Autoresonance versus localization in weakly coupled oscillators

• An analysis of capture into resonance and escape from it in coupled oscillators under the action of a periodic force is provided.
• An effect of slow modulation of the natural and/or external frequency on the emergence of autoresonance is investigated.
• Explicit asymptotic solutions are derived.
• Numerical simulations prove a good agreement between the analytical and numerical (exact) results.

We study formation of autoresonance (AR) in a two-degree of freedom oscillator array including a nonlinear (Duffing) oscillator (the actuator) weakly coupled to a linear attachment. Two classes of systems are studied. In the first class of systems, a periodic force with constant (resonance) frequency is applied to a nonlinear oscillator (actuator) with slowly time-decreasing stiffness. In the systems of the second class a nonlinear time-invariant oscillator is subjected to an excitation with slowly increasing frequency. In both cases, the attached linear oscillator and linear coupling are time-invariant, and the system is initially engaged in resonance. This paper demonstrates that in the systems of the first type AR in the nonlinear actuator entails oscillations with growing amplitudes in the linear attachment while in the system of the second type energy transfer from the nonlinear actuator is insufficient to excite high-energy oscillations of the attachment. It is also shown that a slow change of stiffness may enhance the response of the actuator and make it sufficient to support oscillations with growing energy in the attachment even beyond the linear resonance. Explicit asymptotic approximations of the solutions are obtained. Close proximity of the derived approximations to exact (numerical) results is demonstrated.