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A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o0705001.png" /> of a differential equation
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$#C+1 = 30 : ~/encyclopedia/old_files/data/O070/O.0700500 Oscillating solution
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<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/o/o070/o070500/o0705002.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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with the property: There exists for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o0705003.png" /> a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o0705004.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o0705005.png" /> changes sign on passing through it. In many applied problems there arises the question of the existence of an oscillating solution or the question whether all the solutions of equation (*) oscillate. Many sufficient conditions are known under which equation (*) has an oscillating solution (see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]). For example, any non-trivial solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o0705006.png" /> with constant coefficients is oscillating if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o0705007.png" />; every non-trivial solution of the equation
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A solution $  x ( t) $
 +
of a differential equation
  
<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/o/o070/o070500/o0705008.png" /></td> </tr></table>
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$$ \tag{* }
 +
x  ^ {(} n)  = \
 +
f ( t , x , x  ^  \prime  \dots x  ^ {(} n- 1) ),\ \
 +
t \in [ t _ {0} , \infty ) ,
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o0705009.png" />-periodic coefficients is oscillating if
+
with the property: There exists for any  $  t _ {1} \geq  t _ {0} $
 +
a point  $  t _ {2} > t _ {1} $
 +
such that  $  x ( t) $
 +
changes sign on passing through it. In many applied problems there arises the question of the existence of an oscillating solution or the question whether all the solutions of equation (*) oscillate. Many sufficient conditions are known under which equation (*) has an oscillating solution (see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]). For example, any non-trivial solution of the equation  $  x  ^ {\prime\prime} + 2 \delta x  ^  \prime  + \omega  ^ {2} x = 0 $
 +
with constant coefficients is oscillating if $  \delta  ^ {2} < \omega  ^ {2} $;
 +
every non-trivial solution of the equation
  
<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/o/o070/o070500/o07050010.png" /></td> </tr></table>
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$$
 +
x  ^ {\prime\prime} + p ( t) x  ^  \prime  + q ( t) x  = 0
 +
$$
  
<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/o/o070/o070500/o07050011.png" /></td> </tr></table>
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with  $  \omega $-
 +
periodic coefficients is oscillating if
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050013.png" />.
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
dt \int\limits _ { t } ^ { {t }  + \omega }
 +
q ( s)  \mathop{\rm exp}
 +
\left ( - \int\limits _ { s } ^ { t }  p ( r)  dr \right )  ds \geq
 +
$$
  
In a number of applications there arises the question of the existence of oscillating solutions (in the above sense) of a system of ordinary differential equations. For example, in control theory one studies the oscillation relative to a given hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050014.png" /> of the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050015.png" /> of the system of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050016.png" />, that is, the question whether the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050017.png" /> oscillates. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050018.png" />-oscillating solutions are also studied; a bounded solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050019.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050020.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050022.png" />-oscillating if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050023.png" /> is oscillating and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050024.png" /> there are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050030.png" />. For the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070500/o07050031.png" /> there also exist other definitions of an oscillating solution.
+
$$
 +
\geq  \
 +
-
 +
\frac{1}{2}
 +
\left ( 1 -  \mathop{\rm exp} \int\limits _ { 0 } ^  \omega  p ( t)  dt \right ) \int\limits _ { 0 } ^  \omega  p ( t)  dt
 +
$$
 +
 
