Referee of the submission “INVARIANT MANIFOLDS OF COMPLEX SYSTEMS ” by Jean-Marc Ginoux and Bruno Rossetto

This paper aims to establish the existence of invariant manifolds in complex systems. The framework of this work is the dynamical systems which fulfil the the assumptions of the Cauchy-Lipschitz theorem which could, at first appear as too restricted but allows to assembly classical results to get the claim (Propositions 1 and 2). In the final work, the reader would desire more illustrations and metaphors in order to grasp what is meant by “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” i. e. the main interest of the results for us. Also, the connection with emergence and autoorganisation could be investigated. The paper is well written and I suggest to accept the contribution after expansion with illustrations and examples.

Suggestion for improvement

[LIE’S DERIVATIVE] is never used as such (at least in mathematics and physics) should be replaced by [LIE DERIVATIVE]