Nonlinear oscillations of a submarine via computer, Melnikov and wavelet approaches

Abstract

This is a part of the general program of investigation of nonlinear differential equations by wavelet-homoclinic methods. We consider the problem of description of oscillation of a beam (circular cylindrical shell) contacting with a semi-infinite channel filled with compressible liquid (a model for submarine oscillations). We consider our problem for the case of two-frequency parametric perturbations and also external time-dependent forces in two-mode Galerkin approximation. Computer calculation gives us a possibility to analyse in the parameter space the time evolution of our system and corresponding transition between different regimes: deterministic dynamics — quasiperiodic dynamics — chaotic dynamics. The regimes for our system of equations, which we compute numerically, we shall consider also via Melnikov method. This analysis allows us to take into account the influence of water on type of oscillations. As the main part we consider wavelet methods of computation of explicit time solution.