Synchronization of hyperchaotic dynamical systems with different dimensions

Abstract

The combination synchronization (CS) and combination-combination synchronization (CCS) for chaotic and hyperchaotic dynamical systems with the same dimensions are introduced and studied in the literature. In this paper, we introduce the definition of CS and CCS for those systems with different dimensions. We state two schemes to achieve these kinds of synchronization based on the active control technique. Two theorems are stated and proved to provide us with analytical expressions for the control functions. We state four hyperchaotic dynamical systems with different dimensions which are used as examples to achieve CS and CCS. These examples are hyperchaotic detuned laser, Lorenz, van der Pol and dynamos systems. These systems appeared in many important applications in applied science, e.g., a ring laser system of two-level atoms, vacuum tube circuit and two coupled dynamos system. The validity of the analytical control functions are tested numerically and good agreement is found between them. The numerical solutions of ODE systems are calculated by using the method of Runge-Kutta of order 4. Other systems can be similarly studied.