Control of spatiotemporal chaos by stochastic resetting

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.

💡 Research Summary

The paper investigates how stochastic resetting—periodically returning a dynamical system to its initial condition with a given probability—can be used to control spatiotemporal chaos in classical many‑body systems. The authors first formulate a discrete‑time map with resetting: at each iteration the state xₙ either evolves under a deterministic chaotic map f(x) with probability (1‑r) or is reset to the original state x₀ with probability r. For the zero‑dimensional case they define a renormalized Lyapunov exponent ˜λ that averages over resetting histories before taking the logarithm. By exploiting a renewal equation that relates the growth of infinitesimal perturbations with and without resetting, they obtain an exact expression ˜λ = λ + ln(1‑r) for r below a critical value r_c = 1‑e^{‑λ}. When r reaches r_c the exponent vanishes linearly, signalling a dynamical phase transition from chaotic to non‑chaotic behavior; for r ≥ r_c the system loses ergodicity and trajectories remain confined to a small subset of phase space that repeatedly revisits the initial condition.

To test the theory they study the logistic map f(x)=αx(1‑x) at α=4, where the bare Lyapunov exponent is λ=ln 2. Numerical simulations confirm the analytical prediction: the renormalized exponent follows the linear decay and disappears at r_c=½. The stationary distribution under resetting is a geometric mixture of iterates of the deterministic map, which remains dependent on the chosen x₀ but does not affect the universal behavior of ˜λ.

The analysis is then extended to spatially extended systems by considering a one‑dimensional coupled‑map lattice (CML). Each site evolves according to the logistic map plus diffusive coupling to nearest neighbours. The authors introduce out‑of‑time‑order correlators (OTOCs) Dₙ,ij = ⟨|δxₙ,i/δx₀,j|⟩ to quantify both temporal growth and spatial spread of perturbations. Assuming a standard ballistic front form Dₙ,ij ∝ exp