Structural stability of a dynamical system near a non-hyperbolic fixed point

TL;DR

This paper establishes the structural stability of certain discrete-time dynamical systems near non-hyperbolic fixed points, using an infinite-dimensional ODE reformulation, with applications to models in statistical physics.

Contribution

It introduces a novel approach to analyze stability near non-hyperbolic points via an infinite-dimensional ODE framework, applicable to renormalization group models.

Findings

Proves stability under perturbations for specific dynamical systems.

Develops an infinite-dimensional ODE formulation for stability analysis.

Applies results to models in statistical physics such as self-avoiding walk.

Abstract

We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. The class of dynamical systems we study, and the boundary conditions we impose, arise in a renormalisation group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional n-component $|\varphi|^4$ spin model.