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Interpolating vector fields for near identity maps and averaging
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Gelfreich, V. and Vieiro, A. (2018) Interpolating vector fields for near identity maps and averaging. Nonlinearity, 31 (9). 4263. doi:10.1088/1361-6544/aacb8e ISSN 0951-7715.
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Official URL: https://doi.org/10.1088/1361-6544/aacb8e
Abstract
For a smooth near identity map, we introduce the notion of an interpolating vector field written in terms of iterates of the map. Our construction is based on Lagrangian interpolation and provides an explicit expression for autonomous vector fields which approximately interpolate the map. We study properties of the interpolating vector fields and explore their applications to the study of dynamics. In particular, we construct adiabatic invariants for symplectic near identity maps. We also introduce the notion of a Poincaré section for a near identity map and use it to visualise dynamics of four-dimensional maps. We illustrate our theory with several examples, including the Chirikov standard map, a volume-preserving map and a symplectic map in dimension four. The last example is motivated by the theory of Arnold diffusion.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Vector fields, Hamiltonian systems | |||||||||||||||
| Journal or Publication Title: | Nonlinearity | |||||||||||||||
| Publisher: | Institute of Physics Publishing Ltd. | |||||||||||||||
| ISSN: | 0951-7715 | |||||||||||||||
| Official Date: | 2 August 2018 | |||||||||||||||
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| Volume: | 31 | |||||||||||||||
| Number: | 9 | |||||||||||||||
| Article Number: | 4263 | |||||||||||||||
| DOI: | 10.1088/1361-6544/aacb8e | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
| Date of first compliant deposit: | 4 June 2018 | |||||||||||||||
| Date of first compliant Open Access: | 18 October 2018 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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