A canard mechanism for localization in systems of globally coupled oscillators

SIAM Journal on Applied Mathematics

Volumn 63, Issue 6, 2003, Pages 1998-2019

  Rotstein, Horacio G ^a,b   Kopell, Nancy ^a   Zhabotinsky, Anatol M ^b   Epstein, Irving R ^b  

^a Boston University   (United States)

^b BRANDEIS UNIVERSITY   (United States)

Author keywords

Canard phenomenon; Globally coupled oscillators; Localization of oscillations; Relaxation oscillator

Indexed keywords

BIFURCATION (MATHEMATICS); CHEMICAL REACTIONS; CHEMICAL REACTORS; COMPUTER SIMULATION; OSCILLATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES; RELAXATION OSCILLATORS;

CANARD PHENOMENON; GLOBALLY COUPLED OSCILLATORS; HOPF BIFURCATION; LOCALIZATION OF OSCILLATORS;

MATHEMATICAL MODELS;

EID: 1642544671     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139902411843     Document Type: Article

References (29)

1. J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
2. I. R. Epstein and K. Showalter, Nonlinear chemical dynamics: Oscillations, patterns and chaos, J. Phys. Chem., 100 (1996), pp. 13132-13147.
3. S. H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1994.
4. V. K. Vanag, L. Yang, M. Dolnik, A. M. Zhabotinsky, and I. R. Epstein, Oscillatory cluster patterns in a homogeneous chemical system with global feedback, Nature, 406 (2000), pp. 389-391.
5. V.K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, Pattern formation in the Belousov-Zhabotinsky reaction with photochemical global feedback, J. Phys. Chem., 104A (2000), pp. 11566-11577.
6. L. Yang, M. Dolnik, A. M. Zhabotinsky, and I. R. Epstein, Oscillatory clusters in a model of the photosensitive Belousov-Zhabotinsky reaction system with global feedback, Phys. Rev. E, 62 (2000), pp. 6414-6420.
7. A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin, and I. R. Epstein, Oscillations and waves in metal-ion-catalyzed bromate oscillating reactions in highly oxidized states, J. Phys. Chem., 97 (1993), pp. 7578-7584.
8. R. Kuske and T. Erneux, Localized synchronization of two coupled solid state lasers, Optics Communications, 139 (1997), pp. 125-131.
9. R. Kuske and T. Erneux, Bifurcation to localized oscillations, European J. Appl. Math., 8 (1997), pp. 389-402.
10. M. Boukalouch, J. Elezgaray, A. Arneodo, J. Boissonade, and P. De Kepper, Oscillatory instability induced by mass interchange between two coupled steady-state reactors, J. Phys. Chem., 91 (1987), pp. 5843-5845.
11. F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Mem. Amer. Math. Soc., 121 (1996).
12. 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. 449-497.
13. S.M. Baer and T. Erneux, Singular Hopf bifurcation to relaxation oscillations, SIAM J. Appl. Math., 46 (1986), pp. 721-739.
14. E. Benoit, J. L. Callot, F. Dienner, and M. Dienner, Chasse au Canard, IRMA, Strasbourg, France, 1980.
15. M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), pp. 312-368.
16. 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.
17. M. Brons and K. Bar-Eli, Canard explosion and excitation in a model of the BZ reaction, J. Phys. Chem., 95 (1991), pp. 8706-8713.
18. F. Buchholtz, M. Dolnik, and I. R. Epstein, Diffusion-induced instabilities near a canard, J. Phys. Chem., 99 (1995), pp. 15093-15101.
19. H. G. Rotstein, N. Kopell, A. Zhabotinsky, and I. R. Epstein, Canard phenomenon and localization of oscillations in the Belousov-Zhabotinsky reaction with global feedback, submitted.
20. H. G. Rotstein and R. Kuske, personal communication.
21. H. G. Rotstein, R. Kuske, and N. Kopell, Localized Oscillations in a Coupled Two-Pool Model. A Canard Mechanism, in preparation.
22. R. L. Burden and J. D. Faires, Numerical Analysis, PWS Publishing, Boston, 1980.
23. R. FitzHugh, Impulses and physiological states in models of nerve membrane, Biophys. J., 1 (1961), pp. 445-466.
24. F. C. Hoppensteadt, An Introduction to the Mathematics of Neurons, Cambridge University Press, New York, 1996.
25. E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, new York, London, 1980.
26. J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications, Springer-Verlag, New York, 1986.
27. F. Dumortier, Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations, in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk, ed., Kluwer Academic Press, Dordrecht, The Netherlands, 1993, pp. 19-73.
28. V. K. Vanag, A. M. Zhabotinsky, and I. R. Epstein, Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction, Phys. Rev. Lett., 86 (2001), pp. 552-555.
29. M. Krupa, W. F. Langford, and J. P. Voroney, Canard Explosion in the Oregonator, in preparation.