Oscillations and Stability in Quasiautonomous System: I - Simple Point of the One-parameter Family of Periodic Motions

摘要

Oscillations in the quasiautonomous periodic system were studied for the case where the generating autonomous system admits the one-parameter family σ(h) of periodic motions with period T(h). Generation of a single cycle was shown to be usual at the ordinary point (dT(h) ≠ 0). Conditions for cycle stability were obtained. Dependence of the period T(h) on some parameter h is the distinctive feature of oscillations in the nonlinear autonomous systems. In the case of mathematical pendulum, the function T(h) is monotonic. However, in the general case T'(h] can change its sign at some points h{sup}*. With changes of h along the oscillation family σ(h), a transition occurs in the phase space from the given oscillation to another oscillation, and the points h{sup}* distinguish oscillations differing from the others. Singularity of these points is well illustrated by the action of periodic perturbations. If as a rule the periodic action selects in the family σ(h) only the oscillations that are synchronous to perturbations in period, then resonance effects occur at the points h{sup}*.