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Abstract

We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.

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@techreport{Milik-1997-GMMO,
  title =      "Geometry of Mixed-mode Oscillations in the 3-d
                Autocatalator",
  author =     "Alexandra Milik and Peter Szmolyan and Helwig
               L{\"o}ffelmann and Meister Eduard Gr{\"o}ller",
  year =       "1997",
  abstract =   "We present a geometric explanation of a basic mechanism
               generating mixed-mode oscillations in a prototypical simple
               model of a chemical oscillator.  Our approach is based on
               geometric singular perturbation theory and canard solutions.
                We explain how the small oscillations are generated near a
               special point, which is classified as a folded saddle-node
               for the reduced problem.  The canard solution passing
               through this point separates small oscillations from large
               relaxation type oscillations.  This allows to define a
               one-dimensional return map in a natural way. This bimodal
               map is capable of explaining the observed bifurcation
               sequence convincingly.",
  address =    "Favoritenstrasse 9-11/186, A-1040 Vienna, Austria",
  institution = "Institute of Computer Graphics and Algorithms, Vienna
               University of Technology",
  note =       "human contact: technical-report@cg.tuwien.ac.at",
  month =      aug,
  number =     "TR-186-2-97-14",
  URL =        "http://www.cg.tuwien.ac.at/research/publications/1997/Milik-1997-GMMO/",
}