Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Persistence of Heterodimensional Cycles

Dongchen Li, Dmitry Turaev

Abstract

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least $C^2$, we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class $r=2,\ldots,\infty,ω$. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

Persistence of Heterodimensional Cycles

Abstract

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least , we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class . The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

Paper Structure

This paper contains 42 sections, 37 theorems, 261 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Stabilization of heterodimensional cycles
  3. Heterodimensionality vs. equidimensionality
  4. Applications of Theorem \ref{['thm:persis_hdd_2p']}
  5. Blenders
  6. Robust heterodimensional dynamics in finite-parameter families
  7. Local maps near periodic orbits
  8. Transition maps and geometric non-degeneracy conditions
  9. Local partial linearization and the fourth non-degeneracy condition
  10. Finite-parameter unfoldings
  11. Three types of heterodimensional cycles in the saddle case
  12. Main results for the saddle case
  13. The case of nonreal central multipliers
  14. Local stabilization of heterodimensional cycles
  15. First-return maps in the saddle case
  16. ...and 27 more sections

Key Result

Theorem A

Any dynamical system of class $C^r$ ($r=1,\ldots,\infty,\omega$) having a heterodimensional cycle of coindex 1 can be $C^r$-approximated by a system which has $C^1$-robust heterodimensional dynamics.

Figures (14)

  • Figure 1: A heterodimensional cycle involving two hyperbolic fixed points $O_1$ and $O_2$ (black dots) of a three-dimensional diffeomorphism. The cycle consists of the two fixed points, a fragile heteroclinic orbit $\Gamma^0$ (blue dots) in the non-transverse intersection of the one-dimensional invariant manifolds, and a robust heteroclinic orbit $\Gamma^1$ (red dots) in the transverse intersection (green curves) of the two-dimensional manifolds. See Section \ref{['sec:intro2']} for details.
  • Figure 2: Three cases of a heterodimensional cycle with two hyperbolic fixed points of a three-dimensional diffeomorphism. The central multipliers -- corresponding to the weakest contraction rate at the left fixed point and the weakest expansion rate at the right fixed point -- are both real in the saddle case (a), one real and one complex in the saddle-focus case (b), and both complex in the double-focus case (c).
  • Figure 3: The local maps near $O_1$ in the discrete-time case (a) and the continuous-time case (b).
  • Figure 4: A heterodimensional cycle of coindex $1$ satisfying conditions C1 - C3, which consists of two periodic orbits containing $O_1$ and $O_2$, a fragile heteroclinic orbit containing $M^-_1$ and $M^+_2$, and a robust heteroclinic orbit containing $M^-_2$ and $M^+_1$. Here $\ell^{ss}$ is a strong-stable leaf through $M^+_1$, $\ell^{uu}$ is a strong-unstable leaf through $M^-_2$, $\ell_1=W^s_{loc}(O_1)\cap F_{21}(W^u_{loc}(O_2))$ and $\ell_2=F^{-1}_{21}\ell_1$.
  • Figure 5: An illustration of condition C4.2. The vector $(x^+_1,x^+_2)$ is not parallel to the $x$-vector component (the dashed line) of the tangent to $\ell_1$.
  • ...and 9 more figures

Theorems & Definitions (77)

  • Definition 1.1: Heterodimensional dynamics
  • Definition 1.2: Heterodimensional cycles
  • Definition 1.3: Robust heterodimensional dynamics
  • Theorem A
  • Remark 1.4
  • Corollary A
  • Theorem B
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7: Stabilization of heterodimensional cycles
  • ...and 67 more