Namespaces
Variants
Actions

Difference between revisions of "Autonomous system"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
a0141901.png
 +
$#A+1 = 128 n = 1
 +
$#C+1 = 128 : ~/encyclopedia/old_files/data/A014/A.0104190 Autonomous system
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''of ordinary differential equations''
 
''of ordinary differential equations''
  
A system of ordinary differential equations which does not explicitly contain the independent variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a0141901.png" /> (time). The general form of a first-order autonomous system in normal form is:
+
A system of ordinary differential equations which does not explicitly contain the independent variable $  t $(
 +
time). The general form of a first-order autonomous system in normal form is:
  
<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/a/a014/a014190/a0141902.png" /></td> </tr></table>
+
$$
 +
\dot{x} _ {j}  = f _ {j} ( x _ {1} \dots x _ {n} ) ,
 +
\  j = 1 \dots n,
 +
$$
  
 
or, in vector notation,
 
or, in vector notation,
  
<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/a/a014/a014190/a0141903.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{x}  = f ( x ) .
 +
$$
  
A non-autonomous system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a0141904.png" /> can be reduced to an autonomous one by introducing a new unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a0141905.png" />. Historically, autonomous systems first appeared in descriptions of physical processes with a finite number of degrees of freedom. They are also called dynamical or conservative systems (cf. [[Dynamical system|Dynamical system]]).
+
A non-autonomous system $  \dot{x} = f(t, x) $
 +
can be reduced to an autonomous one by introducing a new unknown function $  x _ {n+1} = t $.  
 +
Historically, autonomous systems first appeared in descriptions of physical processes with a finite number of degrees of freedom. They are also called dynamical or conservative systems (cf. [[Dynamical system|Dynamical system]]).
  
A complex autonomous system of the form (1) is equivalent to a real autonomous system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a0141906.png" /> unknown functions
+
A complex autonomous system of the form (1) is equivalent to a real autonomous system with $  2n $
 +
unknown functions
  
<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/a/a014/a014190/a0141907.png" /></td> </tr></table>
+
$$
  
The essential contents of the theory of complex autonomous systems — unlike in the real case — is found in the case of an analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a0141908.png" /> (cf. [[Analytic theory of differential equations|Analytic theory of differential equations]]).
+
\frac{d}{dt}
 +
(  \mathop{\rm Re}  x )  =   \mathop{\rm Re}  f ( x ) ,
 +
 +
\frac{d}{dt}
 +
(  \mathop{\rm Im}  x )  =   \mathop{\rm Im}  f ( x ).
 +
$$
  
Consider an analytic system with real coefficients and its real solutions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a0141909.png" /> be an (arbitrary) solution of the analytic system (1), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419010.png" /> be the interval in which it is defined, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419011.png" /> be the solution with initial data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419013.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419015.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419016.png" /> is said to be an equilibrium point, or a point of rest, of the autonomous system (1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419017.png" />. The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419019.png" />, corresponds to such an equilibrium point.
+
The essential contents of the theory of complex autonomous systems — unlike in the real case — is found in the case of an analytic  $  f(x) $(
 +
cf. [[Analytic theory of differential equations|Analytic theory of differential equations]]).
 +
 
 +
Consider an analytic system with real coefficients and its real solutions. Let $  x = \phi (t) $
 +
be an (arbitrary) solution of the analytic system (1), let $  \Delta = ( t _ {-} , t _ {+} ) $
 +
be the interval in which it is defined, and let $  x(t;  t _ {0} , x  ^ {0} ) $
 +
be the solution with initial data $  x \mid  _ {t = t _ {0}  } = x  ^ {0} $.  
 +
Let $  G $
 +
be a domain in $  \mathbf R  ^ {n} $
 +
and $  f \in C  ^ {1} (G) $.  
 +
The point $  x  ^ {0} \in G $
 +
is said to be an equilibrium point, or a point of rest, of the autonomous system (1) if $  f ( x  ^ {0} ) \equiv 0 $.  
 +
The solution $  \phi (t) \equiv x  ^ {0} $,
 +
$  t \in \mathbf R = ( - \infty , + \infty ) $,  
 +
corresponds to such an equilibrium point.
  
 
==Local properties of solutions.==
 
==Local properties of solutions.==
  
 +
1) If  $  \phi (t) $
 +
is a solution, then  $  \phi ( t + c ) $
 +
is a solution for any  $  c \in \mathbf R $.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419020.png" /> is a solution, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419021.png" /> is a solution for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419022.png" />.
+
2) Existence: For any  $  t _ {0} \in \mathbf R , x  ^ {0} \in G $,
 +
a solution $  x(t;  t _ {0} , x  ^ {0} ) $
 +
exists in a certain interval  $  \Delta \ni t $.
  
