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Consider an [[Autonomous system|autonomous system]] of ordinary differential equations depending on a parameter
 
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>
+
$$ \tag{a1 }
 +
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {n} ,  \alpha \in \mathbf R  ^ {1} ,
 +
$$
  
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.
+
where $  f $
 +
is a smooth function. Suppose that at $  \alpha = 0 $,  
 +
the system (a1) has an [[Equilibrium position|equilibrium position]] $  x = 0 $
 +
with a simple pair of purely imaginary eigenvalues $  \lambda _ {1,2 }  = \pm  i \omega _ {0} $,  
 +
$  \omega _ {0} > 0 $,  
 +
of its Jacobian matrix $  A = f _ {x} ( 0,0 ) $.  
 +
Then, generically, a unique [[Limit cycle|limit cycle]] bifurcates from the equilibrium while it changes stability, as $  \alpha $
 +
passes through $  \alpha = 0 $.  
 +
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 $  { \mathop{\rm Re} } \lambda _ {1,2 }  = 0 $(
 +
has codimension one) and appears generically in one-parameter families.
  
 
First, consider a smooth planar system
 
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>
+
$$ \tag{a2 }
 +
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {2} ,  \alpha \in \mathbf R  ^ {1} ,
 +
$$
  
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:
+
that has for all sufficiently small $  | \alpha | $
 +
the equilibrium $  x = 0 $
 +
with eigenvalues $  \lambda _ {1,2 }  ( \alpha ) = \mu ( \alpha ) \pm  i \omega ( \alpha ) $,
 +
$  \mu ( 0 ) = 0 $,  
 +
$  \omega ( 0 ) = \omega _ {0} > 0 $.  
 +
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);
+
1) $  l _ {1} ( 0 ) \neq 0 $,  
 +
where $  l _ {1} ( \alpha ) $
 +
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
+
2) $  \mu  ^  \prime  ( 0 ) \neq 0 $,  
 +
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>
+
$$ \tag{a3 }
 +
\left \{
 +
\begin{array}{l}
 +
{ {\dot{y} } _ {1} = \beta y _ {1} - y _ {2} + \sigma y _ {1} ( y _ {1}  ^ {2} + y _ {2}  ^ {2} ) , \  } \\
 +
{ {\dot{y} } _ {2} = y _ {1} + \beta y _ {2} + \sigma y _ {2} ( y _ {1}  ^ {2} + y _ {2}  ^ {2} ) , \  }
 +
\end{array}
 +
\right .
 +
$$
  
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]]]).
+
where $  y = ( y _ {1} ,y _ {2} )  ^ {T} \in \mathbf R  ^ {2} $,  
 +
$  \beta \in \mathbf R  ^ {1} $,  
 +
$  \sigma = { \mathop{\rm sign} } l _ {1} ( 0 ) = \pm  1 $(
 +
see [[#References|[a2]]], [[#References|[a6]]]).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260a.gif" />
Line 25: Line 68:
 
Supercritical Hopf bifurcation on the plane
 
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.
+
Consider the case $  \sigma = - 1 $.  
 +
Then the system (a3) has an equilibrium at the origin $  x = 0 $,  
 +
which is stable for $  \beta \leq  0 $(
 +
weakly at $  \beta = 0 $)  
 +
and unstable for $  \beta > 0 $.  
 +
Moreover, there is a unique and stable circular limit cycle that exists for $  \beta > 0 $
 +
and has radius $  \sqrt \beta $(
 +
see Fig.a1). This is a supercritical Hopf bifurcation.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260b.gif" />
Line 33: Line 83:
 
Subcritical Hopf bifurcation on the plane
 
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.
+
For $  \sigma = 1 $,  
 +
the origin in (a3) is stable for $  \beta < 0 $
 +
and unstable for $  \beta \geq  0 $(
 +
weakly at $  \beta = 0 $),  
 +
while a unique and unstable limit cycle exists for $  \beta < 0 $(
 +
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:
+
In the $  n $-
 +
dimensional case, the Jacobian matrix $  A $
 +
evaluated at the equilibrium $  x = 0 $
 +
has a simple pair of purely imaginary eigenvalues $  \lambda _ {1,2 }  = \pm  \omega _ {0} $,  
 +
$  \omega _ {0} > 0 $,  
 +
as well as $  n _ {s} $
 +
eigenvalues with $  { \mathop{\rm Re} } \lambda _ {j} < 0 $,  
 +
and $  n _ {u} $
 +
eigenvalues with $  { \mathop{\rm Re} } \lambda _ {j} > 0 $(
 +
$  n _ {s} + n _ {u} + 2 = n $).  
 +
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]] $  {\mathcal M} _  \alpha  $
 +
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>
+
$$ \tag{a4 }
 +
\left \{
 +
\begin{array}{l}
 +
{ {\dot{y} } _ {1} = \beta y _ {1} - y _ {2} + \sigma y _ {1} ( y _ {1}  ^ {2} + y _ {2}  ^ {2} ) , \  } \\
 +
{ {\dot{y} } _ {2} = y _ {1} + \beta y _ {2} + \sigma y _ {2} ( y _ {1}  ^ {2} + y _ {2}  ^ {2} ) , \  } \\
 +
{ {\dot{y} } _ {s} = - y _ {s} , \  } \\
 +
{ {\dot{y} } _ {u} = + y _ {u} , \  } \\
 +
 
