### Abstract

Mixed mode oscillations combine features of small oscillations and large oscillations of
relaxation type. We describe a mechanism for mixed mode oscillations based on the presence of
canard solutions, which are trajectories passing from a stable to an unstable slow manifold. An
important ingredient of this mechanism are singularities known as folded nodes. The main focus of
this article is to show how the local dynamics near a folded node can combine with global features,
leading to mixed mode oscillations. We review and extend the results of [26] on the dynamics near
a folded node and state some results on mixed mode periodic orbits with Farey sequences of the
form 1s. We also show how to generalize the context of one fast variable to an arbitrary number of
fast variables.

Original language | English |
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Title of host publication | Bifurcation Theory and Spatio-Temporal Pattern Formation |

Editors | W. Nagata |

Publisher | American Mathematical Society |

Publication date | 2006 |

Pages | 39-64 |

ISBN (Print) | 0-8218-3725-7 |

Publication status | Published - 2006 |

Series | Fields Insititute Communications |
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Number | 49 |

## Cite this

Brøns, M., Krupa, M., & Wechselberger, M. (2006). Mixed Mode Oscillations due to the Generalized Canard Phenomenon. In W. Nagata (Ed.),

*Bifurcation Theory and Spatio-Temporal Pattern Formation*(pp. 39-64). American Mathematical Society. Fields Insititute Communications, No. 49