2) Existence: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419023.png" />, a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419024.png" /> exists in a certain interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419025.png" />.
+
3) Smoothness: If  $  f \in C  ^ {p} (G) , p \geq  1 $,  
 +
then  $  \phi (t) \in C  ^ {p+1} ( \Delta ) $.
  
3) Smoothness: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419027.png" />.
+
4) Dependence on parameters: Let  $  f = f(x, \alpha ) $,
 +
$  \alpha \in G _  \alpha  \subset  \mathbf R  ^ {m} $,
 +
where  $  G _  \alpha  $
 +
is a domain; if  $  f \in C  ^ {p} (G \times G _  \alpha  ) $,  
 +
$  p \geq  1 $,
 +
then $  x(t;  t _ {0} , x  ^ {0} , \alpha ) \in C  ^ {p} ( \Delta \times G _  \alpha  ) $(
 +
for more details see [[#References|[1]]]–[[#References|[4]]]).
  
4) Dependence on parameters: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419030.png" /> is a domain; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419033.png" /> (for more details see [[#References|[1]]]–[[#References|[4]]]).
+
5) Let $  x  ^ {0} $
 +
be a non-equilibrium point; then there exist neighbourhoods  $  V, W $
 +
of the points  $  x  ^ {0} , f( x  ^ {0} ) $,  
 +
respectively, and a [[Diffeomorphism|diffeomorphism]] $  y = h(x) :  V \rightarrow W $
 +
such that the autonomous system has the form  $  \dot{y} = \textrm{ const } $
 +
in  $  W $.
  
5) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419034.png" /> be a non-equilibrium point; then there exist neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419035.png" /> of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419036.png" />, respectively, and a [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419037.png" /> such that the autonomous system has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419039.png" />.
+
A substitution of variables  $  x = \phi (y) $
 +
in the autonomous system (1) yields the system
  
A substitution of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419040.png" /> in the autonomous system (1) yields the system
+
$$ \tag{2 }
 +
\dot{y}  = ( \phi  ^  \prime  ( y ) )  ^ {-1} f ( \phi ( y ) ),
 +
$$
  
<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/a/a014/a014190/a01419041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
where  $  \phi  ^  \prime  (y) $
 
+
is the [[Jacobi matrix|Jacobi matrix]].
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419042.png" /> is the [[Jacobi matrix|Jacobi matrix]].
 
  
 
==Global properties of solutions.==
 
==Global properties of solutions.==
  
 
+
1) Any solution $  x = \phi (t) $
1) Any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419043.png" /> of the autonomous system (1) may be extended to an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419045.png" />, the solution is said to be unboundedly extendable; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419046.png" />, the solution is said to be unboundedly extendable forwards in time (and, in a similar manner, backwards in time). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419047.png" /> then, for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419049.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419050.png" /> such that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419051.png" /> is outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419052.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419053.png" /> (and, analogously, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419054.png" />; cf. [[Prolongation of solutions of differential equations|Prolongation of solutions of differential equations]]).
+
of the autonomous system (1) may be extended to an interval $  \Delta = (t _ {-} , t _ {+} ) $.  
 +
If $  \Delta = \mathbf R $,  
 +
the solution is said to be unboundedly extendable; if $  t _ {+} = + \infty , t _ {-} > - \infty $,  
 +
the solution is said to be unboundedly extendable forwards in time (and, in a similar manner, backwards in time). If $  t _ {+} < + \infty $
 +
then, for any compact set $  K \subset  \Omega $,  
 +
$  x  ^ {0} \in K $,  
 +
there exists a $  \tau = \tau (K) < t _ {+} $
 +
such that the point $  x(t;  t _ {0} , x  ^ {0} ) $
 +
is outside $  K $
 +
for $  t > \tau (K) $(
 +
and, analogously, for $  t _ {-} > - \infty $;  
 +
cf. [[Prolongation of solutions of differential equations|Prolongation of solutions of differential equations]]).
  
 
2) The extension is unique in the sense that any two solutions with common initial data are identical throughout their range of definition.
 
2) The extension is unique in the sense that any two solutions with common initial data are identical throughout their range of definition.
  
