Network synchronization with periodic coupling

The synchronization behavior of networked chaotic oscillators with periodic coupling is investigated. It is observed in simulations that the network synchronizability could be significantly influenced by tuning the coupling frequency, even making the network alternating between the synchronous and nonsynchronous states. Using the master stability function method, we conduct a detailed analysis of the influence of coupling frequency on network synchronizability and find that the network synchronizability is maximized at some characteristic frequencies comparable to the intrinsic frequency of the local dynamics. Moreover, it is found that as the amplitude of the coupling increases, the characteristic frequencies are gradually decreased. Using the finite-time Lyapunov exponent technique, we investigate further the mechanism for the maximized synchronizability and find that at the characteristic frequencies the power spectrum of the finite-time Lyapunov exponent is abruptly changed from the localized to broad distributions. When this feature is absent or not prominent, the network synchronizability is less influenced by the periodic coupling. Our study shows the efficiency of finite-time Lyapunov exponent in exploring the synchronization behavior of temporally coupled oscillators and sheds lights on the interplay between the system dynamics and structure in general temporal networks.

Figure 1

For the model of three globally coupled chaotic Rössler oscillators, the influence of the coupling frequency, $\omega$, on network synchronization. (a) The variation of the averaged synchronization error, $\langle \delta x\rangle$, with respect to $\omega$ for different coupling amplitudes, ${\varepsilon }_0=1.60$ and 1.55. The results are averaged over a time period of $T=1\times {10}^3$ and 1000 realizations. (b1) The trajectory of the first oscillator in the isolated form in the $(x,y)$ plane. (b2) The PSD of the variable $x$ shown in (b1). The dominant component is locating at ${\omega }_R\approx 1.1$. (b3) With the parameters ${\varepsilon }_0=1.60$ and $\omega ={\omega }_2=0.5$, the trajectory of the first oscillator in the $(x,y)$ plane. (b4) With the parameters ${\varepsilon }_0=1.60$ and $\omega ={\omega }_R$, the trajectory of the first oscillator in the $(x,y)$ plane.

Figure 2

The results obtained by the MSF method. (a) The variation of $\mathrm{\Lambda }$ with respect to $\omega$. For $\sigma =4.8$ (${\varepsilon }_0=1.60$), $\mathrm{\Lambda }$ is negative in the regions $\omega \in (0.06,0.44)$ and $\omega \in (0.63,10.08)$. Increasing $\sigma$ to 10, the stable region is shrunk to $\omega \in (0.63,2.32)$. (b) The dependence of $\mathrm{\Lambda }$ on $\sigma$. For static coupling ($\omega =0$), $\mathrm{\Lambda }$ is negative in the region $\sigma \in (0.19,4.61)$. By periodic coupling of $\omega =1.27$, the stable region is enlarged to $\sigma \in (0.19,17.97)$. (c) The contour plot of $\mathrm{\Lambda }$ in the parameter space of $(\omega ,\sigma )$. The boundaries of the stable region are marked by the dashed lines. (d) The variation of ${\omega }_r$, i.e., the characteristic frequency for optimized synchronization, with respect to $\sigma$. The fitted data give ${\omega }_r\propto {\sigma }^{-\gamma }$, with $\gamma \approx 0.5$. Squares: the characteristic frequency, ${\omega }_r^f$, identified by the FLE method. (e) The variation of the network synchronizability, $\beta$, as a function of $\omega$. $\beta$ is diverged at $\omega \approx 0.66$.

Figure 3

For $\sigma =8$, the results obtained by the method of FLE analysis. (a) By static coupling, the PSD of FLE. The dashed line denotes the threshold used to identify the dominant components. (b) The variation of the dominant PSD frequencies, ${\omega }^f$, with respect to the coupling frequency, $\omega$. The vertical dashed lines denote the characteristic frequencies, ${\omega }_1^f\approx 0.25$ and ${\omega }_r^f\approx 1$, where the dominant frequencies are broadly distributed. The diagonal line represents the component of the periodic coupling. (c) The PSD of FLE for $\omega ={\omega }_r^f$. (d) The probability distributions of FLE for $\omega =0$ and ${\omega }_r^f$.

Figure 4

The results for chaotic Lorenz oscillators. (a) The variation of $\mathrm{\Lambda }$ with respect to $\omega$ and $\sigma$. (b) Fixing $\sigma =11$, the variation of the dominant frequencies, as identified from the PSD of FLE, with respect to $\omega$. The diagonal line corresponds to the frequency associated to periodic coupling.