Time-delayed feedback control for random dynamical systems

TL;DR

This work extends time-delayed feedback control (TDFC) to random dynamical systems by introducing an extended stability classification based on the almost-sure Lyapunov exponent $\lambda$ and trajectory fluctuation ratio $\sigma/\epsilon$, yielding SS, WS, and U regimes. The authors apply the framework to a random logistic map and a stochastic Rössler system, demonstrating that increasing noise can induce chaos that overcomes control, producing transitions from pseudo-periodic to thickened and finally to random strange attractor dynamics. They connect these regimes to the geometry of random pullback attractors, offering a stochastic generalization of the deterministic TDFC criterion. The results provide a practical, model-agnostic guideline for applying delayed feedback to noisy nonlinear systems and highlight the need to consider both stability and fluctuations in control design.

Abstract

We extend the Pyragas time-delayed feedback control (TDFC) to apply it to random dynamical systems and introduce an extended classification based on Lyapunov exponents and trajectory fluctuations. We demonstrate the applicability of this framework using the random logistic map and the stochastic Rössler system. Our results reveal that noise-induced chaos triggers a transition from stable to unstable regimes based on a phenomenon inherent to random dynamical systems.

Figures (10)

• Figure 1: $a=1.784, D=0.002, K=0.05, \tau=3$ (left) Stabilized pseudo-period-3 orbit. (right) The density distribution function for time series $x$. The supports of the distribution function is disconnected. The standard deviation of trajectory is $0.0024$, which is approximately the same order of the standard deviation of noise $0.00115$. The largest Lyapunov exponent is $-0.02$. This case corresponds to the strongly stable control.
• Figure 2: $a=1.784, D=0.00265, K=0.05, \tau=3$ (left) Partially chaotic orbit. (right) The density distribution function for time series $x$. The supports of the distribution function is connected. The standard deviation of trajectory is $1.198$, which is larger than the standard deviation of noise 0.00153, still the largest Lyapunov exponent is $-0.007$. This case corresponds to the weakly stable control (WC) with $p=0.66$. The value $p$ represents the proportion of time during which $|x_n - x_{n - \tau}|$ does not exceed the standard deviation of the system noise.
• Figure 3: $a=1.784, D=0.003, K=0.05, \tau=3$ (left) Random strange attractor. (right) The density distribution function for time series $x$. The supports of the distribution function is connected. The standard deviation of trajectory is $1.177$, which is larger than the standard deviation of noise 0.0025, the largest Lyapunov exponent is $0.014$. This case corresponds to the unstable control.
• Figure 4: Standard deviation (SD) of a trajectory (top) and the largest Lyapunov exponent (bottom) in the controlled system in changing the system noise intensity $D$. Standard deviation of the trajectory becomes much bigger than that of the system noise above $D=0.0023$. The largest Lyapunov exponent becomes positive above $D=0.0028$. The system is strongly stable (SS) in $D<0.0023$, is weakly stable (WS), and is unstable (U).
• Figure 5: Bifurcation diagram in changing noise intensity $D$. The transition from SS to WS occurs at $D=0.0023$, which is consistent with Figure 4 (top).