Inertial Manifolds and Limit Cycles of Dynamical Systems in $\mathbb R^n$

TL;DR

This paper demonstrates how two-dimensional inertial manifolds enable the reduction of complex dynamical systems in high dimensions to simpler two-dimensional analysis for identifying stable limit cycles, with applications in celestial mechanics and biochemistry.

Contribution

It introduces a method to reduce high-dimensional ODEs to two-dimensional systems using inertial manifolds, facilitating the analysis of stable limit cycles.

Findings

Reduction of high-dimensional systems to 2D for limit cycle analysis

Application to satellite rotation model

Application to biochemical model

Abstract

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in ${\mathbb R}^{n}$ permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare--Bendixson theory. In the case $n=3$ we implement such a scenario for a model of a satellite rotation around a celestial body of small mass and for a biochemical model.