Figure 1

(Color online) Differences in the synchronization of two Rössler systems with identical coupling strengths and different power spectra. Phase plot trajectories of the variables $x$ vs their Hilbert transform $x^H$ for (a) system (1,2), with $x_1$ corresponding to ${\omega }_1={\omega }_0+\mathrm{\Delta }\omega$, where ${\omega }_0=0.6$ and $\mathrm{\Delta }\omega =0.005$; (b) system (3,4), with $x_3$ corresponding to ${\omega }_3={\omega }_0+\mathrm{\Delta }\omega$, where now ${\omega }_0=1$ and $\mathrm{\Delta }\omega =0.015$. For both Rössler systems $C=0.03$. (c) Power spectra of the time sequence $x_1$ (dashed line) and $x_3$ (solid line). A broader spectrum is observed for system (1,2) compared to system (3,4). (d) Instantaneous phase difference $\mathrm{\Delta }{\psi }_{1,1}\equiv (\varphi x_1\left(t\right)-\varphi x_2\left(t\right))\mathrm{mod}\left(2\pi \right)$ for system (1,2) (dashed line), and $\mathrm{\Delta }{\psi }_{1,1}\equiv (\varphi x_3\left(t\right)-\varphi x_4\left(t\right))\mathrm{mod}\left(2\pi \right)$ for system (3,4) (solid line), and (e) their corresponding distributions $P\left(\mathrm{\Delta }{\psi }_{1,1}\right)$. System (1,2) exhibits larger fluctuations in $\mathrm{\Delta }{\psi }_{1,1}$ and is characterized by a broader distribution $P\left(\mathrm{\Delta }{\psi }_{1,1}\right)$. (f) Synchronization index $\rho$ as a function of the coupling strength $C$. For identical values of $C$, system (3,4) (solid line), which is characterized by a narrower power spectrum, exhibits stronger synchronization (larger index $\rho$) compared to system (1,2) with a broader power spectrum. Specifically, for identical coupling strength $C=C_0=0.03$, the index $\rho ={\rho }_0$ $(◻)$ for system (1,2), while $\rho =0.3>{\rho }_0$ $(\circ )$ for system (3,4) although the frequency mismatch for system (3,4) is much larger. The effect of a Fourier bandpass filter applied to the system (1,2) while keeping $C=0.03$ fixed is equivalent to an increase of the coupling strength of the system leading to a larger index ${\rho }_1>{\rho }_0$ $\left(\Delta \right)$ as also shown in Fig. 2e.Reuse & Permissions

Figure 2

(Color online) Effects of bandpass filtering on synchronization. Time sequence of the variables $x_1$ and $x_2$ of system (1,2) (a) before and (b) after applying a bandpass Fourier filter with bandwidth $\mathrm{\Delta }f=0.01$. After bandpass filtering the sequences $x_1$ and $x_2$ are better aligned in time (with almost matching peaks). (c) Instantaneous phase difference $\mathrm{\Delta }{\psi }_{1,1}$ and (d) the distribution $P\left(\mathrm{\Delta }{\psi }_{1,1}\right)$ before (dashed line) and after (solid line) the Fourier bandpass filtering. After filtering, $\mathrm{\Delta }{\psi }_{1,1}$ is characterized by fewer fluctuations and a much narrower distribution $P\left(\mathrm{\Delta }{\psi }_{1,1}\right)$, indicating a stronger synchronization, although the coupling strength $C=0.03$ remains constant. (e) Dependence of the index $\rho$ on the bandwidth $2\pi \mathrm{\Delta }f$ for fixed $C=0.03$. A filter with a relatively broader bandwidth $(2\pi \mathrm{\Delta }f>1)$ leaves the synchronization index $\rho$ practically unchanged, $\rho ={\rho }_0$, where ${\rho }_0$ characterizes the synchronization between $x_1$ and $x_2$ before filtering. Narrowing $\mathrm{\Delta }f$ leads to a sharp increase in $\rho$, which is an artifact of the Fourier filtering as the coupling $C$ and all other parameters remain unchanged, e.g., for $\mathrm{\Delta }f=0.005$, $\rho ={\rho }_1\approx 4{\rho }_0$.Reuse & Permissions

Figure 3

(Color online) Effect of external additive white noise on phase synchronization for system (1,2) defined in Eq. (1). (a) Dependence of the synchronization index $\rho$ on the noise strength ${\sigma }_{\eta }$ for fixed value of the coupling constant $C$. (b) Dependence of the synchronization index $\rho$ on the coupling strength $C$ for different levels of white noise which are defined through the standard deviation ${\sigma }_{\eta }$.Reuse & Permissions

Figure 4

(Color online) Combined effects of external noise and Fourier bandpass filtering on the synchronization. (a) Cumulative distribution function $F\left(\rho \right)\equiv 1-{\int }_0^{\rho }P\left({\rho }^{\prime }\right)d{\rho }^{\prime }$for the synchronization index $\rho$ obtained from $100$ different realizations of pairs of white noise signals without coupling. The length of the noise signals is $\mathrm{int}[{10}^7∕2\pi ]$. Tails of the distributions for each bandwidth indicate the maximum values of $\rho$ one can obtain simply as a result of bandpass filtering when there is no synchronization between two white noise signals. (b) Synchronization index $\rho$ obtained for system (1,2) defined in Eq. (1) with additive white noise as a function of the bandwidth $\mathrm{\Delta }f$ for $C=0.03$. While the effect of noise is gradually reduced by the Fourier bandpass filter with decreasing bandwidth $\mathrm{\Delta }f$, there is an artificially increased synchronization (sharp increase in $\rho$) when $2\pi \mathrm{\Delta }f<1$, as also shown in Fig. 2e.Reuse & Permissions