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Consider an [[Autonomous system|autonomous system]] of ordinary differential equations
 
Consider an [[Autonomous system|autonomous system]] of ordinary differential equations
  
<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/c/c110/c110130/c1101301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
{\dot{x} } = f ( x ) , \quad x \in \mathbf R  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101302.png" /> is sufficiently smooth, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101303.png" />. Let the eigenvalues of the [[Jacobi matrix|Jacobi matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101304.png" /> evaluated at the [[Equilibrium position|equilibrium position]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101305.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101306.png" />. Suppose the equilibrium is non-hyperbolic, i.e. has eigenvalues with zero real part. Assume also that there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101307.png" /> eigenvalues (counting multiplicities) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c1101309.png" /> eigenvalues with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013011.png" /> eigenvalues with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013013.png" /> denote the linear (generalized) eigenspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013014.png" /> corresponding to the union of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013015.png" /> eigenvalues on the imaginary axis. The eigenvalues with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013016.png" /> are often called critical, as is the eigenspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013017.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013018.png" /> denote the [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] associated with (a1). Under the assumptions stated above, the following centre manifold theorem holds [[#References|[a7]]], [[#References|[a9]]], [[#References|[a3]]], [[#References|[a11]]]: There is a locally defined smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013019.png" />-dimensional invariant manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013021.png" /> that is tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013022.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013023.png" />.
+
where $  f : {\mathbf R  ^ {n} } \rightarrow {\mathbf R  ^ {n} } $
 +
is sufficiently smooth, $  f ( 0 ) = 0 $.  
 +
Let the eigenvalues of the [[Jacobi matrix|Jacobi matrix]] $  A $
 +
evaluated at the [[Equilibrium position|equilibrium position]] $  x _ {0} = 0 $
 +
be $  \lambda _ {1} \dots \lambda _ {n} $.  
 +
Suppose the equilibrium is non-hyperbolic, i.e. has eigenvalues with zero real part. Assume also that there are $  n _ {u} $
 +
eigenvalues (counting multiplicities) with $  { \mathop{\rm Re} } \lambda _ {j} > 0 $,  
 +
$  n _ {c} $
 +
eigenvalues with $  { \mathop{\rm Re} } \lambda _ {j} = 0 $,  
 +
and $  n _ {s} $
 +
eigenvalues with $  { \mathop{\rm Re} } \lambda _ {j} < 0 $.  
 +
Let $  T  ^ {c} $
 +
denote the linear (generalized) eigenspace of $  A $
 +
corresponding to the union of the $  n _ {c} $
 +
eigenvalues on the imaginary axis. The eigenvalues with $  { \mathop{\rm Re} } \lambda _ {j} = 0 $
 +
are often called critical, as is the eigenspace $  T  ^ {c} $.  
 +
Let $  \varphi  ^ {t} $
 +
denote the [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] associated with (a1). Under the assumptions stated above, the following centre manifold theorem holds [[#References|[a7]]], [[#References|[a9]]], [[#References|[a3]]], [[#References|[a11]]]: There is a locally defined smooth $  n _ {c} $-
 +
dimensional invariant manifold $  W  ^ {c} ( 0 ) $
 +
of $  \varphi  ^ {t} $
 +
that is tangent to $  T  ^ {c} $
 +
at $  x = 0 $.
  
The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013024.png" /> is called the centre manifold. The centre manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013025.png" /> need not be unique. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013026.png" /> with finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013028.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013029.png" />-manifold in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013031.png" />. However, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013032.png" /> the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013033.png" /> may shrink, thus resulting in the non-existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013034.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013035.png" /> for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013036.png" /> systems.
+
The manifold $  W  ^ {c} ( 0 ) $
 +
is called the centre manifold. The centre manifold $  W  ^ {c} ( 0 ) $
 +
need not be unique. If $  f \in C  ^ {k} $
 +
with finite $  k $,
 +
$  W  ^ {c} ( 0 ) $
 +
is a $  C  ^ {k} $-
 +
manifold in some neighbourhood $  U $
 +
of $  x _ {0} $.  
 +
However, as $  k \rightarrow \infty $
 +
the neighbourhood $  U $
 +
may shrink, thus resulting in the non-existence of a $  C  ^  \infty  $-
 +
manifold $  W  ^ {c} ( 0 ) $
 +
for certain $  C  ^  \infty  $
 +
systems.
  
