Computation of Hopf bifurcation with bordered matrices

SIAM Journal on Numerical Analysis

Volumn 33, Issue 2, 1996, Pages 435-455

  Werner, Bodo ^a  

^a UNIVERSITY OF HAMBURG   (Germany)

Author keywords

Bordered matrices; Detection of hopf points; Hopf bifurcation; Hopf curves; Naimark Sacker bifurcation; Real and imaginary Hopf points; Test function

Indexed keywords

EID: 0000595530     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/0733023     Document Type: Article

References (21)

1. J. M. ANDERSON, Eutrophication of lakes, in Toward Global Equilibrium, D. L. Maedows and H. Donella, eds., Wright-Allen Press, Cambridge, MA, 1973.
2. W. J. BEYN, Numerical methods for dynamical systems, in Advances in Numerical Analysis, Vol. I: Nonlinear Partial Differential Equations and Dynamical Systems, Clarendon Press, Oxford, 1991, pp. 175-236.
3. K. CHU, W. GOVAERTS, AND A. SPENCE, Matrices with rank deficiency two in eigenvalue problems and dynamical systems, SIAM J. Numer. Anal., 31 (1994), pp. 524-539.
4. T. J. GARRAT, G. MOORE, AND A. SPENCE, Two methods for the numerical detection of Hopf bifurcations, in Bifurcation and Chaos, R. Seydel, F. W. Schneider, T. Küpper, and H. Troger, eds., Internat. Ser. Numer. Math., 97, Birkhäuser, 1991, pp. 119-123.
5. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, Johns Hopkins University Press, Baltimore and London, 1989.
6. W. GOVAERTS AND J. D. PRYCE, Mixed block elimination for linear systems with wider borders, IMA J. Numer. Anal., 13 (1993), pp. 161-180.
7. W. GOVAERTS, Solution of bordered singular systems in numerical continuation and bifurcation, J. Comput. Appl. Math., 50 (1994), pp. 339-347.
8. _, Bordered Matrix Methods in the Computation of Hopf Points, University of Gent, Belgium, 1994, preprint.
9. A. GRIEWANK AND G. W. REDDIEN, Characterization and computation of generalized turning points, SIAM J. Numer. Anal., 21 (1984), pp. 176-184.
10. J. GUCKENHEIMER AND P. J. HOLMES, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer, Berlin, New York, 1983.
11. J. GUCKENHEIMER, M. MYERS, AND B. STURMFELS, Computing Hopf Bifurcations, SIAM J. Sci. Comput., to appear.
12. V. JANOVSKY AND P. PLECHAC, Numerical applications of equivariant reduction techniques, in Bifurcation and Symmetry, E. Allgower, K. Böhmer, and M. Golubitsky, eds., Internat. Ser. Numer. Math., 104, Birkhäuser, Basel, 1992, pp. 203-213.
13. _, Local Numerical Analysis of Hopf Bifurcation, SIAM J. Numer. Anal., to appear.
14. A. KHIBNIK, Y. KUZNETSOV, V. LEVETIN, AND E. NICOLAEV, Continuation techniques and interactive software for bifurcation analysis of ODEs and Iterated maps, Phys. D, 62 (1993), pp. 360-371.
15. M. KUBICEK AND M. HOLODNIOK, Numerical determination of bifurcation points in steady state and periodic solutions - Numerical algorithms, in Numerical Methods for Bifurcation Problems, T. Küpper, H. D. Mittelmann, and H. Weber, eds., Internat. Ser. Numer. Math., 70, Birkhäuser, Basel, 1984, pp. 247-270.
16. YU. A. KUZNETSOV, S. MURATORI, AND S. RINALDI, Bifurcation and chaos in a periodic predator-prey model, Internat. J. Bifur. Chaos Appl. Sci. Eng., 2 (1992), pp. 117-128.
17. D. ROOSE AND V. HLAVACEK, A direct method for the computation of Hopf bifurcation points, SIAM J. Appl. Math., 45 (1985), pp. 879-894.
18. B. WERNER AND V. JANOVSKY, Computation of Hopf branches bifurcating from Takens-Bogdanov points for problems with symmetries, in Bifurcation and Chaos, R. Seydel, F. W. Schneider, T. Küpper, and H. Troger, eds., Internat. Ser. Numer. Math., 97, Birkhäuser, Basel, 1991, pp. 377-388.
19. B. WERNER, Test functions for bifurcation points and Hopf points in problems with symmetries, in Bifurcation and Symmetry, E. Allgower, K. Böhmer, and M. Golubitsky, eds., Internat. Ser. Numer. Math., 104, Birkhäuser, Basel, 1992, pp. 317-327.
20. _, The numerical analysis of bifurcation problems with symmetries based on bordered Jacobians, in Exploiting Symmetry in Applied and Numerical Analysis, E. Allgower, K. Georg, and R. Miranda, eds., 1992 AMS-SIAM Summer Seminar, Lectures in Applied Mathematics, American Mathematical Society, Providence, RI, 1993, pp. 443-457.
21. LIJUN YANG AND B. WERNER, Analysis of Takens-Bogdanov bifurcation by characteristic functions, Nonlinear Anal., to appear.