Various projective synchronization phenomena in two different variable time-delayed systems related to optical bistable devices

We analytically and numerically analyze the existence of various projective (projective-anticipatory, projective and projective-lag) synchronization between two different modulated time-delayed chaotic systems in this paper. The innovation of our work is that we report on projective-anticipatory and projective-lag synchronization of two different variable time-delayed chaotic systems, while other existing work only mentioned projective synchronization in such systems. Based on the theory of Krasovskii–Lyapunov stability, we derive the sufficient stability condition through theoretical analysis. Mackey–Glass and Ikeda models related to optical bistable or hybrid optical bistable devices are taken for examples to verify the validity and effectiveness of the proposed method.