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Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

Dongchen Li, Xiaolong Li, Katsutoshi Shinohara, Dmitry Turaev

TL;DR

The paper establishes a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit in $C^r$ systems with $ ext{dim}\mathcal{M}\geq 3$ (diffeomorphisms) or $ ext{dim}\mathcal{M}\geq 4$ (flows). In two-parameter unfoldings, when at least one central multiplier is non-real and the central dynamics are not sectionally dissipative, $C^1$-robust heterodimensional dynamics of coindex one arise, organized by two center-blenders whose intersections persist under perturbation. The construction hinges on a precise two-parameter normal-form analysis of the first-return map in the saddle-focus case (and its bi-focus reduction), the creation of a higher-index periodic point $Q$, and the realization of a nondegenerate heterodimensional cycle involving $O$ and $Q$, augmented by $C^1$-robust tangencies via blenders. Consequently, systems with such homoclinic tangencies lie in the $C^r$-closure of the $C^1$-open Bonatti-Díaz (and, by corollaries, Newhouse) domains, linking robust nonhyperbolic dynamics to established nonhyperbolic domains and extending blender-based results to higher dimensions.

Abstract

We establish a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit. We prove for $C^r$ ($r=3,\dots,\infty,ω$) dynamical systems on a manifold $\mathcal{M}$, with $\dim \mathcal{M}\geqslant 3$ for diffeomorphisms and with $\dim \mathcal{M}\geqslant 4$ for flows, that $C^1$-robust heterodimensional dynamics of coindex one appear in any generic two-parameter $C^r$ unfolding of a homoclinic tangency to a periodic orbit such that at least one central multiplier is not real and the central dynamics are not sectionally dissipative. The heterodimensional dynamics also involve a blender exhibiting $C^1$-robust homoclinic tangencies. As a corollary, any system with a homoclinic tangency of the class described above belongs to the $C^r$ closure of the $C^1$-open Newhouse domain.

Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

TL;DR

The paper establishes a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit in systems with (diffeomorphisms) or (flows). In two-parameter unfoldings, when at least one central multiplier is non-real and the central dynamics are not sectionally dissipative, -robust heterodimensional dynamics of coindex one arise, organized by two center-blenders whose intersections persist under perturbation. The construction hinges on a precise two-parameter normal-form analysis of the first-return map in the saddle-focus case (and its bi-focus reduction), the creation of a higher-index periodic point , and the realization of a nondegenerate heterodimensional cycle involving and , augmented by -robust tangencies via blenders. Consequently, systems with such homoclinic tangencies lie in the -closure of the -open Bonatti-Díaz (and, by corollaries, Newhouse) domains, linking robust nonhyperbolic dynamics to established nonhyperbolic domains and extending blender-based results to higher dimensions.

Abstract

We establish a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit. We prove for () dynamical systems on a manifold , with for diffeomorphisms and with for flows, that -robust heterodimensional dynamics of coindex one appear in any generic two-parameter unfolding of a homoclinic tangency to a periodic orbit such that at least one central multiplier is not real and the central dynamics are not sectionally dissipative. The heterodimensional dynamics also involve a blender exhibiting -robust homoclinic tangencies. As a corollary, any system with a homoclinic tangency of the class described above belongs to the closure of the -open Newhouse domain.
Paper Structure (35 sections, 22 theorems, 247 equations, 1 figure)

This paper contains 35 sections, 22 theorems, 247 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Effective dimension and robust heterodimensional cycles
  3. Blenders and $C^1$-robust homoclinic tangencies
  4. Heterodimensional dynamics in two-parameter unfolding families
  5. General setup
  6. Proper unfoldings
  7. Genericity conditions
  8. First-return map in the saddle-focus case
  9. Local and global maps
  10. Normal form for the first-return map
  11. Composition.
  12. Shilnikov coordinates.
  13. Removing linear in $Y$ terms.
  14. Shift of the origin.
  15. Selecting the center coordinates.
  16. ...and 20 more sections

Key Result

Theorem 1

Let $f$ be a system of class $C^r$ ($r=1,\dots,\infty,\omega$) which has a generic homoclinic tangency to a hyperbolic periodic orbit $L$ with $d_{\rm eff}>1$. Then, arbitrarily $C^r$-close to $f$, there exists a system $\tilde{f}$ having $C^1$-robust heterodimensional dynamics, which involve two no

Figures (1)

  • Figure 1: A schematic picture for the local manifolds and the strong-stable foliation, where in particular the transversality condition is drawn (i.e., condition C3 of Section \ref{['sec:gc']}).

Theorems & Definitions (44)

  • Definition 1.1: Heterodimensional cycles
  • Definition 1.2: Robust heterodimensional dynamics
  • Definition 1.3: Homoclinic relations
  • Theorem 1
  • Corollary 2
  • Definition 2.1: Effective dimension
  • Theorem 3
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1: cf. GST08
  • ...and 34 more