Emergence and stability of autoresonance in nonlinear oscillators
Agnessa Kovaleva, Leonid Manevich
Formation and control of autoresonance (AR) with persistent growth of energy is based on the inherent property of nonlinear systems to stay in resonance when the driving frequency varies in time. However, the mechanism of transitions from bounded oscillations to AR is still an open question. As this paper demonstrates, and in sharp contrast to previous investigations, the emergence of AR from stable bounded oscillations is basically analogous to the transition from small to large oscillations in the time-invariant oscillator driven by an external harmonic excitation with constant fre-
quency. It is shown that AR results from the loss of stability of the so-called limiting phase trajectory (LPT) of small oscillations. A critical parameter, which determines a boundary between small and large oscillations in the time-invariant system, may be considered
as a lower threshold of the autoresonance excitation in the oscillator with slowly-varying parameters. The values of bifurcation parameters accounting for small sweeping rate and close to numerical results are obtained from the condition of the capture into resonance.
CYBERNETICS AND PHYSICS, Vol. 2, No. 1, 2013 , 25-30