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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.10752 |
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| _version_ | 1866914716057600000 |
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| author | Mendez-Bermudez, J. A. Aguilar-Sanchez, R. |
| author_facet | Mendez-Bermudez, J. A. Aguilar-Sanchez, R. |
| contents | The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. It is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Caputo derivative, respectively. In this work we perform a scaling study of the average squared action $\left< I^2 \right>$ along strongly chaotic orbits, i.e. when $K\gg1$. We numerically prove that $\left< I^2 \right>\propto n^μ$ with $0\leμ(α)\le1$, for large enough discrete times $n$. That is, we demonstrate that the C-fSM displays subdiffusion for $1<α<2$. Specifically, we show that diffusion is suppressed for $α\to1$ since $μ(1)=0$, while standard diffusion is recovered for $α=2$ where $μ(2)=1$. We describe our numerical results with a phenomenological analytical estimation. We also contrast the C-fSM with the Riemann-Liouville fSM and Chirikov's standard map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10752 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tunable subdiffusion in the Caputo fractional standard map Mendez-Bermudez, J. A. Aguilar-Sanchez, R. Chaotic Dynamics Dynamical Systems The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. It is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Caputo derivative, respectively. In this work we perform a scaling study of the average squared action $\left< I^2 \right>$ along strongly chaotic orbits, i.e. when $K\gg1$. We numerically prove that $\left< I^2 \right>\propto n^μ$ with $0\leμ(α)\le1$, for large enough discrete times $n$. That is, we demonstrate that the C-fSM displays subdiffusion for $1<α<2$. Specifically, we show that diffusion is suppressed for $α\to1$ since $μ(1)=0$, while standard diffusion is recovered for $α=2$ where $μ(2)=1$. We describe our numerical results with a phenomenological analytical estimation. We also contrast the C-fSM with the Riemann-Liouville fSM and Chirikov's standard map. |
| title | Tunable subdiffusion in the Caputo fractional standard map |
| topic | Chaotic Dynamics Dynamical Systems |
| url | https://arxiv.org/abs/2403.10752 |