Advertisement

BIT Numerical Mathematics

, Volume 45, Issue 3, pp 605–625 | Cite as

A Newton-Picard Collocation Method for Periodic Solutions of Delay Differential Equations

  • Koen Verheyden
  • Kurt Lust
Article

Abstract

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We exploit the equivalence of the linearized collocation system and the discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using a variant of the Newton-Picard method [Int. J. Bifurcation Chaos, 7 (1997), pp. 2547–2560]. This method combines a direct method in the low-dimensional subspace of the weakly stable and unstable modes with an iterative solver in the high-dimensional orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm.

Key words

delay differential equation periodic solution collocation Newton-Picard numerical bifurcation analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    U. M. Ascher, R. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, vol. 13 of Classics in Applied Mathematics, SIAM, 1995.Google Scholar
  2. 2.
    Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, vol. 11 of Software, Environments, and Tools, SIAM, 2000.Google Scholar
  3. 3.
    J. Bélair and M. Mackey, A model for the regulation of mammalian platelet production, Ann. N. Y. Acad. Sci., 504 (1987), pp. 280–282.Google Scholar
  4. 4.
    O. Diekmann, S. van Gils, S. Verduyn Lunel, and H.-O. Walther, Delay Equations, vol. 110 of Applied Mathematical Sciences, Springer, 1995.Google Scholar
  5. 5.
    E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, and X.-J. Wang, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations, Technical Report, Dept. of Computer Science, Concordia University, 1998.Google Scholar
  6. 6.
    E. Doedel, H. Keller, and J. Kernévez, Numerical analysis and control of bifurcation problems, part II: Bifurcation in infinite dimensions, Int. J. Bifurcation Chaos, 1 (1991), pp. 745–772.Google Scholar
  7. 7.
    E. Doedel and J. Kernévez, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Applied Mathematics Report, California Institute of Technology, Pasadena, U.S.A., 1986.Google Scholar
  8. 8.
    K. Engelborghs, Numerical Bifurcation Analysis of Delay Differential Equations, PhD thesis, Department of Computer Science, K. U. Leuven, Leuven, Belgium, May 2000.Google Scholar
  9. 9.
    K. Engelborghs and E. Doedel, Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations, Numer. Math., 91 (2002), pp. 627–648.Google Scholar
  10. 10.
    K. Engelborghs, K. Lust, and D. Roose, Direct computation of period doubling bifurcation points of large-scale systems of ODEs using a Newton-Picard method, IMA J. Numer. Anal., 19 (1999), pp. 525–547.Google Scholar
  11. 11.
    K. Engelborghs, T. Luzyanina, K. in ’t Hout, and D. Roose, Collocation methods for the computation of periodic solutions of delay differential equations, SIAM J. Sci. Comput., 22 (2000), pp. 1593–1609.Google Scholar
  12. 12.
    K. Engelborghs, T. Luzyanina, and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), pp. 1–21.Google Scholar
  13. 13.
    K. Engelborghs, T. Luzyanina, and G. Samaey, DDE-BIFTOOL v. 2.00 User Manual: A Matlab Package for Numerical Bifurcation Analysis of Delay Differential Equations, Report TW 330, Department of Computer Science, K. U. Leuven, Leuven, Belgium, Oct. 2001.Google Scholar
  14. 14.
    J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, 1993.Google Scholar
  15. 15.
    K. Lust, Numerical Bifurcation Analysis of Periodic Solutions of Partial Differential Equations, PhD thesis, Department of Computer Science, K. U. Leuven, Leuven, Belgium, Dec. 1997.Google Scholar
  16. 16.
    K. Lust, D. Roose, A. Spence, and A. Champneys, An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions, SIAM J. Sci. Comput., 19 (1998), pp. 1188–1209.Google Scholar
  17. 17.
    T. Luzyanina and K. Engelborghs, Computing Floquet multipliers for functional differential equations, Int. J. Bifurcation Chaos, 11 (2002), pp. 737–753.Google Scholar
  18. 18.
    T. Luzyanina, K. Engelborghs, K. Lust, and D. Roose, Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations, Int. J. Bifurcation Chaos, 7 (1997), pp. 2547–2560.Google Scholar
  19. 19.
    Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, 1992.Google Scholar
  20. 20.
    L. Shayer and S. Campbell, Stability, bifurcation and multistability in a system of two coupled neurons with multiple time delays, SIAM J. Appl. Math., 61 (2000), pp. 673–700.Google Scholar
  21. 21.
    R. Weiss, The application of implicit Runge-Kutta and collocation methods to boundary value problems, Math. Comp., 28 (1974), pp. 449–464.Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of Computer ScienceKatholieke Universiteit LeuvenHeverlee-LeuvenBelgium
  2. 2.Institute for Mathematics and Computing ScienceUniversity of GroningenGroningenThe Netherlands

Personalised recommendations