When are synchronization errors small?

Lucas Illing¹, Jochen Bröcker², Ljupco Kocarev¹, Ulrich Parlitz², and Henry D. I. Abarbanel¹

¹Institute for Nonlinear Science, University of California, San Diego, La Jolla,
California 93093-0402
²Third Physical Institute, University of Göttingen, Bürgerstrasse 42-44, 37073 Göttingen, Germany

We address the question of bounds on the synchronization error for the case of nearly identical nonlinear systems. It is pointed out that negative largest conditional Lyapunov exponents of the synchronization manifold are not sufficient to guarantee a small synchronization error and that one has to find bounds for the deformation of the manifold due to perturbations. We present an analytic bound for a simple subclass of systems, which includes the Lur'e systems, showing that the bound for the deformation grows as the largest singular value of the linearized system gets larger. Then, the Lorenz system is taken as an example to demonstrate that the phenomenon is not restricted to Lur'e systems.