ABSTRACT
Periodicity is one of the important fundamental concepts in various mathematical models governed by differential equations in a periodic environment in as diverse fields as economics, model ecosystems, physics and engineering. Later, Markoff recognized that almost periodicity and stability qualitative properties for a differential equation are closely related. In his paper for the fist time it was considered that strong stable bounded solutions are almost periodic. The beginning of the study of almost periodic piecewise continuous functions is in the sixteenth years of XX century. A variety of direct and indirect methods have been applied in the investigation of the existence, uniqueness and stability of almost periodic solutions for numerous classes of differential equations. One of the universal methods for investigating qualitative properties of dynamical systems from different types is the second method of Lyapunov.
