Dynamic-equilibrium solutions of ordinary differential equations and their role in applied problems

The work introduces the notion of an dynamic-equilibrium (DE) solution of an ordinary differential equation (ODE) as the special (limit) version of the ODE general solution. The dynamic equilibrium is understood as independence of the initial point. The work explains the special importance of ODEs which have DE solutions. The criteria for the existence and global attraction of these solutions are developed. A few examples illustrate different aspects of the DE-solution theory and application. The work discusses the role of these solutions in applied problems (related to ODEs in both Euclidean and function Banach spaces) with the emphasis on advanced models for living systems (such as the active-particle generalized kinetic theory). This discussion also concerns a few directions for future research.