Numerical bifurcation analysis of differential equations with state-dependent delay

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 11, Issue 3, 2001, Pages 737-753

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

^a UNIVERSITY OF LEUVEN   (Belgium)

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

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035632530     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127401002407     Document Type: Article

References (34)

1. Aiello, W. G., Freedman, H. I. & Wu, J. [1992] "Analysis of a model representing stage-structured population growth with state-dependent time delay," SIAM J. Appl. Math. 52(3), 855-869.
2. Arino, O., Hadeler, K. P. & Hbid, M. L. [1998a] "Existence of periodic solutions for delay differential equations with state dependent delay," J. Diff. Eqns. 144, 263-301.
3. Arino, O., Hbid, M. L. & de la Parra, R. B. [1998b] "A mathematical model of growth of population of fish in the larval stage: Density-dependence effects," Math. Biosci. 150, 1-20.
4. Ascher, U. M., Mattheij, R. M. M. & Russell, R. D. [1988] Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Prentice Hall).
5. Baker, C. T. H., Bocharov, G. A. & Rihan, F. A. [1999] "A report on the use of delay differential equations in numerical modeling in the biosciences," Technical Report 343, Manchester Centre for Computational Mathematics, University of Manchester.
6. Brokate, M. & Colonius, F. [1990] "Linearizing equations with state-dependent delays," Appl. Math. Optim. 21, 45-52.
7. Cao, Y., Fan, J. & Gard, T. C. [1992] "The effects of state-dependent time delay on a stage-structured population growth model," Nonlin. Anal. 19(2), 95-105.
8. Cooke, K. L. & Huang, W. [1996] "On the problem of linearization for state-dependent delay differential equations," Proc. AMS 124(5), 1417-1426.
9. Corwin, S. P., Sarafyan, D. & Thompson, S. [1997] "DKLAG6: A code based on continuously imbedded sixth order Runge-Kutta methods for the solution of state dependent functional differential equations," Appl. Numer. Math. 24(2 & 3), 319-330.
10. Doedel, E. J., Keller, H. B. & Kernevez, J. P. [1991] "Numerical analysis and control of bifurcation problems (I) bifurcation in finite dimensions," Int. J. Bifurcation and Chaos 1(3), 493-520.
11. Engelborghs, K. & Roose, D. [1999] "Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations," Adv. Comput. Math. 10, 271-289.
12. Engelborghs, K. [2000a] "DDE-BIFTOOL: A Matlab package for bifurcation analysis of delay differential equations," Technical Report TW305, Department of Computer Science, Katholieke Universiteit Leuven, Belgium.
13. Engelborghs, K. [2000b] "Numerical bifurcation analysis of delay differential equations," PhD thesis, Department of Computer Science, Katholieke Universiteit Leuven, Belgium.
14. Engelborghs, K., Luzyanina, T., in't Hout, K. & Roose, D. [2000a] "Collocation methods for the computation of periodic solutions of delay differential equations," Technical Report TW295, Department of Computer Science, Katholieke Universiteit Leuven, Belgium; SIAM J. Sci. Comput. 22, 1593-1609.
15. Engelborghs, K., Luzyanina, T. & Roose, D. [2000b] "Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL," IMA J. Numer. Anal., submitted.
16. Engelborghs, K. & Doedel, E. [2001] "Stability of piecewise polynomial collocation methods for computing periodic solutions of delay differential equations," Numer. Math., accepted.
17. Feldstein, A. & Neves, K. W. [1984] "High order methods for state-dependent delay differential equations with nonsmooth solutions," SIAM J. Numer. Anal. 21(5), 844-863.
18. Ford, N. J. [1999] "Numerical approximation of the characteristic values for a delay differential equation," Technical Report 350, Manchester Centre for Computational Mathematics, University of Manchester.
19. Hale, J. K. [1977] Theory of Functional Differential Equations, Applied Mathematical Science, Vol. 3 (Springer-Verlag, Berlin).
20. Hartung, F. & Turi, J. [1995] "Stability in a class of functional differential equations with state-dependent delay," in Qualitative Problems for Differential Equations and Control Theory, ed. Corduneanu, C. (World Scientific, Singapore), pp. 15-31.
21. Hartung, F. & Turi, J. [1997] "On differentiability of solutions with respect to parameters in state-dependent delay equations," J. Diff. Eqns. 135, 192-237.
22. Hong-Jiong, T. & Jiao-Xun, K. [1996] "The numerical stability of linear multistep methods for delay differential equations with many delays," SIAM J. Numer. Anal. 33(3), 883-889.
23. Kuang, Y. & Smith, H. L. [1992] "Slowly oscillating periodic solutions of autonomous state-dependent delay equations," Nonlin. Anal. 19(9), 855-872.
24. 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.
25. Mahaffy, J. M., Bélair, J. & Mackey, M. C. [1998] "Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis," J. Theor. Biol. 190, 135-146.
26. Mallet-Paret, J. & Nussbaum, R. D. [1992] "Boundary layer phenomena for differential-delay equations with state-dependent time lags, 1," Arch. Rat. Mech. Anal. 120, 99-146.
27. Mallet-Paret, J., Nussbaum, R. D. & Paraskevopoulos, P. [1994] "Periodic solutions for functional differential equations with multiple state-dependent time lags," Topol. Meth. Nonlin. Anal. 3, 101-162.
28. Metz, J. A. J. & Diekmann, O. [1986] The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Vol. 68 (Springer-Verlag, Berlin).
29. Neves, K. W. & Feldstein, A. [1976] "Characterization of jump discontinuities for state dependent delay differential equations," J. Math. Anal. Appl. 56, 689-707.
30. Paul, C. A. H. [1997] "A user-guide to Archi - an explicit Runge-Kutta code for solving delay and neutral differential equations," Technical Report 283, Manchester Center for Computational Mathematics, University of Manchester.
31. Smith, H. L. & Kuang, Y. [1992] "Periodic solutions of differential delay equations with threshold-type delays," Contemp. Math. 129, 153-176.
32. Smith, H. L. [1993] "Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study," Math. Biosci. 113, 1-23.
33. Willé, D. R. & Baker, C. T. H. [1992] "The tracking of derivative discontinuities in systems of delay differential equations," Appl. Num. Math. 9, 209-222.
34. Yebdri, M, [1999] "Special slowly oscillating periodic solution of a state dependent delay differential equation," Technical Report, Institut de Mathématiques, Université de Tlemcen, Tlemcen, Algerie.