Asymptotic analysis of the autoresonance phenomenon

Autoresonance is a phase locking phenomenon occurring in nonlinear oscillatory system, which is forced by oscillating perturbation. Many physical applications of the autoresonance are known in nonlinear physics. The essence of the phenomenon is that the nonlinear oscillator selfadjusts to the varying external conditions so that it remains in resonance with the driver for a long time. This long time resonance leads to a strong increase in the response amplitude under weak driving perturbation. An analytic treatment of a simple mathematical model is done here by means of asymptotic analysis using a small driving parameter.