Singular Hopf bifurcation in systems with two slow variables

SIAM Journal on Applied Dynamical Systems

Volumn 7, Issue 4, 2008, Pages 1355-1377

  Guckenheimer, John ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CANARD EXPLOSIONS; CODIMENSION TWOS; COMPLEX SOLUTIONS; DIMENSIONAL VECTORS; FAST SYSTEMS; GLOBAL FEATURES; HIGHER ORDERS; IT IDENTIFIES; LECTURE NOTES; LYAPUNOV COEFFICIENTS; MIXED MODES; MODE OSCILLATIONS; NETHERLANDS; NEW YORK; NORMAL FORMS; PERIODIC ORBITS; RAPID GROWTHS; SINGULAR PERTURBATIONS; TIME SCALES; TYPE II;

ASYMPTOTIC ANALYSIS; BIFURCATION (MATHEMATICS); CHAOS THEORY; DYNAMICAL SYSTEMS; OSCILLATORS (MECHANICAL); SCHRODINGER EQUATION; TELECOMMUNICATION NETWORKS;

HOPF BIFURCATION;

EID: 56449117107     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/080718528     Document Type: Article

References (33)

1. V. I. ARNOLD, V. S. AFRAJMOVICH, Yu. S. IL'YASHENKO, AND L. P. SHIL'NIKOV, Dynamical Systems V, Encyclopaedia Math. Sci., Springer-Verlag, Berlin, 1994.
2. S. M. BAER AND T. ERNEUX, Singular Hopf bifurcation to relaxation oscillations, SIAM J. Appl. Math., 46 (1986), pp. 721-739.
3. S. M. BAER AND T. ERNEUX, Singular Hopf bifurcation to relaxation oscillations II, SIAM J. Appl. Math., 52 (1992), pp. 1651-1664.
4. D. BARKLEY, SLOW MANIFOLDS AND MIXED MODE OSCILLATIONS IN THE BELOUSOV-ZHABOTINSKII REACTION, J. Chem. Phys., 89 (1988), pp. 3812-3820.
5. D. BARKLEY, J. RINGLAND, AND J. TURNER, Observations of a torus in a model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 87 (1987), pp. 5547-5559.
6. É. BENOÎT, Canards et enlacements, Inst. Hautes Etudes Sci. Publ. Math., 72 (1990), pp. 63-91.
7. É. BENOÎT, J. L. CALLOT, F. DIENER, AND M. DIENER, Chasse au canards, Collect. Math., 31 (1981), pp. 37-119.
8. B. BRAAKSMA, Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlinear Sci., 8 (1998), pp. 457-490.
9. M. BROØNS, M. KRUPA, AND M. WECHSELBERGER, Mixed mode oscillations due to the generalized canard phenomenon, in Bifurcation Theory and Spatio-Temporal Pattern Formation, Fields Inst. Commun. 49, AMS, Providence, RI, 2006, pp. 39-63.
10. A. DHOOGE, W. GOVAERTS, AND YU. A. KUZNETSOV, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), pp. 141-164. Also available online from http:// www.matcont.ugent.be/.
11. E. DOEDEL, AUTO: Software for Continuation and Bifuracation Problems in Ordinary Differential Equations, http://indy.cs.concordia.ca/ auto/.
12. F. DUMORTIER AND R. ROUSSARIE, Canard cycles and center manifolds, Mem. Amer. Math. Soc, 121 (577) (1996).
13. W. ECKHAUS, Relaxation oscillations, including a standard chase on French ducks, in Asymptotic Analysis II, Lecture Notes in Math. 985, Springer-Verlag, Berlin, 1983, pp. 449-494.
14. N. FENICHEL, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), pp. 193-225.
15. J. GUCKENHEIMER, Bifurcations of relaxation oscillations, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem. 137, Kluwer, Dordrecht, The Netherlands, 2004, pp. 295-316.
16. J. GUCKENHEIMER, Return maps of folded nodes and folded saddle-nodes, Chaos, 18 (2008), 015108.
17. J. GUCKENHEIMER, AND R. HAIDUC, Canards at folded nodes, Mosc. Math. J., 5 (2005), pp. 91-103.
18. J. GUCKENHEIMER AND P. J. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
19. J. GUCKENHEIMER AND A. WILLMS, Asymptotic analysis of suhcritical Hopf-homoclinic bifurcation, Phys. D, 139 (2000), pp. 195-216.
20. J. HUDSON, M. HART, AND D. MARINKO, An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 71 (1979), pp. 1601-1606.
21. G. KOZYREFF AND T. ERNEUX, Singular Hopf bifurcation to strongly pulsating oscillations in lasers containing a saturable absorber, European J. Appl. Math., 14 (2003), pp. 407-420.
22. K. KRISCHER, M. LÜBKE, M. EISWIRTH, W. WOLF, J. L. HUDSON, AND G. ERTL, A hierarchy of transitions to mixed mode oscillations in an electrochemical system, Phys. D, 62 (1993), pp. 123-133.
23. M. KRUPA AND P. SZMOLYAN, Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), pp. 286-314.
24. YU. A. KUZNETSOV, Elements of Applied Bifurcation Theory, 3rd ed., Appl. Math. Sci. 112, Springer-Verlag, New York, 2004.
25. Y, LLJUN AND Z, XIANWU, Stability of singular Hopf bifurcations, J. Differential Equations, 206 (2004), pp. 30-54.
26. http://www.maplesoft.com/.
27. A. MILIK AND P. SZMOLYAN, Multiple time scales and canards in a chemical oscillator, in Multiple-Time-Scale Dynamical Systems (Minneapolis, 1997), IMA Vol. Math. Appl. 122, Springer-Verlag, New York, 2001, pp. 117-140.
28. V. PETROV, S. SCOTT, AND K. SHOWALTER, Mixed-mode oscillations in chemical systems, J. Chem. Phys., 97 (1992), pp. 6191-6198.
29. J. RINZEL, A formal classification of bursting mechanisms in excitable systems, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, 1986), AMS, Providence, RI, 1987, pp. 1578-1593.
30. J. RINZEL, AND M. WECHSELBERGER, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biol. Cybernet., 97 (2007), pp. 5-32.
31. 3, J. Differential Equations, 177 (2001), pp. 419-453.
32. F. TAKENS, Unfoldings of certain singularities of vector fields: Generalized Hopf bifurcations, J. Differential Equations, 14 (1973), pp. 476-493.
33. 3 in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101-139.