2024/11/20 by Christian Beck, Beck, Christian, Uǧur Tırnaklı +3 · 3 citations
Physics and Astronomy · #Advanced Mathematical Theories and Applications #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2411.13629
openalex publication_date 2024/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 xt+1=(1)/(xtα) - [(1)/(xtα) ] where α≥ 0 is a parameter and xt ∈ [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=αc ≡ 0.241485141808811… 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 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→ αc+. 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.