TY - JOUR
T1 - On the numerical continuation of isolas of equilibria
AU - Avitabile, D.
AU - Desroches, M.
AU - Rodrigues, S.
PY - 2012/1/1
Y1 - 2012/1/1
N2 - 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 on 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.
AB - 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 on 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.
KW - bifurcation theory
KW - isolas
KW - Numerical continuation
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U2 - 10.1142/S021812741250277X
DO - 10.1142/S021812741250277X
M3 - Article
AN - SCOPUS:84871021243
SN - 0218-1274
VL - 22
JO - International Journal of Bifurcation and Chaos
JF - International Journal of Bifurcation and Chaos
IS - 11
M1 - 1250277
ER -