Control of spatiotemporal chaos by stochastic resetting

Abstract

We study how spatiotemporal chaos in dynamical systems can be controlled by stochastically returning them to their initial conditions. Focusing on discrete nonlinear maps, we analyze how key measures of chaos -- the Lyapunov exponent and butterfly velocity, which quantify sensitivity to initial perturbations and the ballistic spread of information, respectively -- are reduced by stochastic resetting. We identify a critical resetting rate that induces a dynamical phase transition, characterized by the simultaneous vanishing of the Lyapunov exponent and butterfly velocity, effectively arresting the spread of information. These theoretical predictions are validated and illustrated with numerical simulations of the celebrated logistic map and its lattice extension. Beyond discrete maps, our findings are applicable to virtually any chaotic extended classical many-body system.

Figures (4)

• Figure 1: Logistic map [Eq. (\ref{['eq:logistic']}), $\alpha=4$] subject to stochastic resetting. (a) Exponential sensitivity to initial perturbations for the deterministic map ($r=0$) and for a resetting rate $r = 0.2 < r_{\rm c}$. The former validates the exponential ansatz $d_ n:= \log |\delta x_n/\delta x_0| \propto \lambda \,n$, even at early times. The Lyapunov exponents $\lambda = \log 2$ and $\tilde{\lambda}$ can be extracted from the slopes. For $r=0.6 > r_{\rm c}$, $\tilde{d}_n$ saturates to a constant. (b) The renormalized Lyapunov $\tilde{\lambda}$ computed with Eq. (\ref{['eq:cumprod']}) is compared to the analytical predictions of Eq. (\ref{['eq:Lyapunov_resetting']}) as a function of the resetting rate $r$. The critical resetting is located at $r_{\rm c} = 1/2$.
• Figure 2: Coupled logistic map [Eq. (\ref{['eq:logistic_lattice']}] subject to stochastic resetting: the OTOC $\tilde{D}_{n,\,ij}$ as a function of space and time reveals the ballistic spreading of perturbations for the (a) Deterministic case at $r=0$, and (b) Finite resetting rate $r = 0.2 < r_{\rm c} \approx 0.38$. (c) Dynamical arrest at $r = 0.4 > r_{\rm c}$. The butterfly velocities can be extracted from the slopes of the light cones. Lattice size $L=701$, $\alpha=4$, and $c=0.1$. The data are spatially averaged over the initial perturbation site $j$, ranging from 1 to $L$.
• Figure 3: Spatiotemporal chaos in the coupled logistic map subject to stochastic resetting. (a) The velocity-dependent Lyapunov $\lambda(v)$ [Eq. (\ref{['eq:ansatz_lattice']})] is numerically extracted from the collapse of $(\log D_{n,\, ij}) / n$ versus $v:=|i-j|/n$ for various values of $i-j$ in the deterministic case ($r=0$). The butterfly velocity $\tilde{v}_{\rm B}$ at $r>0$ can be determined by solving $\lambda(\tilde{v}_{\rm B}) = -\log(1-r)$. The slope of $\lambda(v)$ at this solution determines the Lyapunov rate $\tilde{\lambda}_{\rm B}$. (b) $\tilde{v}_{\rm B}$ is extracted from the slope of the light cones (see Fig. \ref{['fig:lightcone']}) and is compared to the prediction of Eq. (\ref{['eq:butterfly']}) as a function of the resetting rate $r$. Lattice size $L=701$, $\alpha = 4$, and $c = 0.1$.
• Figure 4: Coupled logistic map with $\alpha=4$ and $c=0.1$ (no resetting, $r=0$): the local marginal of the stationary distribution is identical at all sites, here $i=1$ and $i=(L-1)/2$. The stationary distribution of the corresponding logistic map ($c=0$), $p_{\rm st}(x) = 1/(\pi \sqrt{x(1-x)})$, is plotted for comparison. Lattice size $L=701$.