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''of the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522801.png" /> of a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522802.png" />''
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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522803.png" /> which is the union of entire trajectories, that is, a set satisfying the condition
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{{TEX|auto}}
 +
{{TEX|done}}
  
<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/i/i052/i052280/i0522804.png" /></td> </tr></table>
+
''of the phase space  $  R $
 +
of a dynamical system  $  f ( p , t ) $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522805.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522806.png" /> under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522807.png" /> corresponding to a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522808.png" />.
+
A set  $  M $
 +
which is the union of entire trajectories, that is, a set satisfying the condition
  
The invariant set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522809.png" /> may possess a definite topological structure as a set of the metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228010.png" />; for example, it can be a topological or smooth manifold, a surface, a closed Jordan curve, or an isolated point. One then says that the invariant set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228011.png" /> is an invariant manifold, an invariant surface, an invariant curve, or an invariant point.
+
$$
 +
f ( M , t )  = M ,\  t \in \mathbf R ,
 +
$$
  
An invariant point is usually called a stationary point of the dynamical system, since for this point the motion reduces to rest: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228012.png" /> for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228013.png" />. A closed invariant curve not containing any invariant points of the dynamical system is always formed by the trajectory of a periodic motion, that is, a motion satisfying the condition
+
where  $  f ( M , t ) $
 +
is the image of $  M $
 +
under the transformation  $  p \mapsto f ( p , t ) $
 +
corresponding to a given  $  t $.
  
<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/i/i052/i052280/i05228014.png" /></td> </tr></table>
+
The invariant set  $  M $
 +
may possess a definite topological structure as a set of the metric space  $  R $;  
 +
for example, it can be a topological or smooth manifold, a surface, a closed Jordan curve, or an isolated point. One then says that the invariant set  $  M $
 +
is an invariant manifold, an invariant surface, an invariant curve, or an invariant point.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228015.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228016.png" />. For this reason it is called a periodic trajectory. Examples which occur as invariant manifolds are a sphere, a torus and a disc; invariant surfaces — a cone, a Möbius strip and a sphere with handles; invariant sets — the set of all stationary points, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228018.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228019.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228020.png" />-limit points of the motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228021.png" />, respectively, and also the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228022.png" /> of all wandering points or the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228023.png" /> of all non-wandering points (cf. [[Wandering point|Wandering point]]; [[Non-wandering point|Non-wandering point]]).
+
An invariant point is usually called a stationary point of the dynamical system, since for this point the motion reduces to rest:  $  f ( p , t) = p $
 +
for all values of  $  t $.
 +
A closed invariant curve not containing any invariant points of the dynamical system is always formed by the trajectory of a periodic motion, that is, a motion satisfying the condition
 +
 
 +
$$
 +
f ( p , t + T )  = f ( p , t )
 +
$$
 +
 
 +
for all  $  t \in \mathbf R $
 +
and some $  T > 0 $.  
 +
For this reason it is called a periodic trajectory. Examples which occur as invariant manifolds are a sphere, a torus and a disc; invariant surfaces — a cone, a Möbius strip and a sphere with handles; invariant sets — the set of all stationary points, the sets $  \Omega _ {p} $
 +
and $  A _ {p} $
 +
of all $  \omega $-  
 +
and $  \alpha $-
 +
limit points of the motion $  f ( p , t ) $,  
 +
respectively, and also the set $  W $
 +
of all wandering points or the set $  R \setminus  W $
 +
of all non-wandering points (cf. [[Wandering point|Wandering point]]; [[Non-wandering point|Non-wandering point]]).
  
 
An invariant point of the dynamical system in the plane
 
An invariant point of the dynamical system in the plane
  
<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/i/i052/i052280/i05228024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
 
 +
\frac{dx}{dt}
 +
  = f ( x , y ) ,\ \
 +
 
 +
\frac{dy}{dt}
 +
  = g ( x , y )
 +
$$
  
 
belongs to one of the following four types, according to the nature of the behaviour of the trajectories in a neighbourhood of it: a [[Node|node]], (Fig.a) a [[Focus|focus]], (Fig.b) a [[Saddle|saddle]], (Fig.c) or a [[Centre|centre]] (Fig.d).
 
