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''of a system of ordinary differential equations
 
''of a 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/c020/c020940/c0209401.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\dot{x}  = \
 +
f (t, x),\ \
 +
x \in \mathbf R  ^ {n}
 +
$$
  
 
''
 
''
  
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c0209402.png" />, depending on two parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c0209403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c0209404.png" />, which, given the value of any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c0209405.png" /> of the system
+
The operator $  X ( \theta , \tau ) : \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $,  
at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c0209406.png" />, gives the value of that solution at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c0209407.png" />:
+
depending on two parameters $  \theta $
 +
and $  \tau $,  
 +
which, given the value of any solution $  x (t) $
 +
of the system
 +
at the point $  t = \tau $,  
 +
gives the value of that solution at the point $  t = \theta $:
  
<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/c020/c020940/c0209408.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td> </tr></table>
+
$$
 +
X ( \theta , \tau )
 +
x ( \tau )  = x ( \theta ).
 +
$$
  
 
If (1) is a linear system, i.e.
 
If (1) is a linear system, i.e.
  
<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/c020/c020940/c0209409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{x}  = A (t) x,
 +
$$
 +
 
 +
where  $  A ( \cdot ) $
 +
is a mapping  $  ( \alpha , \beta ) \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $(
 +
or  $  ( \alpha , \beta ) \rightarrow  \mathop{\rm Hom} ( \mathbf C  ^ {n} , \mathbf C  ^ {n} ) $),
 +
summable over every interval, then for any  $  \theta , \tau \in ( \alpha , \beta ) $
 +
the Cauchy operator is a non-singular linear mapping  $  \mathbf R  ^ {n} \rightarrow \mathbf R  ^ {n} $(
 +
or  $  \mathbf C  ^ {n} \rightarrow \mathbf C  ^ {n} $),
 +
and for any  $  \theta , \tau , \eta \in ( \alpha , \beta ) $,
 +
it satisfies
 +
 
 +
$$ \tag{3 }
 +
\left .
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094010.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094011.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094012.png" />), summable over every interval, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094013.png" /> the Cauchy operator is a non-singular linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094014.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094015.png" />), and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094016.png" />, it satisfies
+
\begin{array}{c}
 +
X ( \theta , \theta )  = I,\ \
 +
X ( \theta , \tau )  = \
 +
X  ^ {-1} ( \tau , \theta ),  \\
 +
X ( \theta , \eta )
 +
X ( \eta , \tau )  = \
 +
X ( \theta , \tau )  \\
 +
\end{array}
  
<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/c020/c020940/c02094017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\right \}
 +
$$
  
 
and the inequality
 
and the inequality
  
<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/c020/c020940/c02094018.png" /></td> </tr></table>
+
$$
 +
\| X ( \theta , \tau ) \|
 +
\leq    \mathop{\rm exp} \
 +
\left | \int\limits _  \tau  ^  \theta  \| A (t) \|  dt \right | .
 +
$$
  
 
(The equations (3) are also valid for a non-linear system
 
(The equations (3) are also valid for a non-linear system
Line 26: Line 76:
 
satisfying the conditions of the existence and uniqueness theorem for solutions of the Cauchy problem, with the necessary stipulations concerning the domains of definition of the operators figuring therein.) The general solution of the system
 
satisfying the conditions of the existence and uniqueness theorem for solutions of the Cauchy problem, with the necessary stipulations concerning the domains of definition of the operators figuring therein.) The general solution 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/c/c020/c020940/c02094019.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = \
 +
A (t) x + h (t),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094020.png" /> is a mapping
+
where $  h ( \cdot ) $
 +
is a mapping
  
<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/c020/c020940/c02094021.png" /></td> </tr></table>
+
$$
 +
( \alpha , \beta )  \rightarrow  \mathbf R  ^ {n} \ \
 +
( \textrm{ or } \
 +
( \alpha , \beta )  \rightarrow  \mathbf C  ^ {n} )
 +
$$
  
summable on every interval, is described in terms of the Cauchy operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094022.png" /> of the system (2) by the formula of [[Variation of constants|variation of constants]]:
+
summable on every interval, is described in terms of the Cauchy operator $  X ( \theta , \tau ) $
 +
of the system (2) by the formula of [[Variation of constants|variation of constants]]:
  
