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 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.
Highlights
Bifurcation theory for dynamical systems is concerned with determining and classifying the long-term behaviour of evolution equations upon parameter variation
The paper is organised as follows: in Section 2 we discuss how to set up the isola continuation problem; in Section 3 we show how to discretize the problem and we discuss our numerical implementation; in Section 4 and 5 we continue isolas in two applications, a continuous stirred tank reactor and a model from plasma physics, respectively; in Section 6 we show that our strategy can continue families of isolas that bend around a fold; in Section 7 we present an example in which isolas exist between two formation centers and comment on the computation of initial guesses; in Section 8, we show how to implement node adaptivity; in Section 9, we discuss possible extensions of the method
The main idea behind our method is to interpret isolas as individual objects: each isola is approximated by a polygon, and for fixed values of the secondary parameter we solve an algebraic problem whose unknowns are the nodes of the polygon and its perimeter
Summary
Bifurcation theory for dynamical systems (see, e.g., [23]) is concerned with determining and classifying the long-term behaviour of evolution equations upon parameter variation. The paper is organised as follows: in Section 2 we discuss how to set up the isola continuation problem; in Section 3 we show how to discretize the problem and we discuss our numerical implementation; in Section 4 and 5 we continue isolas in two applications, a continuous stirred tank reactor and a model from plasma physics, respectively; in Section 6 we show that our strategy can continue families of isolas that bend around a fold; in Section 7 we present an example in which isolas exist between two formation centers and comment on the computation of initial guesses; in Section 8, we show how to implement node adaptivity; in Section 9, we discuss possible extensions of the method
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