Intermittency transition to generalized synchronization in coupled time-delay systems

We report the nature of the transition to generalized synchronization (GS) in a system of two coupled scalar piecewise linear time-delay systems using the auxiliary system approach. We demonstrate that the transition to GS occurs via an on-off intermittency route and that it also exhibits characteristically distinct behaviors for different coupling configurations. In particular, the intermittency transition occurs in a rather broad range of coupling strength for the error feedback coupling configuration and in a narrow range of coupling strength for the direct feedback coupling configuration. It is also shown that the intermittent dynamics displays periodic bursts of periods equal to the delay time of the response system in the former case, while they occur in random time intervals of finite duration in the latter case. The robustness of these transitions with system parameters and delay times has also been studied for both linear and nonlinear coupling configurations. The results are corroborated analytically by suitable stability conditions for asymptotically stable synchronized states and numerically by the probability of synchronization and by the transition of sub-Lyapunov exponents of the coupled time-delay systems. We have also indicated the reason behind these distinct transitions by referring to the unstable periodic orbit theory of intermittency synchronization in low-dimensional systems.

Figure 1

The hyperchaotic attractor of the system (1) for the parameter values $a=1.0$, $b=1.2$, and $\tau =15.0$.

Figure 2

The first ten maximal Lyapunov exponents ${\lambda }_{\text{max}}$ of the scalar time-delay system for the parameter values $a=1.0$, $b_1=1.2$, and $\tau \in \left(2,29\right)$ [which is the same as Eq. (1) with $b_1$ replaced by $b$], Eq. (3a).

Figure 3

Dynamics in the phase space of the systems (3a, 3b, 3c). (a) and (b) Approximate GS and CS, respectively, for the value of the coupling strength $b_3=0.4$. (c) and (d) Perfect GS and CS, respectively, for the value of the coupling strength $b_3=0.9$.

Figure 4

The intermittent dynamics of the response $x_2\left(t\right)$ and auxiliary $x_3\left(t\right)$ systems for the value of the coupling strength $b_3=0.4$. (a) Time traces of the difference $x_2\left(t\right)-x_3\left(t\right)$ corresponding to Fig. 3b. (b) Enlarged in $x$ scale to show bursts at periodic intervals when bursts of larger amplitudes $\Delta >\left|0.01\right|$ are considered. (c) Enlarged in $y$ scale to show random bursts when bursts of smaller amplitudes $\Delta <\left|0.01\right|$ are considered.

Figure 5

The statistical distribution of laminar phases corresponding to Fig. 4. (a) For threshold value $\Delta =\left|0.1\right|$ and (b) for the threshold value of $\Delta =\left|0.0001\right|$.

Figure 6

(a) The probability of synchronization, $\Phi \left(b_3\right)$, between the response $x_2\left(t\right)$ and auxiliary $x_3\left(t\right)$ systems. (b) Largest Lyapunov exponents of the coupled drive $x_1\left(t\right)$ and response $x_2\left(t\right)$ systems (3a, 3b).

Figure 7

The probability of synchronization, $\Phi \left(b_3\right)$, in three-dimensional plots (a) as a function of the system parameter $b_2$ and the coupling strength $b_3$ and (b) as a function of the coupling delay ${\tau }_2$ and the coupling strength $b_3$, exhibiting a broad range intermittency transition to GS for linear error feedback coupling.

Figure 8

Dynamics in the phase space of the systems (11a, 11b, 11c): (a) and (b) Approximate GS and CS, respectively, for the value of the coupling strength $b_3=0.64$. (c) and (d) Perfect GS and CS, respectively, for the value of the coupling strength $b_3=0.8$.

Figure 9

The intermittent dynamics of the response $x_2\left(t\right)$, Eq. (11b), and auxiliary $x_3\left(t\right)$, Eq. (11c), systems for the value of the coupling strength $b_3=0.64$. (a) and (b) Time traces of the difference $x_2\left(t\right)-x_3\left(t\right)$ corresponding to Fig. 8b.

Figure 10

The statistical distribution of laminar phases for Fig. 9.

