Computer assisted proofs for transverse heteroclinics by the parameterization method
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911883186929664 |
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| author | Murray, Maxime James, J. D. Mireles |
| author_facet | Murray, Maxime James, J. D. Mireles |
| contents | This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a Fourier-Taylor approximation of the local stable/unstable manifold of the periodic orbit, combined with a numerical method for solving two point boundary value problems via Chebyshev series approximations. The a-posteriori analysis developed provides mathematically rigorous bounds on all approximation errors, providing both abstract existence results and quantitative information about the true heteroclinic solution. Example calculations are given for both the dissipative Lorenz system and the Hamiltonian Hill Restricted Four Body Problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computer assisted proofs for transverse heteroclinics by the parameterization method Murray, Maxime James, J. D. Mireles Dynamical Systems Numerical Analysis Mathematical Physics This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a Fourier-Taylor approximation of the local stable/unstable manifold of the periodic orbit, combined with a numerical method for solving two point boundary value problems via Chebyshev series approximations. The a-posteriori analysis developed provides mathematically rigorous bounds on all approximation errors, providing both abstract existence results and quantitative information about the true heteroclinic solution. Example calculations are given for both the dissipative Lorenz system and the Hamiltonian Hill Restricted Four Body Problem. |
| title | Computer assisted proofs for transverse heteroclinics by the parameterization method |
| topic | Dynamical Systems Numerical Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2405.12446 |