Limit periodic motions in systems with after-effect in a critical case*

Abstract

Systems with after-effect are considered, whose states are described by Volterra integro-differential equations. The critical case of one zero root of the characteristic equation is investigated (where all the other roots have negative real parts) along with the question of the existence in this case of limit periodic motions of the system, i.e., motions which tend exponentially to periodic regimes with unbounded increase of time. A time-dependent, small, piecewise-continuous limit periodic perturbation, generated by external factors, is present in the system. It is shown that in the system under the perturbation, limit periodic motions arise that are represented by power series in fractional powers of a small parameter characterizing the perturbation magnitude. As an example, rotational limit periodic oscillations of a solid plate in an air flow are considered with time dependence of the flow about the plate taken into account by introducing integral terms into the aerodynamic torque.