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| In the [[bifurcation theory|mathematical theory of bifurcations]], a '''Hopf''' or '''Poincaré–Andronov–Hopf bifurcation''', named after [[Henri Poincaré]], [[Eberhard Hopf]], and [[Aleksandr Andronov]], is a local bifurcation in which a [[fixed point (mathematics)|fixed point]] of a [[dynamical system]] loses stability as a pair of [[complex conjugate]] [[eigenvalue]]s of the [[linearization]] around the fixed point cross the imaginary axis of the [[complex plane]]. Under reasonably generic assumptions about the dynamical system, we can expect to see a small-amplitude [[limit cycle]] branching from the fixed point.
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| For a more general survey on Hopf bifurcation and dynamical systems in general, see.<ref name="Strogatz1994"/><ref name="Kuznetsov2004"/><ref name="HaleKocak1991"/><ref name="Guckenheimer1997"/><ref name="HairerNorsettWanner1993"/>
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| == Overview ==
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| === Supercritical / subcritical Hopf bifurcations ===
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| The limit cycle is orbitally stable if a specific quantity called the '''first Lyapunov coefficient''' is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical.
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| The [[normal form (mathematics)|normal form]] of a Hopf bifurcation is:
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| ::<math>\frac{dz}{dt}=z((\lambda + i ) + b |z|^2), </math>
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| where ''z'', ''b'' are both complex and ''λ'' is a parameter. Write
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| : <math> b= \alpha + i \beta. \,</math>
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| The number ''α'' is called the first Lyapunov coefficient.
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| * If ''α'' is negative then there is a stable limit cycle for ''λ'' > 0:
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| ::<math> z(t) = r e^{i \omega t} \,</math>
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| : where
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| ::<math> r=\sqrt{-\lambda/\alpha}\text{ and }\omega= 1 + \beta r^2. \, </math>
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| : The bifurcation is then called '''supercritical.'''
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| * If ''α'' is positive then there is an unstable limit cycle for ''λ'' < 0. The bifurcation is called '''subcritical.'''
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| === Remarks ===
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| The "smallest chemical reaction with Hopf bifurcation" was found in 1995 in Berlin, Germany.<ref name="Wilhelm1995"/> The same biochemical system has been used in order to investigate how the existence of a Hopf bifurcation influences our ability to reverse-engineer dynamical systems.<ref name="KirkToniStumpf2008"/>
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| Under some general hypothesis, in the neighborhood of a Hopf bifurcation, a stable [[fixed point (mathematics)|steady point]] of the system gives birth to a small stable [[limit cycle]]. Remark that looking for Hopf bifurcation is not equivalent to looking for stable limit cycles. First, some Hopf bifurcations (e.g. subcritical ones) do not imply the existence of stable limit cycles; second, there may exist limit cycles not related to Hopf bifurcations.
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| === Example ===
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| [[Image:hopf-bif.gif|thumb|250px|right|The Hopf bifurcation in the Selkov system (see article). As the parameters change, a limit cycle (in blue) appears out of an unstable equilibrium.]]
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| Hopf bifurcations occur in the [[Hodgkin–Huxley model]] for nerve membrane {{citation needed|date=August 2013}}, the Selkov model<ref name="Selkov Model">{{cite web| title=Selkov Model Wolfram Demo| work=[demonstrations.wolfram.com ]|url=http://demonstrations.wolfram.com/HopfBifurcationInTheSelkovModel/|accessdate=30 September 2012}}</ref> of [[glycolysis]], the [[Belousov–Zhabotinsky reaction]], the [[Lorenz attractor]] and in the following simpler chemical system called the '''[[Brusselator]]''' as the parameter ''B'' changes:
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| :<math> \frac{dX}{dt} = A + X^2Y -(B+1)X </math>
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| :<math> \frac{dY}{dt} = B X -X^2Y. </math>
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| The Selkov model is
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| : <math> \frac{dx}{dt} = -x + ay + x^2 y, ~~ \frac{dy}{dt} = b - a y - x^2 y. </math>
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| The phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right. See Strogatz, Steven H. (1994). "Nonlinear Dynamics and Chaos",<ref name="Strogatz1994"/> page 205 for detailed derivation.
