Invariant manifolds of complex systems

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. © 2009 Springer-Verlag Berlin Heidelberg.