In a basis formed by all (generalized) eigenvectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013037.png" /> (or their linear combinations if the corresponding eigenvalues are complex), the system (a1) can be written as
+
In a basis formed by all (generalized) eigenvectors of $  A $(
 +
or their linear combinations if the corresponding eigenvalues are complex), the system (a1) can be written 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/c/c110/c110130/c11013038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\left \{
 +
\begin{array}{l}
 +
{ {\dot{u} } = Bu + g ( u,v ) , \  } \\
 +
{ {\dot{v} } = Cv + h ( u,v ) , \  }
 +
\end{array}
 +
\right .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013041.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013042.png" />-matrix with all its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013043.png" /> eigenvalues on the imaginary axis, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013044.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013045.png" />-matrix with no eigenvalue on the imaginary axis; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013046.png" />. A centre manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013047.png" /> of (a2) can be locally represented as the graph of a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013049.png" />:
+
where $  u \in \mathbf R ^ {n _ {c} } $,  
 +
$  v \in \mathbf R ^ {n _ {u} + n _ {s} } $,  
 +
$  B $
 +
is an $  ( n _ {c} \times n _ {c} ) $-
 +
matrix with all its $  n _ {c} $
 +
eigenvalues on the imaginary axis, while $  C $
 +
is an $  ( ( n _ {u} + n _ {s} ) \times ( n _ {u} + n _ {s} ) ) $-
 +
matrix with no eigenvalue on the imaginary axis; $  g,h = O ( \| {( u,v ) } \|  ^ {2} ) $.  
 +
A centre manifold $  W  ^ {c} $
 +
of (a2) can be locally represented as the graph of a smooth function $  V : {\mathbf R ^ {n _ {c} } } \rightarrow {\mathbf R ^ {n _ {u} + n _ {s} } } $,  
 +
$  V ( u ) = O ( \| u \|  ^ {2} ) $:
  
<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/c/c110/c110130/c11013050.png" /></td> </tr></table>
+
$$
 +
W  ^ {c} = \left \{ {( u,v ) } : {v = V ( u ) ,  \left \| u \right \| < \varepsilon } \right \} .
 +
$$
  
 
The following reduction principle is valid (see [[#References|[a1]]], [[#References|[a8]]]): The system (a2) is locally topologically equivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the system
 
The following reduction principle is valid (see [[#References|[a1]]], [[#References|[a8]]]): The system (a2) is locally topologically equivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the 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/c/c110/c110130/c11013051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a3 }
 +
\left \{
 +
\begin{array}{l}
 +
{ {\dot{u} } = Bu + g ( u,V ( u ) ) , \  } \\
 +
{ {\dot{v} } = Cv. \  }
 +
\end{array}
 +
\right .
 +
$$
  
The equations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013053.png" /> are uncoupled in (a3). The first equation is the restriction of (a2) to its centre manifold. Thus, the dynamics of (a2) near a non-hyperbolic equilibrium are determined by this restriction, since the second equation in (a3) is linear and has exponentially decaying/growing solutions. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013054.png" /> is the asymptotically stable equilibrium of the restriction and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013055.png" /> has no eigenvalue with positive real part, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013056.png" /> is the asymptotically stable equilibrium of (a2). If there is more than one centre manifold, then all the resulting systems (a3) with different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013057.png" /> are locally topologically equivalent (actually, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013058.png" /> differ only by flat functions).
+
The equations for $  u $
 +
and $  v $
 +
are uncoupled in (a3). The first equation is the restriction of (a2) to its centre manifold. Thus, the dynamics of (a2) near a non-hyperbolic equilibrium are determined by this restriction, since the second equation in (a3) is linear and has exponentially decaying/growing solutions. For example, if $  u = 0 $
 +
is the asymptotically stable equilibrium of the restriction and the matrix $  C $
 +
has no eigenvalue with positive real part, then $  ( u,v ) = ( 0,0 ) $
 +
is the asymptotically stable equilibrium of (a2). If there is more than one centre manifold, then all the resulting systems (a3) with different $  V $
 +
are locally topologically equivalent (actually, the $  V $
 +
differ only by flat functions).
  