belongs to one of the following four types, according to the nature of the behaviour of the trajectories in a neighbourhood of it: a [[Node|node]], (Fig.a) a [[Focus|focus]], (Fig.b) a [[Saddle|saddle]], (Fig.c) or a [[Centre|centre]] (Fig.d).
Line 37: Line 79:
 
Figure: i052280d
 
Figure: i052280d
  
The node and focus are asymptotically stable or unstable, the saddle is unstable, and the centre is stable (cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]). The Poincaré index of the node, centre and focus is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228025.png" />; that of the saddle is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228026.png" />.
+
The node and focus are asymptotically stable or unstable, the saddle is unstable, and the centre is stable (cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]). The Poincaré index of the node, centre and focus is equal to $  + 1 $;  
 +
that of the saddle is $  - 1 $.
  
 
In the case when the Jacobi matrix
 
In the case when the Jacobi matrix
  
<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/i/i052/i052280/i05228027.png" /></td> </tr></table>
+
$$
 +
J ( x , y )  = \
 +
\left (
 +
\begin{array}{cc}
 +
 
 +
\frac{\partial  f ( x , y ) }{\partial  x }
 +
  &
 +
\frac{\partial  f ( x , y ) }{\partial  y }
 +
  \\
 +
 
 +
\frac{\partial  g ( x , y ) }{\partial  x }
 +
  &
 +
\frac{\partial  g ( x , y ) }{\partial  y }
 +
  \\
 +
\end{array}
 +
\right )
 +
$$
  
of the right-hand side of (1) has at the stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228028.png" /> eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228030.png" /> with non-zero real parts, the invariant point is: a node if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228032.png" /> are real and of the same sign; a saddle if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228034.png" /> are real with different signs; or a focus if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228036.png" /> are complex conjugates.
+
of the right-hand side of (1) has at the stationary point $  x = x _ {0} , y = y _ {0} $
 +
eigen values $  \lambda _ {1} $,  
 +
$  \lambda _ {2} $
 +
with non-zero real parts, the invariant point is: a node if $  \lambda _ {1} $
 +
and $  \lambda _ {2} $
 +
are real and of the same sign; a saddle if $  \lambda _ {1} $
 +
and $  \lambda _ {2} $
 +
are real with different signs; or a focus if $  \lambda _ {1} $
 +
and $  \lambda _ {2} $
 +
are complex conjugates.
  
In all these cases, the type of the singular point of the system (1) is the same as for the linear system obtained from (1) by expanding its right-hand side in a Taylor series at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228037.png" />, that is, it has the same type as the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228038.png" /> of the system
+
In all these cases, the type of the singular point of the system (1) is the same as for the linear system obtained from (1) by expanding its right-hand side in a Taylor series at the point $  x = x _ {0} , y = y _ {0} $,  
 +
that is, it has the same type as the point $  x _ {1} = 0 , y _ {1} = 0 $
 +
of 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/i/i052/i052280/i05228039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
  
whose matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228040.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228041.png" />. There is a deeper relation between the trajectories of (1) in a neighbourhood of a singular point of one of the above types and the trajectories of (2) than that already mentioned. Namely, whenever the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228043.png" /> are analytic in a neighbourhood of the invariant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228044.png" /> and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228045.png" /> has eigen values with non-zero real parts, there exists a continuously-differentiable change of variables in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228046.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228047.png" />,
+
\frac{d x _ {1} }{dt}
 +
  = a x _ {1} + b y _ {1} ,\ \
  
<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/i/i052/i052280/i05228048.png" /></td> </tr></table>
+
\frac{d y _ {1} }{dt}
 +
  = c x _ {1} + d y _ {1} ,
 +
$$
  
<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/i/i052/i052280/i05228049.png" /></td> </tr></table>
+
whose matrix  $  ( _ {c}  ^ {a} {} _ {d}  ^ {b} ) $
 +
is equal to  $  J ( x _ {0} , y _ {0} ) $.
 +
There is a deeper relation between the trajectories of (1) in a neighbourhood of a singular point of one of the above types and the trajectories of (2) than that already mentioned. Namely, whenever the functions  $  f $
 +
and  $  g $
 +
are analytic in a neighbourhood of the invariant point  $  x = x _ {0} = 0 , y = y _ {0} = 0 $
 +
and the matrix  $  J ( x _ {0,\ } y _ {0} ) $
 +
has eigen values with non-zero real parts, there exists a continuously-differentiable change of variables in a neighbourhood  $  U $
 +
of the point  $  x = 0 , y = 0 $,
 +
 
