Non-Autonomous Inertial Manifold Reduction

TL;DR

This paper introduces a novel method for decoupling dissipative differential equations using orthogonal transformations and boundary value problems, enabling efficient computation of inertial manifolds with practical examples.

Contribution

It presents a new approach combining orthogonal changes of variables and BVP solutions for inertial manifold reduction, improving upon traditional Lyapunov-Perron methods.

Findings

Effective decoupling of dissipative systems demonstrated

Numerical implementation using Householder reflectors shown to be successful

Boundary value problem approach facilitates Newton iteration

Abstract

Techniques are developed for decoupling dissipative differential equations. The approach considered is based upon obtaining a sufficient gap in the time dependent linear portion of the equation that corresponds to the linear variational equation. This is done using an orthogonal change of variables that has proven useful in the computation of Lyapunov to decompose the differential equation in terms of slow and fast variables. Numerically this is accomplished in our implementation using smooth, time dependent Householder reflectors. The the nonlinear decoupling transformation or inertial manifold is obtained by solving a boundary value problem (BVP) which allows for a Newton iteration as opposed to the traditional Lyapunov-Perron approach via a fixed point iteration. Finally, the efficacy of the technique is shown using some challenging examples.