[Paper Review] Nonautonomous equations, generalized dichotomies and stable manifolds

António J. G. Bento, César Silva|ArXiv.org|May 29, 2009

Stability and Controllability of Differential Equations16 references3 citations

TL;DR

This paper establishes the existence of global $C^1$ stable manifolds for nonlinear perturbations of nonautonomous linear differential equations in Banach spaces, under a generalized nonuniform dichotomy condition defined by an arbitrary increasing differentiable function $\mu(t)$. Unlike prior work restricted to exponential or polynomial dichotomies, the authors introduce a broader class of dichotomies that includes nonuniform exponential and polynomial cases, and prove the existence of invariant manifolds via a fixed-point argument in a suitable function space, with the key result being the persistence of stable manifolds under small nonlinear perturbations without relying on Gronwall's lemma.

ABSTRACT

Assuming the existence of a general nonuniform dichotomy for the evolution operator of a non-autonomous ordinary linear differential equation in a Banach space, we establish the existence of invariant stable manifolds for the semiflow generated by sufficiently small nonlinear perturbations of the linear equation. The family of dichotomies considered satisfies a general growth rate given by some increasing differentiable function, allows situations for which the classical Lyapunov exponents are zero, and contains the nonuniform exponential dichotomies as a very particular case. In addition we also give explicit examples of linear equations that admit all the possible considered dichotomies.

Motivation & Objective

• To extend the stable manifold theory beyond nonuniform exponential dichotomies to a broader class of nonuniform growth rates defined by an increasing differentiable function $\mu(t)$.
• To prove the existence of global $C^1$ invariant stable manifolds for small nonlinear perturbations of linear nonautonomous equations under this generalized dichotomy condition.
• To provide explicit examples of linear equations admitting all considered dichotomies, including cases where classical Lyapunov exponents are zero.
• To overcome the limitation of relying on Gronwall’s lemma in estimating solutions by using mathematical induction instead.

Proposed method

• The authors define a generalized nonuniform dichotomy using a growth function $\mu(t)$, generalizing both exponential and polynomial dichotomies.
• They construct a Banach space $\mathcal{X}$ of continuous, bounded, and Lipschitz-continuous maps to represent candidate stable manifolds.
• A nonlinear operator $\Phi$ is defined on $\mathcal{X}$, mapping a function $\phi$ to the solution of a fixed-point equation derived from the perturbed dynamics.
• The existence of a unique fixed point of $\Phi$ is proven via a contraction mapping argument, relying on estimates involving $\mu(t)$, $\mu'(t)$, and the dichotomy bounds.
• The contraction property is established using integral estimates and bounds on the evolution operator and nonlinear terms, avoiding Gronwall’s lemma through induction.
• The resulting fixed point defines a $C^1$ graph over the stable subspace, which is shown to be invariant and globally defined under the semiflow.

Experimental results

Research questions

• RQ1 Can stable manifold theory be extended to nonautonomous linear equations with nonuniform dichotomies that are not exponential?
• RQ2 What conditions on the growth rate function $\mu(t)$ ensure the existence of $C^1$ stable manifolds under small nonlinear perturbations?
• RQ3 How can the contraction argument be constructed without relying on Gronwall’s lemma in the presence of non-exponential growth?
• RQ4 Are there explicit examples of linear equations that admit the full class of generalized dichotomies considered?
• RQ5 Can the theory accommodate cases where classical Lyapunov exponents are zero, yet stable manifolds still exist?

Key findings

• The paper establishes the existence of global $C^1$ stable manifolds for nonlinear perturbations of nonautonomous linear ODEs under a generalized nonuniform dichotomy condition defined by an increasing differentiable function $\mu(t)$.
• The method avoids reliance on Gronwall’s lemma by using mathematical induction to prove the key estimate (41), which is essential for the contraction argument.
• The stable manifold is shown to be globally defined and $C^1$, with the graph of the solution map being a $C^1$ parametrization of the manifold.
• The contraction constant of the operator $\Phi$ is bounded by $\frac{3CD\delta}{|a-b-2\varepsilon|}$, ensuring a unique fixed point when $\delta < \frac{|a-b-2\varepsilon|}{3CD}$.
• The theory includes nonuniform exponential and polynomial dichotomies as special cases, and the authors provide explicit examples of linear equations admitting all such dichotomies.
• The framework allows for cases where the classical Lyapunov exponent is zero, demonstrating that stable manifolds can exist even in the absence of exponential decay or growth.