On the numerical continuation of isolas of equilibria

Rodrigues, S.; Desroches, M.; Avitabile, D.

Abstract

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.

Citation

Rodrigues, S., Desroches, M., & Avitabile, D. (2012). On the numerical continuation of isolas of equilibria. International Journal of Bifurcation and Chaos, 22(11), 1250277. https://doi.org/10.1142/s021812741250277x

• Journal Article Type: Article
• Online Publication Date: Apr 12, 2012
• Publication Date: 2012-11
• Deposit Date: Feb 27, 2013
• Publicly Available Date: Feb 27, 2013
• Journal: International Journal of Bifurcation and Chaos
• Print ISSN: 0218-1274
• Electronic ISSN: 1793-6551
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 22
• Issue: 11
• Article Number: 1250277
• Pages: 1250277
• DOI: https://doi.org/10.1142/s021812741250277x
• Keywords: Modelling and Simulation; Applied Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/1006137
• Publisher URL: http://www.worldscientific.com/doi/abs/10.1142/S021812741250277X
• Additional Information: Electronic version of an article published as International Journal of Bifurcation and Chaos, 22, 11, 2012, 1250277, doi: 10.1142/S021812741250277X © World Scientific Publishing Company, http://www.worldscientific.com/doi/abs/10.1142/S021812741250277X