## 3aSA7. Wavelet and Melnikov function analysis of oscillations of a submarine.

### Session: Wednesday Morning, December 4

### Time: 9:35

**Author: Michael G. Zeitlin**

**Location: Russian Acad. of Sci., Inst. of Problems of Mech. Eng., V. O., Bolshojpr., 61, St. Petersburg 199178, Russia**

**Author: Antonina N. Fedorova**

**Location: Russian Acad. of Sci., Inst. of Problems of Mech. Eng., V. O., Bolshojpr., 61, St. Petersburg 199178, Russia**

**Abstract:**

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.

ASA 132nd meeting - Hawaii, December 1996