Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Lokesh Singh,Dhirendra Bahuguna

doi:10.3390/axioms10020105

Lokesh Singh, Dhirendra Bahuguna

Open Access

https://doi.org/10.3390/axioms10020105

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Abstract

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

Highlights

The invariant manifold theory started with the work of Hadamard [1] in 1901 when he constructed a manifold in the solution space of a differential equation with the property that if the trajectory of a solution starts in the manifold, it will remain in the manifold for all time t > 0
In the article [16], Pan proved the existence of Lipschitz stable invariant manifold for impulsive nonautonomous differential equations with the assumption of ρ-nonuniform exponential dichotomy
We prove the existence of a C1 stable invariant manifold for the delay differential Equation (1) following the approach of Perron and Lyapunov

Summary

Introduction

The invariant manifold theory started with the work of Hadamard [1] in 1901 when he constructed a manifold in the solution space of a differential equation with the property that if the trajectory of a solution starts in the manifold, it will remain in the manifold for all time t > 0. Perron introduced (and assumed) the notion of (uniform) exponential dichotomy for the solution operators and proved the existence of Lipschitz stable invariant manifolds for the small nonlinear perturbation of autonomous differential equations The smoothness of these invariant manifolds is proved by Pesin [7]. In the article [16], Pan proved the existence of Lipschitz stable invariant manifold for impulsive nonautonomous differential equations with the assumption of ρ-nonuniform exponential dichotomy. Satisfies ρ-nonuniform exponential dichotomy and the nonlinear perturbation f (t, xt ) is sufficiently small and smooth With these assumptions, we prove the existence of a C1 stable invariant manifold for the delay differential Equation (1) following the approach of Perron and Lyapunov. We present a few more examples satisfying the assumptions of our main theorem

Preliminaries

Stable Manifold Theorem

Stable Manifold and Perturbations

Examples

Conclusions