Abstract
We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.
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References
U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, NJ, 1988).
A. Bellen, One-step collocation for delay differential equations, J. Comput. Appl. Math. 10 (1984) 275–283.
W.J. Beyn, The effect of discretization on homoclinic orbits, in: Bifurcation Analysis, Algorithms, Applications, eds. T. Küpper, R. Seydel and H. Troger, International Series of Numerical Mathematics, Vol. 79 (Birkhäuser, Stuttgart, 1987) pp. 1–8.
W.J. Beyn, Global bifurcations and their numerical computation, in: Continuation and Bifurcations: Numerical Techniques and Applications, eds. D. Roose, A. Spence and B. De Dier (Kluwer, Dordrecht, 1990) pp. 169–181.
W.J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal. 9 (1990) 379–405.
A.R. Champneys, Y.A. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, University of Bristol, Applied Nonlinear Mathematics Research Report No. 95.1 (1995).
O. Diekmann, S.A. Van Gils, S.M. Verduyn Lunel and H.-O.Walther, Delay Equations: Functional-, Complex-and Nonlinear Analysis, Applied Mathematical Sciences, Vol. 110 (Springer, New York, 1995).
E. Doedel, Auto, a program for the automatic bifurcation analysis of autonomous systems, Congr. Numer. (1981) 265–384.
E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede and X.Wang, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont) (March 1998).
E.J. Doedel, M.J. Friedman and B.I. Kunin, Successive continuation for locating connecting orbits, Numer. Algorithms 17 (1997) 103–124.
K. Engelborghs, Numerical bifurcation analysis of delay differential equations, Ph.D. thesis, Katholieke Universiteit Leuven (2000).
K. Engelborghs and E. Doedel, Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations, Numer. Math. (2001) accepted.
K. Engelborghs, T. Luzyanina, K.J. Hout and D. Roose, Collocation methods for the computation of periodic solutions of delay differential equations, SIAM J. Sci. Comput. 22(5) (2000) 1593–1609.
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL (2001) accepted.
K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v2.00: a Matlab package for bifurcation analysis of delay differential equations, Technical Report TW-330, Department of Computer Science, K.U. Leuven (October 2001).
F. Giannakopoulos and O. Oster, Bifurcation properties of a planar system modelling neural activity, J. Diff. Equations Dynam. Systems 5(3/4) (1997).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectorfields (Springer, Berlin, 1993).
J.K. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 3 (Springer, New York, 1977).
J.K. Hale and X.-B. Lin, Heteroclinic orbits for retarded functional differential equations, J. Differential Equations 65 (1986).
A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) 500–544.
H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in: Applications of Bifurcation Theory, ed. P. Rabanowitz (Academic Press, New York, 1997) pp. 359–384.
A.I. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity 11(6) (1998) 1505–1519.
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Samaey, G., Engelborghs, K. & Roose, D. Numerical Computation of Connecting Orbits in Delay Differential Equations. Numerical Algorithms 30, 335–352 (2002). https://doi.org/10.1023/A:1020102317544
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DOI: https://doi.org/10.1023/A:1020102317544