3) Any solution of an autonomous system belongs to one of the following three types: a) aperiodic, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419055.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419057.png" />; b) periodic, non-constant; or c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419058.png" />.
+
3) Any solution of an autonomous system belongs to one of the following three types: a) aperiodic, with $  \phi (t _ {1} ) \neq \phi ( t _ {2} ) $
 +
for all $  t _ {1} \neq t _ {2} $,  
 +
$  t _ {j} \in \mathbf R $;  
 +
b) periodic, non-constant; or c) $  \phi (t) \equiv \textrm{ const } $.
  
 
==Geometric interpretation of an autonomous system.==
 
==Geometric interpretation of an autonomous system.==
To each solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419059.png" /> is assigned a corresponding curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419060.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419062.png" />, inside the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419063.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419064.png" /> is then said to be the [[Phase space|phase space]] of the autonomous system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419065.png" /> is its trajectory in the phase space, and the solution is interpreted as motion along this trajectory in the phase space. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419066.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419067.png" /> (i.e. each point moves along the phase trajectory during time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419068.png" />) is called the phase flow. In its domain of definition the phase flow satisfies the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419069.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419070.png" />; and 2) the group property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419071.png" />.
+
To each solution $  x = \phi (t) $
 +
is assigned a corresponding curve $  \Gamma $:  
 +
$  x = \phi (t) $,  
 +
$  t \in \Delta $,  
 +
inside the domain $  G $.  
 +
$  G $
 +
is then said to be the [[Phase space|phase space]] of the autonomous system, $  \Gamma $
 +
is its trajectory in the phase space, and the solution is interpreted as motion along this trajectory in the phase space. The mapping $  g  ^ {t} : G \rightarrow G $
 +
defined by the formula $  g  ^ {t} x  ^ {0} = x (t;  0, x  ^ {0} ) $(
 +
i.e. each point moves along the phase trajectory during time $  t $)  
 +
is called the phase flow. In its domain of definition the phase flow satisfies the following conditions: 1) $  g  ^ {t} x $
 +
is continuous in $  (t, x) $;  
 +
and 2) the group property $  g ^ {t _ {1} + t _ {2} } x = g ^ {t _ {1} } g ^ {t _ {2} } x $.
  
The Liouville theorem is valid: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419072.png" /> be a domain with a finite volume and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419073.png" /> be the volume of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419074.png" />, then
+
The Liouville theorem is valid: Let $  D \subset  G $
 +
be a domain with a finite volume and let $  v _ {t} $
 +
be the volume of the domain $  g  ^ {t} D \subset  G $,  
 +
then
  
<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/a/a014/a014190/a01419075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\left .
 +
\frac{d v _ {t} }{dt}
 +
\right | _ {t = 0 }
 +
= \int\limits _ { D }  \mathop{\rm div}  f ( x ) dx .
 +
$$
  
For a Hamiltonian system, a consequence of (3) is the conservation of the phase volume by the phase flow. A second variant of (3) is obtained as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419076.png" /> be a family of solutions of (1), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419077.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419078.png" /> be a domain and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419079.png" />, then
+
For a Hamiltonian system, a consequence of (3) is the conservation of the phase volume by the phase flow. A second variant of (3) is obtained as follows. Let $  x = \phi (t, \alpha ) $
 +
be a family of solutions of (1), $  \alpha = ( \alpha _ {1} \dots \alpha _ {n-1} ) \in G _  \alpha  $,  
 +
let $  G _  \alpha  $
 +
be a domain and let $  \phi \in C  ^ {1} ( \Delta \times G _  \alpha  ) $,  
 +
then
  
<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/a/a014/a014190/a01419080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3prm)</td></tr></table>
+
$$ \tag{3\prime }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419081.png" />.
+
\frac{d}{dt}
 +
  \mathop{\rm ln}  I ( t , \alpha )  =  \mathop{\rm div}  f ( x ) ,
 +
$$
 +
 
 +
where $  I(t, \alpha ) = \mathop{\rm det}  \partial  x / \partial  (t, \alpha ) $.
  
 
==Structure of phase trajectories.==
 
==Structure of phase trajectories.==
 
  
 
1) Any two phase trajectories have either no points in common or coincide.
 
1) Any two phase trajectories have either no points in common or coincide.
  
