Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866914525175873536 |
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| author | Zolkin, Tim Nagaitsev, Sergei Morozov, Ivan Kladov, Sergei |
| author_facet | Zolkin, Tim Nagaitsev, Sergei Morozov, Ivan Kladov, Sergei |
| contents | This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only two are irreducible: the linearized rotation number at the fixed point and a nonlinear parameter representing the ratio of terms in the biquadratic invariant. Through a detailed analysis, we classify regimes of stable motion, provide exact solutions to the mapping equations, and derive a canonical set of action-angle variables, offering analytical expressions for the rotation number and nonlinear tune shift. We further establish connections between general standard-form mappings and the symmetric McMillan map, using the area-preserving Hénon map and accelerator lattices with thin sextupole magnet as representative case studies. Our results show that, despite being a second-order approximation, the symmetric McMillan map provides a highly accurate depiction of dynamics across a wide range of system parameters, demonstrating its practical relevance in both theoretical and applied contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10380 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity Zolkin, Tim Nagaitsev, Sergei Morozov, Ivan Kladov, Sergei Exactly Solvable and Integrable Systems Chaotic Dynamics Accelerator Physics This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only two are irreducible: the linearized rotation number at the fixed point and a nonlinear parameter representing the ratio of terms in the biquadratic invariant. Through a detailed analysis, we classify regimes of stable motion, provide exact solutions to the mapping equations, and derive a canonical set of action-angle variables, offering analytical expressions for the rotation number and nonlinear tune shift. We further establish connections between general standard-form mappings and the symmetric McMillan map, using the area-preserving Hénon map and accelerator lattices with thin sextupole magnet as representative case studies. Our results show that, despite being a second-order approximation, the symmetric McMillan map provides a highly accurate depiction of dynamics across a wide range of system parameters, demonstrating its practical relevance in both theoretical and applied contexts. |
| title | Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity |
| topic | Exactly Solvable and Integrable Systems Chaotic Dynamics Accelerator Physics |
| url | https://arxiv.org/abs/2410.10380 |