Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point.
Andronov-Hopf bifurcationAndronov–Hopf bifurcationBernd NoackBicycle and motorcycle dynamicsBifurcation theoryBiological applications of bifurcation theoryBogdanov–Takens bifurcationBurstingCellular modelCenter manifoldCompetitive Lotka–Volterra equationsDehaene–Changeux modelDicke modelDissipative systemEberhard HopfHodgkin–Huxley modelHopf BifurcationHopf bifurcation theoremIntermittencyKuramoto modelLorenz systemMangalore Anantha PaiMathematical and theoretical biologyMorris–Lecar modelNutation (botany)Paradox of enrichmentPauline van den DriesschePeriodic travelling wavePest insect population dynamicsPitchfork bifurcationReaction–diffusion systemSaddle-node bifurcationSelf-oscillationSpeed wobbleStellar pulsationStuart–Landau equationSupercriticalThomas' cyclically symmetric attractor
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Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point.
has abstract
Dans la théorie des bifurcatio ...... ique et en électronique, voir.
@fr
De Hopf-bifurcatie of Andronov ...... e oscillatie zeer klein wordt.
@nl
Eine Hopf-Bifurkation oder Hop ...... igenwert der Jacobimatrix aus.
@de
In matematica, in particolare ...... mite (attrattivo o repulsivo).
@it
In the mathematical theory of ...... dr Andronov and Eberhard Hopf.
@en
Біфуркація Гопфа — біфуркація, ...... ть математика Ебергарда Гопфа.
@uk
В теории динамических систем, ...... днопараметрических семействах.
@ru
力学系においてホップ分岐（ホップぶんき、英: Hopf bi ...... 、ポアンカレ・アンドロノフ・ホップ分岐と呼ばれることもある。
@ja
在數學的分岔理論中，霍普夫分岔（Hopf bifurcati ...... ré–Andronov–Hopf bifurcation"。
@zh
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Dans la théorie des bifurcatio ...... ique et en électronique, voir.
@fr
De Hopf-bifurcatie of Andronov ...... ing, maar die is niet stabiel.
@nl
Eine Hopf-Bifurkation oder Hop ...... arameter
* ist ein Eigenwert.
@de
In matematica, in particolare ...... mite (attrattivo o repulsivo).
@it
In the mathematical theory of ...... branches from the fixed point.
@en
Біфуркація Гопфа — біфуркація, ...... ть математика Ебергарда Гопфа.
@uk
В теории динамических систем, ...... (жёсткая потеря устойчивости).
@ru
力学系においてホップ分岐（ホップぶんき、英: Hopf bi ...... 、ポアンカレ・アンドロノフ・ホップ分岐と呼ばれることもある。
@ja
在數學的分岔理論中，霍普夫分岔（Hopf bifurcati ...... ré–Andronov–Hopf bifurcation"。
@zh
label
Biforcazione di Hopf
@it
Bifurcation de Hopf
@fr
Hopf bifurcation
@en
Hopf-Bifurkation
@de
Hopf-bifurcatie
@nl
Бифуркация Андронова — Хопфа
@ru
Біфуркація Андронова — Гопфа
@uk
ホップ分岐
@ja
霍普夫分岔
@zh