Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points - Fold and Canard Points in Two Dimensions

By Martin Krupa, Peter Szmolyan. SIAM Journal on Mathematical Analysis, 2001

554 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/d6096d4e1b2c7b57dd908083d5696a2e1a89f4e1

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. Stephen Schecter
7. Martin Krupa
8. John M. Guckenheimer

In Geometry and Topology

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

In Statistical and Nonlinear Physics

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

Citations in the corpus

• Geometric singular perturbation analysis of an autocatalator model (2009, methodology): “For another introduction to the method in the context of singularly perturbed planar fold points we refer to [13].”
• Local analysis near a folded saddle-node singularity (2010, background): “The transition itself is denoted as a canard explosion because of the exponentially small sensitivity in parameter variation; see e.g. [9, 10, 24, 25 ].”
• Mixed-Mode Oscillations with Multiple Time Scales (2012, methodology): “A detailed dynamical analysis of canard explosion and the associated singular Hopf bifurcation using geometric or asymptotic methods exists for planar slow-fast systems [12, 13, 57, 58, 141, 143]; we summarize these results as follows.”
• Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system (2011, background): “) ∈ R3 : x 1 = 1 2, x 2 = −λ + 1 4}, its phase portrait can be constructed by rescaling time dt dτ = −(2x 1 −1) ⇒ t = − Zτ 0 (2x 1(s) −1)ds, producing the phase portrait for the desingularised system [14]: dx 1 dτ = r − XN n=1 xn 1, (31) dλ dτ = −r(2x 1 −1), (32) and then reversing the direction of ”
• A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics (2011, background): “ection ∆in = {(x,−ρ2) : x ∈ J} transverse to Ca and define a section ∆out = {(−ρ,y) : y ∈ R} transverse to the fast subsystems. The next result describes the resulting flow map between the two sections [48]. 11 Theorem 3.2. Near a generic fold bifurcation of the fast subsystem the extension of Ca ǫ un”
• The entry-exit function and geometric singular perturbation theory (2015, methodology): “ solution for small ">0 is shown in Figure 1.1. Pontryagin delay in this context has been studied using nonstandard analysis [1], asymptotic expansions [12], complex analysis [3], and blow-up [7, 11]. In contrast to the system (1.1){(1.3), the system (1.5){(1.6) for "6= 0 does not have ”
• Sliding Vector Fields via Slow--Fast Systems (2008, background): “Needless to say that in this area very good surveys are available (see [2, 3, 5, 6] for instance).”
• Excitability in ramped systems: the compost-bomb instability (2011, background): “The analysis is greatly facilitated by the following scaling also known as desingularization (Dumortier & Roussarie 1996; Krupa & Szmolyan 2001), t = −t̂gSz (z , pr , p), (4.8) Proc.”
• Neural Cross-Frequency Coupling: Connecting Architectures, Mechanisms, and Functions. (2015, unknown): “Novel perturbation methods to investigate continuous strong coupling are being developed [16,112], although a general theory is still lacking.”