Canards in R3

By Peter Szmolyan, Martin Wechselberger. 2001

357 citations recorded by the source corpus.

Subfields: Computational Mechanics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7a50ab8612785d1344ca7b048b5c9746a7cfe8e0

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In Computational Mechanics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Excitability in ramped systems: the compost-bomb instability (2011, background): “Specifically, Szmolyan & Wechselberger (2001) prove that a singular canard through a folded saddle for the limiting problem perturbs to a maximal canard for sufficiently small e.”
Local analysis near a folded saddle-node singularity (2010, methodology): “In particular, we show how results from the folded node theory on canards [ 35 , 41] extend into the FSN II case as well as results on delayed loss of stability due to the slow passage through a Hopf bifurcation.”
Canards at Folded Nodes (2005, positive): “Smolyan and Wechselberger [9] give a presentation of some of Benôıt’s results using geometric methods that avoid nonstandard analysis.”
Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system (2011, neutral): “n=1 x1 , (35) dλ dτ = −e ρ (2x1 − 1), (36) has a saddle equilibrium for all ρ > 0, corresponding to a folded-saddle singularity [1, 24]: F = (x1,F , λF (ρ)) = (”
Mixed-Mode Oscillations with Multiple Time Scales (2012, background): “Wechselberger and collaborators [31, 212, 233] gave a detailed analysis of folded nodes while Guckenheimer and Haiduc [86] and Guckenheimer [84] computed intersections of slow manifolds near a folded node and maps along trajectories passing through these regions.”
A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics (2011, background): “One example are fold bifurcations with two slow variables and one fast variable [33, 81] which occur generically on one-dimensional curves.”
Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems (2012, background): “16) are not only consistent with the canards familiar in singularly perturbed systems [99,100], indeed they can be derived as approximations to them, and furthermore reveal that canard explosions are examples of catastrophic sliding bifurcations.”