<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/c020/c020940/c02094023.png" /></td> </tr></table>
+
$$
 +
x (t)  = \
 +
X (t, \tau ) x ( \tau ) +
 +
\int\limits _  \tau  ^ { t }
 +
X (t, \theta )
 +
h ( \theta )  d \theta .
 +
$$
  
 
The Cauchy operator of the system (2) satisfies the [[Liouville–Ostrogradski formula|Liouville–Ostrogradski formula]]:
 
The Cauchy operator of the system (2) satisfies the [[Liouville–Ostrogradski formula|Liouville–Ostrogradski formula]]:
  
<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/c020/c020940/c02094024.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm det}  X ( \theta , \tau )  = \
 +
\mathop{\rm exp}  \int\limits _  \tau  ^  \theta 
 +
\mathop{\rm tr}  A ( \xi )  d \xi ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094025.png" /> is the trace of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094026.png" />.
+
where $  \mathop{\rm tr}  A ( \xi ) $
 +
is the trace of the operator $  A ( \xi ) $.
  
The derivative of the Cauchy operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094027.png" /> of the system
+
The derivative of the Cauchy operator $  X ( \theta , \tau ) $
 +
of the system
  
at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094028.png" /> is equal to the Cauchy operator of the system of equations in variations along the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094029.png" /> of the system
+
at a point $  x \in \mathbf R  ^ {n} $
 +
is equal to the Cauchy operator of the system of equations in variations along the solution $  x (t) $
 +
of the system
  
the value of which at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094031.png" /> (on the assumption that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094032.png" /> in the interval with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094034.png" />, the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094035.png" /> lies in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094037.png" /> is a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094038.png" /> with continuous derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094040.png" />; this is one formulation of a theorem asserting the differentiability of the solution with respect to the initial value).
+
the value of which at $  t = \tau $
 +
is $  x $(
 +
on the assumption that, for all $  t $
 +
in the interval with end points $  \theta $
 +
and $  \tau $,  
 +
the graph of $  x (t) $
 +
lies in a domain $  G \subset  \mathbf R ^ {n + 1 } $
 +
such that $  f $
 +
is a continuous mapping $  G \rightarrow \mathbf R  ^ {n} $
 +
with continuous derivative $  f _ {x} ^ { \prime } $
 +
in $  G $;  
 +
this is one formulation of a theorem asserting the differentiability of the solution with respect to the initial value).
  
For a linear system (2) with constant coefficients (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094041.png" />), the Cauchy operator is defined by
+
For a linear system (2) with constant coefficients ( $  A (t) \equiv A $),  
 +
the Cauchy operator is defined by
  
<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/c020/c020940/c02094042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
X ( \theta , \tau )  = \
 +
\mathop{\rm exp} ( ( \theta - \tau ) A)
 +
$$
  
(given a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094044.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094045.png" />; adopting a different approach, one can define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094046.png" /> via formula (4), putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094047.png" />). It is evident from (4) that the Cauchy operator depends only on the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094048.png" /> of the parameters:
+
(given a linear operator $  B $,  
 +
$  \mathop{\rm exp}  B $
 +
is defined as $  \sum _ {k = 0 }  ^  \infty  B  ^ {k} /k! $;  
 +
adopting a different approach, one can define $  \mathop{\rm exp}  A $
 +
via formula (4), putting $  \theta = \tau + 1 $).  
 +
It is evident from (4) that the Cauchy operator depends only on the difference $  \theta - \tau $
 +
of the 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/c020/c020940/c02094049.png" /></td> </tr></table>
+
$$
 +
X ( \theta + t, \tau + t)  = \
 +
X ( \theta , \tau ).
 +
$$
  
 
This equation is a consequence of the autonomy of the system — a property valid for every [[Autonomous system|autonomous system]]
 
This equation is a consequence of the autonomy of the system — a property valid for every [[Autonomous system|autonomous 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/c020/c020940/c02094050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\dot{x}  = f (x),\ \
 +
x \in \mathbf R  ^ {n} .
 +
$$
  
Denoting the Cauchy operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094051.png" /> of the system (5) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094052.png" />, one obtains the following formulas from (3):
+
Denoting the Cauchy operator $  X ( \theta , \tau ) $
 +
of the system (5) by $  f ^ {\theta - \tau } $,  
 +
one obtains the following formulas from (3):
  
<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/c020/c020940/c02094053.png" /></td> </tr></table>
+
$$
 +
f  ^ {0}  = I; \ \
 +
(f  ^ {t} )  ^ {-1}  = \
 +
f  ^ {-t;} \ \
 +
f  ^ {t} f  ^  \tau  = \
 +
f ^ {t + \tau }
 +
$$
  
 
(see also [[Dynamical system|Dynamical system]]; [[Action of a group on a manifold|Action of a group on a manifold]]).
 
