Bifurcations of Planar Vector Fields

Dumortier, Freddy; Roussarie, Robert; Sotomayor, Jorge; Żołądek, Henryk

doi:10.1007/bfb0098353

Cited by 111 publications

(127 citation statements)

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“…The relation between the vector fields T H (22) and V [42], as well as the completion of the bifurcation diagram of T H in Figure 4 top panels, are still under investigation by the authors. Another point of interest to us is whether there exists a relation between the cone-like structure found for the HSN bifurcation and the nilpotent singularity analyzed in [29,30].…”

Section: Discussionmentioning

confidence: 99%

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Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Broer

Simó

Vitolo

2008

Physica D: Nonlinear Phenomena

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The dynamics near a Hopf-saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf-saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.

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Section: Discussionmentioning

confidence: 99%

“…The vertex of the cone is the point (γ, µ, δ) = (0, 1, 0), where the derivative DT H at the equilibrium (w, z) = (1, 0) is equal to zero. This is a special case of the three-dimensional nilpotent singularity studied in [29,30]. Also see [28] for a detailed study of the HSN for vector fields.…”

Section: The Cylindrical Coordinates Of Pmentioning

confidence: 99%

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Broer

Simó

Vitolo

2008

Physica D: Nonlinear Phenomena

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“…It is a well-studied example of a bifurcation with codimension two and named after Bogdanov [4,5] and Takens [41,42], who independently and simultaneously described this bifurcation. For more results about Bogdanov-Takens bifurcation, see, for example, Chow and Hale [6], Dumortier et al [16], Arnold [1], Chow et al [7], Guckenheimer and Holmes [22], Kuznetsov [27], Xiao and Ruan ([38,51]), and others. Bogdanov-Takens bifurcation occurs also in infinite dimensional differential equations associated with functional differential equations and a few other partial differential equations.…”

Section: (T)u (T)mentioning

confidence: 99%

Bogdanov–Takens bifurcation in a predator–prey model

Liu

Magal

Xiao

2016

Z. Angew. Math. Phys.

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Abstract. In this paper, we investigate a class of predator-prey model with age structure and discuss whether the model can undergo Bogdanov-Takens bifurcation. The analysis is based on the normal form theory and the center manifold theory for semilinear equations with non-dense domain combined with integrated semigroup theory. Qualitative analysis indicates that there exist some parameter values such that this predator-prey model has an unique positive equilibrium which is Bogdanov-Takens singularity. Moreover, it is shown that under suitable small perturbation, the system undergoes the Bogdanov-Takens bifurcation in a small neighborhood of this positive equilibrium.

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“…The singularity is of nilpotent saddle, focus or elliptic type (see [15]), depending on parameters (s 2 ,s 3 , r 3 ).…”

Section: Bifurcation From Nilpotent Cuspsmentioning

confidence: 99%

Cyclicity of a fake saddle inside the quadratic vector fields

Maesschalck

Rebollo-Perdomo

Torregrosa

2015

Journal of Differential Equations

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This paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as a impassable grain. The normal form of the unperturbed vector field is like a degenerate flow box. Near the singularity,the phase portrait consists of parallel fibers, all of which but one have no singular points, and at the singular fiber, there is one node. We study different techniques in order to show that the cyclicity is bigger or equal than two when the normal form is quadratic.

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Bifurcations of Planar Vector Fields

Cited by 111 publications

References 0 publications

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Bogdanov–Takens bifurcation in a predator–prey model

Cyclicity of a fake saddle inside the quadratic vector fields

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