On the numerical continuation of isolas of equilibria
We present a numerical strategy to compute one-parameter families of isolas of equilibrium solutions in ODEs. Isolas are solution branches closed in parameter space. Numerical continuation is required to compute one single isola since it contains at least one unstable segment. We show how to use ps...
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| Format: | Article |
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World Scientific
2012
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| Online Access: | https://eprints.nottingham.ac.uk/1652/ |
| _version_ | 1848790647090184192 |
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| author | Avitabile, Daniele Desroches, Mathieu Rodrigues, Serafim |
| author_facet | Avitabile, Daniele Desroches, Mathieu Rodrigues, Serafim |
| author_sort | Avitabile, Daniele |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | We present a numerical strategy to compute one-parameter families of isolas of equilibrium solutions in ODEs. Isolas are solution branches closed in parameter space. Numerical
continuation is required to compute one single isola since it contains at least one unstable segment. We show how to use pseudo-arclength predictor-corrector schemes in order to follow an entire isola in parameter space, as an individual object, by posing a suitable algebraic problem. We continue isolas of equilibria in a two-dimensional dynamical system, the so-called continuous stirred tank reactor model, and also in a three-dimensional model related to plasma physics. We then construct a toy model and follow a family of isolas past a fold and illustrate how to initiate the computation close to a formation center, using approximate ellipses in a model inspired by the Van der Pol equation. We also show how to introduce node adaptivity in the discretization of the isola, so as to concentrate nodes in region with higher curvature. We conclude by commenting on the extension of the proposed numerical strategy to the case of isolas of periodic orbits. |
| first_indexed | 2025-11-14T18:15:56Z |
| format | Article |
| id | nottingham-1652 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:15:56Z |
| publishDate | 2012 |
| publisher | World Scientific |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-16522020-05-04T20:21:09Z https://eprints.nottingham.ac.uk/1652/ On the numerical continuation of isolas of equilibria Avitabile, Daniele Desroches, Mathieu Rodrigues, Serafim We present a numerical strategy to compute one-parameter families of isolas of equilibrium solutions in ODEs. Isolas are solution branches closed in parameter space. Numerical continuation is required to compute one single isola since it contains at least one unstable segment. We show how to use pseudo-arclength predictor-corrector schemes in order to follow an entire isola in parameter space, as an individual object, by posing a suitable algebraic problem. We continue isolas of equilibria in a two-dimensional dynamical system, the so-called continuous stirred tank reactor model, and also in a three-dimensional model related to plasma physics. We then construct a toy model and follow a family of isolas past a fold and illustrate how to initiate the computation close to a formation center, using approximate ellipses in a model inspired by the Van der Pol equation. We also show how to introduce node adaptivity in the discretization of the isola, so as to concentrate nodes in region with higher curvature. We conclude by commenting on the extension of the proposed numerical strategy to the case of isolas of periodic orbits. World Scientific 2012-11 Article PeerReviewed Avitabile, Daniele, Desroches, Mathieu and Rodrigues, Serafim (2012) On the numerical continuation of isolas of equilibria. International Journal of Bifurcation and Chaos, 22 (11). 1250277/1- 1250277/12. ISSN 0218-1274 (In Press) http://www.worldscientific.com/doi/abs/10.1142/S021812741250277X doi:10.1142/S021812741250277X doi:10.1142/S021812741250277X |
| spellingShingle | Avitabile, Daniele Desroches, Mathieu Rodrigues, Serafim On the numerical continuation of isolas of equilibria |
| title | On the numerical continuation of isolas of equilibria |
| title_full | On the numerical continuation of isolas of equilibria |
| title_fullStr | On the numerical continuation of isolas of equilibria |
| title_full_unstemmed | On the numerical continuation of isolas of equilibria |
| title_short | On the numerical continuation of isolas of equilibria |
| title_sort | on the numerical continuation of isolas of equilibria |
| url | https://eprints.nottingham.ac.uk/1652/ https://eprints.nottingham.ac.uk/1652/ https://eprints.nottingham.ac.uk/1652/ |