# Linear system of differential equations with almost-periodic coefficients

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). 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 (see [1]). Interest in a narrower class of systems (where $A ( t)$ and $f ( t)$ are quasi-periodic mappings, cf. 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), 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 $x = L ( t) y$; 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 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

 [1] H. Bohr, "Almost-periodic functions" , Chelsea, reprint (1947) (Translated from German) [2] J. Favard, "Leçons sur les fonctions presque-périodiques" , Gauthier-Villars (1933) [3] N.P. Erugin, "Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients" , Acad. Press (1966) (Translated from Russian) [4] J.L. Massera, J.J. Shäffer, "Linear differential equations and function spaces" , Acad. Press (1986) [5] 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 [6] Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146