View source for Hopf bifurcation

Consider an [[Autonomous system|autonomous system]] of ordinary differential equations depending on a parameter

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102602.png" /> is a smooth function. Suppose that at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102603.png" />, the system (a1) has an [[Equilibrium position|equilibrium position]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102604.png" /> with a simple pair of purely imaginary eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102606.png" />, of its Jacobian matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102607.png" />. Then, generically, a unique [[Limit cycle|limit cycle]] bifurcates from the equilibrium while it changes stability, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102608.png" /> passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h1102609.png" />. This phenomenon is called the Hopf (or Andronov–Hopf) bifurcation [[#References|[a1]]], [[#References|[a7]]], [[#References|[a2]]], [[#References|[a3]]]. It is characterized by a single bifurcation condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026010.png" /> (has codimension one) and appears generically in one-parameter families.

First, consider a smooth planar system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

that has for all sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026012.png" /> the equilibrium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026013.png" /> with eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026016.png" />. If the following non-degeneracy (genericity) conditions hold:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026018.png" /> is the first Lyapunov coefficient (see below);

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026019.png" />, then (a2) is locally topologically equivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the normal form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026023.png" /> (see [[#References|[a2]]], [[#References|[a6]]]).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260a.gif" />

Figure: h110260a

Supercritical Hopf bifurcation on the plane

Consider the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026024.png" />. Then the system (a3) has an equilibrium at the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026025.png" />, which is stable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026026.png" /> (weakly at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026027.png" />) and unstable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026028.png" />. Moreover, there is a unique and stable circular limit cycle that exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026029.png" /> and has radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026030.png" /> (see Fig.a1). This is a supercritical Hopf bifurcation.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260b.gif" />

Figure: h110260b

Subcritical Hopf bifurcation on the plane

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026031.png" />, the origin in (a3) is stable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026032.png" /> and unstable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026033.png" /> (weakly at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026034.png" />), while a unique and unstable limit cycle exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026035.png" /> (see Fig.a2). This is a subcritical Hopf bifurcation.

In the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026036.png" />-dimensional case, the Jacobian matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026037.png" /> evaluated at the equilibrium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026038.png" /> has a simple pair of purely imaginary eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026040.png" />, as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026041.png" /> eigenvalues with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026043.png" /> eigenvalues with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026044.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026045.png" />). According to the centre manifold theorem (cf. [[Centre manifold|Centre manifold]]) [[#References|[a5]]], [[#References|[a7]]], [[#References|[a2]]], there is an invariant two-dimensional [[Centre manifold|centre manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026046.png" /> near the origin, the restriction of (a1) to which has the form (a2). Moreover, [[#References|[a2]]], under the non-degeneracy conditions 1) and 2), the system (a1) is locally topologically equivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the suspension of the normal form (a3) by the standard saddle:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026051.png" />. Fig.a3 shows the phase portraits of the system (a4) in the three-dimensional case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026054.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026055.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260c.gif" />

Figure: h110260c

Hopf bifurcation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026056.png" />

The first Lyapunov coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026057.png" /> can be computed (to within a scalar multiple) in terms of the right-hand side of (a1) at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026058.png" />. Represent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026059.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026060.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026062.png" /> are multilinear functions (cf. also [[Multilinear mapping|Multilinear mapping]]). In coordinates one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026063.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026064.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026065.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026066.png" /> be a complex eigenvector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026067.png" /> corresponding to the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026068.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026069.png" /></td> </tr></table>

Introduce also the adjoint eigenvector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026070.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026071.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026072.png" /> is the [[Inner product|inner product]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026073.png" />. Then (see, for example, [[#References|[a6]]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026074.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026075.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026076.png" /></td> </tr></table>

There is an analogue of the Hopf bifurcation for discrete-time dynamical systems, called the Neimark–Sacker bifurcation [[#References|[a7]]], [[#References|[a4]]], [[#References|[a2]]], [[#References|[a8]]], [[#References|[a6]]]. Under certain non-degeneracy conditions, it generates a closed invariant curve around a fixed point which changes stability due to the transition of a pair of its complex eigenvalues through the unit circle.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Andronov,   E.A. Leontovich,   I.I. Gordon,   A.G. Maier,   "Theory of bifurcations of dynamical systems on a plane" , Israel Program Sci. Transl.  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,   "Geometrical methods in the theory of ordinary differential equations" , ''Grundlehren math. Wiss.'' , '''250''' , Springer  (1983)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Guckenheimer,   Ph. Holmes,   "Nonlinear oscillations, dynamical systems and bifurcations of vector fields" , Springer  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Iooss,   "Bifurcations of maps and applications" , North-Holland  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Kelley,   "The stable, center stable, center, center unstable and unstable manifolds"  ''J. Diff. Eq.'' , '''3'''  (1967)  pp. 546–570</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Yu.A. Kuznetsov,   "Elements of applied bifurcation theory" , Springer  (1995)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Marsden,   M. McCracken,   "Hopf bifurcation and its applications" , Springer  (1976)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D.C. Whitley,   "Discrete dynamical systems in dimensions one and two"  ''Bull. London Math. Soc.'' , '''15'''  (1983)  pp. 177–217</TD></TR></table>