Smooth Persistence of Attractors for Set-Valued Dynamical Systems: A Boundary Map Approach

TL;DR

The paper addresses the problem of smooth persistence of attractors with $C^r$-smooth boundaries for set-valued dynamical systems of the form $F_{f,\varepsilon}$, which arise in random and control dynamics. It introduces the boundary map $\beta_{f,\varepsilon}$, a $C^{r-1}$-contactomorphism on the unit tangent bundle $T_1\mathbb{R}^d$, establishing a bijection between invariant sets with smooth boundaries and invariant Legendrian manifolds given by outer unit normal bundles $N_1^+\partial M$, and uses this to apply normal hyperbolicity theory. The main contribution is a persistence theorem: if $\beta_{f,\varepsilon}$ is normally hyperbolic at $N_1^+\partial M$, then $M$ is a minimal attractor that persists under small perturbations $(\tilde{f},\tilde{\varepsilon})$, yielding a unique nearby attractor $\tilde{M}$ with $\partial \tilde{M}$ $C^2$-close to $\partial M$; the framework handles both normally attracting and normally repelling cases (via a dual map $F^*_{f,\varepsilon}$). The approach leverages contact geometry and a boundary-graph transform, offering a robust geometric method for analyzing attractor persistence in broader set-valued dynamics and suggesting extensions to manifolds and numerical bifurcation analysis.

Abstract

We study the problem of persistence of attractors with smooth boundary for a class of set-valued dynamical systems that naturally arise in the context of random and control dynamical systems, as well as in systems modeling the dynamical propagation of uncertainty. In order to tackle the inherent difficulties associated to the multi-valued structure of such dynamical systems, we introduce a single-valued map, the so-called boundary map, which is a contactomorphism of the unit-tangent bundle of the state space, with the following characteristic property: boundaries of attractors of the set-valued dynamical system correspond in a unique way to invariant Legendrian manifolds of this map. We show how the underlying contact geometry guarantees the smooth persistence of such attractors under perturbations of the set-valued dynamical system, provided that the associated boundary map is normally hyperbolic at the unit normal bundle of the boundary.

Theorems & Definitions (23)

• Theorem A
• Theorem B
• Example 1.1
• proof
• Definition 2.1: Boundary map
• Proposition 2.2
• proof
• Corollary 2.3
• Lemma 2.4
• proof