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Hopf bifurcation and the Hopf fibration
Nonlinearity, 1994Summary: We present techniques for studying the local dynamics generated by an equivariant Hopf bifurcation. We show that under general hypothesis we can expect the formation of a branch of attracting invariant spheres with capture all the local dynamics.
Field, Mike, Swift, James W.
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1985
The term Hopf bifurcation refers to a phenomenon in which a steady state of an evolution equation evolves into a periodic orbit as a bifurcation parameter is varied. The Hopf bifurcation theorem (Theorem 3.2) provides sufficient conditions for determining when this behavior occurs.
Martin Golubitsky, David G. Schaeffer
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The term Hopf bifurcation refers to a phenomenon in which a steady state of an evolution equation evolves into a periodic orbit as a bifurcation parameter is varied. The Hopf bifurcation theorem (Theorem 3.2) provides sufficient conditions for determining when this behavior occurs.
Martin Golubitsky, David G. Schaeffer
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Turing Instabilities at Hopf Bifurcation
Journal of Nonlinear Science, 2009A simple procedure for deriving a uniform asymptotic expansion for the limit cycle in the vicinity of the Hopf bifurcation point for a two dimensional reaction system \[ u_{t} =D_{u}\Delta u+f\left( u,v;a\right) , \] \[ v_{t} =D_{v}\Delta v+g\left( u,v;a\right) \tag{b} \] is suggested. First, an algorithm allowing reduction of the system (ref {b}) to a
Ricard, M.R., Mischler, Stéphane
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2014
In this chapter we study a bifurcation characterized by a zero eigenvalue and a pair of nonzero purely imaginary eigenvalues of the linearization transverse to a plane of equilibria. It turns out that instead we can study a one-parameter family of lines in a system depending on one parameter.
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In this chapter we study a bifurcation characterized by a zero eigenvalue and a pair of nonzero purely imaginary eigenvalues of the linearization transverse to a plane of equilibria. It turns out that instead we can study a one-parameter family of lines in a system depending on one parameter.
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2011
Jusqu’a present, nous avons etudie des bifurcations stationnaires correspondant a des changements de solutions stationnaires. Ce chapitre decrit quelques exemples de systemes non lineaires presentant des bifurcations de Hopf, du nom de l’astronome mathematicien autrichien Eberhard Frederich Ferdinand Hopf (1902–1983), caracteristiques d’une transition ...
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Jusqu’a present, nous avons etudie des bifurcations stationnaires correspondant a des changements de solutions stationnaires. Ce chapitre decrit quelques exemples de systemes non lineaires presentant des bifurcations de Hopf, du nom de l’astronome mathematicien autrichien Eberhard Frederich Ferdinand Hopf (1902–1983), caracteristiques d’une transition ...
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2018
The Hopf bifurcation theorem provides an effective criterion for finding periodic solutions for ordinary differential equations. Although various proofs of this classical theorem are known, there seems to be no easy way to arrive at the goal.
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The Hopf bifurcation theorem provides an effective criterion for finding periodic solutions for ordinary differential equations. Although various proofs of this classical theorem are known, there seems to be no easy way to arrive at the goal.
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2014
The final bifurcation of codimension 2 is characterized by the intersection of 2 curves of Poincare-Andronov-Hopf points on a two-dimensional surface of equilibria. As we shall see, the drift direction at the Hopf lines play an important role. In the case of a parameter-dependent fixed line of equilibria, drifts at both Hopf-lines can be opposite and ...
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The final bifurcation of codimension 2 is characterized by the intersection of 2 curves of Poincare-Andronov-Hopf points on a two-dimensional surface of equilibria. As we shall see, the drift direction at the Hopf lines play an important role. In the case of a parameter-dependent fixed line of equilibria, drifts at both Hopf-lines can be opposite and ...
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1979
Let X = Σ Xi∂i = Σ piξi∂i be a vectorfield on Δ. X is a function from Δ to RI. We define the Hessian of X at p, HPX: Tp Δ × Tp Δ → R to be the bilinear form defined by: $$ {H_P}X\left( {{Y^1}{Y^2}} \right) = {\left( {{d_P}X\left( {{Y^1}} \right),{Y^2}} \right)_P}. $$ (1.1) .
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Let X = Σ Xi∂i = Σ piξi∂i be a vectorfield on Δ. X is a function from Δ to RI. We define the Hessian of X at p, HPX: Tp Δ × Tp Δ → R to be the bilinear form defined by: $$ {H_P}X\left( {{Y^1}{Y^2}} \right) = {\left( {{d_P}X\left( {{Y^1}} \right),{Y^2}} \right)_P}. $$ (1.1) .
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Asterisque, 2003
In this note examples are presented of vector fields depending on parameters and determined by the 3-jet, which display persistent occurrence of n-quasiperiodicity. In the parameter space this occurrence has relatively large measure. A leading example consists of weakly coupled Hopf bifurcations. This example, however, is extended to full generality in
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In this note examples are presented of vector fields depending on parameters and determined by the 3-jet, which display persistent occurrence of n-quasiperiodicity. In the parameter space this occurrence has relatively large measure. A leading example consists of weakly coupled Hopf bifurcations. This example, however, is extended to full generality in
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