Analytical approach to solving the problems of nonlinear dynamics of systems with time-varying parameters under the conditions of the external environment reaction

Purpose. The development and application of analytical (including approximate) methods of analyzing nonlinear mathematical models of system dynamics, which have complex behavior due to the presence of time-dependent characteristics, is relevant for solving various classes of engineering problems. Therefore, the purpose of this work is mathematical modeling of problems of nonlinear dynamics of such systems, which allows determining the trajectory of the system's movement over time and other dynamic characteristics in accordance with their setting. Method. Based on the modern achievements of analytical, in particular asymptotic and numerical research methods based on existing software complexes, the possibility of non-local research and the formation of a sufficiently complete representation of the peculiarities of the behavior of nonlinear systems is considered. To achieve the goal, a mathematical model of the nonlinear dynamics of a system with time-varying properties is considered, provided that the reaction of the environment depends on the function of the speed of movement of the system to the n degree. The results. The solution of some engineering problems of the nonlinear dynamics of systems with time-varying characteristics of dependence for n=2 and systems whose response to the external environment can be a function of both whole and fractional degrees allow to determine the trajectory of the system's movement over time and other dynamic characteristics in accordance with before their production. Scientific novelty. The analytical dependence of the speed function of the dynamic process of the system with time-varying parameters was obtained for the nonlinearity parameter of the studied system most used in practice, n=2, for the reaction function of the external environment. Practical significance. The obtained analytical dependencies can be applied in a sufficiently wide range of research. Approximate analytical methods based on asymptotic approaches based on hybrid methods (perturbation, phase integrals in combination with Galerkin's principle) are applied. The use of asymptotic and numerical methods of research based on existing software complexes opens up the possibility of non-local research and the formation of a sufficiently complete representation of the peculiarities of the behavior of nonlinear systems with variable characteristics of materials, in particular, composite and functionally gradient ones.