 +
$$
 +
x _ {1}  =  x + F _ {1} ( x , y ) ,\ \
 +
y _ {1}  = y + F _ {2} ( x , y ) ,
 +
$$
 +
 
 +
$$
 +
F _ {i} ( 0 , 0 )  =
 +
\frac{\partial  F _ {i} ( 0 , 0 ) }{\partial
 +
x }
 +
  =
 +
\frac{\partial  F _ {i} ( 0 , 0 ) }{\partial  y }
 +
  =  0 ,\  i = 1 , 2 ,
 +
$$
  
 
that reduces the system (1) to the system (2).
 
that reduces the system (1) to the system (2).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228051.png" /> are imaginary, then the invariant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228052.png" /> can either be a focus or a centre. In this case, the question of the classification of the type of the singular point is a separate and difficult problem, namely the [[Centre and focus problem|centre and focus problem]], and requires more refined criteria for distinguishing between the centre and the focus (see [[#References|[1]]], [[#References|[7]]]). Similar difficulties occur in the determination of the type of the singular point in the case when the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228053.png" /> is singular.
+
If $  \lambda _ {1} $,  
 +
$  \lambda _ {2} $
 +
are imaginary, then the invariant point $  x _ {0} , y _ {0} $
 +
can either be a focus or a centre. In this case, the question of the classification of the type of the singular point is a separate and difficult problem, namely the [[Centre and focus problem|centre and focus problem]], and requires more refined criteria for distinguishing between the centre and the focus (see [[#References|[1]]], [[#References|[7]]]). Similar difficulties occur in the determination of the type of the singular point in the case when the matrix $  J ( x _ {0} , y _ {0} ) $
 +
is singular.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) {{MR|0121520}} {{ZBL|0089.29502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927) {{MR|1555257}} {{ZBL|53.0733.03}} {{ZBL|53.0732.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) {{MR|0087811}} {{ZBL|0070.30603}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) {{MR|0658490}} {{ZBL|0476.34002}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.G. Malkin, "Theorie der Stabilität einer Bewegung" , R. Oldenbourg , München (1959) (Translated from Russian) {{MR|0104029}} {{ZBL|0124.30003}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.M. Lyapunov, "Collected works" , '''2''' , Moscow-Leningrad (1956) (In Russian) {{MR|0121296}} {{MR|0106812}} {{MR|0086006}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> K.S. Sibirskii, "Algebraic invariance of differential equations and matrices" , Kishinev (1976) (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) {{MR|0121520}} {{ZBL|0089.29502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927) {{MR|1555257}} {{ZBL|53.0733.03}} {{ZBL|53.0732.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) {{MR|0087811}} {{ZBL|0070.30603}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) {{MR|0658490}} {{ZBL|0476.34002}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.G. Malkin, "Theorie der Stabilität einer Bewegung" , R. Oldenbourg , München (1959) (Translated from Russian) {{MR|0104029}} {{ZBL|0124.30003}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.M. Lyapunov, "Collected works" , '''2''' , Moscow-Leningrad (1956) (In Russian) {{MR|0121296}} {{MR|0106812}} {{MR|0086006}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> K.S. Sibirskii, "Algebraic invariance of differential equations and matrices" , Kishinev (1976) (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The reduction of the system (1) to the system (2) in a neighbourhood of a singular point is often called linearization of the equation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228055.png" /> have non-zero real parts (in that case the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228056.png" /> is said to be of hyperbolic type) such a linearization can always be performed by means of a topological (non-smooth) coordinate transformation (i.e., the transformation and its converse are merely continuous), provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228058.png" /> in (1) are of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228059.png" />. This result holds also for autonomous differential equations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228060.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228061.png" />. If certain "non-resonance" conditions between the eigen values are fulfilled, then one can also obtain smooth (under additional conditions even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228062.png" /> or analytic) transformations (work of H. Poincaré, C.L. Siegel, S. Sternberg). See [[#References|[4]]], pp. 256-271 and the references given in [[#References|[4]]] on pp. 271-272. These results are related to the theory of normal forms for differential equations (cf. [[Normal form|Normal form]]) and involve problems on [[Small denominators|small denominators]].
+
The reduction of the system (1) to the system (2) in a neighbourhood of a singular point is often called linearization of the equation. If $  \lambda _ {1} $
 +
and $  \lambda _ {2} $
 +
have non-zero real parts (in that case the singular point $  ( x _ {0} , y _ {0} ) $
 +
is said to be of hyperbolic type) such a linearization can always be performed by means of a topological (non-smooth) coordinate transformation (i.e., the transformation and its converse are merely continuous), provided $  f $
 +
and $  g $
 +
in (1) are of class $  C  ^ {1} $.  
 +
This result holds also for autonomous differential equations in $  \mathbf R  ^ {n} $
 +
for $  n \geq  2 $.  
 +
If certain "non-resonance" conditions between the eigen values are fulfilled, then one can also obtain smooth (under additional conditions even $  C  ^  \infty  $
 +
or analytic) transformations (work of H. Poincaré, C.L. Siegel, S. Sternberg). See [[#References|[4]]], pp. 256-271 and the references given in [[#References|[4]]] on pp. 271-272. These results are related to the theory of normal forms for differential equations (cf. [[Normal form|Normal form]]) and involve problems on [[Small denominators|small denominators]].
  
