Anticipated synchronization in coupled complex Ginzburg-Landau systems

We study the occurrence of anticipated synchronization in two complex Ginzburg-Landau systems coupled in a master-slave configuration. Master and slave systems are ruled by the same autonomous function, but the slave system receives the injection from the master and is subject to a negative delayed self-feedback loop. We give evidence that the magnitude of the largest anticipation time, obtained for complex-valued coupling constants, depends on the dynamical regime where the system operates (defect turbulence, phase turbulence, or bichaos) and scales with the linear autocorrelation time of the system. We also provide analytical conditions for the stability of the anticipated synchronization manifold that are in qualitative agreement with those obtained numerically. Finally, we report on the existence of anticipated synchronization in coupled two-dimensional complex Ginzburg-Landau systems.

Figure 1

Autocorrelation function versus time for the three chaotic regimes of the complex Ginzburg-Landau equation: defect turbulence (solid line), phase turbulence (dashed line), and bichaos (dotted line). The system parameters used are $\epsilon =1$ (in all cases), $(c_1,c_2)=(3,-2.5)$ for defect turbulence, $(c_1,c_2)=(1.5,-0.9)$ for phase turbulence, and $(c_1,c_2)=(1.1,-1.2)$ for bichaos regime, as given in Ref. [28].

Figure 2

Spatiotemporal dynamics of (a) amplitude and (b) phase of the master and the slave systems for two unidimensional complex Ginzburg-Landau systems coupled as in Eqs. (5) and (6). Parameter values: $\tau =0.6, \theta =\frac{\pi }{4}, K=0.6, N=64, \delta x=0.2, \delta t=2\times {10}^{-3}, c_1=3, c_2=-2.5$ (in a range of defect turbulence regime). The difference in time between the horizontal dashed lines is the anticipation time $\tau =0.6$.

Figure 3

Time series of the master amplitude $\left|A\right|$ (dotted line) and the slave $\left|B\right|$ (dashed line) for the complex Ginzburg-Landau equation in the defect turbulence regime. Horizontal arrows mark the anticipation time. Same parameters as described in the caption of Fig. 2.

Figure 4

Amplitude correlations between master $\left|A\right|$ and slave $|B_{\tau }|$ time series in the defect turbulence regime with $(c_1,c_2)=(3,-2.5)$ at the spatial position $x=L/2$ for (a) $\theta =-0.55$, (b) $\theta =0$, and (c) $\theta =0.55$. Purple areas correspond to high correlations ($C>0.99$). Solid lines represent analytical results obtained from Eq. (18) using a linear stability analysis.

Figure 5

Purple areas indicate the projections in the planes (a) $(\theta ,\tau )$ and (b) $(\theta ,K)$ of the region for which stable anticipated synchronization solutions have been found numerically in the defect turbulence regime with parameters $(c_1,c_2)=(3,-2.5)$. The solid lines are the analytical results obtained from an approximate linear stability analysis; see Eq. (18).

Figure 6

Amplitude evolution of two-dimensional complex Ginzburg-Landau equations, Eqs. (9) and (10), master (upper row) and slave (lower row) for coupling parameters $\tau =0.37, K=1.35$, and $\theta =-\frac{\pi }{4}$. Snapshots are shown at times separated by a time unit $\tau$. Anticipated synchronization is apparent since each frame in the dynamics of the master, upper row, has been reached in an earlier frame of the slave system, lower row, as indicated by the arrows. System size is $N\times N=64\times 64$ with $\delta x=0.3$ and periodic boundary conditions. Defect turbulence regime is considered with $(c_1,c_2)=(3,-2.5)$.