 
The second equation in (a3) can be replaced by the standard saddle:
 
The second equation in (a3) can be replaced 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/c/c110/c110130/c11013059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
$$ \tag{a4 }
 +
\left \{
 +
\begin{array}{l}
 +
{ {\dot{v} } = - v \  } \\
 +
{ {\dot{w} } = w, \  }
 +
\end{array}
 +
\right .
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013060.png" />. In other words, near a non-hyperbolic equilibrium the system is locally topologically equivalent to the suspension of its restriction to the centre manifold by the standard saddle.
+
with $  ( v,w ) \in \mathbf R ^ {n _ {s} } \times \mathbf R ^ {n _ {u} } $.  
 +
In other words, near a non-hyperbolic equilibrium the system is locally topologically equivalent to the suspension of its restriction to the centre manifold by the standard saddle.
  
 
Consider now a system that depends smoothly on parameters:
 
Consider now a system that depends smoothly on parameters:
  
<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/c/c110/c110130/c11013061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
$$ \tag{a5 }
 +
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {n} ,  \alpha \in \mathbf R  ^ {m} .
 +
$$
  
Suppose that at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013062.png" /> the system has a non-hyperbolic equilibrium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013063.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013064.png" /> eigenvalues on the imaginary axis and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013065.png" /> eigenvalues with non-zero real part. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013066.png" /> of them have negative real part, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013067.png" /> have positive real part. Applying the centre manifold theorem to the following extended system:
+
Suppose that at $  \alpha = 0 $
 +
the system has a non-hyperbolic equilibrium $  x = 0 $
 +
with $  n _ {c} $
 +
eigenvalues on the imaginary axis and $  ( n - n _ {c} ) $
 +
eigenvalues with non-zero real part. Let $  n _ {s} $
 +
of them have negative real part, while $  n _ {u} $
 +
have positive real part. Applying the centre manifold theorem to the following extended 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/c/c110/c110130/c11013068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
$$ \tag{a6 }
 +
\left \{
 +
\begin{array}{l}
 +
{ {\dot \alpha  } = 0, \  } \\
 +
{ {\dot{x} } = f ( x, \alpha ) , \  }
 +
\end{array}
 +
\right .
 +
$$
  
one can prove the existence of a parameter-dependent local invariant manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013069.png" /> in (a5). The manifold has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013071.png" /> coincides with a centre manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013072.png" /> of the (a5) at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013073.png" />. Often, the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013074.png" /> is called the centre manifold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013075.png" />. For each small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013076.png" /> one can restrict system (a5) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013077.png" />. Introducing a coordinate system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013078.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013079.png" /> as the coordinate, this restriction will be represented by a smooth system:
+
one can prove the existence of a parameter-dependent local invariant manifold $  {\mathcal M} _  \alpha  $
 +
in (a5). The manifold has dimension $  n _ {c} $
 +
and $  {\mathcal M} _ {0} $
 +
coincides with a centre manifold $  W  ^ {c} ( 0 ) $
 +
of the (a5) at $  \alpha = 0 $.  
 +
Often, the manifold $  {\mathcal M} _  \alpha  $
 +
is called the centre manifold for all $  \alpha $.  
 +
For each small $  | \alpha | $
 +
one can restrict system (a5) to $  {\mathcal M} _  \alpha  $.  
 +
Introducing a coordinate system on $  {\mathcal M} _  \alpha  $
 +
with $  u \in \mathbf R ^ {n _ {c} } $
 +
as the coordinate, this restriction will be represented by a smooth 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/c/c110/c110130/c11013080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
$$ \tag{a7 }
 +
{\dot{u} } = \Phi ( u, \alpha ) .
 +
$$
  
