Noise-induced chaos: a conditioned random dynamics perspective

TL;DR

The work addresses how increasing bounded additive noise can induce chaos in a low-dimensional map, bridging noise-driven bifurcations with conditioned random dynamics. It introduces a two-compartment model that partitions the attractor into expanding and contracting regions and links the overall growth rate to a weighted mix of conditioned rates via $\overline{\Lambda} = (\tau_{\text{exp}}\Lambda_{\text{exp}} + \tau_{\text{cont}}\Lambda_{\text{cont}})/(\tau_{\text{exp}}+\tau_{\text{cont}})$, where the escape times arise from quasi-stationary and quasi-ergodic densities computed with Ulam’s method. A key finding is that the transition to chaos is driven by a rapid decay of the escape time from the contracting region ($\tau_{\text{cont}}$), while $\Lambda_{\text{exp}}$ and $\Lambda_{\text{cont}}$ stay roughly constant, leading to a sign change in $\overline{\Lambda}$ consistent with the full map’s Lyapunov exponent. The approach provides a noise-strength independent framework for bifurcations in random systems and is suggested to be universal, with analogous behavior observed in higher-dimensional maps such as the Hénon map with bounded noise.

Abstract

We consider transitions to chaos in random dynamical systems induced by an increase of noise amplitude. We show how the emergence of chaos (indicated by a positive Lyapunov exponent) in a logistic map with bounded additive noise can be analyzed in the framework of conditioned random dynamics through expected escape times and conditioned Lyapunov exponents for a compartmental model representing the competition between contracting and expanding behavior. In contrast to the existing literature, our approach does not rely on small noise assumptions, nor refers to deterministic paradigms. We find that the noise-induced transition to chaos is caused by a rapid decay of the expected escape time from the contracting compartment, while all other order parameters remain approximately constant.

Figures (6)

• Figure 1: On the left, bifurcation diagram for the deterministic logistic map $x_{n+1} = ax_n(1-x_n)$, with $a \in [3.6,4]$. The value $a = 3.83$ shows to be inside the three periodic window, where almost all orbits converge towards a three periodic orbit. On the right, bifurcation diagram for the random logistic map $x_{n+1} = 3.83x_n(1-x_n) + \omega_n$, together with the system's Lyapunov exponent, $\Lambda$, with $\omega_n \sim \text{Unif}(-\varepsilon, \varepsilon)$ and $\varepsilon \in [0,0.0025]$. The three phases are separated by two dashed vertical lines at the bifurcation points $\varepsilon_D$ and $\varepsilon_L$.
• Figure 2: Time series evolution of $u_n$ for an initial condition $(x_0,y_0)$ of the two-point motion for phase II ($\varepsilon_D < \varepsilon = 0.00125 < \varepsilon_L, \Lambda <0$), at the transition point ($\varepsilon = 0.00146 \simeq \varepsilon_L, \Lambda = 0$), and phase III ($\varepsilon_L < \varepsilon = 0.00175, \Lambda >0$).
• Figure 3: Lyapunov exponent $\Lambda$ of Eq. \ref{['eq:rlm']} as a function of noise amplitude obtained from the stationary density, and $\overline{\Lambda}$ obtained from the quasi-stationary and quasi-ergodic densities.
• Figure 4: From top to bottom, the quasi-stationary density $m$, the function $v$ and the quasi-ergodic density $q$, on the expanding region $M_{\text{exp}}$ for $\varepsilon = 0.00125$. These have been obtained using Ulam's method with $2^{14}$ components. The compartment $M_{\text{exp}}$ has been highlighted in bold on the bottom horizontal axis.
• Figure 5: Conditioned Lyapunov exponent on the expanding and contracting components, $\Lambda_\text{exp}$ and $\Lambda_\text{cont}$, as a function of $\varepsilon$. The Lyapunov exponent $\overline{\Lambda}$ is presented to illustrate the role of the escape times in Eq. \ref{['eq:Lyap_model']} as $\Lambda_\text{exp}$ and $\Lambda_\text{cont}$ remain almost constant.