View source for Linear system of differential equations with almost-periodic coefficients

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A system of ordinary differential equations

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

where  $  A ( \cdot ) :  \mathbf R \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $, 
$  f ( \cdot ) :  \mathbf R \rightarrow \mathbf R  ^ {n} $
are almost-periodic mappings (cf. [[Almost-periodic function|Almost-periodic function]]). In coordinate form one has

$$ 
\dot{x}   ^ {i}  =  \sum_{j=1}^ { n }  a _ {j}  ^ {i} ( t) x
 ^ {j} + f ^ { i } ( t) ,\  i = 1 \dots n ,
$$

where  $  a _ {j}  ^ {i} ( t) $
and  $  f ^ { i } ( t) $, 
$  i , j = 1 \dots n $, 
are almost-periodic real-valued functions. Such systems arise in connection with [[Bohr almost-periodic functions|Bohr almost-periodic functions]] (see [[#References|[1]]]). Interest in a narrower class of systems (where  $  A ( t) $
and  $  f ( t) $
are quasi-periodic mappings, cf. [[Quasi-periodic function|Quasi-periodic function]]) arose much earlier in connection with the examination of variational equations along conditionally-periodic solutions of the equations of celestial mechanics.

If the homogeneous system

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

is a system with integral separation (see [[Integral separation condition|Integral separation condition]]), then it reduces to a diagonal system  $  \dot{y} = B ( t) y $
with almost-periodic coefficients by an almost-periodic (with respect to  $  t $) 
[[Lyapunov transformation|Lyapunov transformation]]  $  x = L ( t) y $; 
that is, it reduces to a system for which there is a basis of  $  \mathbf R  ^ {n} $, 
independent of  $  t $, 
consisting of vectors that are eigen vectors of the operator  $  B ( t) $
for every  $  t \in \mathbf R $. 
In coordinates with respect to this basis the system  $  \dot{y} = B ( t) y $
is written in diagonal form:

$$ 
\dot{y}   ^ {i}  =  b _ {i}  ^ {i} ( t) y  ^ {i} ,\  i = 1
\dots n .
$$

The set of systems with integral separation is open in the space of systems (2) with almost-periodic coefficients, endowed with the metric

$$ 
d ( A _ {1} , A _ {2} )  =  \sup _ {t \in \mathbf R } \ 
\| A _ {1} ( t) - A _ {2} ( t) \| .
$$

The following theorem holds. Let  $  A ( t) = C + \epsilon D ( t) $, 
where  $  C \in  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $, 
let the eigen values of  $  C $
all be real and distinct, and let  $  D ( \cdot ) $
be an almost-periodic mapping  $  \mathbf R \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $. 
Then there is an  $  \eta > 0 $
such that for all  $  \epsilon $
with  $  | \epsilon | < \eta $
the system (2) reduces to a diagonal system with almost-periodic coefficients, by an almost-periodic (with respect to  $  t $) 
Lyapunov transformation.

For an almost-periodic mapping  $  A ( t) :  \mathbf R \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $
the following four assertions are equivalent: 1) for every almost-periodic mapping  $  f ( \cdot ) :  \mathbf R \rightarrow \mathbf R  ^ {n} $
there is an almost-periodic solution of the system (1); 2) there is exponential [[Dichotomy|dichotomy]] of solutions of the system (2); 3) none of the systems  $  \dot{x} = \widetilde{A}  ( t) x $, 
where  $  \widetilde{A}  ( t) = \lim\limits _ {k \rightarrow \infty }   A ( t _ {k} + t) $, 
has non-zero bounded solutions; and 4) for every bounded mapping  $  f ( t) :  \mathbf R \rightarrow \mathbf R  ^ {n} $
there is a bounded solution of the system (1).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,   "Almost-periodic functions" , Chelsea, reprint  (1947)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Favard,   "Leçons sur les fonctions presque-périodiques" , Gauthier-Villars  (1933)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.P. Erugin,   "Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.L. Massera,   J.J. Shäffer,   "Linear differential equations and function spaces" , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Mukhamadiev,   "On invertibility of differential operators in the space of continuous functions bounded on the real axis"  ''Soviet Math. Dokl.'' , '''12'''  (1971)  pp. 49–52  ''Dokl. Akad. Nauk SSSR'' , '''196''' :  1  (1971)  pp. 47–49</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 71–146</TD></TR></table>

====Comments====
See also [[Differential equation, ordinary]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1969)</TD></TR></table>