Local numerical analysis of Hopf bifurcation

SIAM Journal on Numerical Analysis

Volumn 33, Issue 3, 1996, Pages 1150-1168

  Janovský, Vladimír ^a   Plecháč, Petr ^b,c  

^a CHARLES UNIVERSITY   (Czech Republic)

^b UNIVERSITY OF BATH   (United Kingdom)

^c HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

Bordered matrices; Equivariant bifurcation; Hopf bifurcation; Liapunov Schmidt reduction; Stability exchange

Indexed keywords

EID: 0004401031     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/0733057     Document Type: Article

References (20)

1. E. ALLGOWER AND K. GEORG, Numerical Continuation Techniques, Springer-Verlag, New York, 1990.
2. W. GOVAERTS Stable solvers and block elimination for bordered systems, SIAM J. Numer. Anal., 12 (1991), pp. 469-483.
3. K.-W. E. 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. W. GOVAERTS AND A. SPENCE, Detection of Hopf points by counting sectors in the complex plane, Numer. Math., to appear.
5. T. GARRAT, G. MOORE, AND A. SPENCE, Two Methods for Numerical Detection of Hopf Bifurcation, Internat. Ser. Numer. Math., 97 (1991), pp. 129-133.
6. M. GOLUBITSKY AND D. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. 1, Springer-Verlag, New York, 1985.
7. M. GOLUBITSKY, I. STEWART, AND D. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. 2, Springer-Verlag, New York, 1988.
8. A. GRIEWANK AND G. W. REDDIEN, The calculation of Hopf points by a direct method, IMA J. Numer. Anal., 3 (1983), pp. 295-303.
9. M. GRINFELD, Dynamics of a model equation in viscoelasticity, in Proc. Equadiff' 91, Vol. II, World Scientific, River Edge, NJ, 1993, pp. 568-572.
10. P. FREITAS, M. GRINFELD, AND P. A. KNIGHT, Stabilisation in systems with distributed damping, Adv. Math. Sci. Appl., 1997, to appear.
11. J. GUCKENHEIMER, M. MYERS, AND B. STURMFELS, Computing Hopf bifurcations, SIAM J. Numer. Anal., 33 (1996), to appear.
12. V. JANOVSKÝ AND P. PLECHÁČ, Computer aided analysis of imperfect bifurcation diagrams I. Simple bifurcation point and isola formation centre, SIAM J. Numer. Anal., 21 (1992), pp. 498-512.
13. _, Numerical applications of equivariant reduction techniques, in Bifurcation and Symmetry, M. Golubitsky, E. Allgower, K. Böhmer, eds., Birkhäuser-Verlag, Basel, Switzerland, 1992.
14. A. D. JEPSON AND A. SPENCE, On a reduction process for nonlinear equations, SIAM J. Math. Anal., 20 (1989), pp. 39-56.
15. A. D. JEPSON, A. SPENCE, AND K. A. CLIFFE, The numerical solution of nonlinear equations having several parameters, SIAM J. Numer. Anal., 22 (1985), pp. 736-759.
16. A. I. KHIBNIK, YU. A. KUZNETSOV, V. V. LEVITIN, AND E. V. NIKOLAEV, Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps, Physica D, 62 (1993), pp. 360-371.
17. W. E. OLMSTEAD, S. H. DAVIS, S. ROSENBLAT, AND W. L. KATH, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), pp. 171-188.
18. D. ROOSE AND V. HLAVACEK, A direct method for computation of Hopf bifurcation points, SIAM J. Appl. Anal., 45 (1985), pp. 879-894.
19. G. M. SHROFF AND H. B. KELLER, Stabilization of unstable procedures: The recursive projection method, SIAM J. Numer. Anal., 30 (1993), pp. 1099-1120.
20. B. WERNER, Computation of Hopf bifurcation with bordered matrices, SIAM J. Numer. Anal., 33 (1996), pp. 435-455.