Wavelet Approach to Nonlinear Problems

We give via variational approach and multiresolution expansion in the base of compactly supported wavelets the explicit time description of four following problems: dynamics and optimal dynamics for some important electromechanical system, Galerkin approximation for beam equation, computations of Melnikov functions for perturbed Hamiltonian systems. We also consider wavelet parametrization in Floer variational approach for periodic loop solutions. We give the explicit time description of the following problems: dynamics and optimal dynamics for nonlinear dynamical systems, Galerkin approximations for some class of partial differential equations, computations of Melnikov functions for perturbed Hamiltonian systems. All these problems 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 the 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 type equations the solution as a series on shifted Legendre polynomials, which is parametrized by the solution of reduced algebraical (also Riccati) system of equations 1]{{5]. In the second case (general polynomial systems) we have the solution in a compactly supported wavelet basis 6]{{8]. Multiresolution expansion is the second main part of our construction. In this case the solution is parametrized by solutions of two reduced algebraic 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 (FWT) 9], Stationary Subdivision Schemes (SSS) 10], the method of Connection Coeecients (CC) 11]. As the last point we consider the symplectization of our variational approach which gives us the possibility to construct wavelet parametrization of periodic loops (Arnold-Weinstein curves 19]) in Hamiltonian systems. We use our general construction for solution of important technical problems: minimization of energy and detecting signals from oscillations of a submarine. Our initial problem comes from very important technical problem { minimization of energy in electromechanical system with enormous expense of energy. That is synchronous drive of the mill{the electrical machine with the mill as load. It is described by Park system of equations 1], 2]: A rs i r i s + A k (t); where A ` ; (k; `; r; s = …