Generalization of the Gauss Map: A jump into chaos with universal features
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916885601189888 |
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| author | Beck, Christian Tirnakli, Ugur Tsallis, Constantino |
| author_facet | Beck, Christian Tirnakli, Ugur Tsallis, Constantino |
| contents | 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 $x_{t+1}=\frac{1}{x_t^α} - \Bigl[\frac{1}{x_t^α} \Bigr]$ where $α\geq 0$ 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 $α=1$. The system exhibits a sudden `jump into chaos' at the critical parameter value $α=α_c \equiv 0.241485141808811\dots$ 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 $α$. 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 $α\to α_c^+$. 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 $α\to \infty$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13629 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalization of the Gauss Map: A jump into chaos with universal features Beck, Christian Tirnakli, Ugur Tsallis, Constantino Statistical Mechanics Chaotic Dynamics 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 $x_{t+1}=\frac{1}{x_t^α} - \Bigl[\frac{1}{x_t^α} \Bigr]$ where $α\geq 0$ 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 $α=1$. The system exhibits a sudden `jump into chaos' at the critical parameter value $α=α_c \equiv 0.241485141808811\dots$ 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 $α$. 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 $α\to α_c^+$. 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 $α\to \infty$ 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. |
| title | Generalization of the Gauss Map: A jump into chaos with universal features |
| topic | Statistical Mechanics Chaotic Dynamics |
| url | https://arxiv.org/abs/2411.13629 |