Resonance-induced energy localization in a weakly dissipative nonlinear chain

This work investigates the emergence of resonance and resonance-induced localization in a weakly dissipative chain of coupled anharmonic oscillators under the influence of a harmonic force applied at one end of the chain. The dynamics of the chain is studied assuming 1:1 (fundamental) resonance, when the response of each nonlinear oscillator has a dominant harmonic component with the frequency close to the frequency of the external excitation. It is shown that weak dissipation in a strongly nonlinear chain may be a key factor preventing large-amplitude resonance. The resulting process in the dissipative system represents resonance-induced large-amplitude oscillations of a part of the chain adjacent to the actuator and escape from resonance of the distant oscillators. Conditions of the emergence of resonance and energy localization are derived. The obtained solutions indicate that the maximal concentration of energy on the excited oscillator arises together with equipartition of energy among the other resonant oscillators. An agreement between the analytical and numerical results is demonstrated.