Universal Critical Behavior of Transition to Chaos: Intermittency Route

TL;DR

This paper analytically demonstrates that the critical exponent of the Lyapunov exponent during the transition to chaos is universally 1/2 across a broad class of generalized one-dimensional maps, confirming the robustness of universality in chaotic transitions.

Contribution

The authors generalized one-dimensional chaotic maps and proved the universality of the critical Lyapunov exponent's value as 1/2, regardless of the density function's power exponent.

Findings

Critical exponent of Lyapunov exponent is 1/2 for generalized maps.

Universality holds for a countably infinite class of maps.

Analytical proof extends understanding of chaos transition universality.

Abstract

The robustness of the universality class concept of the chaotic transition was investigated by analytically obtaining its critical exponent for a wide class of maps. In particular, we extended the existing one-dimensional chaotic maps, thereby generalising the invariant density function from the Cauchy distribution by adding one parameter. This generalisation enables the adjustment of the power exponents of the density function and superdiffusive behavior. We proved that these generalised one-dimensional chaotic maps are exact (stronger condition than ergodicity) to obtain the critical exponent of the Lyapunov exponent from the phase average. Furthermore, we proved that the critical exponent of the Lyapunov exponent is $\frac{1}{2}$ regardless of the power exponent of the density function and is thus universal. This result can be considered as rigorous proof of the universality of the critical exponent of the Lyapunov exponent for a countably infinite number of maps.