Based on the given conceptual models of real phenomena from the disparate scientific fields the main task of my nine-year research was to establish corresponding mathematical models with acceptable approximations and to give a satisfactory prediction of system dynamics behavior. The relationship of main variables has predominantly non-linear properties thus we use nonlinear analysis and Krylov-Bogoliubov-Mitropolsky method to obtain first asymptotic approximations of system dynamics behavior and to be able to study parameter changes influence analytically and numerically. The multi-mode mutual coupling and transition through resonant regimes of system dynamics were analyzed and explained. Since the number of contributed parameters was considerable the multi-parametric analysis was exploited to examine the synchronization of system parts with external periodic loading. The mathematical analogy between discrete and continuous systems dynamics was detected and used in disparate phenomena behavior description. Continuous mechanical system nonlinear coupling, nonlinear lattice of orthogonal chains of discrete material particles representing biological systems of zona pellucid, and population models of bone cell behavior were all analyzed by using the same mathematical formalism and mapping. The mathematical analogy was detected in time-domain of solutions for disparate natural phenomena so that the same dynamics behavior can be detected and explained.