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

Comments

See also Differential equation, ordinary.

References

[a1]	J.K. Hale, "Ordinary differential equations" , Wiley (1969)