At <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013081.png" />, the system (a7) is equivalent to the restriction of (a5) to its centre manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110130/c11013082.png" />. The following results are known as the Šošitaišvili theorem [[#References|[a10]]] (see also [[#References|[a1]]], [[#References|[a2]]], [[#References|[a8]]]): The system (a5) is locally topologically equivalent near the origin to the suspension of (a7) by the standard saddle (a4). Moreover, (a7) can be replaced by any locally topologically equivalent system.
+
At $  \alpha = 0 $,  
 +
the system (a7) is equivalent to the restriction of (a5) to its centre manifold $  W  ^ {c} ( 0 ) $.  
 +
The following results are known as the Šošitaišvili theorem [[#References|[a10]]] (see also [[#References|[a1]]], [[#References|[a2]]], [[#References|[a8]]]): The system (a5) is locally topologically equivalent near the origin to the suspension of (a7) by the standard saddle (a4). Moreover, (a7) can be replaced by any locally topologically equivalent system.
  
 
This theorem reduces the study of bifurcations of non-hyperbolic equilibria (cf. also [[Bifurcation|Bifurcation]]) to those on the corresponding centre manifold of dimension equal to the number of critical eigenvalues. There are analogues of the reduction principle and Šošitaišvili's theorem for discrete-time dynamical systems defined by iterations of diffeomorphisms (see, for example, [[#References|[a1]]], [[#References|[a8]]]). Existence of centre manifolds has also been proved for certain infinite-dimensional dynamical systems defined by partial differential equations [[#References|[a9]]], [[#References|[a3]]], [[#References|[a6]]] and delay differential equations [[#References|[a5]]], [[#References|[a4]]].
 
This theorem reduces the study of bifurcations of non-hyperbolic equilibria (cf. also [[Bifurcation|Bifurcation]]) to those on the corresponding centre manifold of dimension equal to the number of critical eigenvalues. There are analogues of the reduction principle and Šošitaišvili's theorem for discrete-time dynamical systems defined by iterations of diffeomorphisms (see, for example, [[#References|[a1]]], [[#References|[a8]]]). Existence of centre manifolds has also been proved for certain infinite-dimensional dynamical systems defined by partial differential equations [[#References|[a9]]], [[#References|[a3]]], [[#References|[a6]]] and delay differential equations [[#References|[a5]]], [[#References|[a4]]].

Latest revision as of 16:43, 4 June 2020


Consider an autonomous system of ordinary differential equations

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

where $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $ is sufficiently smooth, $ f ( 0 ) = 0 $. Let the eigenvalues of the Jacobi matrix $ A $ evaluated at the equilibrium position $ x _ {0} = 0 $ be $ \lambda _ {1} \dots \lambda _ {n} $. Suppose the equilibrium is non-hyperbolic, i.e. has eigenvalues with zero real part. Assume also that there are $ n _ {u} $ eigenvalues (counting multiplicities) with $ { \mathop{\rm Re} } \lambda _ {j} > 0 $, $ n _ {c} $ eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} = 0 $, and $ n _ {s} $ eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} < 0 $. Let $ T ^ {c} $ denote the linear (generalized) eigenspace of $ A $ corresponding to the union of the $ n _ {c} $ eigenvalues on the imaginary axis. The eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} = 0 $ are often called critical, as is the eigenspace $ T ^ {c} $. Let $ \varphi ^ {t} $ denote the flow (continuous-time dynamical system) associated with (a1). Under the assumptions stated above, the following centre manifold theorem holds [a7], [a9], [a3], [a11]: There is a locally defined smooth $ n _ {c} $- dimensional invariant manifold $ W ^ {c} ( 0 ) $ of $ \varphi ^ {t} $ that is tangent to $ T ^ {c} $ at $ x = 0 $.