 
For a description of the Poincaré index see [[Singular point|Singular point]].
 
For a description of the Poincaré index see [[Singular point|Singular point]].

Latest revision as of 22:13, 5 June 2020


of the phase space $ R $ of a dynamical system $ f ( p , t ) $

A set $ M $ which is the union of entire trajectories, that is, a set satisfying the condition

$$ f ( M , t ) = M ,\ t \in \mathbf R , $$

where $ f ( M , t ) $ is the image of $ M $ under the transformation $ p \mapsto f ( p , t ) $ corresponding to a given $ t $.

The invariant set $ M $ may possess a definite topological structure as a set of the metric space $ R $; for example, it can be a topological or smooth manifold, a surface, a closed Jordan curve, or an isolated point. One then says that the invariant set $ M $ is an invariant manifold, an invariant surface, an invariant curve, or an invariant point.

An invariant point is usually called a stationary point of the dynamical system, since for this point the motion reduces to rest: $ f ( p , t) = p $ for all values of $ t $. A closed invariant curve not containing any invariant points of the dynamical system is always formed by the trajectory of a periodic motion, that is, a motion satisfying the condition

$$ f ( p , t + T ) = f ( p , t ) $$

for all $ t \in \mathbf R $ and some $ T > 0 $. For this reason it is called a periodic trajectory. Examples which occur as invariant manifolds are a sphere, a torus and a disc; invariant surfaces — a cone, a Möbius strip and a sphere with handles; invariant sets — the set of all stationary points, the sets $ \Omega _ {p} $ and $ A _ {p} $ of all $ \omega $- and $ \alpha $- limit points of the motion $ f ( p , t ) $, respectively, and also the set $ W $ of all wandering points or the set $ R \setminus W $ of all non-wandering points (cf. Wandering point; Non-wandering point).

An invariant point of the dynamical system in the plane

$$ \tag{1 } \frac{dx}{dt} = f ( x , y ) ,\ \ \frac{dy}{dt} = g ( x , y ) $$

belongs to one of the following four types, according to the nature of the behaviour of the trajectories in a neighbourhood of it: a node, (Fig.a) a focus, (Fig.b) a saddle, (Fig.c) or a centre (Fig.d).

Figure: i052280a

Figure: i052280b

Figure: i052280c

Figure: i052280d

The node and focus are asymptotically stable or unstable, the saddle is unstable, and the centre is stable (cf. Asymptotically-stable solution). The Poincaré index of the node, centre and focus is equal to $ + 1 $; that of the saddle is $ - 1 $.

In the case when the Jacobi matrix

$$ J ( x , y ) = \ \left ( \begin{array}{cc} \frac{\partial f ( x , y ) }{\partial x } & \frac{\partial f ( x , y ) }{\partial y } \\ \frac{\partial g ( x , y ) }{\partial x } & \frac{\partial g ( x , y ) }{\partial y } \\ \end{array} \right ) $$

of the right-hand side of (1) has at the stationary point $ x = x _ {0} , y = y _ {0} $ eigen values $ \lambda _ {1} $, $ \lambda _ {2} $ with non-zero real parts, the invariant point is: a node if $ \lambda _ {1} $ and $ \lambda _ {2} $ are real and of the same sign; a saddle if $ \lambda _ {1} $ and $ \lambda _ {2} $ are real with different signs; or a focus if $ \lambda _ {1} $ and $ \lambda _ {2} $ are complex conjugates.

