Synchronization of Chaotic Systems with Coexisting Attractors

Synchronization of coupled oscillators exhibiting the coexistence of chaotic attractors is investigated, both numerically and experimentally. The route from the asynchronous motion to a completely synchronized state is characterized by the sequence of type-I and on-off intermittencies, intermittent phase synchronization, anticipated synchronization, and period-doubling phase synchronization.

Figure 1

Dynamics of the slave system at $\epsilon =0.0051$. (a) Time series demonstrating intermittent switches between coexisting attractors. (b) Temporal behavior of phase difference. $\Delta \varphi$ increases linearly when the systems stay in different attractors and fluctuates around a certain value when the systems stay in the same attractor. (c) Random walk of phase difference inside window $1.33\mathrm{s}<t<1.95\mathrm{s}$ where the systems stay in the same attractor. (d) Probability distribution of the phase difference inside the windows. The line is the Gaussian fit. The negative phase difference means anticipated synchronization.

Figure 2

Average phase difference (filled dots) and fluctuation range (open dots) versus coupling strength inside windows of IPS. While $\epsilon$ is increased, both the anticipation time and $\delta \varphi$ decrease leading to almost perfect IPS.

Figure 3

Power law dependences of mean duration of IPS windows on coupling strength. The straight lines have slopes of (a) $-1/2$ and (b) $-1$ that characterize type-I and on-off intermittencies.

Figure 4

Dependence of natural frequency of slave system on coupling strength. $f_s$ is locked by the master oscillations and approaches $f_m/2$. The solid line is the sigmoidal fit.

Figure 5

Experimental time series demonstrated (a) anticipated synchronization within IPS window at $\epsilon =0.07$ and (b) period-doubling phase synchronization at $\epsilon =0.92$. Master and slave oscillations are shown, respectively, by the dotted and solid lines.