2) Any phase trajectory belongs to one of the following types: a) a smooth, simple, non-closed Jordan arc; b) a cycle, i.e. a curve diffeomorphic to a circle; or c) a point (an equilibrium point). The local structure of phase trajectories in a small neighbourhood of a point other than an equilibrium point is trivial (cf. local property 5) of the solutions): The family of phase trajectories is diffeomorphic to a family of parallel straight lines. For a linear autonomous system the structure of phase trajectories in a neighbourhood of an equilibrium point is known, since the autonomous system is integrable [[#References|[5]]]. For non-linear autonomous systems this problem has not yet been completely solved, even for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419082.png" /> (cf. [[Qualitative theory of differential equations|Qualitative theory of differential equations]]). One aspect of this problem is the question of stability of an equilibrium point (cf. [[Stability theory|Stability theory]]). A few results will be given below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419083.png" /> be equilibrium points of the system (1), let
+
2) Any phase trajectory belongs to one of the following types: a) a smooth, simple, non-closed Jordan arc; b) a cycle, i.e. a curve diffeomorphic to a circle; or c) a point (an equilibrium point). The local structure of phase trajectories in a small neighbourhood of a point other than an equilibrium point is trivial (cf. local property 5) of the solutions): The family of phase trajectories is diffeomorphic to a family of parallel straight lines. For a linear autonomous system the structure of phase trajectories in a neighbourhood of an equilibrium point is known, since the autonomous system is integrable [[#References|[5]]]. For non-linear autonomous systems this problem has not yet been completely solved, even for $  n = 2 $(
 +
cf. [[Qualitative theory of differential equations|Qualitative theory of differential equations]]). One aspect of this problem is the question of stability of an equilibrium point (cf. [[Stability theory|Stability theory]]). A few results will be given below. Let $  x  ^ {0} , y  ^ {0} $
 +
be equilibrium points of the system (1), let
  
<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/a/a014/a014190/a01419084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
+
$$ \tag{1\prime }
 +
\dot{y}  = g ( y )
 +
$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419085.png" /> be neighbourhoods of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419086.png" />. The systems (1) and (1prm) are said to be equivalent in neighbourhoods of their equilibrium points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419087.png" /> if there exist neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419088.png" /> and a bijective mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419089.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419090.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419093.png" />), i.e. as a result of the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419094.png" /> the trajectories of the autonomous system (1) go into trajectories of the autonomous system (1prm). The equivalence is said to be differentiable (topological) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419095.png" /> is a diffeomorphism (homeomorphism). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419096.png" /> be an equilibrium point of the autonomous system (1), let the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419097.png" /> be non-degenerate, and let it not possess any pure imaginary eigen values. Then the autonomous system (1) in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419098.png" /> is topologically equivalent to its linear part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a01419099.png" />. An important example is the autonomous system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190100.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190101.png" /> are constant matrices with pure imaginary eigen values and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190102.png" />; it is not known when these autonomous systems are topologically equivalent. One of the most fundamental problems in the theory of autonomous systems is that of the structure of the entire family of phase trajectories. The most complete results have been obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190103.png" />, but even in this case the solution is far from complete.
+
and let $  U, V $
 +
be neighbourhoods of the points $  x  ^ {0} , y  ^ {0} $.  
 +
The systems (1) and (1prm) are said to be equivalent in neighbourhoods of their equilibrium points $  x  ^ {0} , y  ^ {0} $
 +
if there exist neighbourhoods $  U, V $
 +
and a bijective mapping $  h: U \rightarrow V $
 +
such that $  ( h \circ f  ^ {t} ) x = (g  ^ {t} \circ h ) x $(
 +
for $  x \in U $,  
 +
$  f  ^ {t} x \in U $,  
 +
$  ( g  ^ {t} \circ h)x \in V $),  
 +
i.e. as a result of the substitution $  y = h(x) $
 +
the trajectories of the autonomous system (1) go into trajectories of the autonomous system (1prm). The equivalence is said to be differentiable (topological) if $  h $
 +
is a diffeomorphism (homeomorphism). Let $  x  ^ {0} $
 +
be an equilibrium point of the autonomous system (1), let the matrix $  f ^ { \prime } (x  ^ {0} ) $
 +
be non-degenerate, and let it not possess any pure imaginary eigen values. Then the autonomous system (1) in a neighbourhood of $  x  ^ {0} $
 +
is topologically equivalent to its linear part $  \dot{y} = f ^ { \prime } (x  ^ {0} ) y $.  
 +
An important example is the autonomous system $  \dot{x} = Ax , \dot{y} = By $
 +
where $  A, B $
 +
are constant matrices with pure imaginary eigen values and $  n > 2 $;  
 +
it is not known when these autonomous systems are topologically equivalent. One of the most fundamental problems in the theory of autonomous systems is that of the structure of the entire family of phase trajectories. The most complete results have been obtained for $  n = 2 $,  
 +
but even in this case the solution is far from complete.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.G. Petrovskii,  "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.G. Petrovskii,  "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The question of the topological equivalence (homeomorphic equivalence of orbit systems) of two linear dynamical systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190105.png" /> has been examined and solved [[#References|[a2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190106.png" /> be the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190107.png" /> corresponding to (generalized) eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190108.png" /> with positive, negative and zero real parts respectively, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190109.png" /> be the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190110.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190111.png" />. Then the orbit systems of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190113.png" /> are homeomorphic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190117.png" /> are linearly equivalent (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190118.png" /> for some constant invertible matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190119.png" />).
+
The question of the topological equivalence (homeomorphic equivalence of orbit systems) of two linear dynamical systems $  \dot{x} = Ax $
 +
and $  \dot{y} = By $
 +
has been examined and solved [[#References|[a2]]]. Let $  V _ {A}  ^ {+} , V _ {A}  ^ {-} , V _ {A}  ^ {0} $
 +
be the decomposition of $  \mathbf R  ^ {n} $
 +
corresponding to (generalized) eigen values of $  A $
 +
with positive, negative and zero real parts respectively, and let $  A  ^ {0} $
 +
be the restriction of $  A $
 +
to $  V _ {A}  ^ {0} $.  
 +
Then the orbit systems of $  \dot{x} = Ax $
 +
and $  \dot{y} = By $
 +
are homeomorphic if $  \mathop{\rm dim}  V _ {A}  ^ {+} = \mathop{\rm dim}  V _ {B}  ^ {+} $,  
 +
$  \mathop{\rm dim}  V _ {A}  ^ {-} = \mathop{\rm dim}  V _ {B}  ^ {-} $
 +
and $  A  ^ {0} $
 +
and $  B  ^ {0} $
 +
are linearly equivalent (i.e. $  A  ^ {0} = S  ^ {-1} B  ^ {0} S $
 +
for some constant invertible matrix $  S $).
  