In all these cases, the type of the singular point of the system (1) is the same as for the linear system obtained from (1) by expanding its right-hand side in a Taylor series at the point $ x = x _ {0} , y = y _ {0} $, that is, it has the same type as the point $ x _ {1} = 0 , y _ {1} = 0 $ of the system

$$ \tag{2 } \frac{d x _ {1} }{dt} = a x _ {1} + b y _ {1} ,\ \ \frac{d y _ {1} }{dt} = c x _ {1} + d y _ {1} , $$

whose matrix $ ( _ {c} ^ {a} {} _ {d} ^ {b} ) $ is equal to $ J ( x _ {0} , y _ {0} ) $. There is a deeper relation between the trajectories of (1) in a neighbourhood of a singular point of one of the above types and the trajectories of (2) than that already mentioned. Namely, whenever the functions $ f $ and $ g $ are analytic in a neighbourhood of the invariant point $ x = x _ {0} = 0 , y = y _ {0} = 0 $ and the matrix $ J ( x _ {0,\ } y _ {0} ) $ has eigen values with non-zero real parts, there exists a continuously-differentiable change of variables in a neighbourhood $ U $ of the point $ x = 0 , y = 0 $,

$$ x _ {1} = x + F _ {1} ( x , y ) ,\ \ y _ {1} = y + F _ {2} ( x , y ) , $$

$$ F _ {i} ( 0 , 0 ) = \frac{\partial F _ {i} ( 0 , 0 ) }{\partial x } = \frac{\partial F _ {i} ( 0 , 0 ) }{\partial y } = 0 ,\ i = 1 , 2 , $$

that reduces the system (1) to the system (2).

If $ \lambda _ {1} $, $ \lambda _ {2} $ are imaginary, then the invariant point $ x _ {0} , y _ {0} $ can either be a focus or a centre. In this case, the question of the classification of the type of the singular point is a separate and difficult problem, namely the centre and focus problem, and requires more refined criteria for distinguishing between the centre and the focus (see [1], [7]). Similar difficulties occur in the determination of the type of the singular point in the case when the matrix $ J ( x _ {0} , y _ {0} ) $ is singular.

References

[1] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502
[2] G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927) MR1555257 Zbl 53.0733.03 Zbl 53.0732.01
[3] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) MR0087811 Zbl 0070.30603
[4] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) MR0658490 Zbl 0476.34002
[5] I.G. Malkin, "Theorie der Stabilität einer Bewegung" , R. Oldenbourg , München (1959) (Translated from Russian) MR0104029 Zbl 0124.30003
[6] A.M. Lyapunov, "Collected works" , 2 , Moscow-Leningrad (1956) (In Russian) MR0121296 MR0106812 MR0086006
[7] K.S. Sibirskii, "Algebraic invariance of differential equations and matrices" , Kishinev (1976) (In Russian)

Comments

The reduction of the system (1) to the system (2) in a neighbourhood of a singular point is often called linearization of the equation. If $ \lambda _ {1} $ and $ \lambda _ {2} $ have non-zero real parts (in that case the singular point $ ( x _ {0} , y _ {0} ) $ is said to be of hyperbolic type) such a linearization can always be performed by means of a topological (non-smooth) coordinate transformation (i.e., the transformation and its converse are merely continuous), provided $ f $ and $ g $ in (1) are of class $ C ^ {1} $. This result holds also for autonomous differential equations in $ \mathbf R ^ {n} $ for $ n \geq 2 $. If certain "non-resonance" conditions between the eigen values are fulfilled, then one can also obtain smooth (under additional conditions even $ C ^ \infty $ or analytic) transformations (work of H. Poincaré, C.L. Siegel, S. Sternberg). See [4], pp. 256-271 and the references given in [4] on pp. 271-272. These results are related to the theory of normal forms for differential equations (cf. Normal form) and involve problems on small denominators.

For a description of the Poincaré index see Singular point.

How to Cite This Entry:
Invariant set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_set&oldid=24478
This article was adapted from an original article by A.M. Samoilenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article