Wavelets in dynamics, optimal control and Galerkin approximations

We give the explicit time description of the following nonlinear (polynomial) problems: dynamics and optimal dynamics for some important electromechanical system, Galerkin approximation for beam equation, and detecting chaos in the Melnikov approach. We present the solution in a compactly supported wavelet basis. The solution is parameterized by solutions of two reduced algebraical problems, the first is nonlinear (polynomial), the second is a linear problem, which is obtained from one of the next wavelet construction: fast wavelet transform, stationary subdivision schemes, the method of connection coefficients.