Relaxation Oscillation and Canard Explosion

By Martin Krupa, Peter Szmolyan. 2001

528 citations recorded by the source corpus.

Abstract We give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields. The transition from small Hopf-type cycles to large relaxation cycles, which occurs in an exponentially thin parameter interval, is described as a perturbation of a family of singular cycles. The results are obtained by means of two blow-up transformations combined with standard tools of dynamical systems theory. The efficient use of various charts is emphasized. The results are applied to the van der Pol equation.

Subfields: Computer Networks and Communications, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dcc36ab44c791d0f0649701c407ae37ff1ceaec1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

1. Peter De Maesschalck
2. Ilona Gucwa
3. Christian Kuehn
4. Martin Wechselberger
5. Peter Szmolyan
6. Freddy Dumortier
7. Martin Krupa
8. John M. Guckenheimer

In Geometry and Topology

1. Ilona Gucwa
2. Peter Szmolyan
3. Freddy Dumortier
4. Peter De Maesschalck
5. Christian Kuehn
6. Martin Wechselberger
7. John M. Guckenheimer
8. Martin Krupa

In Statistical and Nonlinear Physics

1. Freddy Dumortier
2. Ilona Gucwa
3. Peter De Maesschalck
4. Christian Kuehn
5. Peter Szmolyan
6. Martin Wechselberger
7. John M. Guckenheimer
8. Martin Krupa

Citations in the corpus

• Canard cycles for predator–prey systems with Holling types of functional response☆ (2013, background): “4, are often referred to as canard slow–fast cycle without a head and respectively canard slow–fast cycle with a head, see [9] or [19] for example.”
• A singular approach to discontinuous vector fields on the plane (2006, background): “There is a distinguished history of investigations on this context [1,8].”
• Canard solutions at non-generic turning points (2005, methodology): “It has also been used by Krupa and Szmolyan in e.g. [KS1], [KS2], [KS3] and [KS4].”
• Mixed-Mode Oscillations with Multiple Time Scales (2012, methodology): “There are other important techniques that are part of GSPT, such as the Exchange Lemma [110, 112], the blow-up method [55, 142, 233] and slow-fast normal form theory [10] that are not described in this paper.”
• Geometric singular perturbation analysis of an autocatalator model (2009, methodology): “The blow-up method pioneered by Dumortier and Roussarie [4] has proven to be a powerful tool in the geometric analysis of such problems [2], [5], [6], [10], [12], [13], [14], [15], [21].”
• Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points - Fold and Canard Points in Two Dimensions (2001, methodology): “Bifurcations from the center equilibrium and the family of periodic orbits and their relevance to canard explosion are studied in [14].”