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November  2001, 1(4): 421-442. doi: 10.3934/dcdsb.2001.1.421

Synchronization in directionally coupled systems: Some rigorous results

V. Afraimovich 1, , J.-R. Chazottes 1, and  A. Cordonet 1,

1. 

IICO-UASLP, A. Obregón 64, 78000 San Luis Postosí, SLP, Mexico, Mexico

Received  March 2001 Revised  May 2001 Published  September 2001

We prove, under some general assumptions, that master-slave synchronization implies generalized synchronization, that is we show the existence and continuity of the functional dependence between the “slave” coordinates and the “master” ones. Then, we prove that this function may be Lipschitz continuous and even less “smooth”, that is only Hölder continuous, depending on the coupling strength. We go beyond the above mentioned assumptions by coupling two identical maps of the interval that are neither continuous nor invertible to prove ‘almost-everywhere’ synchronization instead of global synchronization. Then we relate the Hausdorff dimension and the dimension for Poincaré recurrence of the attractor of master and slave systems. We provide some examples illustrating these results.
Keywords: generalized synchronization, fractal dimensions., Master-slave synchronization, almost-everywhere synchronization.
Mathematics Subject Classification: 37A9.
Citation: V. Afraimovich, J.-R. Chazottes, A. Cordonet. Synchronization in directionally coupled systems: Some rigorous results. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 421-442. doi: 10.3934/dcdsb.2001.1.421
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