Continuation of bifurcations in periodic delay-differential equations using characteristic matrices

SIAM Journal on Scientific Computing

Volumn 28, Issue 4, 2006, Pages 1301-1317

  Szalai, Róbert ^a   Stépán, Gábor ^b   Hogan, S John ^a,b  

^a BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

^b UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Bifurcations; Continuation; Delay differential equations; Periodic solutions

Indexed keywords

ALGORITHMS; BIFURCATION (MATHEMATICS); COMPUTATIONAL METHODS; COMPUTER AIDED SOFTWARE ENGINEERING; ITERATIVE METHODS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS;

BIFURCATIONS; CHARACTERISTIC MATRIX; CONTINUATION; DELAY-DIFFERENTIAL EQUATIONS; PERIODIC SOLUTIONS; PSEUDO-ARCLENGTH METHOD;

DIFFERENTIAL EQUATIONS;

EID: 34547141626     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040618709     Document Type: Article

References (27)

1. W. J. BEYN, A. CHAMPNEYS, E. J. DOEDEL, W. GOVAERTS, B. SANDSTEDE, AND Y. A. KUZNETSOV, Numerical continuation and computation of normal forms, in Handbook of Dynamical Systems, B. Fiedler, ed., North-Holland, Amsterdam, 2002, pp. 149-219.
2. P. CLEMENT, O. DIEKMANN, M. GYLLENBERG, H. J. A. M. HEIMANS, AND H. R. THIEME, Perturbation theory for dual semigroups. I. The sun-reflexive case, Math. Ann., 277 (1987), pp. 709-725.
3. P. CLEMENT, O. DIEKMANN, M. GYLLENBERG, H. J. A. M. HEIMANS, AND H. R. THIEME, Perturbation theory for dual semigroups. II. Time-dependent perturbations in the sunreflexive case, Proc. Roy. Soc. Edinburgh Sect. A, 109 (1988), pp. 145-172.
4. T. A. DAVIES, UMFPACK Version 4.1 User Guide, Technical report, Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 2003; available online from http://www.cise.ufl.edu/research/sparse/umfpack/.
5. C. DE BOOR AND B. SWARTZ, Collocation at Gaussian points, SIAM J. Numer. Anal., 10 (1973), pp. 582-606.
6. A. DHOOGE, W. GOVAERTS, AND Y. KUZNETSOV, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), pp. 141-164.
7. O. DIEKMANN, S. A. VAN GILS, S. M. VERDUYN LUNEL, AND H. O. WALTHER, Delay Equations, Springer-Verlag, New York, 1995.
8. E. J. DOEDEL, W. GOVAERTS, AND YU. A. KUZNETSOV, Computation of periodic solution bifurcations in ODEs using bordered systems, SIAM J. Numer. Anal., 41 (2003), pp. 401-435.
9. E. J. DOEDEL, A. R. CHAMPNEYS, T. F. FAIRGRIEVE, Y. A. KUZNETSOV, B. SANDSTEDE, AND X.-J. WANG, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations, Technical report, Department of Computer Science, Concordia University, Montreal, 1997; available via anonymous ftp from ftp.cs.concordia.ca from the directory pub/doedel/auto.
10. E. J. DOEDEL, H. B. KELLER, J. P. KERNÉ VEZ, Numerical analysis and control of bifurcation Problems. I. Bifurcation infinite dimensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1 (1991), pp. 493-520.
11. K. ENGELBORGHS AND E. J. DOEDEL, Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations, Numer, Math., 91 (2002), pp. 627-648.
12. K. ENGELBORGHS, T. LUZYANINA, K. J. 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.
13. K. ENGELBORGHS, T. LUZYANINA, AND D. ROOSE, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), pp. 1-21.
14. W. GOVAERTS, Bordered augmented linear systems in numerical continuation and bifurcation, Numer. Math., 58 (1990), pp. 353-368.
15. W. GOVAERTS, Solution of bordered singular systems in numerical continuation and bifurcation, J. Comput. Appl. Math., 50 (1994), pp. 339-347.
16. W. GOVAERTS AND J. D. PRYCE, Mixed block elimination for linear systems with wider borders, IMA J. Numer. Anal., 13 (1993), pp. 161-180.
17. J. K. HALE, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
18. J. K. HALE AND S. M. VERDUYN LUNEL, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
19. E. HILLE AND R. S. PHILLIPS, Functional Analysis and Semigroups, AMS, Providence, RI, 1957.
20. M. A. KAASHOEK AND S. M. VERDUYN LUNEL, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), pp. 479-517.
21. Y. A. KUZNETSOV, CONTENT-Integrated Environment for Analysis of Dynamical Systems: Tutorial, Rapport de Recherche UPMA-98-224, Ecole Normale Superieure de Lyon, Lyon, France, 1998.
22. S. M. VERDUYN LUNEL, Small solutions and completeness for linear functional differential equations, in Oscillations and Dynamics in Delay Equations, Contemp. Math. 129, AMS, Providence, RI, 1992, pp. 127-152.
23. G. OROSZ, R. E. WILSON, AND B. KRAUSKOPF, Global bifurcation investigation of an optimal velocity traffic model with driver reaction time, Phys. Rev. E (3), 70 (2004), article 026207.
24. R. SZALAI, PDDE-CONT: A Continuation and Bifurcation Software for Delay-Differential Equations, Technical report, Budapest University of Technology and Economics, Budapest, Hungary, 2005; available online from http://www.mm.bme.hu/~szalai/pdde/.
25. R. SZALAI AND G. STÉPÁN, Stability boundaries of high-speed milling corresponding to period doubling are essentially closed curves, in Proceedings of IMECE2003, ASME International Mechanical Engineering Congress and R&D Expo, 2003, pp. 1-6 (CD-ROM).
26. R. SZALAI AND G. STÉPÁN, Period Doubling Bifurcation and Center Manifold Reduction in a Time-Periodic and Time-Delayed Model of Machining, preprint, 2006; available online from http://www.mm.bme.hu/~szalai/publications.html.
27. J. TLUSTY, Manufacturing Process and Equipment, Prentice-Hall, Englewood Cliffs, NJ, 2000.