Wavelet approach to accelerator problems. III. Melnikov functions and symplectic topology

This is the third part of a series of talks in which we present applications of methods of wavelet analysis to polynomial approximations for a number of accelerator physics problems. We consider the generalization of our variational wavelet approach to nonlinear polynomial problems to the case of Hamiltonian systems for which we need to preserve underlying symplectic or Poissonian or quasicomplex structures in any type of calculations. We use our approach for the problem of explicit calculations of Arnold-Weinstein curves via a Floer variational approach from symplectic topology. The loop solutions are parametrized by the solutions of reduced algebraical problem-matrix quadratic mirror filter equations. Also we consider wavelet approach to the calculations of Melnikov functions in the theory of homoclinic chaos in perturbed Hamiltonian systems