Defining functions for multiple hopf bifurcations

SIAM Journal on Numerical Analysis

Volumn 34, Issue 3, 1997, Pages 1269-1288

  Govaerts, W ^a   Guckenheimer, J ^b   Khibnik, A ^b  

^a GHENT UNIVERSITY   (Belgium)

^b Department of Obstetrics Gynecology   (United States)

Author keywords

Biproduct matrix; Dynamical system; Multiple hopf

Indexed keywords

EID: 0001101472     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142995282182     Document Type: Article

References (20)

1. V. ARNOLD, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), pp. 29-43.
2. A. BACK, J. GUCKENHEIMER, M. MYERS, F. WICKLIN, AND P. WORFOLK, Ds Tool: Computer assisted exploration of dynamical systems, AMS Notices, 39 (1992), pp. 303-309.
3. F. BUCHHOLTZ, J. GOLOWASCH, I. R. EPSTEIN, AND E. MARDER, Mathematical model of an identified stomatogastric ganglion neuron, J. Neurophysiology, 67 (1992), pp. 332 340.
4. E. DOEDEL AND X. WANG, AUTO94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, California Institute of Technology, Pasadena, CA, 1994.
5. A.T. FULLER, Conditions for a matrix to have only characteristic roots with negative real parts, J. Math. Anal. Appl., 23 (1968), pp. 71-98.
6. J. GOLOASCH AND E. MARDER. Ionic currents of the lateral pyloric neuron of the stomatogastric ganglion of the crab, J. Neurophysiology, 67 (1992). pp. 318-31.
7. M. GOLUBITSKY AND V. GUILLEMIN, Stable mappings and their singularities, Graduate Texts in Mathematics 14, Springer-Verlag, New York, 1973.
8. W. GOVAERTS AND J. D. PRYCE, A singular value inequality for block matrices, Linear Algebra Appl., 125 (1989), pp. 141-148.
9. W. GOVAERTS, Computation of Takens-Bogdanov type bifurcations with arbitrary codimension, SIAM J. Numer. Anal. 30 (1993), pp. 1121-1133.
10. A. GRIEWANK AND G.W. REDDIEN, Computation of cusp singularities for operator equations and their discretizations, J. Comput. Appl. Math., 26 (1989), pp. 133-153.
11. J. GUCKENHEIMER AND P. HOLMES, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sci., 42, Springer-Verlag, New York, 1983.
12. J. GUCKENHEIMER, S. GUERON, AND R. M. HARRIS-WARRICK, Mapping the dynamics of a bursting neuron, Philos. Trans. Roy. Soc. London Series B, 341 (1993), pp. 345-359.
13. J. GUCKENHEIMER, M. MYERS, AND B. STURMFELS, Computing Hopf bifurcations I. SIAM J. Numer. Anal., 34 (1997). pp. 1-21.
14. J. GUCKENHEIMER AND M. MYERS, Computing Hopf Bifurcations II; Three examples from neurophysiology, SIAM J. Sci. Comput., 17 (1996), pp. 1275-1301.
15. J. HARRIS. Algebraic geometry, Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1992.
16. A. KHIBNIK, Y. KUZNETSOV, V. LEVITIN, V. NIKOLAEV, AND E. NIKOLAEV, LOCBIF, version 2: Interactive LOCal BIFurcation analyzer, CAN Expertise Centre, Amsterdam, 1993.
17. A. KHIBNIK, Y. KUZNETSOV, V. LEVITIN, AND E. NIKOLAEV, Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps, Phys. D, 62 (1993), pp. 360-371.
18. K. MEERBERGEN, A. SPENCE, AND D. ROOSE, Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices, BIT, 34 (1994), pp. 409-423.
19. C. STÉPHANOS, Sur une extension du calcul des substitutions linéaires, J. Math. Pures Appl., 6 (1900), pp. 73-128.
20. S. VAN GILS, M. KRUPA, AND W. LANGFORD, Hopf bifurcation with non-semisimple 1:1 resonance. Nonlinearity, 3 (1990), pp. 825-850.