Invariant Manifolds of Complex Systems

TL;DR

This paper demonstrates the existence of invariant manifolds in complex multi-scale dynamical systems, such as predator-prey and neuronal models, highlighting their role in system stability and determinism restoration.

Contribution

It establishes the existence of invariant manifolds in two- and three-dimensional complex systems, extending understanding of their stability properties.

Findings

Invariant manifolds exist in predator-prey and neuronal models.

Invariant manifolds help restore determinism in complex systems.

The manifolds are curves or surfaces in phase space.

Abstract

The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal bursting models) it is shown that there exists in the phase space a curve (resp. a surface) which is invariant with respect to the flow of such systems. These invariant manifolds are playing a very important role in the stability of complex systems in the sense that they are "restoring" the determinism of trajectory curves.