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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.17673 |
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| _version_ | 1866916141084966912 |
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| author | Mendez-Bermudez, J. A. Aguilar-Sanchez, R. Sigarreta, J. M. Leonel, E. D. |
| author_facet | Mendez-Bermudez, J. A. Aguilar-Sanchez, R. Sigarreta, J. M. Leonel, E. D. |
| contents | The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. The RL-fSM is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work, we present a scaling study of the average squared action $\left< I^2 \right>$ of the RL-fSM along strongly chaotic orbits, i.e. for $K\gg1$. We observe two scenarios depending on the initial action $I_0$, $I_0\ll K$ or $I_0\gg K$. However, we can show that $\left< I^2 \right>/I_0^2$ is a universal function of the scaled discrete time $nK^2/I_0^2$ ($n$ being the $n$th iteration of the RL-fSM). In addition, we note that $\left< I^2 \right>$ is independent of $α$ for $K\gg1$. Analytical estimations support our numerical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17673 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scaling properties of the action in the Riemann-Liouville fractional standard map Mendez-Bermudez, J. A. Aguilar-Sanchez, R. Sigarreta, J. M. Leonel, E. D. Chaotic Dynamics The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. The RL-fSM is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work, we present a scaling study of the average squared action $\left< I^2 \right>$ of the RL-fSM along strongly chaotic orbits, i.e. for $K\gg1$. We observe two scenarios depending on the initial action $I_0$, $I_0\ll K$ or $I_0\gg K$. However, we can show that $\left< I^2 \right>/I_0^2$ is a universal function of the scaled discrete time $nK^2/I_0^2$ ($n$ being the $n$th iteration of the RL-fSM). In addition, we note that $\left< I^2 \right>$ is independent of $α$ for $K\gg1$. Analytical estimations support our numerical results. |
| title | Scaling properties of the action in the Riemann-Liouville fractional standard map |
| topic | Chaotic Dynamics |
| url | https://arxiv.org/abs/2402.17673 |