 +
\end{array}
 +
\right .
 +
$$
  
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" />.
+
where $  \beta \in \mathbf R  ^ {1} $,
 +
$  y = ( y _ {1} ,y _ {2} )  ^ {T} \in \mathbf R  ^ {2} $,  
 +
$  y _ {s} \in \mathbf R ^ {n _ {s} } $,  
 +
$  y _ {u} \in \mathbf R ^ {n _ {u} } $.  
 +
Fig.a3 shows the phase portraits of the system (a4) in the three-dimensional case, when $  n = 3 $,  
 +
$  n _ {s} = 1 $,  
 +
$  n _ {u} = 0 $,  
 +
and $  \sigma = - 1 $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260c.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h110260c.gif" />
Line 45: Line 128:
 
Figure: h110260c
 
Figure: h110260c
  
Hopf bifurcation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026056.png" />
+
Hopf bifurcation in $  \mathbf R  ^ {3} $
  
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
+
The first Lyapunov coefficient $  l _ {1} ( 0 ) $
 +
can be computed (to within a scalar multiple) in terms of the right-hand side of (a1) at $  \alpha = 0 $.  
 +
Represent $  f ( x,0 ) $
 +
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>
+
$$
 +
f ( x,0 ) = Ax + {
 +
\frac{1}{2}
 +
} B ( x,x ) + {
 +
\frac{1}{6}
 +
} C ( x,x,x ) + O ( \left \| x \right \|  ^ {4} ) ,
 +
$$
  
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
+
where $  B ( x,y ) $
 +
and $  C ( x,y,z ) $
 +
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>
+
$$
 +
B _ {i} ( x,y ) = \sum _ {j,k = 1 } ^ { n }  \left . {
 +
\frac{\partial  ^ {2} f _ {i} ( \xi,0 ) }{\partial  \xi _ {j} \partial  \xi _ {k} }
 +
} \right | _ {\xi = 0 }  x _ {j} y _ {k} ,
 +
$$
  
<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>
+
$$
 +
C _ {i} ( x,y,z ) = \sum _ {j,k,l = 1 } ^ { n }  \left . {
 +
\frac{\partial  ^ {3} f _ {i} ( \xi,0 ) }{\partial  \xi _ {j} \partial  \xi _ {k} \partial  \xi _ {l} }
 +
} \right | _ {\xi = 0 }  x _ {j} y _ {k} z _ {l} ,
 +
$$
  
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" />:
+
where $  i = 1 \dots n $.  
 +
Let $  q $
 +
be a complex eigenvector of $  A $
 +
corresponding to the eigenvalue $  i \omega _ {0} $:
  
<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>
+
$$
 +
Aq = i \omega _ {0} q, \quad q \in \mathbf C  ^ {n} .
 +
$$
  
Introduce also the adjoint eigenvector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110260/h11026070.png" />:
+
Introduce also the adjoint eigenvector $  p $:
  
<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>
+
$$
 +
A  ^ {T} p = - i \omega _ {0} p, \quad \left \langle  {p,q } \right \rangle = 1,
 +
$$
  
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]]])
+
where $  \langle  {p,q } \rangle = \sum _ {j = 1 }  ^ {n} {\overline{p}\; } _ {j} q _ {j} $
 +
is the [[Inner product|inner product]] in $  \mathbf C  ^ {n} $.  
 +
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>
+
$$
 +
l _ {1} ( 0 ) =
 +
$$
  
<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>
+
$$
 +
=  
 +
{
 +
\frac{1}{2 \omega _ {0} }
 +
} { \mathop{\rm Re} } \left [ \left \langle  {p,C ( q,q, {\overline{q}\; } ) } \right \rangle - 2 \left \langle  {p,B ( q,A ^ {- 1 } B ( q, {\overline{q}\; } ) ) } \right \rangle \right . +
 +
$$
  
<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>
+
$$
 +
+
 +
\left . \left \langle  {p,B ( {\overline{q}\; } , ( 2i \omega _ {0} I _ {n} - A ) ^ {- 1 } B ( q,q ) ) } \right \rangle \right ] .
 +
$$
  
 
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.
 
