Difference between revisions of "Linear system of differential equations with almost-periodic coefficients"
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A system of ordinary differential equations | A system of ordinary differential equations | ||
− | + | $$ \tag{1 } | |
+ | \dot{x} = A ( t) x + f ( t) ,\ x \in \mathbf R ^ {n} , | ||
+ | $$ | ||
− | where | + | 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 | + | 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 | 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 | + | 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 | 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 | + | 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 | + | 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==== | ====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> | <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> | ||
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− | |||
====Comments==== | ====Comments==== |
Revision as of 22:17, 5 June 2020
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) |
Linear system of differential equations with almost-periodic coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_system_of_differential_equations_with_almost-periodic_coefficients&oldid=17504