Finite differences for design of nonlinear discrete time oscillators

The method of finite differences is used for sampling of time in mathematical models - the differential motion equations of self-oscillatory systems of Thomson type. It is shown that both the right, and left differences allow to keep conservatism of a linear oscillatory contour of self-oscillatory system, but the iterated differential motion equations (discrete mappings) give only the left differences. The method of slow-changing amplitudes is applied to approximate analysis of the finite differences equations. Violations in dynamics of a phase of self-oscillations upon transition to discrete time are noted. The mode of generation of chaotic self-oscillations at high levels of excitement of the discrete oscillator of Van der Pol is shown. New approach to design of self-oscillatory systems with discrete time is offered. Parameters of the difference operators are selected from it taking into account invariance of the shortened equations for slow complex amplitudes of rather temporal sampling.