Numerical integration of high-order variational equations of ODEs
This paper discusses the numerical integration of high-order variational equationsof ODEs. It is proved that, given a numerical method (say, any Runge-Kutta or Taylor method), to use automatic differentiation on this method (that is, using jet transport up to order $p$ with a time step $h$ for the n...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/195266 |
| Acceso en línea: | https://hdl.handle.net/2445/195266 |
| Access Level: | acceso abierto |
| Palabra clave: | Anàlisi numèrica Equacions diferencials ordinàries Sistemes dinàmics diferenciables Problemes de valor inicial Numerical analysis Ordinary differential equations Differentiable dynamical systems Initial value problems |
| Sumario: | This paper discusses the numerical integration of high-order variational equationsof ODEs. It is proved that, given a numerical method (say, any Runge-Kutta or Taylor method), to use automatic differentiation on this method (that is, using jet transport up to order $p$ with a time step $h$ for the numerical integration) produces exactly the same results as integrating the variational equationsup to of order $p$ with the same method and time step $h$ as before. This allows to design step-size control strategies based on error estimates of the orbit and of the jets. Finally, the paper discusses how to use jet transport to obtain power expansions of Poincaré maps (either with spatial or temporal Poincaré sections) and invariant manifolds. Some examples are provided. |
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