On the approach to identifying periodic and bounded solutions of linear dynamic systems

Background. The purpose of the study is to simplify the expressibility criterion for all functions $x_1(t),x_2(t),...,x_n(t)$ included in a given system $x'(t)=A \cdot x(t)$, in the form of linear combinations of derivatives of only one unknown function $x_k(t)$ included in this system and apply it to identify a periodic and limited solution of the system $x'(t)=A \cdot x(t)$. Materials and methods. The essence of the proposed approach is that the construction and study of a solution to the system $x'(t)=A \cdot x(t)$ is equivalently reduced to one high-order scalar differential equation. Results. A simplified criterion for the expressibility of all functions of the system $x'(t)=A \cdot x(t)$ in the form of linear combinations of derivatives $x_k(t)$ is formulated, and its correctness is proved. It is also argued that when the expressibility criterion is satisfied, the periodicity and boundedness of the solution vector $x(t)$ of the system $x'(t)=A \cdot x(t)$ follow only from the periodicity and boundedness of one coordinate $x_k(t)$, respectively. Conclusions. When the expressibility criterion is met, the proposed approach can be used to identify a periodic and bounded solution of the system $x'(t)=A \cdot x(t)$, since it allows us to identify a periodic and bounded solution of the system $x'(t)=A \cdot x(t)$ based on the periodicity and limitation of only one coordinate $x_k(t)$, respectively.