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 x_{t+1}=1/x_{t}^{α}-[1/x_{t}^{α}] where α≥0 is a parameter and x_{t}∈[0,1] (t=0,1,2,3,...). The symbol [⋯] denotes the integer part. This map reduces to the ordinary Gauss map for α=1. The system exhibits a sudden "jump into chaos" at the critical parameter value α=α_{c}≡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 α. 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 α→α_{c}^{+}. Moreover, in the chaotic region for large values of the parameter α we analytically derive approximate formulas for the invariant density, by solving the corresponding Perron-Frobenius equation. For α→∞ 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.