Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 7, Issue 11, 1997, Pages 2547-2560

  Luzyanina, Tatyana ^a   Engelborghs, Koen ^b   Lust, Kurt ^b   Roose, Dirk ^b  

^a INSTITUTE OF MATHEMATICAL PROBLEMS OF BIOLOGY   (Russian Federation)

^b UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031279807     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127497001709     Document Type: Article

References (34)

1. Baker, C. T. H., Paul, C. A. H. & Willé, D. R. [1995] "A bibliography on the numerical solution of delay differential equations," Technical Report 269 (University of Manchester, Manchester Centre for Computational Mathematics).
2. Bank, H. T. & Burns, J. A. [1978] "Hereditary control problems: Numerical methods based on averaging approximations," SIAM J. Control Optim. 16(2), 169-209.
3. Bellman, R. & Cooke, K. L. [1963] Differential-Difference Equations, Mathematics in Science and Engineering, Vol. 6 (Academic Press).
4. Castelfranco, A. M. & Stech, H. W. [1987] "Periodic solutions in a model of recurrent neural feedback," SIAM J. Appl. Math. 47(3), 573-588.
5. Chow, S.-N. [1974] "Existence of periodic solutions of autonomous functional differential equations," J. Diff. Eqns. 15, 350-378.
6. Chow, S.-N. & Walther, H.-O. [1988] "Characteristic multipliers and stability of symmetric periodic solutions of x(t) = g(x(t - 1))," Trans. Amer. Math. Soc. 370(1), 127-142.
7. Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M. & Walther, O.-H. [1995] Delay Equations: Functional-, Complex-, and Nonlinear Analysis (Springer-Verlag, Berlin).
8. Doedel, E. J. & Leung, P. P. C. [1982] "A numerical technique for bifurcation problems in delay differential equations," Cong. Num. 34, 225-237 (Proc. 11th Manitoba Conf. Num. Math. Comput., Univ. Manitoba, Winnipeg, Canada).
9. Driver, R. D. [1977] Ordinary and Delay Differential Equations, Applied Mathematical Science, Vol. 20 (Springer-Verlag, Berlin).
10. El'sgol'ts, L. E. & Norkin, S. B. [1973] Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Mathematics in Science and Engineering, Vol. 105 (Academic Press).
11. Engelborghs, K., Lust, K. & Roose, D. [1996] "A Newton-Picard method for accurate computation of period doubling bifurcation points of large-scale systems of ODEs," Technical Report TW251 (Katholieke Universiteit Leuven, Department of Computer Science, Belgium).
12. Erbe, L. & Liu, X. [1991] "Monotone iterative methods for differential systems with finite delay," Appl. Math. Comput. 43, 43-64.
13. Giannakopoulos, F. & Zapp, A. [1997] "Local and global Hopf-Bifurcation in a scalar differential equation with time delay," in preparation.
14. Hadeler, K. P. [1980] " Effective computation of periodic orbits and bifurcation diagrams in delay equations," Numer. Math. 34, 457-467.
15. Hale, J. K. [1977] Theory of Functional Differential Equations, Applied Mathematical Science, Vol. 3 (Springer-Verlag, Berlin).
16. Hale, J. K. & Verduyn Lunel, S. M. [1993] Introduction to Functional Differential Equations (Springer-Verlag, Berlin).
17. Kaplan, J. & Yorke J. [1974] "Ordinary differential equations which yield periodic solutions of differential-delay equations," J. Math. Anal. Appl. 48, 317-324.
18. Kaplan, J. & Yorke J. [1975] "On the stability of a periodic solution of a DDE," SIAM J. Math. Anal. 6(2), 268-282.
19. Keller, H. B. [1977] "Numerical solution of bifurcation and nonlinear eigenvalue problems," in Applications of Bifurcation Theory, ed. Rabinowitz, P. H. (Academic Press, New York), pp. 359-385.
20. Kolmanovskii, V. B. & Myshkis, A. [1992] Applied Theory of Functional Differential Equations, Mathematics and its Applications, Vol. 85 (Kluwer Academic Publishers, Dordrecht).
21. Liu, X. [1994] "Periodic boundary value problem for delay equations with finite delay," Dynam. Syst. Appl. 3, 357-368.
22. Lust, K., Roose D., Spence, A. & Champneys, A. R. [1997] "An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions," SIAM J. Sci. Comput.
23. Luzyanina, T. & Roose, D. [1996] "Numerical stability analysis and computation of Hopf bifurcation points for delay differential equations," J. Comput. Appl. Math. 72, 379-392.
24. Moiola, J. L., Chiacchiarini, H. G. & Desages, A. C. [1996] "Bifurcations and Hopf degeneracies in nonlinear feedback systems with time delay," Int. J. Bifurcation and Chaos 6(4), 661-672.
25. Nussbaum, R. D. [1974] "Periodic solutions of some nonlinear autonomous functional differential equations," I. Ann. Mat. Pura Appl. 101, 236-306.
26. Nussbaum, R. D. [1978] "Periodic solutions of nonlinear autonomous functional differential equations," in Functional Differential Equations and Approximation of Fixed Points, eds. Peitgen, H.-O. & Walther, H.-O. (Springer-Verlag, Berlin), pp. 283-325.
27. Olmstead, W. E., Davis, S. H., Rosenblat, S. & Kath, W. L. [1986] "Bifurcation with memory," SIAM J. Appl. Math. 46(2), 171-188.
28. Roose, D., Lust, K., Spence, A. & Champneys, A. R. [1995] "A Newton-Picard shooting method for computing periodic solutions of large-scale dynamical system," Chaos, Solitons and Fractals 5(10), 1913-1925.
29. Saad, Y. [1992] Numerical Methods for Large Eigenvalue Problems, Algorithms and Architectures for Advanced Scientific Computing (Manchester University Press).
30. Schmitt, K. [1978] "Periodic solutions of delay-differential equations," in Functional Differential Equations and Approximation of Fixed Points, eds. Peitgen, H.-O. & Walther, H.-O. (Springer-Verlag, Berlin), pp. 455-469.
31. Shroff, G. M. & Keller, H. B. [1993] "Stabilization of unstable the: Procedures recursive projection method," SIAM J. Numer. Anal. 30(4), 1099-1120.
32. Stech, H. W. [1987] "A numerical analysis of the structure of periodic orbits in autonomous functional differential equations," in Dynamics of Infinite Dimensional Systems, eds. Chow, S.-N. & Hale, J. K. (Springer-Verlag, Berlin), pp. 325-337.
33. Walther, H.-O. [1978] "A theorem on the amplitudes of periodic solutions of differential delay equations with application to bifurcation," J. Diff. Eqns. 39, 369-404.
34. Zapp, A. [1997] "Verzweigungseigenschaften bei nichtlinearen Differentialyleichungen mit einer Zeitverzögerung," Diplomarbeit, University of Cologne, Germany.