Limit periodic motions in some systems with aftereffect under resonance condition

Abstract

Volterra-type integrodifferential equations and their solutions are considered which, when the time increases without limit, exponentially tend to periodic modes. In the critical case of stability, when the characteristic equation has a pair of pure imaginary roots and the remaining roots have negative real parts, the problem of the existence of limit periodic solutions with resonance, caused by coincidence between the periodic part of the limit external periodic perturbation and the natural frequency of the linearized system, is solved. It is shown that, if the right-hand side of the equation is an analytic function and the existence of limit periodic solutions is determined by terms of the (2m+1)-th order, these solutions are represented by power series in the arbitrary initial values of the non-critical variables and the parameter μ1/(2m+1), where μ is a small parameter, characterizing the magnitude of the maximum external periodic perturbation. The amplitude equations are presented.