Nonzero error synchronization of chaotic systems via dynamic coupling

When feedback coupling is used to synchronize arbitrary chaotic systems large enough constant interaction gains lead to nearly complete synchronization at quasi-zero error. This forced oscillatory regime takes place in a region of phase space that, although natural for the guiding system, can result to be impracticable as an operating region for the guided system. However, we show that a dynamic feedback coupling with the appropriate variable gain can lead to a fully synchronized regime at a given nonzero synchronization error, that is, with the guided system operating on a desired region of the phase space. Computational results for oscillators of the Lorenz and Rِssler families are shown. The cost of maintaining a couple of oscillatory Lorenz systems synchronized at different constant values of the synchronization error has been evaluated. To do so, an energy-like function associated to the state of the guided system has been defined.