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| == Definition of a Hopf bifurcation ==
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| The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a steady point is known as the Hopf bifurcation. The following theorem works with steady points with one pair of conjugate nonzero purely imaginary [[eigenvalue]]s. It tells the conditions under which this bifurcation phenomenon occurs.
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| '''Theorem''' (see section 11.2 of <ref name="HaleKocak1991"/>). Let <math>J_0</math> be the [[Jacobian]] of a continuous parametric [[dynamical system]] evaluated at a steady point <math>Z_e</math> of it. Suppose that all eigenvalues of <math>J_0</math> have negative real parts except one conjugate nonzero purely imaginary pair <math>\pm i\beta</math>. A ''Hopf bifurcation'' arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.
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| == Routh–Hurwitz criterion ==
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| [[Routh–Hurwitz stability criterion|Routh–Hurwitz criterion]] (section I.13 of <ref name="HairerNorsettWanner1993"/>) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea.<ref name="KahouiWeber2000"/>
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| === Sturm series ===
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| Let <math>p_0,p_1,\dots,p_k</math> be [[Sturm series]] associated to a [[characteristic polynomial]] <math>P</math>. They can be written in the form:
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| :<math>
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| p_i(\mu)= c_{i,0} \mu^{k-i} + c_{i,1} \mu^{k-i-2} + c_{i,2} \mu^{k-i-4}+\cdots
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| </math>
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| The coefficients <math>c_{i,0}</math> for <math>i</math> in <math>\{1,\dots,k\}</math> correspond to what is called [[Hurwitz determinants]].<ref name="KahouiWeber2000"/> Their definition is related to the associated [[Hurwitz matrix]].
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| === Propositions ===
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| '''Proposition 1'''. If all the Hurwitz determinants <math>c_{i,0}</math> are positive, apart perhaps <math>c_{k,0}</math> then the associated Jacobian has no pure imaginary eigenvalues.
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| '''Proposition 2'''. If all Hurwitz determinants <math>c_{i,0}</math> (for all <math>i</math> in <math>\{0,\dots,k-2\}</math> are positive, <math>c_{k-1,0}=0</math> and <math>c_{k-2,1}<0</math> then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair.
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| The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition.
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| == Example ==
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| Let us consider the classical [[Van der Pol oscillator]] written with ordinary differential equations:
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| :<math>
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| \left \{
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| \begin{array}{l}
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| \dfrac{dx}{dt} = \mu (1-y^2)x - y, \\
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| \dfrac{dy}{dt} = x.
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| \end{array}
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| \right .
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| </math>
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| The Jacobian matrix associated to this system follows:
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| :<math>
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| J =
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| \begin{pmatrix}
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| -\mu (-1+y^2) & -2 \mu y x -1 \\
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| 1 & 0
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| \end{pmatrix}.
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| </math>
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| The characteristic polynomial (in <math>\lambda</math>) of the linearization at (0,0) is equal to:
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| :<math>
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| P(\lambda) = \lambda^2 - \mu \lambda + 1.
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| </math>
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| The coefficients are:
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| <math>a_0=1, a_1=-\mu, a_2=1</math> <br>
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| The associated [[Sturm series]] is:
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| :<math>
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| \begin{array}{l}
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| p_0(\lambda)=a_0 \lambda^2 -a_2 \\
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| p_1(\lambda)=a_1 \lambda
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| \end{array}
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| </math>
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| The [[Sturm's theorem|Sturm]] polynomials can be written as (here <math>i=0,1</math>):
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| :<math>
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| p_i(\mu)= c_{i,0} \mu^{k-i} + c_{i,1} \mu^{k-i-2} + c_{i,2} \mu^{k-i-4}+\cdots
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| </math>
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| The above proposition 2 tells that one must have:
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| :<math>
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| c_{0,0} = 1 >0, c_{1,0}=- \mu = 0, c_{0,1}=-1 <0.