The analogous question for discrete dynamical systems: when does there exist for two linear endomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190120.png" /> a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190121.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190122.png" />, is much deeper (topological similarity of matrices) and has to do with such things as lens spaces and the P.A. Smith conjecture. See [[#References|[a1]]] for a full account. A theorem of G. de Rham says that if the orthogonal matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190124.png" /> are topologically similar and the topological (non-linear) similarity preserves the unit sphere and restricts to a diffeomorphism in it, then the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190126.png" /> are also linearly similar. Results of S.E. Cappell and J.L. Shaneson imply that for matrices of eigen values of absolute value 1 (the crucial case) topological and linear equivalence are the same for dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190127.png" />. Combined with results of Kuiper–Robbin [[#References|[a3]]] this gives a complete topological classification of matrices of dimension at most 5. In higher dimensions it is definitely not true in general that for matrices of eigen values of modulus 1 topological similarity implies linear similarity [[#References|[a1]]]. These results can be used to tell at what times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190128.png" /> the phase flows for continuous-time systems become topologically equivalent. This can happen repeatedly even for systems that are not topologically equivalent.
+
The analogous question for discrete dynamical systems: when does there exist for two linear endomorphisms $  A, B: \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $
 +
a homeomorphism $  \phi $
 +
such that $  A \phi = \phi B $,  
 +
is much deeper (topological similarity of matrices) and has to do with such things as lens spaces and the P.A. Smith conjecture. See [[#References|[a1]]] for a full account. A theorem of G. de Rham says that if the orthogonal matrices $  A $
 +
and $  B $
 +
are topologically similar and the topological (non-linear) similarity preserves the unit sphere and restricts to a diffeomorphism in it, then the matrices $  A $
 +
and $  B $
 +
are also linearly similar. Results of S.E. Cappell and J.L. Shaneson imply that for matrices of eigen values of absolute value 1 (the crucial case) topological and linear equivalence are the same for dimensions $  \leq  5 $.  
 +
Combined with results of Kuiper–Robbin [[#References|[a3]]] this gives a complete topological classification of matrices of dimension at most 5. In higher dimensions it is definitely not true in general that for matrices of eigen values of modulus 1 topological similarity implies linear similarity [[#References|[a1]]]. These results can be used to tell at what times $  t $
 +
the phase flows for continuous-time systems become topologically equivalent. This can happen repeatedly even for systems that are not topologically equivalent.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.E. Cappell,  J.L. Shaneson,  "Non-linear similarity"  ''Ann. of Math.'' , '''113'''  (1981)  pp. 315–355</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.H. Kuiper,  "The topology of the solutions of a linear differential equation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190129.png" />" , ''Proc. Internat. Congress on Manifolds (Tokyo, 1973)''  pp. 195–203</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.H. Kuiper,  J.W. Robbin,  "Topological classification of linear endomorphisms"  ''Inv. Math.'' , '''19'''  (1973)  pp. 83–106</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.E. Cappell,  J.L. Shaneson,  "Non-linear similarity"  ''Ann. of Math.'' , '''113'''  (1981)  pp. 315–355</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.H. Kuiper,  "The topology of the solutions of a linear differential equation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014190/a014190129.png" />" , ''Proc. Internat. Congress on Manifolds (Tokyo, 1973)''  pp. 195–203</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.H. Kuiper,  J.W. Robbin,  "Topological classification of linear endomorphisms"  ''Inv. Math.'' , '''19'''  (1973)  pp. 83–106</TD></TR></table>