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.

Latest revision as of 22:11, 5 June 2020


Consider an autonomous system of ordinary differential equations depending on a parameter

$$ \tag{a1 } {\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {n} , \alpha \in \mathbf R ^ {1} , $$

where $ f $ is a smooth function. Suppose that at $ \alpha = 0 $, the system (a1) has an equilibrium position $ x = 0 $ with a simple pair of purely imaginary eigenvalues $ \lambda _ {1,2 } = \pm i \omega _ {0} $, $ \omega _ {0} > 0 $, of its Jacobian matrix $ A = f _ {x} ( 0,0 ) $. Then, generically, a unique limit cycle bifurcates from the equilibrium while it changes stability, as $ \alpha $ passes through $ \alpha = 0 $. This phenomenon is called the Hopf (or Andronov–Hopf) bifurcation [a1], [a7], [a2], [a3]. It is characterized by a single bifurcation condition $ { \mathop{\rm Re} } \lambda _ {1,2 } = 0 $( has codimension one) and appears generically in one-parameter families.

First, consider a smooth planar system

$$ \tag{a2 } {\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {2} , \alpha \in \mathbf R ^ {1} , $$

that has for all sufficiently small $ | \alpha | $ the equilibrium $ x = 0 $ with eigenvalues $ \lambda _ {1,2 } ( \alpha ) = \mu ( \alpha ) \pm i \omega ( \alpha ) $, $ \mu ( 0 ) = 0 $, $ \omega ( 0 ) = \omega _ {0} > 0 $. If the following non-degeneracy (genericity) conditions hold:

1) $ l _ {1} ( 0 ) \neq 0 $, where $ l _ {1} ( \alpha ) $ is the first Lyapunov coefficient (see below);

2) $ \mu ^ \prime ( 0 ) \neq 0 $, then (a2) is locally topologically equivalent (cf. Equivalence of dynamical systems) near the origin to the normal form

$$ \tag{a3 } \left \{ \begin{array}{l} { {\dot{y} } _ {1} = \beta y _ {1} - y _ {2} + \sigma y _ {1} ( y _ {1} ^ {2} + y _ {2} ^ {2} ) , \ } \\ { {\dot{y} } _ {2} = y _ {1} + \beta y _ {2} + \sigma y _ {2} ( y _ {1} ^ {2} + y _ {2} ^ {2} ) , \ } \end{array} \right . $$

where $ y = ( y _ {1} ,y _ {2} ) ^ {T} \in \mathbf R ^ {2} $, $ \beta \in \mathbf R ^ {1} $, $ \sigma = { \mathop{\rm sign} } l _ {1} ( 0 ) = \pm 1 $( see [a2], [a6]).

Figure: h110260a

Supercritical Hopf bifurcation on the plane

Consider the case $ \sigma = - 1 $. Then the system (a3) has an equilibrium at the origin $ x = 0 $, which is stable for $ \beta \leq 0 $( weakly at $ \beta = 0 $) and unstable for $ \beta > 0 $. Moreover, there is a unique and stable circular limit cycle that exists for $ \beta > 0 $ and has radius $ \sqrt \beta $( see Fig.a1). This is a supercritical Hopf bifurcation.

Figure: h110260b

Subcritical Hopf bifurcation on the plane

For $ \sigma = 1 $, the origin in (a3) is stable for $ \beta < 0 $ and unstable for $ \beta \geq 0 $( weakly at $ \beta = 0 $), while a unique and unstable limit cycle exists for $ \beta < 0 $( see Fig.a2). This is a subcritical Hopf bifurcation.

In the $ n $- dimensional case, the Jacobian matrix $ A $ evaluated at the equilibrium $ x = 0 $ has a simple pair of purely imaginary eigenvalues $ \lambda _ {1,2 } = \pm \omega _ {0} $, $ \omega _ {0} > 0 $, as well as $ n _ {s} $ eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} < 0 $, and $ n _ {u} $ eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} > 0 $( $ n _ {s} + n _ {u} + 2 = n $). According to the centre manifold theorem (cf. Centre manifold) [a5], [a7], [a2], there is an invariant two-dimensional centre manifold $ {\mathcal M} _ \alpha $ near the origin, the restriction of (a1) to which has the form (a2). Moreover, [a2], under the non-degeneracy conditions 1) and 2), the system (a1) is locally topologically equivalent (cf. Equivalence of dynamical systems) near the origin to the suspension of the normal form (a3) by the standard saddle:

