Canard solutions at non-generic turning points
By Peter De Maesschalck, Freddy Dumortier. 2005
64 citations recorded by the source corpus.
This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds. "Canard solutions" are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a "turning point" and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the "control curve".
Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/38a54c9347aacbc4cda039a0ec9c10e675536e57
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In Applied Mathematics
In Geometry and Topology
- Ilona Gucwa
- Chengzhi Li
- Peter Szmolyan
- Freddy Dumortier
- Huaiping Zhu
- Peter De Maesschalck
- Martin Krupa
- Martin Wechselberger
In Statistical and Nonlinear Physics
- Ilona Gucwa
- Freddy Dumortier
- Chengzhi Li
- Peter Szmolyan
- Huaiping Zhu
- Peter De Maesschalck
- Martin Krupa
- Martin Wechselberger
Citations in the corpus
- 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].”
- Canard cycles for predator–prey systems with Holling types of functional response☆ (2013, background): “3) is negative, and by Theorem 4, Theorems 1 and 2 of [7] for a suitable choice of B0 = B0(ε,λ) there is a family of canard cycles {γε}, bifurcating from ΓV .”
- Slow Divergence Integral and Balanced Canard Solutions (2011, result): “C ∞ . We also consider their continuous extension at e = 0. It has been proven in [7], based on results of [ 4 ], that these continuous extensions, that we still denote as W f”