Noise-Induced Chaos: A Consequence of Long Deterministic Transients

Abstract

We argue that transient chaos in deterministic dynamical systems is a major source of noiseinduced chaos. The line of arguments is based on the fractal properties of the dynamical invariant sets responsible for transient chaos, which were not taken into account in previous works. We point out that noise-induced chaos is a weak noise phenomenon since intermediate noise strengths destroy fractality. The existence of a deterministic nonattracting chaotic set, and of chaotic transients, underlying noise-induced chaos is illustrated by examples, among others by a population dynamical model.

Fractal snapshot components in chaos induced by strong noise

“…It has been shown in Refs. [6,7] that noise makes the dynamics space filling on small phase-space scales, less than ε c = 0.018 for σ = 0.01. For weak noise, there is always a scaling region, although short, with a slope given by the noise-free fractal dimension of the unstable manifold, which is D 0 = 1.5 in this case.…”

“…With the average Lyapunov exponents determined numerically in the noisy system, the Lyapunov dimension (6) yields as D L = 1.117 for our snapshot attractors at strong noise (σ = 0.1). The dimension of the noise-induced chaotic attractor at weak noise, however, is provided by another expression [6,7]. The relation and the crossover between these expressions is discussed in the appendix.…”

“…A typical trajectory then visits both the original periodic attractors and the nonattracting chaotic set. This phenomenon is known as noise-induced chaos [1][2][3], which is fundamental in nonlinear and statistical physics and has received prolonged attention [4][5][6][7]. The field of potential applications is broad, ranging from electronic circuits [8] to epidemics [9] or population dyanmics [10].…”

Transient chaos in a globally coupled system of nearly conservative Hamiltonian Duffing oscillators

“…diffusion and other transport phenomena [17,18]. Recently, a noise induced system which exhibits a interesting phenomenon of transient chaos has also been reported [19].…”

Chaos and noise

“…Nevertheless, such situations are not only common, but also of significant interest in many applications. These include such diverse areas as noise-induced chaos [7][8][9][10], ecology [11], galactic dynamics [12], and the quantum-classical transition [13,14].…”