Smooth inertial manifolds for neutral differential equations with small delays

TL;DR

This paper investigates the existence and smoothness of inertial manifolds in neutral differential equations with small delays, providing new insights into their dynamical behavior and stability, especially in models like the van der Pol oscillator.

Contribution

It establishes the existence and smoothness of global inertial manifolds for neutral differential equations with small delays and demonstrates the smooth dependence on delay parameters.

Findings

Existence of global inertial manifolds for small delay neutral differential equations

Smoothness of inertial manifolds with respect to small delays

Application to the van der Pol oscillator model with delay

Abstract

In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general small-delay systems. Finally, we apply our main results to the van der Pol oscillator model with small delay.