On Phase and Anti-Phase Combination Synchronization of Time Delay Nonlinear Systems

Abstract

Extensive studies have been done on the phenomenon of phase and anti-phase synchronization (APS) between one drive and one response systems. As well as, combination synchronization for chaotic and hyperchaotic systems without delay also has been investigated. Thus, this paper aims to introduce the concept of phase and anti-phase combination synchronization (PCS and APCS) between two drive and one response time delay systems, which are not studied in the literature as far as we know. The analysis of PCS and APCS are carried out using active control technique. An example is given to test the validity of the expressions of control forces to achieve the PCS and APCS of time delay systems. This example is between three different systems. When there is no control, the PCS does not occur where the phase difference is unbounded. The bounded phase difference appears when the control is applied which means that PCS is achieved. The special case which is the combination synchronization is studied as well.