(see also [[Dynamical system|Dynamical system]]; [[Action of a group on a manifold|Action of a group on a manifold]]).
Line 68: Line 176:
 
For a linear system (2) with periodic coefficients, i.e.
 
For a linear system (2) with periodic coefficients, i.e.
  
<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/c020/c020940/c02094054.png" /></td> </tr></table>
+
$$
 +
A (t + T)  = \
 +
A (t)
 +
$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094055.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094056.png" />, one has the identity
+
for some $  T > 0 $
 +
and all $  t \in \mathbf R $,  
 +
one has the identity
  
<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/c020/c020940/c02094057.png" /></td> </tr></table>
+
$$
 +
X ( \theta + T, \tau + T)  = \
 +
X ( \theta , \tau )
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094058.png" />; in this case the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094059.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094060.png" /> is arbitrary, is called the monodromy operator. The matrix defining the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094061.png" /> (or, say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094062.png" />) relative to some basis is called the [[Monodromy matrix|monodromy matrix]]. All monodromy operators of a fixed linear system with periodic coefficients are similar to one another:
+
for all $  \theta , \tau \in \mathbf R $;  
 +
in this case the operator $  X ( \tau + T, \tau ) $,  
 +
where $  \tau \in \mathbf R $
 +
is arbitrary, is called the monodromy operator. The matrix defining the operator $  X ( \tau + T, \tau ) $(
 +
or, say, $  X (T, 0) $)
 +
relative to some basis is called the [[Monodromy matrix|monodromy matrix]]. All monodromy operators of a fixed linear system with periodic coefficients are similar to one another:
  
<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/c020/c020940/c02094063.png" /></td> </tr></table>
+
$$
 +
X ( \theta + T, \theta )  = \
 +
X ( \theta , \tau )
 +
X ( \tau + T, \tau )
 +
X  ^ {-1} ( \theta , \tau ),
 +
$$
  
therefore, the spectrum of the monodromy operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094064.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094065.png" />. The eigen values of the monodromy operator are called the [[Multipliers|multipliers]] of the system; one can express conditions for stability and conditional stability of the system in terms of the multipliers (see [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]; [[Lyapunov stability|Lyapunov stability]]; [[Stability theory|Stability theory]]). If the system (2) has periodic complex coefficients,
+
therefore, the spectrum of the monodromy operator $  X ( \tau + T, \tau ) $
 +
is independent of $  \tau $.  
 +
The eigen values of the monodromy operator are called the [[Multipliers|multipliers]] of the system; one can express conditions for stability and conditional stability of the system in terms of the multipliers (see [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]]; [[Lyapunov stability|Lyapunov stability]]; [[Stability theory|Stability theory]]). If the system (2) has periodic complex coefficients,
  
<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/c020/c020940/c02094066.png" /></td> </tr></table>
+
$$
 +
A ( \cdot ): \mathbf R  \rightarrow  \mathop{\rm Hom}
 +
( \mathbf C  ^ {n} , \mathbf C  ^ {n} ),\ \
 +
A (t + T)  = A (t)
 +
$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094067.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094068.png" />, one has the theorem of Lyapunov
+
for some $  T > 0 $
 +
and all $  t \in \mathbf R $,  
 +
one has the theorem of Lyapunov
  
<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/c020/c020940/c02094069.png" /></td> </tr></table>
+
$$
 +
X ( \theta , \tau )  = \
 +
S _  \tau  ( \theta )
 +
\mathop{\rm exp} (( \theta - \tau )
 +
B _  \tau  ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094070.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094071.png" /> is a non-singular linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094072.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094073.png" />, which is a periodic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094074.png" />:
+
where $  B _  \tau  = (1/T)  \mathop{\rm ln}  X ( \tau + T, \tau ) $,  
 +
and $  S _  \tau  ( \theta ) $
 +
is a non-singular linear operator $  \mathbf C  ^ {n} \rightarrow \mathbf C  ^ {n} $
 +
for any $  \theta , \tau \in \mathbf R $,  
 +
which is a periodic function of $  \theta $:
  
<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/c020/c020940/c02094075.png" /></td> </tr></table>
+
$$
 +
S _  \tau  ( \theta + T)  = \
 +
S _  \tau  ( \theta ).
 +
$$
  
 
Different names are sometimes used for the Cauchy operator (e.g.  "matrizant"  for a linear system, or  "operator of translation along trajectories" ).
 