Revision as of 10:01, 25 April 2020


of ordinary differential equations

A system of ordinary differential equations which does not explicitly contain the independent variable $ t $( time). The general form of a first-order autonomous system in normal form is:

$$ \dot{x} _ {j} = f _ {j} ( x _ {1} \dots x _ {n} ) , \ j = 1 \dots n, $$

or, in vector notation,

$$ \tag{1 } \dot{x} = f ( x ) . $$

A non-autonomous system $ \dot{x} = f(t, x) $ can be reduced to an autonomous one by introducing a new unknown function $ x _ {n+1} = t $. Historically, autonomous systems first appeared in descriptions of physical processes with a finite number of degrees of freedom. They are also called dynamical or conservative systems (cf. Dynamical system).

A complex autonomous system of the form (1) is equivalent to a real autonomous system with $ 2n $ unknown functions

$$ \frac{d}{dt} ( \mathop{\rm Re} x ) = \mathop{\rm Re} f ( x ) , \ \frac{d}{dt} ( \mathop{\rm Im} x ) = \mathop{\rm Im} f ( x ). $$

The essential contents of the theory of complex autonomous systems — unlike in the real case — is found in the case of an analytic $ f(x) $( cf. Analytic theory of differential equations).

Consider an analytic system with real coefficients and its real solutions. Let $ x = \phi (t) $ be an (arbitrary) solution of the analytic system (1), let $ \Delta = ( t _ {-} , t _ {+} ) $ be the interval in which it is defined, and let $ x(t; t _ {0} , x ^ {0} ) $ be the solution with initial data $ x \mid _ {t = t _ {0} } = x ^ {0} $. Let $ G $ be a domain in $ \mathbf R ^ {n} $ and $ f \in C ^ {1} (G) $. The point $ x ^ {0} \in G $ is said to be an equilibrium point, or a point of rest, of the autonomous system (1) if $ f ( x ^ {0} ) \equiv 0 $. The solution $ \phi (t) \equiv x ^ {0} $, $ t \in \mathbf R = ( - \infty , + \infty ) $, corresponds to such an equilibrium point.

Local properties of solutions.

1) If $ \phi (t) $ is a solution, then $ \phi ( t + c ) $ is a solution for any $ c \in \mathbf R $.

2) Existence: For any $ t _ {0} \in \mathbf R , x ^ {0} \in G $, a solution $ x(t; t _ {0} , x ^ {0} ) $ exists in a certain interval $ \Delta \ni t $.

3) Smoothness: If $ f \in C ^ {p} (G) , p \geq 1 $, then $ \phi (t) \in C ^ {p+1} ( \Delta ) $.

4) Dependence on parameters: Let $ f = f(x, \alpha ) $, $ \alpha \in G _ \alpha \subset \mathbf R ^ {m} $, where $ G _ \alpha $ is a domain; if $ f \in C ^ {p} (G \times G _ \alpha ) $, $ p \geq 1 $, then $ x(t; t _ {0} , x ^ {0} , \alpha ) \in C ^ {p} ( \Delta \times G _ \alpha ) $( for more details see [1][4]).

5) Let $ x ^ {0} $ be a non-equilibrium point; then there exist neighbourhoods $ V, W $ of the points $ x ^ {0} , f( x ^ {0} ) $, respectively, and a diffeomorphism $ y = h(x) : V \rightarrow W $ such that the autonomous system has the form $ \dot{y} = \textrm{ const } $ in $ W $.