The manifold $ W ^ {c} ( 0 ) $ is called the centre manifold. The centre manifold $ W ^ {c} ( 0 ) $ need not be unique. If $ f \in C ^ {k} $ with finite $ k $, $ W ^ {c} ( 0 ) $ is a $ C ^ {k} $- manifold in some neighbourhood $ U $ of $ x _ {0} $. However, as $ k \rightarrow \infty $ the neighbourhood $ U $ may shrink, thus resulting in the non-existence of a $ C ^ \infty $- manifold $ W ^ {c} ( 0 ) $ for certain $ C ^ \infty $ systems.

In a basis formed by all (generalized) eigenvectors of $ A $( or their linear combinations if the corresponding eigenvalues are complex), the system (a1) can be written as

$$ \tag{a2 } \left \{ \begin{array}{l} { {\dot{u} } = Bu + g ( u,v ) , \ } \\ { {\dot{v} } = Cv + h ( u,v ) , \ } \end{array} \right . $$

where $ u \in \mathbf R ^ {n _ {c} } $, $ v \in \mathbf R ^ {n _ {u} + n _ {s} } $, $ B $ is an $ ( n _ {c} \times n _ {c} ) $- matrix with all its $ n _ {c} $ eigenvalues on the imaginary axis, while $ C $ is an $ ( ( n _ {u} + n _ {s} ) \times ( n _ {u} + n _ {s} ) ) $- matrix with no eigenvalue on the imaginary axis; $ g,h = O ( \| {( u,v ) } \| ^ {2} ) $. A centre manifold $ W ^ {c} $ of (a2) can be locally represented as the graph of a smooth function $ V : {\mathbf R ^ {n _ {c} } } \rightarrow {\mathbf R ^ {n _ {u} + n _ {s} } } $, $ V ( u ) = O ( \| u \| ^ {2} ) $:

$$ W ^ {c} = \left \{ {( u,v ) } : {v = V ( u ) , \left \| u \right \| < \varepsilon } \right \} . $$

The following reduction principle is valid (see [a1], [a8]): The system (a2) is locally topologically equivalent (cf. Equivalence of dynamical systems) near the origin to the system

$$ \tag{a3 } \left \{ \begin{array}{l} { {\dot{u} } = Bu + g ( u,V ( u ) ) , \ } \\ { {\dot{v} } = Cv. \ } \end{array} \right . $$

The equations for $ u $ and $ v $ are uncoupled in (a3). The first equation is the restriction of (a2) to its centre manifold. Thus, the dynamics of (a2) near a non-hyperbolic equilibrium are determined by this restriction, since the second equation in (a3) is linear and has exponentially decaying/growing solutions. For example, if $ u = 0 $ is the asymptotically stable equilibrium of the restriction and the matrix $ C $ has no eigenvalue with positive real part, then $ ( u,v ) = ( 0,0 ) $ is the asymptotically stable equilibrium of (a2). If there is more than one centre manifold, then all the resulting systems (a3) with different $ V $ are locally topologically equivalent (actually, the $ V $ differ only by flat functions).

The second equation in (a3) can be replaced by the standard saddle:

$$ \tag{a4 } \left \{ \begin{array}{l} { {\dot{v} } = - v \ } \\ { {\dot{w} } = w, \ } \end{array} \right . $$

with $ ( v,w ) \in \mathbf R ^ {n _ {s} } \times \mathbf R ^ {n _ {u} } $. In other words, near a non-hyperbolic equilibrium the system is locally topologically equivalent to the suspension of its restriction to the centre manifold by the standard saddle.