$$ \tag{a4 } \left \{ \begin{array}{l} { {\dot{y} } _ {1} = \beta y _ {1} - y _ {2} + \sigma y _ {1} ( y _ {1} ^ {2} + y _ {2} ^ {2} ) , \ } \\ { {\dot{y} } _ {2} = y _ {1} + \beta y _ {2} + \sigma y _ {2} ( y _ {1} ^ {2} + y _ {2} ^ {2} ) , \ } \\ { {\dot{y} } _ {s} = - y _ {s} , \ } \\ { {\dot{y} } _ {u} = + y _ {u} , \ } \\ \end{array} \right . $$

where $ \beta \in \mathbf R ^ {1} $, $ y = ( y _ {1} ,y _ {2} ) ^ {T} \in \mathbf R ^ {2} $, $ y _ {s} \in \mathbf R ^ {n _ {s} } $, $ y _ {u} \in \mathbf R ^ {n _ {u} } $. Fig.a3 shows the phase portraits of the system (a4) in the three-dimensional case, when $ n = 3 $, $ n _ {s} = 1 $, $ n _ {u} = 0 $, and $ \sigma = - 1 $.

Figure: h110260c

Hopf bifurcation in $ \mathbf R ^ {3} $

The first Lyapunov coefficient $ l _ {1} ( 0 ) $ can be computed (to within a scalar multiple) in terms of the right-hand side of (a1) at $ \alpha = 0 $. Represent $ f ( x,0 ) $ as

$$ f ( x,0 ) = Ax + { \frac{1}{2} } B ( x,x ) + { \frac{1}{6} } C ( x,x,x ) + O ( \left \| x \right \| ^ {4} ) , $$

where $ B ( x,y ) $ and $ C ( x,y,z ) $ are multilinear functions (cf. also Multilinear mapping). In coordinates one has

$$ B _ {i} ( x,y ) = \sum _ {j,k = 1 } ^ { n } \left . { \frac{\partial ^ {2} f _ {i} ( \xi,0 ) }{\partial \xi _ {j} \partial \xi _ {k} } } \right | _ {\xi = 0 } x _ {j} y _ {k} , $$

$$ C _ {i} ( x,y,z ) = \sum _ {j,k,l = 1 } ^ { n } \left . { \frac{\partial ^ {3} f _ {i} ( \xi,0 ) }{\partial \xi _ {j} \partial \xi _ {k} \partial \xi _ {l} } } \right | _ {\xi = 0 } x _ {j} y _ {k} z _ {l} , $$

where $ i = 1 \dots n $. Let $ q $ be a complex eigenvector of $ A $ corresponding to the eigenvalue $ i \omega _ {0} $:

$$ Aq = i \omega _ {0} q, \quad q \in \mathbf C ^ {n} . $$

Introduce also the adjoint eigenvector $ p $:

$$ A ^ {T} p = - i \omega _ {0} p, \quad \left \langle {p,q } \right \rangle = 1, $$

where $ \langle {p,q } \rangle = \sum _ {j = 1 } ^ {n} {\overline{p}\; } _ {j} q _ {j} $ is the inner product in $ \mathbf C ^ {n} $. Then (see, for example, [a6])

$$ l _ {1} ( 0 ) = $$

$$ = { \frac{1}{2 \omega _ {0} } } { \mathop{\rm Re} } \left [ \left \langle {p,C ( q,q, {\overline{q}\; } ) } \right \rangle - 2 \left \langle {p,B ( q,A ^ {- 1 } B ( q, {\overline{q}\; } ) ) } \right \rangle \right . + $$

$$ + \left . \left \langle {p,B ( {\overline{q}\; } , ( 2i \omega _ {0} I _ {n} - A ) ^ {- 1 } B ( q,q ) ) } \right \rangle \right ] . $$

There is an analogue of the Hopf bifurcation for discrete-time dynamical systems, called the Neimark–Sacker bifurcation [a7], [a4], [a2], [a8], [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

[a1] 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)
[a2] V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Grundlehren math. Wiss. , 250 , Springer (1983) (In Russian)
[a3] J. Guckenheimer, Ph. Holmes, "Nonlinear oscillations, dynamical systems and bifurcations of vector fields" , Springer (1983)
[a4] G. Iooss, "Bifurcations of maps and applications" , North-Holland (1979)
[a5] A. Kelley, "The stable, center stable, center, center unstable and unstable manifolds" J. Diff. Eq. , 3 (1967) pp. 546–570
[a6] Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995)
[a7] J. Marsden, M. McCracken, "Hopf bifurcation and its applications" , Springer (1976)
[a8] D.C. Whitley, "Discrete dynamical systems in dimensions one and two" Bull. London Math. Soc. , 15 (1983) pp. 177–217
How to Cite This Entry:
Hopf bifurcation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_bifurcation&oldid=18291
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article