MULTI-SCALE ANALYSIS of APPLIED DYNAMICAL SYSTEMS with PERIODIC COEFFICIENTS

The project belongs to an active interdisciplinary research direction - Vibrodynamics, which has been developed by the PI since 2003-4. We are going to apply the methods of Vibrodynamics to the studies of Applied Dynamical Systems (ADS). Surprisingly, the ADS with periodic coefficients (ADS-PC) have escaped almost any attention of researchers. At the same time in many practical situations, say, under the periodic changing of external condition, the coefficients in the equations must be periodic. Vibrodynamics provides an efficient method of studying of ADS-PC with the use of the multi-scale approach, averaging method, and asymptotic analysis. The main point of Vibrodynamics is the creating of an algorithm for the choosing of the slow time-scale, which is different for different equations and situations. In this project we apply Vibrodynamics to various ADS-PC. Our approach is conceptually new for many classes of ADS-PC. We will derive the averaged forms of ADS-PC including the Hamiltonian, Lagrangian equations, etc. As our main examples we will target the most popular ADS such as the equations of Classical Mechanics (vibrational transport), the Predator-Prey equations, Lorenz equations, and equations of astrophysics (including motions of planets, moons, and cosmic dust). In Astrophysics we will concentrate at the descriptions of motions in the fields of rotating dipoles and quadruples, which can have great applications. The obtained new equations will be solved numerically and results will be compared with the available experimental situations. Another direction of our activity will be a broad search among the available ADS, where we will introduce realistically justified periodic coefficients with the aim of obtaining new results in Physics, Chemistry, Engineering, and Biology.