A substitution of variables $ x = \phi (y) $ in the autonomous system (1) yields the system

$$ \tag{2 } \dot{y} = ( \phi ^ \prime ( y ) ) ^ {-1} f ( \phi ( y ) ), $$

where $ \phi ^ \prime (y) $ is the Jacobi matrix.

Global properties of solutions.

1) Any solution $ x = \phi (t) $ of the autonomous system (1) may be extended to an interval $ \Delta = (t _ {-} , t _ {+} ) $. If $ \Delta = \mathbf R $, the solution is said to be unboundedly extendable; if $ t _ {+} = + \infty , t _ {-} > - \infty $, the solution is said to be unboundedly extendable forwards in time (and, in a similar manner, backwards in time). If $ t _ {+} < + \infty $ then, for any compact set $ K \subset \Omega $, $ x ^ {0} \in K $, there exists a $ \tau = \tau (K) < t _ {+} $ such that the point $ x(t; t _ {0} , x ^ {0} ) $ is outside $ K $ for $ t > \tau (K) $( and, analogously, for $ t _ {-} > - \infty $; cf. Prolongation of solutions of differential equations).

2) The extension is unique in the sense that any two solutions with common initial data are identical throughout their range of definition.

3) Any solution of an autonomous system belongs to one of the following three types: a) aperiodic, with $ \phi (t _ {1} ) \neq \phi ( t _ {2} ) $ for all $ t _ {1} \neq t _ {2} $, $ t _ {j} \in \mathbf R $; b) periodic, non-constant; or c) $ \phi (t) \equiv \textrm{ const } $.

Geometric interpretation of an autonomous system.

To each solution $ x = \phi (t) $ is assigned a corresponding curve $ \Gamma $: $ x = \phi (t) $, $ t \in \Delta $, inside the domain $ G $. $ G $ is then said to be the phase space of the autonomous system, $ \Gamma $ is its trajectory in the phase space, and the solution is interpreted as motion along this trajectory in the phase space. The mapping $ g ^ {t} : G \rightarrow G $ defined by the formula $ g ^ {t} x ^ {0} = x (t; 0, x ^ {0} ) $( i.e. each point moves along the phase trajectory during time $ t $) is called the phase flow. In its domain of definition the phase flow satisfies the following conditions: 1) $ g ^ {t} x $ is continuous in $ (t, x) $; and 2) the group property $ g ^ {t _ {1} + t _ {2} } x = g ^ {t _ {1} } g ^ {t _ {2} } x $.

The Liouville theorem is valid: Let $ D \subset G $ be a domain with a finite volume and let $ v _ {t} $ be the volume of the domain $ g ^ {t} D \subset G $, then

$$ \tag{3 } \left . \frac{d v _ {t} }{dt} \right | _ {t = 0 } = \int\limits _ { D } \mathop{\rm div} f ( x ) dx . $$

For a Hamiltonian system, a consequence of (3) is the conservation of the phase volume by the phase flow. A second variant of (3) is obtained as follows. Let $ x = \phi (t, \alpha ) $ be a family of solutions of (1), $ \alpha = ( \alpha _ {1} \dots \alpha _ {n-1} ) \in G _ \alpha $, let $ G _ \alpha $ be a domain and let $ \phi \in C ^ {1} ( \Delta \times G _ \alpha ) $, then

$$ \tag{3\prime } \frac{d}{dt} \mathop{\rm ln} I ( t , \alpha ) = \mathop{\rm div} f ( x ) , $$

where $ I(t, \alpha ) = \mathop{\rm det} \partial x / \partial (t, \alpha ) $.

Structure of phase trajectories.

1) Any two phase trajectories have either no points in common or coincide.

2) Any phase trajectory belongs to one of the following types: a) a smooth, simple, non-closed Jordan arc; b) a cycle, i.e. a curve diffeomorphic to a circle; or c) a point (an equilibrium point). The local structure of phase trajectories in a small neighbourhood of a point other than an equilibrium point is trivial (cf. local property 5) of the solutions): The family of phase trajectories is diffeomorphic to a family of parallel straight lines. For a linear autonomous system the structure of phase trajectories in a neighbourhood of an equilibrium point is known, since the autonomous system is integrable [5]. For non-linear autonomous systems this problem has not yet been completely solved, even for $ n = 2 $( cf. Qualitative theory of differential equations). One aspect of this problem is the question of stability of an equilibrium point (cf. Stability theory). A few results will be given below. Let $ x ^ {0} , y ^ {0} $ be equilibrium points of the system (1), let

