Chaotic synchronization of mutually coupled systems–arbitrary proportional linear relations

Abstract

Considering a system combining two generalized Boolean transformations, we found that depending on the parameters, we can generate generalized synchronization such that the two chaotic orbits have arbitrary proportional linear relations. We rigorously determined its synchronization conditions by the explicit computing conditional Lyapunov exponent using the ergodic property and stable property of the Cauchy distribution. We found that a phenomenon similar to chaotic synchronization occurs even when the synchronization conditions are not strictly satisfied, which exhibits some degree of structural stability of chaotic synchronization. Our model can be further extended to systems with more degrees of freedom and, in the future, can be applied to reservoir computing.