Effective synchronization amid noise-induced chaos

Abstract

Two remote agents with synchronized clocks may use them to act in concert and communicate. This necessitates some means of creating and maintaining synchrony. One method, not requiring any direct interaction between the agents, is to expose them to a common, environmental, stochastic forcing. This "noise-induced synchronization" only occurs under sufficiently mild forcing; stronger forcing disrupts synchronization. We investigate the regime of strong noise, where the clocks' relative phases evolve chaotically. Using a simple realization of disruptive noise, we demonstrate effective synchronization. First, although the relative phases of the two clocks varied erratically, we confirm that they became statistically independent of initial conditions and hence equivalent after a well-defined timescale. Second, we show that an agent can estimate an effective phase that closely agrees with the other's phase. Thus, synchronization is practically attainable beyond the regime of conventional noise-induced synchronization. We finally discuss how it might be used in living systems.

Figures (13)

• Figure 1: Illustration of a phase map $\psi(\varphi)$ determination for a given limit cycle. A toy dynamical system is considered whose limit cycle is depicted by the closed black line in some dynamical phase space. Phase-position labels $\varphi$ along the loop are marked for the assigned phase origin $\varphi = 0$ and for phase positions displaced by $1/4$, $1/2$, and $3/4$ cycle from this origin. An open square marks the phase position of the oscillator at a moment of a kick, when its phase position was $\varphi'$. The subsequent trajectory of this oscillator is sketched in blue, showing its departure from the limit cycle and its eventual return to it. Before and after the kick, the oscillator gains phase at a constant speed (along the dashed trajectories). Also shown (in red) is the unperturbed trajectory of an oscillator which was at the phase origin at the moment of the kick. The position of the kicked trajectory at some arbitrary later time is marked with a filled square. The position of the unkicked trajectory at this same moment is marked with a filled circle. Since both trajectories advance around the loop at the same rate, their phase difference, indicated in orange, is independent of time. This difference for the arbitrary kick position $\varphi'$ is defined as the phase map, $\psi(\varphi')$.
• Figure 2: A sequence of two, initially-different phase distributions $q^{(k)}_{\mathrm{A}}(\varphi)$ (red) and $q^{(k)}_{\mathrm{B}}(\varphi)$ (blue) subjected to common noise in a typical realization of Eq. \ref{['eq:dynamics']} and Eq. \ref{['eq:phase_map']} with either $\Lambda=-0.141$ ($A=7.32$) or $\Lambda=0.141$ ($A=9.32$), as indicated. The initial distributions are uniform, $\varphi\in[0.0,0.05)$ and $\varphi\in[0.5,0.55)$, respectively. The distributions are shown on a circle so the periodicity $\,\mathrm{mod}\,1$ of the phase circle is apparent. The radial axis shows the distributions' values $q(\varphi)$ on a log scale, where $q(\varphi)=1$ for the inner full circle and as indicated for the outer dotted circles. The kick numbers $k$ are shown inside the inner circle. The $\Lambda<0$ dynamics are synchronizing, so their distributions become centered around the same phase value and their width decreases over time on average. For $\Lambda>0$, the distributions evolve erratically. However, remarkably, the red and blue points co-locate; they converge unto the same set of sharply-multimodal distributions under common noise --- this property enables the effective synchronization demonstrated below. The distributions are sampled by tracking $N=500$ initial phases via the nearest-neighbor distances; see Eq. \ref{['eq:est_q-phi']}. The entropies of the two distributions and the KLDs among them are shown in Fig. \ref{['fig:LposS&KLD']} for $\Lambda=0.141$ and Appendix \ref{['sec:convergenceL<0']} for $\Lambda=-0.141$. A detailed view of the blue distribution at $k=100$ for $\Lambda=0.141$ is shown in Appendix \ref{['sec:misidentification']}.
• Figure 3: (a) Kullback-Leibler divergences (KLDs) $D({\mathrm{A}}\Vert{\mathrm{B}})$ and $D({\mathrm{B}}\Vert{\mathrm{A}})$ among two distributions and (b) entropies $S_{\mathrm{A}}$ and $S_{\mathrm{B}}$ of each distribution, as obtained for the waiting time realization $\{\beta^k\}$ of Fig. \ref{['fig:circ_q-phi']} with $\Lambda=0.141$. Inset: The KLDs on a log-scale, where the numerical error in the KLD's estimation is indicates by the horizontal dotted line. The number of phase samples is $N=500$. Both initial distributions are of width $u=0.05$. The numerical errors in the estimation of the KLDs and entropies are $\Delta D=0.05$ and $\Delta S=0.2$, respectively.
• Figure 4: The average Kullback-Leibler divergence (KLDs) $D({\mathrm{A}}\Vert{\mathrm{B}})$ and $D({\mathrm{B}}\Vert{\mathrm{A}})$ among the two distributions, averaged over $500$ waiting-time realizations. We used phase maps of Eq. \ref{['eq:phase_map']} with various values of $\Lambda$ obtained by varying the gain parameter $A$. We show the results for (a) $\Lambda=0.405$ and (b) $\Lambda=0.121$. The number of phase samples is $N=500$ in each ensemble, and they are initially uniform-distributed with width $u=10^{-4}$. Dashed orange lines depict the exponential fit from which we extract the mixing-kick number. (c) The mixing-kick number $K_\mathrm{m}$ versus Lyapunov exponent $\Lambda$. The two enlarged orange points correspond to $K_\mathrm{m}$ for the $\Lambda$'s shown in panels (a) and (b). $K_\mathrm{m}$ is identified as minus the inverse slope in the exponential decay of $\langle D\rangle$. The exponential regime begins after a transient where the distributions spread and overlap, and terminates when the KLD is comparable with its numerical estimation error, $\sim0.05$. Mixing occurs faster as $\Lambda$ increases, so the two agents typically converge earlier. The decay is consistent with the powerlaw $K_\mathrm{m}=2.82\Lambda^{-1.49}$.
• Figure 5: (a) A density plot of the rescaled discrepancy, $\ell=\sqrt N|\Delta\varphi_{\mathrm{f}}|/e^S$ versus $S=(S_{\mathrm{A}}+S_{\mathrm{B}})/2$. The shade of blue represents the indicated values of the joint probability density function (PDF) to obtain $S$ and $\log_{10}\ell$, relative to its maximum. It was computed with kernel-density estimation from the $k=10^4$ data points shown in Appendix \ref{['sec:sigma_vary']} (Fig. \ref{['fig:deviation_vs_entropySM']}(a)), using a Gaussian kernel of width $0.4$ in $S$ and $0.2$ in $\log_{10}\ell$. The maximal value is $\mathrm{PDF}(-1.94,-0.0052)=0.135$, whose position is indicated by the black point. The puzzling lobe of probability at the upper right is addressed in Appendix \ref{['sec:sigma_vary']}. (b) The marginal cumulative distribution function (CDF) of the discrepancy $\log_{10}\ell$, which was arranged into bins of size $0.25$. The dashed line is the error function of width unity; its close agreement with the data suggests that the discrepancy is predominantly normal-distributed with the expected scaling $|\Delta\varphi_{\mathrm{f}}|\sim e^{S}/\sqrt{N}$. Inset: The marginal CDF of the entropy $S$, which was arranged into bins of size $0.5$. The dashed linear line implies, up to the low- and high-entropy outliers, that the entropy is exponentially distributed SongPRE2022.