An initial boundary value problem to a system of linear Schrodinger equations with nonlinear boundary conditions is considered. It is shown that attractor of the problem lies on circles in complex plane. Trajectories tend to xed points of hyperbolic type with unstable manifold which is formed by saddle points of codimension one. Each element of the attractor are periodic piecewise constant function on pase and amplitude of a wave function in WKB -approximation with nite or innite points of discontinuities on a period of the Julia type. More exactly, it has been obtained limit solutions of the problem which with accuracy O(h2) match the exact attractor of the boundary problem, which is independent on h > 0 in the zero WKB - approximation. The presented mathematical result are applied to the study of dynamics of two charged particles with opposite impulses, which are conned by two at walls with surface potentials of double-well type. It is shown that asymptotic behaviour of particles is similar to the behaviour of orbits that arise to well-known logistic map in complex plane. As example, there exist limit periodic nearly piecewise constant distributions of wave functions of Mandelbrot type with Julia type points of 'jumps' for amplitudes and phases of given free charged particles in a conned box with surface nonlinear double-well potential at walls in magnetic eld.