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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 [[#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

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

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====
<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====

====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>