Abstract
We give a numerical method for the computation of heteroclinic orbits connecting two saddle points in ℝ2. These can be computed to very high period due to an integral phase condition and an adaptive discretization. We can also compute entire branches (one-dimensional continua) of such orbits. The method can be extended to compute an invariant manifold that connects two fixed points in ℝn. As an example we compute branches of traveling wave front solutions to the Huxley equation. Using weighted Sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period.
Supported in part by NSERC (Canada), A4274 and FCAC (Quebec) EQ1438.
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
D.G. Aronson, Density dependent interaction-diffusion systems, in: Dynamics and Modelling of Reactive Systems (Academic Press, New York, 1980) 161–176.
I. Babuška, The finite element method for infinite domains, Math. Comp. 26 (1972) 1–11.
M. Bieterman and I. Babuška, An adaptive method of lines with error control for parabolic equations of the reaction-diffusion type, J. Comput. Physics 63 (1986) 33–66.
J. Buckmaster and G.S.S. Ludford, Theory of Laminar Flames (Cambridge University Press, Cambridge, 1982).
J. Descloux and J. Rappaz, Approximation of solution branches of nonlinear equations, R.A.I.R.O. 16 (1982) 319–349.
E.J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Numer. 30 (1981) 265–284.
E.J. Doedel and J.P. Kernévez, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, Applied Mathematics Report, California Institute of Technology, 1986, 226 pages.
E.J. Doedel and M.J. Friedman, Numerical computation and continuation of invariant manifolds connecting fixed points, in preparation.
E.J. Doedel and J.P. Kernévez, A numerical analysis of wave phenomena in a reaction diffusion model, in: H.G. Othmer, Ed., Nonlinear Oscillations in Biology and Chemistry, Lecture Notes Biomath. 66 (Springer, Berlin, 1986) 261–273.
B.D. Hassard, Computation of invariant manifolds, in: P.J. Holmes, Ed., New Approaches to Nonlinear Problems in Dynamics (SIAM, Philadelphia, PA, 1980) 27–42.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math. 840 (Springer, Berlin, 1981).
T.M. Hagstrom and H.B. Keller, Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains, Math. Comput. 48 (1987) 449–470.
T.M. Hagstrom and H.B. Keller, The numerical calculation of traveling wave solutions of nonlinear parabolic equations, SIAM J. Sci. Stat. Comput. 7 (1986) 978–988.
H.B. Keller, Approximation methods for nonlinear problems with application to two-point boundary value problems, Math. Comput. 29 (1975) 464–474.
M. Lentini and H.B. Keller, Boundary value problems over semi infinite intervals and their numerical solution, SIAM J. Numer. Anal 17 (1980) 557–604.
R.M. Miura, Accurate computation of the stable solitary wave for the Fitz-Hugh-Nagumo equation, J. Math. Biology 13 (1982) 247–269.
H.G. Othmer, Nonlinear wave propagation in reacting systems, J. Math. Biology 2 (1975) 133–163.
J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion system, SIAM J. Appl. Math. 42 (1982) 1111–1137.
R.D. Russell and J. Christiansen, Adaptive mesh selection strategies for solving boundary value problems, SIAM J. Numer. Anal. 15 (1978) 59–890.
D.H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Mathematics 22 (1976) 312–355.
D. Terman, Traveling wave solutions arising from a combustion model, IMA Preprint Series 216, University of Minnesota, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Basel AG
About this chapter
Cite this chapter
Doedel, E.J., Friedman, M.J. (1990). Numerical computation of heteroclinic orbits. In: Mittelmann, H.D., Roose, D. (eds) Continuation Techniques and Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 92. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5681-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5681-2_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-2397-4
Online ISBN: 978-3-0348-5681-2
eBook Packages: Springer Book Archive