 +
and  $  q ( t) \not\equiv 0 $
 +
on  $  [ 0 , \omega ] $.
 +
 
 +
In a number of applications there arises the question of the existence of oscillating solutions (in the above sense) of a system of ordinary differential equations. For example, in control theory one studies the oscillation relative to a given hyperplane $  \sum _ {i=} 1  ^ {n} c _ {i} x _ {i} = 0 $
 +
of the solutions $  x ( t) = ( x _ {1} ( t) \dots x _ {n} ( t) ) $
 +
of the system of equations $  x  ^  \prime  = f ( t , x ) $,  
 +
that is, the question whether the function $  \sigma ( t) = \sum _ {i=} 1  ^ {n} c _ {i} x _ {i} ( t) $
 +
oscillates. $  [ \alpha , \beta ] $-
 +
oscillating solutions are also studied; a bounded solution $  x ( t) $
 +
of the system $  x  ^  \prime  = f ( t , x ) $
 +
is called $  [ \alpha , \beta ] $-
 +
oscillating if $  \sigma ( t) $
 +
is oscillating and for any $  t _ {1} \geq  t _ {0} $
 +
there are points $  t _ {2} $
 +
and $  t _ {3} $
 +
such that $  t _ {1} < t _ {2} < t _ {3} $,  
 +
$  \sigma ( t _ {2} ) < \alpha $,  
 +
$  \sigma ( t _ {3} ) > \beta $,  
 +
where $  \alpha < 0 < \beta $.  
 +
For the system $  x  ^  \prime  = f ( x , t ) $
 +
there also exist other definitions of an oscillating solution.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.A. Swanson,  "Comparison and oscillation theory of linear differential equations" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.T. Kiguradze,  "Some singular boundary value problems for ordinary differential equations" , Tbilisi  (1975)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.A. Swanson,  "Comparison and oscillation theory of linear differential equations" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.T. Kiguradze,  "Some singular boundary value problems for ordinary differential equations" , Tbilisi  (1975)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR></table>

Revision as of 08:04, 6 June 2020


A solution $ x ( t) $ of a differential equation

$$ \tag{* } x ^ {(} n) = \ f ( t , x , x ^ \prime \dots x ^ {(} n- 1) ),\ \ t \in [ t _ {0} , \infty ) , $$

with the property: There exists for any $ t _ {1} \geq t _ {0} $ a point $ t _ {2} > t _ {1} $ such that $ x ( t) $ changes sign on passing through it. In many applied problems there arises the question of the existence of an oscillating solution or the question whether all the solutions of equation (*) oscillate. Many sufficient conditions are known under which equation (*) has an oscillating solution (see [1], [2], [3]). For example, any non-trivial solution of the equation $ x ^ {\prime\prime} + 2 \delta x ^ \prime + \omega ^ {2} x = 0 $ with constant coefficients is oscillating if $ \delta ^ {2} < \omega ^ {2} $; every non-trivial solution of the equation

$$ x ^ {\prime\prime} + p ( t) x ^ \prime + q ( t) x = 0 $$

with $ \omega $- periodic coefficients is oscillating if

$$ \int\limits _ { 0 } ^ \infty dt \int\limits _ { t } ^ { {t } + \omega } q ( s) \mathop{\rm exp} \left ( - \int\limits _ { s } ^ { t } p ( r) dr \right ) ds \geq $$

$$ \geq \ - \frac{1}{2} \left ( 1 - \mathop{\rm exp} \int\limits _ { 0 } ^ \omega p ( t) dt \right ) \int\limits _ { 0 } ^ \omega p ( t) dt $$

and $ q ( t) \not\equiv 0 $ on $ [ 0 , \omega ] $.

In a number of applications there arises the question of the existence of oscillating solutions (in the above sense) of a system of ordinary differential equations. For example, in control theory one studies the oscillation relative to a given hyperplane $ \sum _ {i=} 1 ^ {n} c _ {i} x _ {i} = 0 $ of the solutions $ x ( t) = ( x _ {1} ( t) \dots x _ {n} ( t) ) $ of the system of equations $ x ^ \prime = f ( t , x ) $, that is, the question whether the function $ \sigma ( t) = \sum _ {i=} 1 ^ {n} c _ {i} x _ {i} ( t) $ oscillates. $ [ \alpha , \beta ] $- oscillating solutions are also studied; a bounded solution $ x ( t) $ of the system $ x ^ \prime = f ( t , x ) $ is called $ [ \alpha , \beta ] $- oscillating if $ \sigma ( t) $ is oscillating and for any $ t _ {1} \geq t _ {0} $ there are points $ t _ {2} $ and $ t _ {3} $ such that $ t _ {1} < t _ {2} < t _ {3} $, $ \sigma ( t _ {2} ) < \alpha $, $ \sigma ( t _ {3} ) > \beta $, where $ \alpha < 0 < \beta $. For the system $ x ^ \prime = f ( x , t ) $ there also exist other definitions of an oscillating solution.

References

[1] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[2] C.A. Swanson, "Comparison and oscillation theory of linear differential equations" , Acad. Press (1968)
[3] I.T. Kiguradze, "Some singular boundary value problems for ordinary differential equations" , Tbilisi (1975) (In Russian)

Comments

References

[a1] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
How to Cite This Entry:
Oscillating solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillating_solution&oldid=48085
This article was adapted from an original article by Yu.V. KomlenkoE.L. Tonkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article