A newton-picard collocation method for periodic solutions of delay differential equations

BIT Numerical Mathematics

Volumn 45, Issue 3, 2005, Pages 605-625

  Verheyden, Koen ^a   Lust, Kurt ^a,b  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

Collocation; Delay differential equation; Newton Picard; Numerical bifurcation analysis; Periodic solution

Indexed keywords

EID: 28144446250     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-005-0013-4     Document Type: Article

References (21)

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.
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.
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.
4. O. Diekmann, S. van Gils, S. Verduyn Lunel, and H.-O. Walther, Delay Equations, vol. 110 of Applied Mathematical Sciences, Springer, 1995.
5. E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, and X.-J. Wang, A UTO97: Continuation and Bifurcation Software- for Ordinary Differential, Equations, Technical Report, Dept. of Computer Science, Concordia University, 1998.
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.
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.
8. K. Engelborghs, Numerical Bifurcation Analysis of Delay Differential Equations, PhD thesis, Department of Computer Science, K. U. Leuven, Leuven, Belgium, May 2000.
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.
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.
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.
12. K. Engelborghs, T. Luzyanina, and D. Roose, Numerical bifurcation analysis of delay differential equations usina DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), pp. 1-21.
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.
14. J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, 1993.
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.
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.
17. T. Luzyanina and K. Engelborghs, Computing Floquet multipliers for functional differential equations, Int. J. Bifurcation Chaos, 11 (2002), pp. 737-753.
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.
19. Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, 1992.
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.
21. R. Weiss, The application of implicit Runge-Kutta and collocation methods to boundary value problems, Math. Comp., 28 (1974), pp. 449-464.