Generalization of the Gauss Map: A jump into chaos with universal features

TL;DR

This paper introduces a generalized Gauss map with a parameter that induces a transition to chaos at a critical value, revealing universal features in its invariant density and chaotic behaviour.

Contribution

The authors define a new parameterized map generalizing the Gauss map, analyze the chaos transition at a critical parameter, and derive invariant densities, highlighting universal features of such dynamical systems.

Findings

Chaotic transition occurs at a specific critical parameter value.

Invariant density approaches a Cauchy distribution at the transition.

For large parameters, the invariant density becomes uniform.

Abstract

The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behaviour and which generates a symbolic dynamics consisting of infinitely many different symbols. Here we introduce a generalization of the Gauss map which is given by $x_{t+1}=\frac{1}{x_t^\alpha} - \Bigl[\frac{1}{x_t^\alpha} \Bigr]$ where $\alpha \geq 0$ is a parameter and $x_t \in [0,1]$ ($t=0,1,2,3,\ldots$). The symbol $[\dots ]$ denotes the integer part. This map reduces to the ordinary Gauss map for $\alpha=1$. The system exhibits a sudden `jump into chaos' at the critical parameter value $\alpha=\alpha_c \equiv 0.241485141808811\dots$ which we analyse in detail in this paper. Several analytical and numerical results are established for this new map as a function of the parameter $\alpha$. In particular, we show that, at the critical point, the invariant density approaches a $q$-Gaussian with $q=2$ (i.e., the Cauchy distribution), which becomes infinitely narrow as $\alpha \to \alpha_c^+$. Moreover, in the chaotic region for large values of the parameter $\alpha$ we analytically derive approximate formulas for the invariant density, by solving the corresponding Perron-Frobenius equation. For $\alpha \to \infty$ the uniform density is approached. We provide arguments that some features of this transition scenario are universal and are relevant for other, more general systems as well.