Bifurcations at $\infty$ in a model for 1:4 resonance

Krauskopf, Bernd

doi:10.1017/s0143385797085039

Cited by 9 publications

(9 citation statements)

References 15 publications

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“…7. The local bifurcation analysis at infinity for the model of 1:4 resonance is in agreement with the bifurcation diagrams obtained for the parameter space of the 1:4 resonance (see section 8 and [Kra97]).…”

Section: Resultssupporting

confidence: 85%

“…which is a small perturbation of (4) for the particular choice of constants (a, b) = ( 1 2 , 0). We remark that the perturbation O(u) does not affect the values of a and b; compare [Kra97].…”

Section: Application To Z 4 -Equivariant Vector Fieldsmentioning

confidence: 90%

“…The line β = 1, ϕ = 3π/2, α ∈ (−π, π] plays an important role in the bifurcation set of (26) as several bifurcation surfaces accumulate on it; compare [Kra94,Kra97]. Indeed, along this line there is a singularity at ∞ of codimension at least two, the type of which depends on the value of α.…”

Section: Application To Z 4 -Equivariant Vector Fieldsmentioning

confidence: 99%

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Codimension-three unfoldings of reflectionally symmetric planar vector fields

Krauskopf

Rousseau

1997

Nonlinearity

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We consider unfoldings of a codimension-three singularity of reflectionally symmetric planar vector fields that provide a link between the different codimension-two unfoldings known in the literature. This work has been motivated by and can be applied to the study of bifurcations at infinity of planar vector fields with an even rotational symmetry. The codimension-three singularity is determined by its 4-jet, which gives eight classes. We analyse the 4-jet of the unfoldings, of which there are 22 classes. The unfoldings are (conjecturally) versal for 16 classes. The other six classes of unfoldings contain centres, and it is necessary to consider a jet of higher order, which is only indicated here. In our analysis we use that the 4-jet can be reduced to a quadratic vector field.

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Section: Resultssupporting

confidence: 85%

Section: Application To Z 4 -Equivariant Vector Fieldsmentioning

confidence: 90%

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Codimension-three unfoldings of reflectionally symmetric planar vector fields

Krauskopf

Rousseau

1997

Nonlinearity

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“…In this way, we combine systems (1), ( 3) and ( 4) into one compactified system that is autonomous, but not necessarily C 1 -smooth. The compactification approach presented in this paper is akin to the Poincaré-type phase-space compactification (of the xdimensions) that enables analysis of dynamical behaviour at infinity [10,13,16,21,29,31], collisions in many-body problems [30], stability of nonlinear waves [1,17], as well as highercodimension bifurcations [12] and canard solutions in slow-fast systems [14,20,[23][24][25]43] via blow up of singularities of vector fields. The difference between our work and these studies is twofold.…”

Section: The Basic Settingmentioning

confidence: 99%

Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

2021

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We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.

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“…In this way, we combine systems (1), ( 3) and ( 4) into one compactified system that is autonomous, but not necessarily C 1 -smooth. The compactification approach presented in this paper is akin to the Poincaré-type phase-space compactification (of the x-dimensions) that enables analysis of dynamical behaviour at infinity [10,20,13,30,28,15], collisions in many-body problems [29], stability of nonlinear waves [16,1], as well as higher-codimension bifurcations [12] and canard solutions in slow-fast systems [23,41,19,24,22] via blow up of singularities of vector fields. The difference between our work and these studies is twofold.…”

Section: The Basic Settingmentioning

confidence: 99%

Compactification for Asymptotically Autonomous Dynamical Systems: Theory, Applications and Invariant Manifolds

Wieczorek,

Xie,

Jones

2020

Preprint

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We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyse the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework.

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Bifurcations at $\infty$ in a model for 1:4 resonance

Cited by 9 publications

References 15 publications

Codimension-three unfoldings of reflectionally symmetric planar vector fields

Codimension-three unfoldings of reflectionally symmetric planar vector fields

Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

Compactification for Asymptotically Autonomous Dynamical Systems: Theory, Applications and Invariant Manifolds

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