Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory

Stability of synchronization in unidirectionally coupled time-delay systems is studied using the Krasovskii-Lyapunov theory. We have shown that the same general stability condition is valid for different cases, even for the general situation (but with a constraint) where all the coefficients of the error equation corresponding to the synchronization manifold are time dependent. These analytical results are also confirmed by the numerical simulation of paradigmatic examples.

Figure 1

(Color online) The time trajectory plot of the variables $x\left(t\right)$ and $y\left(t\right)$ of the coupled time-delay systems (1) indicating complete synchronization between them. (a) Piecewise linear time-delay systems (1, 14), for the parameters $a=1.0,b=1.2,\tau =25.0$ and for the constant coupling $k_1=k_2=0.9$. (b) Ikeda time-delay systems (1, 15), for the parameters $a=1.0,b=5,\tau =2.0$ along with the constant coupling $k_1=k_2=5.0$. (c) Piecewise linear time-delay system for the same values of the system parameters as in (a) with the square wave coupling rate $k_1=0.9$ and $k_2=1.0$. (d) Ikeda system for the same values of the system parameters and with the square wave coupling rate $k_1=5.0$ and $k_2=6.0$.