Wavelet and Melnikov function analysis of oscillations of a submarine

The explicit time description, computer modeling, and Melnikov function approach for the problem of detecting signals from oscillations of submarines are given. This problem in the Galerkin approximation is reduced to the problem of solving the system of differential equations with polynomial nonlinearities and variable coefficients. The first part of the construction is a variational approach to this problem, which reduces the initial problem to the problems of the solution of functional equations and algebraical problems. In the general case the solution is represented in a compactly supported wavelet basis. Multiresolution expansion is the second part of the construction. The solution is parametrized by solutions of two reduced algebraical problems: One is a polynomial algebraical system of equations, and the second is some linear problem, which is obtained from one of the following wavelet constructions: the fast wavelet transform, stationary subdivision schemes, and the method of connection coefficients. This solution, computer calculations, and the Melnikov function analysis offer the possibility of analyzing, in the parameter space, the time evolution of the system and the corresponding transition between different regimes: deterministic, quasiperiodic, and chaotic dynamics.