Conditions for Phase Locking and Dephasing of Autoresonant Pumping

Abstract

We study the asymptotic behavior of nonlinear oscillators under an external driver with slowly changing frequency and amplitude. As a result, we obtain formulas for properties of the amplitude and frequency of the driver when the autoresonant behavior of the nonlinear oscillator is observed. Also, we find the measure of autoresonant asymptotic behaviors for such a driven nonlinear oscillator.

Resonances in asymptotically autonomous systems with a decaying chirped-frequency excitation

“…Moreover, in the general case, this method is not applicable to systems with decaying perturbations due to the appearance of metastable states (see, for example, [48]). In addition, the asymptotic approximations obtained for similar systems but with a small parameter in front of the perturbations were justified only in weakly nonlinear cases on asymptotically long but finite time intervals [20,26,29]. In the present paper, a small parameter is not introduced and the behavior of solutions at infinity is discussed.…”

“…Later it was found that the autoresonance has a wide range of applications and can be used in many systems (see, for example, [37,19,18]). Mathematical models associated with the autoresonance have been considered in many papers [17,26,29,32,46], where mainly weakly nonlinear systems were analyzed. In this paper, the presence of a small parameter is not assumed.…”

“…The decreasing perturbations with power-law asymptotics appear, for example, in the study of Painlevé equations [15,8], phase-locking phenomena [27,11], stochastic perturbations [2,47], and in a wide range of other problems associated with nonlinear non-autonomous systems [7,33]. The influence of perturbations with slowly varying chirped-frequency on a finite time behavior of small oscillations is discussed in [29], where the system of the form (1) is considered with…”