Generalization of the Gauss Map: a Jump Into Chaos With Universal Features

Abstract

The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behavior, and it generates a symbolic dynamics consisting of infinitely many different symbols. Here we introduce a generalization of the Gauss map, which is given by xt+1 = 1 where alpha 0 is a parameter and xt is an element of [0, 1] (t = 0, 1, 2, 3, ...). The symbol [... ] 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 equivalent to 0.241485141808811 ... which we analyze 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 -* 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 -* infinity 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.