Matrix-free numerical torus bifurcation of periodic orbits
We consider systems ϕ˙ = f(ϕ, λ) where f : R^n×R → R^n. Such systems often arise from space discretizations of parabolic PDEs. We are interested in branches (with respect to λ) of periodic solutions of such systems. In the present paper we describe a numerical continuation method for tracing such br...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Sapienza Università Editrice
2005-01-01
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| Series: | Rendiconti di Matematica e delle Sue Applicazioni |
| Subjects: | |
| Online Access: | https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2005(1)/33-52.pdf |
| Summary: | We consider systems ϕ˙ = f(ϕ, λ) where f : R^n×R → R^n. Such systems often arise from space discretizations of parabolic PDEs. We are interested in branches (with respect to λ) of periodic solutions of such systems. In the present paper we describe a numerical continuation method for tracing such branches. Our methods are matrix-free, i.e., Jacobians are only implemented as actions, this enables us to allow for large n. Of particular interest is the detection and precise numerical approximation of bifurcation points along such branches: especially period-doubling and torus bifurcation points. This will also be done in a matrix-free context combining Arnoldi iterations (to obtain coarse information) with the calculation of suitable test functions (for precise approximations). We illustrate the method with the one- and two-dimensional Brusselator. |
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| ISSN: | 1120-7183 2532-3350 |