Mixed-mode oscillations in three time-scale systems: A prototypical example

SIAM Journal on Applied Dynamical Systems

Volumn 7, Issue 2, 2008, Pages 361-420

  Krupa, Martin ^a   Popovic, Nikola ^b   Kopell, Nancy ^c  

^a NEW MEXICO STATE UNIVERSITY   (United States)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

^c Boston University   (United States)

Author keywords

Canard mechanism; Geometric desingularization; Mixed mode oscillations; Singular perturbations; Three time scales

Indexed keywords

CANARD MECHANISM; GEOMETRIC DESINGULARIZATION; MIXED-MODE OSCILLATIONS; SINGULAR PERTURBATIONS; THREE TIME-SCALES;

ASYMPTOTIC ANALYSIS; CONTROL SYSTEM ANALYSIS; CONTROL THEORY; MOTION ESTIMATION; OSCILLATORS (MECHANICAL); PARAMETER ESTIMATION;

PERTURBATION TECHNIQUES;

EID: 55949135112     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/070688912     Document Type: Article

References (38)

1. E. BENOIT, J.-L. CALLOT, F. DIENER, AND M. DIENER, Chasse au canard, Collect. Math., 32 (1981), pp. 37-119.
2. M. BRØNS, 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.
3. M. BRØNS, M. KRUPA, AND M. WECHSELBERGER, Dynamics Near a Folded Saddle-Node of Type II, in preparation.
4. K. M. BRUCKS AND C. TRESSER, A Farey tree organization of locking regions for simple circle maps, Proc. Amer. Math. Soc, 124 (1996), pp. 637-647.
5. M. DIENER, The canard unchained or how fast/slow dynamical systems bifurcate, Math. Intelligencer, 6 (1984), pp. 38-49.
6. J. DROVER, J. RUBIN, J. SU, AND B. ERMENTROUT, Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies, SIAM J. Appl. Math., 65 (2004), pp. 69-92.
7. I. DUMORTIER, Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations, in Bifurcations and Periodic Orbits of Vector Fields, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 408, D. Schlomiuk, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993, pp. 19-73.
8. F. DUMORTIER AND R. R.OUSSARIE, Canard cycles and center manifolds, Mem. Amer. Math. Soc, 121 (577) (1996).
9. 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.
10. I. R. EPSTEIN AND K. SHOWALTER, Nonlinear chemical dynamics: Oscillations, patterns, and chaos, J. Phys. Chem., 100 (1996), pp. 13132-13147.
11. N. FENICHEL, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), pp. 53-98.
12. J. GUCKENHEIMER, R. HARRIS-WARWICK, J. PECK, AND A. WILMS, Bifurcation, bursting and frequency spike adaptation, J. Comput. Neurosci, 4 (1997), pp. 255-277.
13. J. GUCKENHEIMER AND A. WILMS, Asymptotic analysis of subcritical Hopf-homoclinic bifurcation, Phys. D, 139 (2000), pp. 196-216.
14. C. K. R. T. JONES, Geometric singular perturbation theory, in Dynamical Systems (Montecatini Terme,1994), Lecture Notes in Math. 1609, Springer-Verlag, Berlin, 1995, pp. 44-118.
15. N. KOPELL AND L. N. HOWARD, Bifurcations and trajectories joining critical points, Advances in Math., 18 (1975), pp. 306-358.
16. M. T. M. KOPER, Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram, Phys. D, 80 (1995), pp. 72-94.
17. M. KRUPA, Mixed-Mode Oscillations in Systems with More than Two Slow Dimensions, in preparation.
18. M. KRUPA, N. POPOVIĆ, N. KOPELL, AND H. G. ROTSTEIN, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron, Chaos, to appear.
19. 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.
20. M. KRUPA AND P. SZMOLYAN, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), pp. 312-368.
21. R. LARTER AND C. G. STEINMETZ, Chaos via mixed-mode oscillations, Philos. Trans. Roy. Soc. London Ser. A, 337 (1991), pp. 291-298.
22. R. LARTER, C. G. STEINMETZ, AND B. D. AGUDA, Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in the peroxidase reaction, J. Phys. Chem., 89 (1988), pp. 6506-6514.
23. G. S. MEDVEDEV AND J. E. CISTERNAS, Multimodal regimes in a compartmental model of the dopamine neuron, Phys. D, 194 (2004), pp. 333-356.
24. G. S. MEDVEDEV, C. J. WILSON, J. C. CALLAWAY, AND N. KOPELL, Dendritic synchrony and transient dynamics in a coupled oscillator model of the dopaminergic neuron, J. Comput. Neurosci., 15 (2003), pp. 53-69.
25. W. DE MELO AND S. VAN STRIEN, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. (3) 25, Springer-Verlag, Berlin, 1993.
26. A. MILIK AND P. SZMOLYAN, Multiple time scales and canards in a chemical oscillator, in Multiple-Time-Scale Dynamical Systems, IMA Vol. Math. Appl. 122, C. K. R. T. Jones and A. I. Khibnik, eds., Springer-Verlag, New York, 2001, pp. 117-140.
27. A. MILIK, P. SZMOLYAN, H. LÖFFELMANN, AND E. GRÖLLER, Geometry of mixed-mode oscillations in the 3-d autocatalator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), pp. 505-519.
28. J. MOEHLIS, Canards in a surface oxidation reaction, J. Nonlinear Sci., 12 (2002), pp. 319-345.
29. C. A. DEL NEGRO, C. G. WILSON, R. J. BUTERA, H. RIGATTO, AND J. C. SMITH, Periodicity, mixed-mode oscillations, and quasiperiodicity in a rhythm-generating neural network, Biophys. J., 82 (2002), pp. 206-214.
30. H. G. ROTSTEIN AND R. KUSKE, Localized and asynchronous patterns via canards in coupled calcium oscillators, Phys. D, 215 (2006), pp. 46-61.
31. H. G. ROTSTEIN, T. OPPERMANN, J. A. WHITE, AND N. KOPELL, The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells, J. Comput. Neurosci., 21 (2006), pp. 271-292.
32. J. RUBIN AND M. WECHSELBERGER, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biol. Cybernet., 97 (2007), pp. 5-32.
33. 3, J. Differential Equations, 177 (2001), pp. 419-453.
34. 3, J. Differential Equations, 200 (2004), pp. 69-104.
35. M. WECHSELBERGER, personal communication.
36. 3 in the case of a folded node, SIAM J.Appl. Dyn. Syst., 4 (2005), pp. 101-139.
37. C. J. WILSON AND J. C. CALLAWAY, Coupled oscillator model of the dopaminergic neuron of the substantia nigra, J. Neurophysiol., 83 (2000), pp. 3084-3100.
38. A. M. ZHABOTINSKY, Periodic kinetics of oxidation of malonic acid in solution, Biofizika, 9 (1964), pp. 306-311.