[Paper Review] Nonuniform $(\mu, u)$-dichotomies and local dynamics of difference equations

António J. G. Bento, César M. Silva|arXiv (Cornell University)|May 11, 2011

Mathematical and Theoretical Epidemiology and Ecology Models3 references3 citations

TL;DR

This paper establishes a local stable manifold theorem for perturbations of nonautonomous linear difference equations with nonuniform $(\mu, \nu)$-dichotomies, allowing for arbitrary growth rates in uniform and nonuniform parts. It proves existence and Hölder-type decay estimates for stable manifolds under general nonlinear perturbations, extending classical results to cases where Lyapunov exponents may be zero and including polynomial and non-exponential dichotomies as special cases.

ABSTRACT

We obtain a local stable manifold theorem for perturbations of nonautonomous linear difference equations possessing a very general type of nonuniform dichotomy, possibly with different growth rates in the uniform and nonuniform parts. We note that we consider situations were the classical Lyapunov exponents can be zero. Additionally, we study how the manifolds decay along the orbit of a point as well as the behavior under perturbations and give examples of nonautonomous linear difference equations that admit the dichotomies considered.

Motivation & Objective

• To establish a local stable manifold theorem for perturbations of nonautonomous linear difference equations under a very general class of nonuniform dichotomies.
• To extend existing results beyond exponential dichotomies to include polynomial and other non-exponential growth rates.
• To analyze the decay behavior of solutions along stable manifolds and their dependence on perturbations.
• To provide a framework valid even when classical Lyapunov exponents are zero, thus covering broader hyperbolic-like dynamics.
• To quantify the continuity of stable manifolds under perturbations using a suitable metric on the perturbation space.

Proposed method

• Introduces a generalized nonuniform $(\mu, \nu)$-dichotomy allowing distinct growth rates $\mu_n$ for the uniform part and $\nu_n$ for the nonuniform part.
• Uses a fixed-point argument in a Banach space of sequences of functions to construct the stable manifold as a graph of a function $\phi$ satisfying a contraction condition.
• Applies a graph transform technique via a contraction mapping principle on a space of Hölder-like functions with weights $\beta_n$.
• Establishes decay estimates for solutions on the stable manifold using the dichotomy bounds and the perturbation structure.
• Derives bounds on the distance between stable manifolds associated with different perturbations using a metric $\|f - \bar{f}\|''' = \sup \frac{\|f_n(u) - \bar{f}_n(u)\|}{\|u\|^{q+1}}$.
• Employs estimates involving operator norms $\|A_{k+1,n}^{-1} Q_{k+1}\|$, weighted by $\mu_n^a \nu_n^\varepsilon$, to control the decay and contraction rates.

Experimental results

Research questions

• RQ1 Can a local stable manifold theorem be established for nonautonomous difference equations with nonuniform, non-exponential dichotomies?
• RQ2 How do the stable manifolds decay along the orbit when the dichotomy has different uniform and nonuniform growth rates?
• RQ3 What is the dependence of the stable manifold on the perturbation, and how can this be quantified in a suitable metric?
• RQ4 Can the theory include cases where classical Lyapunov exponents are zero, such as in polynomial dichotomies?
• RQ5 Under what conditions does the fixed-point argument for the stable manifold converge in the presence of general nonlinear perturbations?

Key findings

• A local stable manifold exists for perturbations of nonautonomous linear difference equations admitting a nonuniform $(\mu, \nu)$-dichotomy with arbitrary growth rates.
• The stable manifold decays along the orbit with a rate bounded by $C \left( \frac{\mu_m}{\mu_n} \right)^a \nu_\varepsilon^n$ for $a < 0$, $\varepsilon \geq 0$, ensuring decay even when $a=0$.
• The theorem includes the classical nonuniform exponential dichotomy and nonuniform polynomial dichotomy as special cases.
• The stable manifold is Hölder continuous with respect to the perturbation, with $\|\phi - \bar{\phi}\|' \leq 4 \cdot 3^{q+1} C^{q+1} D \delta^q \|f - \bar{f}\|'''$.
• The existence result holds for $\delta > 0$ sufficiently small, ensuring the contraction condition in the fixed-point argument is satisfied.
• The framework allows for perturbations satisfying $\|f_m(u) - f_m(v)\| \leq c \|u - v\|(\|u\| + \|v\|)^q$ with $q > 1$, covering superlinear nonlinearities.