Synchronization of networked chaotic oscillators under external periodic driving

The dynamical responses of a complex system to external perturbations are of both fundamental interest and practical significance. Here, by the model of networked chaotic oscillators, we investigate how the synchronization behavior of a complex network is influenced by an externally added periodic driving. Interestingly, it is found that by a slight change of the properties of the external driving, e.g., the frequency or phase lag between its intrinsic oscillation and external driving, the network synchronizability could be significantly modified. We demonstrate this phenomenon by different network models and, based on the method of master stability function, give an analysis on the underlying mechanisms. Our studies highlight the importance of external perturbations on the collective behaviors of complex networks, and also provide an alternate approach for controlling network synchronization.

Figure 1

For an ER network of $N=100$ chaotic Rössler oscillators and average degree $\langle k\rangle =10$, by the coupling strength $\varepsilon =0.9$, the time evolutions of the network synchronization error, $\delta$, for the cases of no external driving (solid line), and external driving with amplitude $b=4$ but different frequencies: ${\omega }_1=1.235$ (dashed line) and ${\omega }_2=1.231$ (dotted line). The relevant eigenvalues for the coupling matrix are ${\lambda }_2=0.180$ and ${\lambda }_N=1.783$. All the following simulations of synchronization dynamics are carried out on the same ER random network.

Figure 2

For the same network model used in Fig. 1, the influence of the driving frequency on the MSF curve. (a) The MSF curves for the cases of no external driving (solid line), and external driving with amplitude $b=4$ but for different frequencies: ${\omega }_1=1.235$ (dashed line) and ${\omega }_2=1.231$ (dotted line). (b) ${\sigma }_1$ vs $\omega$. (c) ${\sigma }_2$ vs $\omega$. ${\sigma }_1$ and ${\sigma }_2$ denote, respectively, the left and right boundaries of the stable region.

Figure 3

(a) For $b=4$, the variation of the largest Lyapunov exponent of the synchronous manifold, ${\Lambda }_s$, as a function of the driving frequency, $\omega$, calculated according to Eq. (4). (b) The periodic motion at ${\omega }_2=1.231$, with ${\Lambda }_s=-0.0367$. (c) The chaotic motion at ${\omega }_1=1.235$, with ${\Lambda }_s=0.088$.

Figure 4

For $\omega =0.7134$ and $0.8998$, the variation of the largest Lyapunov exponent of the synchronous manifold, ${\Lambda }_s$, as a function of the driving amplitude.

Figure 5

For the model of networked chaotic HR oscillator, the dependence of network synchronization on the driving frequency, $\omega$. The amplitude of the driving is fixed as $b=0.5$. (a) The variation of the boundary ratio $R_c={\sigma }_2/{\sigma }_1$ as a function of $\omega$. (b) By increasing $\omega$ adiabatically from 0 to $0.285$ (with the speed $1\times {10}^{-9}$, the time evolution of the network synchronization error, $\delta$.

Figure 6

For the model of networked chaotic Duffing oscillators, the dependence of network synchronization on the phase of the external driving. (a) The variation of ${\sigma }_1$ as a function of the phase lag, ${\varphi }_0$. (b) By the frequency mismatch $\Delta \omega =1\times {10}^{-3}$, the evolution of the network synchronization error, $\delta$, as a function of time.