Figure 11

(a) The probability of synchronization, $\Phi \left(b_3\right)$, between the response $x_2\left(t\right)$, Eq. (11b), and the auxiliary $x_3\left(t\right)$, Eq. (11c), systems. (b) Largest Lyapunov exponents of the coupled drive $x_1\left(t\right)$ and response $x_2\left(t\right)$ systems (11a, 11b).

Figure 12

The probability of synchronization, $\Phi \left(b_3\right)$, in three-dimensional plots (a) as a function of the system parameter $b_2$ and the coupling strength $b_3$ and (b) as a function of the coupling delay ${\tau }_2$ and the coupling strength $b_3$, exhibiting a narrow range intermittency transition to GS for linear direct feedback coupling.

Figure 13

Dynamics in the phase space of the systems (13a, 13b, 13c). (a) and (b) Approximate GS and CS, respectively, for the value of the coupling strength $b_3=0.37$. (c) and (d): Perfect GS and CS, respectively, for the value of the coupling strength $b_3=0.8$.

Figure 14

The intermittent dynamics of the response $x_2\left(t\right)$, Eq. (13b), and auxiliary $x_3\left(t\right)$, Eq. (13c), systems for the value of the coupling strength $b_3=0.37$ for nonlinear error feedback coupling. (a) Time traces of the difference $x_2\left(t\right)-x_3\left(t\right)$ corresponding to Fig. 13b. (b) Enlarged in $x$ scale to show bursts at periodic intervals when bursts of larger amplitudes $\Delta >\left|0.01\right|$ are considered. (c) Enlarged in $y$ scale to show random bursts when bursts of smaller amplitudes $\Delta <\left|0.01\right|$ are considered.

Figure 15

The statistical distribution of laminar phases corresponding to Fig. 14. (a) For threshold value $\Delta =\left|0.1\right|$ and (b) for threshold value of $\Delta =\left|0.0001\right|$.

Figure 16

(a) The probability of synchronization, $\Phi \left(b_3\right)$, between the response $x_2\left(t\right)$, Eq. (13b), and the auxiliary $x_3\left(t\right)$, Eq. (13c), systems. (b) Largest Lyapunov exponents of the coupled drive $x_1\left(t\right)$ and response $x_2\left(t\right)$ systems (13a, 13b).

Figure 17

The probability of synchronization, $\Phi \left(b_3\right)$, in three-dimensional plots (a) as a function of the system parameter $b_2$ and the coupling strength $b_3$ and (b) as a function of the coupling delay ${\tau }_2$ and the coupling strength $b_3$ for the case of the nonlinear error feedback coupling configuration given by Eqs. (13a, 13b, 13c), showing a broad range intermittency transition.

Figure 18

Dynamics in the phase space of the systems (17a, 17b, 17c). (a) and (b) Approximate GS and CS, respectively, for the value of the coupling strength $b_3=0.78$. (c) and (d) Perfect GS and CS, respectively, for the value of the coupling strength $b_3=0.9$.

Figure 19

The intermittent dynamics of the response $x_2\left(t\right)$, Eq. (17b), and auxiliary $x_3\left(t\right)$, Eq. (17c), systems for the value of the coupling strength $b_3=0.78$. (a) and (b) Time traces of the difference $x_2\left(t\right)-x_3\left(t\right)$ corresponding to Fig. 18b.

Figure 20

The statistical distribution of laminar phase for Fig. 19.

Figure 21

(a) The probability of synchronization, $\Phi \left(b_3\right)$, between the response $x_2\left(t\right)$, Eq. (17b), and the auxiliary $x_3\left(t\right)$, Eq. (17c), systems. (b) Largest Lyapunov exponents of the coupled drive $x_1\left(t\right)$ and response $x_2\left(t\right)$ systems (17a, 17b).

Figure 22

The probability of synchronization $\Phi \left(b_3\right)$ in three-dimensional plots (a) as a function of the system parameter $b_2$ and the coupling strength $b_3$ and (b) as a function of the coupling delay ${\tau }_2$ and the coupling strength $b_3$ for the case of the nonlinear direct feedback coupling configuration given by Eqs. (17a, 17b, 17c) showing a narrow range intermittency transition to GS.