Multiple Time Scales and Canards in a Chemical Oscillator
We present a geometric singular perturbation analysis of a chemical oscillator. Although the studied three-dimensional model is rather simple, its dynamics are quite complex. In the original scaling the problem has a folded critical manifold which additionally becomes tangent to the fast fibers in a region relevant to the dynamics. Thus normal hyperbolicity of the critical manifold is lost in two regions. The dynamics depends crucially on effects due to the loss of normal hyperbolicity. In particular, canard solutions play an essential role. We outline how rescalings and blow-up techniques can be used to prove the existence of canards in this problem and to explain other qualitative aspects of the dynamics.
Key wordsGeometric singular perturbation theory chemical oscillator canard solutions blow-up
Unable to display preview. Download preview PDF.
- E. Benoit, J.-L. Callot, F. Diener, and M. Diener, Chasse au canard, Collect. Math., 31 (1981), pp. 37–119.Google Scholar
- B. Braaksma, Critical phenomena in dynamical systems of van der Pol type, Thesis, Rijksuniversiteit te Utrecht, Nederlands, 1993.Google Scholar
- F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Memoirs of the American Mathematical Society, 121 (1996).Google Scholar
- W. Eckhaus, Relaxation oscillations including a standard chase on french ducks, Vol. 985 of Springer Lecture Notes Math., Springer, 1983, pp. 449–494.Google Scholar
- C. Jones, Geometric singular perturbation theory, Vol. 1609 of Springer Lecture Notes Math., Springer, 1995, pp. 44–120.Google Scholar
- M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points, in preparation (1998).Google Scholar
- A. Milik, P. Szmolyan, H. Löffelmann, and E. Gröller, The geometry of mixed-mode oscillations in the 3d-autocatalator, International Journal of Bifurcation and Chaos, 8 (1998).Google Scholar
- E. Mishchenko, Y. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic methods in singularly perturbed systems, Monographs in Contemporary Mathematics, Consultants Bureau, New York, A Division of Plenum Publishing Corporation 233 Spring Street, New York, N. Y. 10013, 1994.Google Scholar
- P. Szmolyan, Geometry of singular perturbations: a case study, in preparation (1998).Google Scholar
- M. Wechselberger, Singularly perturbed folds and canards in ℝ3, Thesis, Technische Universität Wien, Austria, 1998.Google Scholar