A unified approach to compute foliations, inertial manifolds, and tracking initial conditions

TL;DR

This paper introduces algorithms for accurately computing foliations, inertial manifolds, and tracking initial conditions in ODEs near hyperbolic fixed points, demonstrated on the Kuramoto-Sivashinsky equation.

Contribution

It presents a unified contraction mapping approach to compute different types of leaves in foliations and tracking initial conditions in dynamical systems.

Findings

Algorithms successfully compute foliations and inertial manifolds.

Demonstrations on Kuramoto-Sivashinsky equation validate effectiveness.

Unified approach simplifies computation of complex dynamical structures.

Abstract

Several algorithms are presented for the accurate computation of the leaves in the foliation of an ODE near a hyperbolic fixed point. They are variations of a contraction mapping method in [25] to compute inertial manifolds, which represents a particular leaf in the unstable foliation. Such a mapping is combined with one for the leaf in the stable foliation to compute the tracking initial condition for a given solution. The algorithms are demonstrated on the Kuramoto-Sivashinsky equation.