Consider now a system that depends smoothly on parameters:

$$ \tag{a5 } {\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {n} , \alpha \in \mathbf R ^ {m} . $$

Suppose that at $ \alpha = 0 $ the system has a non-hyperbolic equilibrium $ x = 0 $ with $ n _ {c} $ eigenvalues on the imaginary axis and $ ( n - n _ {c} ) $ eigenvalues with non-zero real part. Let $ n _ {s} $ of them have negative real part, while $ n _ {u} $ have positive real part. Applying the centre manifold theorem to the following extended system:

$$ \tag{a6 } \left \{ \begin{array}{l} { {\dot \alpha } = 0, \ } \\ { {\dot{x} } = f ( x, \alpha ) , \ } \end{array} \right . $$

one can prove the existence of a parameter-dependent local invariant manifold $ {\mathcal M} _ \alpha $ in (a5). The manifold has dimension $ n _ {c} $ and $ {\mathcal M} _ {0} $ coincides with a centre manifold $ W ^ {c} ( 0 ) $ of the (a5) at $ \alpha = 0 $. Often, the manifold $ {\mathcal M} _ \alpha $ is called the centre manifold for all $ \alpha $. For each small $ | \alpha | $ one can restrict system (a5) to $ {\mathcal M} _ \alpha $. Introducing a coordinate system on $ {\mathcal M} _ \alpha $ with $ u \in \mathbf R ^ {n _ {c} } $ as the coordinate, this restriction will be represented by a smooth system:

$$ \tag{a7 } {\dot{u} } = \Phi ( u, \alpha ) . $$

At $ \alpha = 0 $, the system (a7) is equivalent to the restriction of (a5) to its centre manifold $ W ^ {c} ( 0 ) $. The following results are known as the Šošitaišvili theorem [a10] (see also [a1], [a2], [a8]): The system (a5) is locally topologically equivalent near the origin to the suspension of (a7) by the standard saddle (a4). Moreover, (a7) can be replaced by any locally topologically equivalent system.

This theorem reduces the study of bifurcations of non-hyperbolic equilibria (cf. also Bifurcation) to those on the corresponding centre manifold of dimension equal to the number of critical eigenvalues. There are analogues of the reduction principle and Šošitaišvili's theorem for discrete-time dynamical systems defined by iterations of diffeomorphisms (see, for example, [a1], [a8]). Existence of centre manifolds has also been proved for certain infinite-dimensional dynamical systems defined by partial differential equations [a9], [a3], [a6] and delay differential equations [a5], [a4].

References

[a1] V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Grundlehren math. Wiss. , 250 , Springer (1983) (In Russian)
[a2] V.I. Arnol'd, V.S. Afraimovich, Yu.S. Il'yashenko, L.P. Shil'nikov, "Bifurcation theory" V.I. Arnol'd (ed.) , Dynamical Systems V , Encycl. Math. Sci. , Springer (1994) (In Russian) Zbl 0791.00009
[a3] J. Carr, "Applications of center manifold theory" , Springer (1981)
[a4] O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, "Delay equations" , Springer (1995) MR1345150 Zbl 0826.34002
[a5] J. Hale, S.M. Verduyn Lunel, "Introduction to functional differential equations" , Springer (1993) MR1243878 Zbl 0787.34002
[a6] D. Henry, "Geometric theory of semilinear parabolic equations" , Springer (1981) MR0610244 Zbl 0456.35001
[a7] A. Kelley, "The stable, center stable, center, center unstable and unstable manifolds" J. Diff. Eq. , 3 (1967) pp. 546–570 MR0221044
[a8] Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995) MR1344214 Zbl 0829.58029
[a9] J. Marsden, M. McCracken, "Hopf bifurcation and its applications" , Springer (1976) MR0494309 Zbl 0346.58007
[a10] A.N. Šošitaišvili, "Bifurcations of topological type of a vector field near a singular point" , Proc. Petrovskii Sem. , 1 , Moscow Univ. (1975) pp. 279–309 (In Russian)
[a11] A. Vanderbauwhede, "Centre manifolds, normal forms and elementary bifurcations" Dynamics Reported , 2 (1989) pp. 89–169 MR1000977 Zbl 0677.58001
How to Cite This Entry:
Centre manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre_manifold&oldid=46296
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article