The geometry of mixed-mode oscillations in the olsen model for the peroxidase-oxidase reaction

Discrete and Continuous Dynamical Systems - Series S

Volumn 2, Issue 4, 2009, Pages 807-827

  Desroches, Mathieu ^a   Krauskopf, Bernd ^a   Osinga, Hinke M ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Delayed Hopf bifurcation; Invariant manifolds; Mixed mode oscillations

Indexed keywords

EID: 77149151935     PISSN: 19371632     EISSN: 19371179     Source Type: Journal    
DOI: 10.3934/dcdss.2009.2.807     Document Type: Article

References (35)

1. Baltazar D. Aguda, Raima Larter and Bruce L. Clarke, Dynamic elements of mixed-mode oscillations and chaos in a peroxidase-oxidase model network, The Journal of Chemical Physics, 90 (1989), 4168-4175.
2. Steven M. Baer, Thomas Erneux and John Rinzel, The slow passage through a Hopf bifurcation: delay, memory effects, and resonance, SIAM J. Appl. Math., 49 (1989), 55-71.
3. Tatiana V. Bronnikova, William M. Schaffer and Lars F. Olsen, Nonlinear dynamics of the peroxidase-oxidase reaction: I. bistability and bursting oscillations at low enzyme concentrations, Journal of Physical Chemistry B, 105 (2001), 310-321.
4. Morten Brøns, Martin Krupa and Martin Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon, In "Bifurcation Theory and Spatio-Temporal Pattern Formation," volume 49 of "Fields Inst. Commun.," 39-63. Amer. Math. Soc., Providence, RI, 2006.
5. Morten Brøns and Martin Krupa, "Mixed Mode Oscillations in the Olsen Model for Peroxidase Reaction," SIAM Conference on Applications of Dynamical Systems, Snowbird (Utah), 2007.
6. Robert Clewley, Horacio G. Rotstein and Nancy Kopell, A computational tool for the reduction of nonlinear ODE systems possessing multiple scales, Multiscale Modeling and Simulation, 4 (2005), 732-759. (Pubitemid 44340286)
7. Mathieu Desroches, Bernd Krauskopf and Hinke M. Osinga, Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system, Chaos, 18 (2008), 015107.
8. Mathieu Desroches, Bernd Krauskopf and Hinke M. Osinga, The geometry of slow manifolds near a folded node, SIAM J. Appl. Dyn. Syst., 7 (2008), 1131-1162.
9. Hans Degn, Lars F. Olsen and John W. Perram, Bistability, oscillation, and chaos in an enzyme reaction, Annals of the New York Academy of Sciences, 316 (1979), 623-637.
10. Eusebius J. Doedel, Randy C. Paffenroth, Alan R. Champneys, Thomas F. Fairgrieve, Yuri A. Kuznetsov, Bart E. Oldeman, Björn Sandstede and Xiao J. Wang, "AUTO2000: Continuation and Bifurcation Software for Ordinary Differential Equations," Available at http://cmvl.cs.concordia.ca/.
11. Simon J. Fraser, The steady state and equilibrium approximations: A geometrical picture, The Journal of Chemical Physics, 88 (1988), 4732-4738.
12. John Guckenheimer, Return maps of folded nodes and folded saddle-nodes, Chaos, 18 (2008), 015108.
13. Marcus J. Hauser and Lars F. Olsen, Mixed-mode oscillations and homoclinic chaos in an enzyme reaction, J. Chem. Soc., Faraday Trans., 92 (1996), 2857-2863. (Pubitemid 126793379)
14. Marcus J. B. Hauser, Lars F. Olsen, Tatiana V. Bronnikova and William M. Schaffer, Routes to chaos in the Peroxidase-Oxidase reaction: Period-doubling and period-adding, Journal of Physical Chemistry B, 101 (1997), 5075-5083. (Pubitemid 127617312)
15. Ale Jan Homburg and Bernd Krauskopf, Resonant homoclinic flip bifurcations, Journal of Dynamics and Differential Equations, 12 (2000), 807-850.
16. Bernd Krauskopf, Hinke M. Osinga, Eusebius J. Doedel, Michael E. Henderson, John Gucken-heimer, Alexander Vladimirsky, Michael Dellnitz and Oliver Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos, 15 (2005), 763-791. (Pubitemid 40878383)
17. Bernd Krauskopf and Hinke M. Osinga, Computing invariant manifolds via the continuation of orbit segments, In B. Krauskopf, H. M. Osinga and J. Gáan-Vioque, editors, "Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems," Underst. Complex Syst., pages 117-154, Springer-Verlag, New-York, 2007.
18. Martin Krupa, "Private Communication," February 2008.
19. Martin Krupa, Nikola Popovíc, Nancy Kopell and Horacio G. Rotstein, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron, Chaos, 18 (2008), 015106.
20. Raima Larter, Christopher L. Bush, Timothy R. Lonis and Baltazar D. Aguda, Multiple steady states, complex oscillations, and the devil's staircase in the peroxidase-oxidase reaction, The Journal of Chemical Physics, 87 (1987), 5765-5771.
21. Raima Larter and Curtis G. Steinmetz, Chaos via mixed-mode oscillations, Philosophical Transactions: Physical Sciences and Engineering, 337 (1991), 291-298.
22. Raima Larter, Curtis G. Steinmetz and Baltazar D. Aguda, Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in the peroxidase reaction, The Journal of Chemical Physics, 89 (1988), 6506-6514.
23. Georgi S. Medvedev and Jaime E. Cisternas, Multimodal regimes in a compartmental model of the dopamine neuron, Physica D, 194 (2004), 333-356.
24. Georgi Medvedev and Yun Yoo, Multimodal oscillations in systems with strong contraction, Physica D, 228(2):87-106, 2007. (Pubitemid 46601626)
25. Anatoly I. Neishtadt, On stability loss delay for dynamical bifurcations I, Differentsial nye Uravneniya, 23, 2060-2067, (Russian)
26. English translation in Differential Equations 23(12):1385-1390, 1987.
27. Anatoly I. Neishtadt, On stability loss delay for dynamical bifurcations II, Differentsial nye Uravneniya, 24, 226-233, 364, (Russian)
28. English translation in Differential Equations 24 (1988), 171-176.
29. Bart E. Oldeman, Bernd Krauskopf and Alan R. Champneys, Death of period-doublings: Locating the homoclinic-doubling cascade, Physica D, 146 (2000), 100-120. (Pubitemid 33592764)
30. Lars F. Olsen, An enzyme reaction with a strange attractor, Physics Letters A, 94 (1983), 454-457.
31. William M. Schaffer, Tatiana V. Bronnikova and Lars F. Olsen, Nonlinear dynamics of the peroxidase-oxidase reaction. II. compatibility of an extended model with previously reported model-data correspondences, Journal of Physical Chemistry B, 105 (2001), 5331-5340. (Pubitemid 35372635)
32. David R. Thompson and Raima Larter, Multiple time scale analysis of two models for the peroxidase-oxidase reaction, Chaos, 5 (1995), 448-457.
33. 3 in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), 101-139. (Pubitemid 43385188)
34. Isao Yamazaki, Kennosuke Yokota and R. Nakajima, Oscillatory oxidations of reduced pyri-dine nucleotide by peroxidase, Biochem. Biophys. Res. Commun., 21 (1965), 582-586.
35. Antonios Zagaris, Hans G. Kaper and Tasso J. Kaper, Analysis of the computational singular perturbation reduction method for chemical kinetics, Journal of Nonlinear Science, 14 (2004), 59-91. (Pubitemid 40881194)