Synchronisation and scaling properties of chaotic networks with multiple delays

¹ Institute of Theoretical Physics, University of Würzburg - 97074 Würzburg, Germany, EU
² Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (UIB - CSIC), Campus Universitat de les Illes Balears - 07122 Palma de Mallorca, Spain, EU
³ Institute of Mathematics, Humboldt University of Berlin - 10099 Berlin, Germany, EU

Received: 23 April 2013
Accepted: 3 July 2013

Abstract

We study chaotic systems with multiple time delays that range over several orders of magnitude. We show that the spectrum of Lyapunov exponents (LEs) in such systems possesses a hierarchical structure, with different parts scaling with the different delays. This leads to different types of chaos, depending on the scaling of the maximal LE. Our results are relevant, in particular, for the synchronisation properties of hierarchical networks (networks of networks) where the nodes of subnetworks are coupled with shorter delays and couplings between different subnetworks are realised with longer delay times. Units within a subnetwork can synchronise if the maximal exponent scales with the shorter delay, long-range synchronisation between different subnetworks is only possible if the maximal exponent scales with the longer delay. The results are illustrated analytically for Bernoulli maps and numerically for tent maps and semiconductor lasers.