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The property that for a linear system of ordinary differential equations
 
The property that for a linear 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/d/d031/d031600/d0316001.png" /></td> </tr></table>
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$$
 +
\dot{x}  = A ( t) x ,\  x \in E  ^ {n} ,\  t \geq  0 ,
 +
$$
  
with bounded continuous coefficients, there are positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316004.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316005.png" /> such that there exists a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316006.png" /> for which
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with bounded continuous coefficients, there are positive constants $  K $,  
 +
$  L $,  
 +
$  \alpha $,  
 +
and $  \beta $
 +
such that there exists a decomposition $  E  ^ {n} = E  ^ {m} + E  ^ {n-} m $
 +
for which
  
<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/d/d031/d031600/d0316007.png" /></td> </tr></table>
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$$
 +
x ( 0) \in E  ^ {m}  \Rightarrow  \| x ( t) \| \leq  K  \| x ( \tau ) \| \
 +
\mathop{\rm exp} [ - \alpha ( t - \tau ) ] ,
 +
$$
  
<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/d/d031/d031600/d0316008.png" /></td> </tr></table>
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$$
 +
t  \geq  \tau  \geq  0 ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d0316009.png" /></td> </tr></table>
+
$$
 +
x ( 0) \in E  ^ {n-} m  \Rightarrow  \| x ( t) \| \leq  L
 +
\| x ( \tau ) \|  \mathop{\rm exp} [ - \beta ( \tau - t ) ] ,
 +
$$
  
<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/d/d031/d031600/d03160010.png" /></td> </tr></table>
+
$$
 +
\tau  \geq  t  \geq  0
 +
$$
  
(exponential dichotomy; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160011.png" />, one has ordinary dichotomy). The presence of exponential dichotomy is equivalent to saying that the inhomogeneous system
+
(exponential dichotomy; if $  \alpha = \beta = 0 $,  
 +
one has ordinary dichotomy). The presence of exponential dichotomy is equivalent to saying that the inhomogeneous 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/d/d031/d031600/d03160012.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = A ( t) x + f ( t)
 +
$$
  
has, for any bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160014.png" />, at least one bounded solution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031600/d03160015.png" /> [[#References|[1]]]. The theory of dichotomy [[#References|[2]]], transferred to equations in Banach spaces, is also employed in the study of flows and cascades on smooth manifolds [[#References|[4]]].
+
has, for any bounded continuous function $  f ( t) $,  
 +
$  t \geq  0 $,  
 +
at least one bounded solution on $  [ 0 , \infty ) $[[#References|[1]]]. The theory of dichotomy [[#References|[2]]], transferred to equations in Banach spaces, is also employed in the study of flows and cascades on smooth manifolds [[#References|[4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Perron,  "Stability of differential equations"  ''Math. Z.'' , '''32''' :  5  (1930)  pp. 703–728</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Scheffer,  "Linear differential equations and function spaces" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.V. Anosov,  "Geodesic flows on closed Riemann manifolds with negative curvature"  ''Proc. Steklov Inst. Math.'' , '''90'''  (1969)  ''Trudy Mat. Inst. Steklov.'' , '''90'''  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Perron,  "Stability of differential equations"  ''Math. Z.'' , '''32''' :  5  (1930)  pp. 703–728</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Scheffer,  "Linear differential equations and function spaces" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.V. Anosov,  "Geodesic flows on closed Riemann manifolds with negative curvature"  ''Proc. Steklov Inst. Math.'' , '''90'''  (1969)  ''Trudy Mat. Inst. Steklov.'' , '''90'''  (1967)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Oseledec,  "A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1969)  pp. 197–232  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 179–210</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Oseledec,  "A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems"  ''Trans. Moscow Math. Soc.'' , '''19'''  (1969)  pp. 197–232  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 179–210</TD></TR></table>

Latest revision as of 17:33, 5 June 2020


The property that for a linear system of ordinary differential equations

$$ \dot{x} = A ( t) x ,\ x \in E ^ {n} ,\ t \geq 0 , $$

with bounded continuous coefficients, there are positive constants $ K $, $ L $, $ \alpha $, and $ \beta $ such that there exists a decomposition $ E ^ {n} = E ^ {m} + E ^ {n-} m $ for which

$$ x ( 0) \in E ^ {m} \Rightarrow \| x ( t) \| \leq K \| x ( \tau ) \| \ \mathop{\rm exp} [ - \alpha ( t - \tau ) ] , $$

$$ t \geq \tau \geq 0 ; $$

$$ x ( 0) \in E ^ {n-} m \Rightarrow \| x ( t) \| \leq L \| x ( \tau ) \| \mathop{\rm exp} [ - \beta ( \tau - t ) ] , $$

$$ \tau \geq t \geq 0 $$

(exponential dichotomy; if $ \alpha = \beta = 0 $, one has ordinary dichotomy). The presence of exponential dichotomy is equivalent to saying that the inhomogeneous system

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

has, for any bounded continuous function $ f ( t) $, $ t \geq 0 $, at least one bounded solution on $ [ 0 , \infty ) $[1]. The theory of dichotomy [2], transferred to equations in Banach spaces, is also employed in the study of flows and cascades on smooth manifolds [4].

References

[1] O. Perron, "Stability of differential equations" Math. Z. , 32 : 5 (1930) pp. 703–728
[2] H.H. Scheffer, "Linear differential equations and function spaces" , Acad. Press (1966)
[3] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)
[4] D.V. Anosov, "Geodesic flows on closed Riemann manifolds with negative curvature" Proc. Steklov Inst. Math. , 90 (1969) Trudy Mat. Inst. Steklov. , 90 (1967)

Comments

References

[a1] V.I. Oseledec, "A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems" Trans. Moscow Math. Soc. , 19 (1969) pp. 197–232 Trudy Moskov. Mat. Obshch. , 19 (1968) pp. 179–210
How to Cite This Entry:
Dichotomy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dichotomy&oldid=12757
This article was adapted from an original article by R.A. Prokhorova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article