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Mixed mode oscillations due to the generalized canard. (English) Zbl 1228.34063

Nagata, Wayne (ed.) et al., Bifurcation theory and spatio-temporal pattern formation. Proceedings ot the workshop on bifurcation theory and spatio-temporal pattern formation in partial differential equations, Toronto, Canada, December 11–13, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3725-7/hbk). Fields Institute Communications 49, 39-63 (2006).
The authors consider a class of singularly perturbed tridimensional systems, namely those having a folded two-dimensional critical manifold. Moreover, it is assumed that the reduced system possesses a folded node singularity.
The paper presents some results concerning the presence of periodic orbits with one large and several small oscillations, due to the presence of canard solutions. The results are corroborated numerically in a particular case corresponding to a van der Pol-like model.
Finally, the results are extended to systems with arbitrarily many fast dimensions, again assuming that the singularly perturbed system has a folded two-dimensional critical manifold.
For the entire collection see [Zbl 1099.34004].

MSC:

34C26 Relaxation oscillations for ordinary differential equations
34D15 Singular perturbations of ordinary differential equations
34E17 Canard solutions to ordinary differential equations
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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