Abstract
We demonstrate a new algorithm for computing one-dimensional stable and unstable manifolds of (Poincaré) maps that we have implemented in DsTool [1]. As an example we investigate the most complicated sequence of bifurcations in the forced Van der Pol oscillator as the amplitude of the forcing is increased. This bifurcation sequence has recently been used to test algorithms for the computation of invariant tori.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A. Back J. Guckenheimer M.R. Myers F.J. Wicklin and P.A. Worfolk Ds Tool Computer assisted exploration of dynamical systems tices Amer. Math. Soc 39 4), (1992), pp. 303–309.
C. Baesens, J. Guckenheimer, S. Kim and R.S. Mackay, Three coupled oscillators: mode-locking, global bifurcations and toroidal chaosy Physica D, 49, (1991), pp. 387–475.
H.W. Broer, H.M. Osinga and G. VEGTER, Algorithms for computing normally hyperbolic invariant manifolds, Z. angew. Math. Phys., 48, (1997), pp. 480–524.
H.W. Broer, R. Roussarie, C. Simó, On the Bogdanov-Takens bifurcation for planar diffeomorphisms, in Proc. Equadiff 91, pp. 81–92, World Scientific, 1993.
H.W. Broer, R. Roussarie, C. Simó, Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms, Ergodic Th. & Dynam. Sys., 16, (1996), pp. 1147–1172.
L. Dieci and J. Lorenz, Computation of invariant tori by the method of characteristics, SIAM J. Numer. Anal., 32, (5), (1995), pp. 1436–1474.
L. Dieci and J. Lorenz, Lyapunov-type numbers and torus breakdown: Numerical aspects and a case study, Numer. Algorithms, 14, (1-3), (1997), pp. 79–102.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983.
M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.
D. Hobson, An efficient method for computing invariant manifolds of planar maps, J. Comp. Phys., 104, (1), (1993), pp. 14–22.
B. Krauskopf and H.M. Osinga, Globalizing two-dimensional unstable manifolds of maps, Int. J. Bifurcation & Chaos, 8, (3), (1998), pp. 483–504. (http://www.geom.umn.edu/docs/research/manifolds/)
B. Krauskopf and H.M. Osinga, Growing unstable manifolds of planar maps, IMA preprint, 1517, (1997). (http://www.ima.umn.edu/preprints/0CT97/1517.ps.gz)
B. Krauskopf and H.M. Osinga, Growing 1D and quasi 2D unstable manifolds of maps, J. Comp. Phys., 146, (1), (1998), pp. 404–419.
H.M. Osinga, Global Manifolds 1D, software for use with DsTool (1998), http://www.cds.caltech.edu/~hinke/dss/map/ko_1D/).
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C: the Art of Scientific computing, Cambridge Univ. Press, second edition, 1992.
V. Reichelt, Computing invariant tori and circles in dynamical systems, this volume, IMA Volumes in Mathematics and its Applications, 119, Springer-Verlag, 1998, pp. 407–437.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this paper
Cite this paper
Krauskopf, B., Osinga, H.M. (2000). Investigating Torus Bifurcations in the Forced Van Der Pol Oscillator. In: Doedel, E., Tuckerman, L.S. (eds) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 119. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1208-9_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1208-9_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7044-7
Online ISBN: 978-1-4612-1208-9
eBook Packages: Springer Book Archive