Abstract
The present article considers one-parameter families of second-order linear differential systems with a coefficient matrix depending on the real parameter, which is a diagonal matrix at each odd time interval of unit length. The Cauchy matrix is the rotation matrix at each odd time interval, whereas the angle is the sum of a parameter value and some real number. Earlier, it has been has proved that the upper Lyapunov exponent of each such a system, which is considered to be the function of parameter, is positive on the set of the positive Lebesque measure if the diagonal part of the coefficient matrix is independent on a parameter and separated from zero. The proof of this result essentially uses a complex matrix of special type. In recent article, the author has given another way to prove this theorem based on implementing the Parseval equality for trygonometric sums. Besides, the author considers the special case of the above systems. Now the diagonal part of the coefficient matrix is time-independent and is sufficiently big, whereas the rotation angle is defined by a maximum degree of two that divides the number of the corresponding time interval. For such a system, in the case of a continious coefficient dependence on a parameter it is proved that such a value exists, at which the corresponding system is unstable.
Highlights
The present article considers one-parameter families of second-order linear differential systems with a coefficient matrix depending on the real parameter, which is a diagonal matrix at each odd time interval of unit length
The Cauchy matrix is the rotation matrix at each odd time interval, whereas the angle is the sum of a parameter value and some real number
It has been has proved that the upper Lyapunov exponent of each such a system, which is considered to be the function of parameter, is positive on the set of the positive Lebesque measure if the diagonal part of the coefficient matrix is independent on a parameter and separated from zero
Summary
И вещественным параметром μ; условия, которым удовлетворяют числа bk ∈ и функции d k (⋅) : → , будут указаны ниже. Для любых α n ∈ , n ∈ и любой непрерывной функции d (⋅) при выполнении условий (2) найдется μ ∈ такое, что старший характеристический показатель системы (1μ) положителен. В силу (19k) для любого μ ∈ M k имеет место включение ξ k (μ) ∈ \ V2−k−1 (α k − 2−1π)), вле кущее за собой неравенства cos ξ k (μ) ≥ sin 2 −k−1 ≥ 2 −k−2. Для любого i ∈ I k верно включение V2−k−1 (Li,k+1) ⊂ Wk. Тогда, поскольку Li,k+1 – отрезок, найдется ji ∈ I k−1 такое, что выполняется соотношение. Что найдутся матрицы Bk+1, j , Ck+1, j ∈ H , j= k, k −1, для которых выполняется равенство (33k+1). Учитывая следующее из (331) соотношение (331), по индукции получаем, что равенство (33n) Для любой справедливо матрицы B =. B U (μ(1 − k)=) ∑ (β jU (γ j )U (μ(1− k))=) ∑ (β jU (μ(1− k))U (γ j )=) U (μ(1− k))B . (36)
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