Different names are sometimes used for the Cauchy operator (e.g.  "matrizant"  for a linear system, or  "operator of translation along trajectories" ).
 
 
  
 
====Comments====
 
====Comments====
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020940/c02094076.png" /> does not usually have the name Cauchy attached to it in the Western literature, and in fact is usually not given any particular name at all. In Section 2.1 of [[#References|[a2]]], Cauchy's role in the analysis of
+
The operator $  X ( \theta , \tau ) $
 +
does not usually have the name Cauchy attached to it in the Western literature, and in fact is usually not given any particular name at all. In Section 2.1 of [[#References|[a2]]], Cauchy's role in the analysis of
 
(1)
 
(1)
 
is sketched. The Liouville–Ostrogradski formula is better known as Liouville's formula. [[#References|[a1]]] contains a proof of this formula.
 
is sketched. The Liouville–Ostrogradski formula is better known as Liouville's formula. [[#References|[a1]]] contains a proof of this formula.

Revision as of 15:35, 4 June 2020


of a system of ordinary differential equations

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

The operator $ X ( \theta , \tau ) : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $, depending on two parameters $ \theta $ and $ \tau $, which, given the value of any solution $ x (t) $ of the system at the point $ t = \tau $, gives the value of that solution at the point $ t = \theta $:

$$ X ( \theta , \tau ) x ( \tau ) = x ( \theta ). $$

If (1) is a linear system, i.e.

$$ \tag{2 } \dot{x} = A (t) x, $$

where $ A ( \cdot ) $ is a mapping $ ( \alpha , \beta ) \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $( or $ ( \alpha , \beta ) \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} ) $), summable over every interval, then for any $ \theta , \tau \in ( \alpha , \beta ) $ the Cauchy operator is a non-singular linear mapping $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $( or $ \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $), and for any $ \theta , \tau , \eta \in ( \alpha , \beta ) $, it satisfies

$$ \tag{3 } \left . \begin{array}{c} X ( \theta , \theta ) = I,\ \ X ( \theta , \tau ) = \ X ^ {-1} ( \tau , \theta ), \\ X ( \theta , \eta ) X ( \eta , \tau ) = \ X ( \theta , \tau ) \\ \end{array} \right \} $$

and the inequality

$$ \| X ( \theta , \tau ) \| \leq \mathop{\rm exp} \ \left | \int\limits _ \tau ^ \theta \| A (t) \| dt \right | . $$

(The equations (3) are also valid for a non-linear system

satisfying the conditions of the existence and uniqueness theorem for solutions of the Cauchy problem, with the necessary stipulations concerning the domains of definition of the operators figuring therein.) The general solution of the system

$$ \dot{x} = \ A (t) x + h (t), $$

where $ h ( \cdot ) $ is a mapping

$$ ( \alpha , \beta ) \rightarrow \mathbf R ^ {n} \ \ ( \textrm{ or } \ ( \alpha , \beta ) \rightarrow \mathbf C ^ {n} ) $$

summable on every interval, is described in terms of the Cauchy operator $ X ( \theta , \tau ) $ of the system (2) by the formula of variation of constants:

$$ x (t) = \ X (t, \tau ) x ( \tau ) + \int\limits _ \tau ^ { t } X (t, \theta ) h ( \theta ) d \theta . $$

The Cauchy operator of the system (2) satisfies the Liouville–Ostrogradski formula:

$$ \mathop{\rm det} X ( \theta , \tau ) = \ \mathop{\rm exp} \int\limits _ \tau ^ \theta \mathop{\rm tr} A ( \xi ) d \xi , $$

where $ \mathop{\rm tr} A ( \xi ) $ is the trace of the operator $ A ( \xi ) $.

The derivative of the Cauchy operator $ X ( \theta , \tau ) $ of the system

at a point $ x \in \mathbf R ^ {n} $ is equal to the Cauchy operator of the system of equations in variations along the solution $ x (t) $ of the system

the value of which at $ t = \tau $ is $ x $( on the assumption that, for all $ t $ in the interval with end points $ \theta $ and $ \tau $, the graph of $ x (t) $ lies in a domain $ G \subset \mathbf R ^ {n + 1 } $ such that $ f $ is a continuous mapping $ G \rightarrow \mathbf R ^ {n} $ with continuous derivative $ f _ {x} ^ { \prime } $ in $ G $; this is one formulation of a theorem asserting the differentiability of the solution with respect to the initial value).

