Wavelet Approach to Polynomialdynamicsa

We give the explicit time description of the following problems: dynamics and optimal dynamics for nonlinear (polynomial) dynamical systems, Galerkin approximations for some class of partial diierential equations and routes to chaos in Melnikov function approach to the perturbations of Hamiltonian systems. The rst three problems and a part of the fourth one are reduced to the problem of the solving of the systems of diierential equations with polynomial nonlinearities with or without some constraints. The rst main part of our construction is some variational approach to this problem, which reduces initial problem to the problem of solution of functional equations at the rst stage and some algebraical problems at the second stage. We consider also two private cases of our general construction. In the rst case (particular) we have for Riccati equations the solution as a series on shifted Legendre polynomials, which is parameterized by the solution of reduced algebraical (also Riccati) system of equations. In the second case (general) we have the solution in a compactly supported wavelet basis. Multiresolution expansion is the second main part of our construction. The solution is parameterized by solutions of two reduced algebraical problems, one as in the rst case and the second is some linear problem, which is obtained from one of the next wavelet construction: Fast Wavelet Transform, Stationary Subdivision Schemes, the method of Connection Coeecients. Our initial problems come from very important technical problems: minimization of energy in electromechanical system with enormous expense of energy and detecting chaotic regimes in Galerkin approximation for beam equation (submarine oscillations) in Melnikov function approach to the perturbations of Hamiltonian systems.