange Attractor
A (Hénon-type) strange attractor of a two-dimensional dissipative diffeomorphism R defined in a Riemannian manifold \(\mathcal {M}\), is a compact invariant set \(\Lambda \) with the following properties:
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\(\Lambda \) equals the closure of the unstable manifold of a hyperbolic periodic point;
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the basin of attraction of \(\Lambda \) contains an open set;
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there is a dense orbit in \(\Lambda \) with a positive Lyapounov exponent (exponential growth of the derivative along its orbit).
A vector field possesses a strange attractor if the first return map to a cross section does.
1.6 SRB Measure
Given an attracting set \({\Omega }\) for a continuous map \(R: \mathcal {M} \rightarrow \mathcal {M}\) where \( \mathcal {M}\) is a compact smooth manifold, consider the Birkhoff average with respect to the continuous function \(T: \mathcal {M} \rightarrow {\mathbb {R}}\) on the R-orbit starting at \(x\in \mathcal {M}\):
Suppose that, for Lebesgue almost all points \(x\in \mathbf{B} ({\Omega })\), the limit (A.2) exists and is independent on x. Then L is a continuous linear functional in the set of continuous maps from \(\mathcal {M}\) to \({\mathbb {R}}\) (denoted by \(C(\mathcal {M}, {\mathbb {R}})\)). By the Riesz Representation Theorem, it defines a unique probability measure \(\mu \) such that:
for all \(T\in C(\mathcal {M}, {\mathbb {R}})\) and for Lebesgue almost all points \(x\in \mathbf{B} ({\Omega })\). If there exists an ergodic measure \(\mu \) supported in \({\Omega }\) such that (A.3) is satisfied for all continuous maps \(T\in C(\mathcal {M}, {\mathbb {R}})\) for Lebesgue almost all points \(x\in \mathbf{B} ({\Omega })\), where \(\mathbf{B} ({\Omega })\) has positive Lebesgue measure, then \(\mu \) is called a SRB (Sinai-Ruelle-Bowen) measure and \({\Omega }\) is a SRB attractor. More details in [55].
1.7 Non-trivial Wandering Domains
A non-trivial wandering domain for a given map R on a Riemannian manifold \( \mathcal {M}\) is a non-empty connected open set \(D \subset \mathcal {M}\) which satisfies the following conditions:
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\(R^i(D)\cap R^j(D)=\emptyset \) for every \(i,j\ge 0\) (\(i\ne j\))
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the union of the \(\omega \)-limit sets of points in D for R, denoted by \(\Omega (D,R)\), is not equal to a single periodic orbit.
A wandering domain D is called contracting if the diameter of \(R^n(D)\) converges to zero as \(n \rightarrow +\infty \).
1.8 Rotational Horseshoe
Let \(\mathcal {H} \) stand for the infinite annulus \(\mathcal {H} = {\mathbb {S}}^1 \times {\mathbb {R}}\) (endowed with the usual inner product from \({\mathbb {R}}^2\)). We denote by \(Homeo^+(\mathcal {H} )\) the set of homeomorphisms of the annulus which preserve orientation. Given a homeomorphism \(f :X \rightarrow X\) and a partition of \(m\in {\mathbb {N}}\backslash \{1\}\) elements \(R_0,..., R_{m-1}\) of \(X\subset \mathcal {H}\), the itinerary function \(\xi : X \rightarrow \{0, ..., m-1\}^{\mathbb {Z}}= \Sigma _m\) is defined by:
Following [43], we say that a compact invariant set \(\Lambda \subset \mathcal {H} \) of \(f \in Homeo^+(\mathcal {H} )\) is a rotational horseshoe if it admits a finite partition \(P =\{R_0, ..., R_{m-1} \}\) by sets \(R_i\) with non empty interior in \(\Lambda \) so that:
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the itinerary \(\xi \) defines a semi-conjugacy between \(f|_\Lambda \) and the full-shift \(\sigma : \Sigma _m \rightarrow \Sigma _m\), that is \(\xi \circ f = \sigma \circ \xi \) with \(\xi \) continuous and onto;
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for any lift \(F: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2\) of f, there exist \(k>0\) and m vectors \(v_0, ...,v_{m-1} \in {\mathbb {Z}}\times \{0\}\) so that:
$$\begin{aligned} \left\| (F^n(\hat{x})-\hat{x}) - \sum _{i=0}^n v_{\xi (x)(i)}\right\| <k \qquad \text {for every} \qquad \hat{x}\in \pi ^{-1}(\Lambda ), \quad n\in {\mathbb {N}}, \end{aligned}$$where \(\Vert \star \Vert \) is the usual norm on \({\mathbb {R}}^2\), \(\pi :{\mathbb {R}}^2\rightarrow \mathcal {H}\) denotes the usual projection map and \(\hat{x} \in \pi ^{-1}(\Lambda )\) is the lift of x; more details in the proof of Lemma 3.1 of [43]. The existence of a rotational horseshoe for a map implies positive topological entropy at least \( \log m \).
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Rodrigues, A.A.P. Dissecting a Resonance Wedge on Heteroclinic Bifurcations. J Stat Phys 184, 25 (2021). https://doi.org/10.1007/s10955-021-02811-4
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DOI: https://doi.org/10.1007/s10955-021-02811-4