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| </math>
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| Because 1 > 0 and −1 < 0 are obvious, one can conclude that a Hopf bifurcation may occur for Van der Pol oscillator if <math>\mu = 0</math>.
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| == References ==
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| {{reflist|refs=
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| <ref name="KirkToniStumpf2008">{{cite journal |last1 = Kirk| first1= P. D. W. | last2= Toni | first2 = T. |last3= Stumpf|first= M. P. H. |year = 2008 |title = Parameter inference for biochemical systems that undergo a Hopf bifurcation | journal = Biophysical Journal |volume = 95 |issue = 2 |pages = 540–549 |doi = 10.1529/biophysj.107.126086 |url=http://download.cell.com/biophysj/pdf/PIIS0006349508702315.pdf |pmid = 18456830 |first3 = MP |pmc = 2440454}}</ref>
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| <ref name="Wilhelm1995">{{cite journal | last1 = Wilhelm | first1 = T.| last2= Heinrich | first2= R. |year = 1995 |title = Smallest chemical reaction system with Hopf bifurcation |journal = Journal of Mathematical Chemistry |volume = 17 |issue = 1 |pages = 1–14 |doi = 10.1007/BF01165134 |url=http://www.fli-leibniz.de/~wilhelm/JMC1995.pdf}}</ref> | |
| <ref name="HairerNorsettWanner1993">{{cite book |title= Solving ordinary differential equations I: nonstiff problems|last1= Hairer |first1= E. | last2= Norsett| first2 =S. P.|last3= Wanner | first3=G. |year= 1993|publisher= Springer-Verlag|location= New York|edition=Second }}</ref>
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| <ref name="HaleKocak1991">{{cite book |title= Dynamics and Bifurcations|last1= Hale|first1= J.|last2= Koçak|first2= H. |year= 1991|publisher= Springer-Verlag|location= New York|series= Texts in Applied Mathematics|volume= 3}}</ref>
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| <ref name="Strogatz1994">{{cite book |title= Nonlinear Dynamics and Chaos |last= Strogatz |first= Steven H. |year= 1994 |publisher= Addison Wesley publishing company}}</ref>
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| <ref name="Kuznetsov2004">{{cite book |title= Elements of Applied Bifurcation Theory |last= Kuznetsov|first= Yuri A. |year= 2004|publisher= Springer-Verlag|location= New York |isbn= 0-387-21906-4 }}</ref>
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| <ref name="KahouiWeber2000">{{cite journal |last1= Kahoui|first1= M. E. |last2= Weber |first2= A. |year=2000 |title= Deciding Hopf bifurcations by quantifier elimination in a software component architecture|journal=Journal of Symbolic Computation |volume=30 |issue= 2 |pages= 161–179 |doi= 10.1006/jsco.1999.0353}}</ref>
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| <ref name="Guckenheimer1997">{{cite journal |last1= Guckenheimer |first1= J.|last2=Myers |first2= M. |last3= Sturmfels| first3= B.|year= 1997 |title= Computing Hopf Bifurcations I|journal= SIAM Journal on Numerical Analysis }}</ref>
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| }}
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| == External links==
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| {{commons category|Hopf bifurcations}}
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| * [[Reaction-diffusion]] systems
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| * [http://www.egwald.com/nonlineardynamics/bifurcations.php#hopfbifurcation The Hopf Bifurcation]
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| * [http://www.scholarpedia.org/article/Andronov-Hopf_bifurcation Andronov–Hopf bifurcation page] at [[Scholarpedia]]
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| [[Category:Bifurcation theory]]
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| [[Category:Circuit theorems]]
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Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. Some time in the past she selected to reside in Alaska and her mothers and fathers reside nearby. Invoicing is my profession. As a lady what she truly likes is fashion and she's been doing it for quite a while.
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