Defect-enhanced anomaly in frequency synchronization of asymmetrically coupled spatially extended systems

We analytically establish and numerically show that anomalous frequency synchronization occurs in a pair of asymmetrically coupled chaotic space extended oscillators. The transition to anomalous behaviors is crucially dependent on asymmetries in the coupling configuration, while the presence of phase defects has the effect of enhancing the anomaly in frequency synchronization with respect to the case of merely time chaotic oscillators.

Figure 1

Frequency mismatch $\mid \Delta \Omega \mid$ vs the parameter space $(c,\theta )$ for Eq. (1). All parameters are specified in the text. The arrow pointing at ${\theta }_{\mathit{cr}}\approx -0.09$ discriminates between regular $({\theta }_{\mathit{cr}}<\theta ⩽1)$ and anomalous $(-1⩽\theta <{\theta }_{\mathit{cr}})$ frequency synchronization.

Figure 2

(a) Time evolution of ${⟨{\varphi }_x{(x,t)}^2⟩}_x$ from Eq. (2) with $\alpha =2$ and $\beta =-0.75$. For a better statistics, nine different random initial conditions have been considered, and the integration lasts until $t={10}^6$. (b) The probability distribution function of the data shown in (a) (dotted line) and its fit by a Gaussian PDF (solid line). (c) The time correlation function of the signal shown in (a).

Figure 3

Mean frequencies ${\Omega }_{1,2}$ calculated for the full CGLE (1) (solid line) and the effective SDE (5) model (dashed line). (a) $\theta =0.88$ (regular FS). (b) $\theta =-0.88$ (AFS) (${\beta }_1=-0.75$, ${\beta }_2=-0.9$).

Figure 4

Total number of defects $N$ generated in Eq. (1) in the system (a) $A_1$ and (b) $A_2$ vs the parameter space $(c,\theta )$. Other parameters are $\alpha =2$, ${\beta }_1=-0.75$, ${\beta }_2=-0.9$. The defects at time $t$ have been considered at those points $x_i$, where the $\rho (x_i,t)$ is smaller than $2.5\times {10}^{-2}$ and that are, furthermore, local minima for the function $\rho (x,t)$.

Figure 5

Number of defects in system 1 (solid line—left ordinate) and frequency mismatch $\Delta {\Omega }_1^e$ (dashed line—right ordinate, see text for definition) vs $c$ for $\theta =-0.88$ (AFS).