Combination complex synchronization of three chaotic complex systems

In this paper, we firstly design a chaotic complex system and study its dynamical properties including invariance, dissipativity, equilibria, Lyapunov exponents, chaotic behavior, as well as chaotic attractors. What is more, the scaling matrices are always chosen as real matrices in previous combination synchronization schemes within two drive real systems and one response real system evolving in the same or inverse directions simultaneously. However, in many real-life applications, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, combination synchronization with regard to the complex scaling matrices, referred as combination complex synchronization, will be made the further research about three chaotic complex systems. Based on Lyapunov stability theory, three identical chaotic complex systems are considered and the corresponding controllers are designed to achieve the complex combination synchronization. The corresponding theoretical proofs and numerical simulations are given to demonstrate the validity and feasibility of the presented control technique.