Phase synchronization in time-delay systems

Though the notion of phase synchronization has been well studied in chaotic dynamical systems without delay, it has not been realized yet in chaotic time-delay systems exhibiting non-phase-coherent hyperchaotic attractors. In this paper we report identification of phase synchronization in coupled time-delay systems exhibiting hyperchaotic attractor. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. These transitions are characterized by recurrence quantification analysis, by phase differences based on a transformation of the attractors, and also by the changes in the Lyapunov exponents. We have found these transitions in coupled piecewise linear and in Mackey-Glass time-delay systems.

Figure 1

Phase synchronization between the systems (1a, 1b). (a) The non-phase-coherent hyperchaotic attractor of the uncoupled drive (1a). (b) Transformed attractor in the $x_1(t+\tau )$ and $z(t+\tau )$ space along with the Poincaré points represented as open circles. (c) The phases of the drive system $\left({\varphi }_1^z\right)$, the response system $\left({\varphi }_2^z\right)$ calculated from the new state variable $z(t+\tau )$, and their difference $\left(\Delta \varphi \right)$ for $b_3=1.5$.

Figure 2

Generalized autocorrelation functions of both the drive $P_1\left(t\right)$ and the response $P_2\left(t\right)$ systems. (a) Non-phase-synchronization for $b_3=0.6$, (b) phase synchronization for $b_3=1.5$, and (c) generalized synchronization for $b_3=2.3$.

Figure 3

(a) Spectrum of first nine largest Lyapunov exponents of the coupled systems (1) as a function of coupling strength $b_3∊(0,3)$; (b) spectrum of CPR and JPR as a function of coupling strength $b_3∊(0,3)$.