Research ArticleDelay-range-dependent chaos synchronization approach under varying time-lags and delayed nonlinear coupling

- Synchronization of nonlinearly coupled chaotic systems with both the intrinsic and the coupling time-varying delays is explored. A Lyapunov-Krasovskii functional based less conservative synchronization condition, ensuring asymptotic convergence of the synchronization error to the origin, considering non-zero lower bounds of the time-delays, is derived by utilizing the state feedback delay-range-dependent control approach.
- To the best of our knowledge, the delay-range-dependent synchronization methodology for coupled chaotic oscillators with intrinsic and coupling time-delays has been addressed for the first time.
- The nonlinear characteristic of coupling terms as well as the time-varying nature of delays is incorporated to investigate a matter-of-fact synchronization problem. Furthermore, the type of systems considered in the present approach differs from the traditionalistic synchronization schemes.
- A delay-dependent approach is derived as a specific case of the present delay-range-dependent synchronization control action. A delay-rate-independent synchronization condition, owing to its importance in the field of chaos synchronization, is provided by application of the proposed LK functional treatment.
- The proposed control scheme is modified by determining an upper bound on the L2 gain from the disturbance vector to the synchronization error to achieve robustness against perturbations.

This paper proposes a novel state feedback delay-range-dependent control approach for chaos synchronization in coupled nonlinear time-delay systems. The coupling between two systems is esteemed to be nonlinear subject to time-lags. Time-varying nature of both the intrinsic and the coupling delays is incorporated to broad scope of the present study for a better-quality synchronization controller synthesis. Lyapunov-Krasovskii (LK) functional is employed to derive delay-range-dependent conditions that can be solved by means of the conventional linear matrix inequality (LMI)-tools. The resultant control approach for chaos synchronization of the master-slave time-delay systems considers non-zero lower bound of the intrinsic as well as the coupling time-delays. Further, the delay-dependent synchronization condition has been established as a special case of the proposed LK functional treatment. Furthermore, a delay-range-dependent condition, independent of the delay-rate, has been provided to address the situation when upper bound of the delay-derivative is unknown. A robust state feedback control methodology is formulated for synchronization of the time-delay chaotic networks against the L2 norm bounded perturbations by minimizing the L2 gain from the disturbance to the synchronization error. Numerical simulation results are provided for the time-delay chaotic networks to show effectiveness of the proposed delay-range-dependent chaos synchronization methodologies.