For a linear system (2) with constant coefficients ( $ A (t) \equiv A $), the Cauchy operator is defined by

$$ \tag{4 } X ( \theta , \tau ) = \ \mathop{\rm exp} ( ( \theta - \tau ) A) $$

(given a linear operator $ B $, $ \mathop{\rm exp} B $ is defined as $ \sum _ {k = 0 } ^ \infty B ^ {k} /k! $; adopting a different approach, one can define $ \mathop{\rm exp} A $ via formula (4), putting $ \theta = \tau + 1 $). It is evident from (4) that the Cauchy operator depends only on the difference $ \theta - \tau $ of the parameters:

$$ X ( \theta + t, \tau + t) = \ X ( \theta , \tau ). $$

This equation is a consequence of the autonomy of the system — a property valid for every autonomous system

$$ \tag{5 } \dot{x} = f (x),\ \ x \in \mathbf R ^ {n} . $$

Denoting the Cauchy operator $ X ( \theta , \tau ) $ of the system (5) by $ f ^ {\theta - \tau } $, one obtains the following formulas from (3):

$$ f ^ {0} = I; \ \ (f ^ {t} ) ^ {-1} = \ f ^ {-t;} \ \ f ^ {t} f ^ \tau = \ f ^ {t + \tau } $$

(see also Dynamical system; Action of a group on a manifold).

For a linear system (2) with periodic coefficients, i.e.

$$ A (t + T) = \ A (t) $$

for some $ T > 0 $ and all $ t \in \mathbf R $, one has the identity

$$ X ( \theta + T, \tau + T) = \ X ( \theta , \tau ) $$

for all $ \theta , \tau \in \mathbf R $; in this case the operator $ X ( \tau + T, \tau ) $, where $ \tau \in \mathbf R $ is arbitrary, is called the monodromy operator. The matrix defining the operator $ X ( \tau + T, \tau ) $( or, say, $ X (T, 0) $) relative to some basis is called the monodromy matrix. All monodromy operators of a fixed linear system with periodic coefficients are similar to one another:

$$ X ( \theta + T, \theta ) = \ X ( \theta , \tau ) X ( \tau + T, \tau ) X ^ {-1} ( \theta , \tau ), $$

therefore, the spectrum of the monodromy operator $ X ( \tau + T, \tau ) $ is independent of $ \tau $. The eigen values of the monodromy operator are called the multipliers of the system; one can express conditions for stability and conditional stability of the system in terms of the multipliers (see Lyapunov characteristic exponent; Lyapunov stability; Stability theory). If the system (2) has periodic complex coefficients,

$$ A ( \cdot ): \mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf C ^ {n} , \mathbf C ^ {n} ),\ \ A (t + T) = A (t) $$

for some $ T > 0 $ and all $ t \in \mathbf R $, one has the theorem of Lyapunov

$$ X ( \theta , \tau ) = \ S _ \tau ( \theta ) \mathop{\rm exp} (( \theta - \tau ) B _ \tau ), $$

where $ B _ \tau = (1/T) \mathop{\rm ln} X ( \tau + T, \tau ) $, and $ S _ \tau ( \theta ) $ is a non-singular linear operator $ \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $ for any $ \theta , \tau \in \mathbf R $, which is a periodic function of $ \theta $:

$$ S _ \tau ( \theta + T) = \ S _ \tau ( \theta ). $$

Different names are sometimes used for the Cauchy operator (e.g. "matrizant" for a linear system, or "operator of translation along trajectories" ).

Comments

The operator $ X ( \theta , \tau ) $ does not usually have the name Cauchy attached to it in the Western literature, and in fact is usually not given any particular name at all. In Section 2.1 of [a2], Cauchy's role in the analysis of (1) is sketched. The Liouville–Ostrogradski formula is better known as Liouville's formula. [a1] contains a proof of this formula.

References

[a1] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[a2] E. Hille, "Ordinary differential equations in the complex domain" , Wiley (Interscience) (1976)
[a3] M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974)
How to Cite This Entry:
Cauchy operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_operator&oldid=43147
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article