Invariant Manifolds of Complex Systems

Jean-Marc Ginoux (1) , Bruno Rossetto

1 PROTEE - Processus de Transfert et d'Echanges dans l'Environnement - EA 3819

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.

Keywords: Invariant curves; invariant surfaces; multiple time scales dynamical systems; complex systems

Domains: Mathematics [math]; Dynamical Systems [math.DS]

Jean-Marc Ginoux, Bruno Rossetto. Invariant Manifolds of Complex Systems. Cyrille Bertelle, Gérard H.E. Duchamp, Hakima Kadri-Dahmani. Complex Systems and Self-organization Modelling, Understanding Complex System, Springer Berlin Heidelberg, pp.41-49, 2009, Understanding Complex Systems, 978-3-540-88072-1. ⟨10.1007/978-3-540-88073-8_4⟩. ⟨hal-01056170v2⟩