$$ \tag{1\prime } \dot{y} = g ( y ) $$

and let $ U, V $ be neighbourhoods of the points $ x ^ {0} , y ^ {0} $. The systems (1) and (1prm) are said to be equivalent in neighbourhoods of their equilibrium points $ x ^ {0} , y ^ {0} $ if there exist neighbourhoods $ U, V $ and a bijective mapping $ h: U \rightarrow V $ such that $ ( h \circ f ^ {t} ) x = (g ^ {t} \circ h ) x $( for $ x \in U $, $ f ^ {t} x \in U $, $ ( g ^ {t} \circ h)x \in V $), i.e. as a result of the substitution $ y = h(x) $ the trajectories of the autonomous system (1) go into trajectories of the autonomous system (1prm). The equivalence is said to be differentiable (topological) if $ h $ is a diffeomorphism (homeomorphism). Let $ x ^ {0} $ be an equilibrium point of the autonomous system (1), let the matrix $ f ^ { \prime } (x ^ {0} ) $ be non-degenerate, and let it not possess any pure imaginary eigen values. Then the autonomous system (1) in a neighbourhood of $ x ^ {0} $ is topologically equivalent to its linear part $ \dot{y} = f ^ { \prime } (x ^ {0} ) y $. An important example is the autonomous system $ \dot{x} = Ax , \dot{y} = By $ where $ A, B $ are constant matrices with pure imaginary eigen values and $ n > 2 $; it is not known when these autonomous systems are topologically equivalent. One of the most fundamental problems in the theory of autonomous systems is that of the structure of the entire family of phase trajectories. The most complete results have been obtained for $ n = 2 $, but even in this case the solution is far from complete.

References

[1] I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)
[2] L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)
[3] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[4] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)
[5] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)

Comments

The question of the topological equivalence (homeomorphic equivalence of orbit systems) of two linear dynamical systems $ \dot{x} = Ax $ and $ \dot{y} = By $ has been examined and solved [a2]. Let $ V _ {A} ^ {+} , V _ {A} ^ {-} , V _ {A} ^ {0} $ be the decomposition of $ \mathbf R ^ {n} $ corresponding to (generalized) eigen values of $ A $ with positive, negative and zero real parts respectively, and let $ A ^ {0} $ be the restriction of $ A $ to $ V _ {A} ^ {0} $. Then the orbit systems of $ \dot{x} = Ax $ and $ \dot{y} = By $ are homeomorphic if $ \mathop{\rm dim} V _ {A} ^ {+} = \mathop{\rm dim} V _ {B} ^ {+} $, $ \mathop{\rm dim} V _ {A} ^ {-} = \mathop{\rm dim} V _ {B} ^ {-} $ and $ A ^ {0} $ and $ B ^ {0} $ are linearly equivalent (i.e. $ A ^ {0} = S ^ {-1} B ^ {0} S $ for some constant invertible matrix $ S $).

The analogous question for discrete dynamical systems: when does there exist for two linear endomorphisms $ A, B: \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $ a homeomorphism $ \phi $ such that $ A \phi = \phi B $, is much deeper (topological similarity of matrices) and has to do with such things as lens spaces and the P.A. Smith conjecture. See [a1] for a full account. A theorem of G. de Rham says that if the orthogonal matrices $ A $ and $ B $ are topologically similar and the topological (non-linear) similarity preserves the unit sphere and restricts to a diffeomorphism in it, then the matrices $ A $ and $ B $ are also linearly similar. Results of S.E. Cappell and J.L. Shaneson imply that for matrices of eigen values of absolute value 1 (the crucial case) topological and linear equivalence are the same for dimensions $ \leq 5 $. Combined with results of Kuiper–Robbin [a3] this gives a complete topological classification of matrices of dimension at most 5. In higher dimensions it is definitely not true in general that for matrices of eigen values of modulus 1 topological similarity implies linear similarity [a1]. These results can be used to tell at what times $ t $ the phase flows for continuous-time systems become topologically equivalent. This can happen repeatedly even for systems that are not topologically equivalent.

References

[a1] S.E. Cappell, J.L. Shaneson, "Non-linear similarity" Ann. of Math. , 113 (1981) pp. 315–355
[a2] N.H. Kuiper, "The topology of the solutions of a linear differential equation on " , Proc. Internat. Congress on Manifolds (Tokyo, 1973) pp. 195–203
[a3] N.H. Kuiper, J.W. Robbin, "Topological classification of linear endomorphisms" Inv. Math. , 19 (1973) pp. 83–106
How to Cite This Entry:
Autonomous system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Autonomous_system&oldid=19025
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article