Zero-dispersion nonlinear resonance

Underdamped oscillators that possess a maximum or minimum in their dependence of eigenfrequency on energy have recently been shown to exhibit a range of unusual phenomena. Because they are associated with the presence of an extremum in whose vicinity the eigenfrequency is almost energy-independent, they have been named zero-dispersion phenomena. They manifest themselves both in the deterministic dynamics and in the presence of noise. When the oscillator is driven by a weak periodic force at a frequency close to that of the extremum, a novel type of nonlinear resonance, zero-dispersion nonlinear resonance (ZDNR) can occur. A giant response then arises even in the absence of resonance between the drive frequency and any eigenoscillation of the system. The properties of ZDNR, the nature of the transition from ZDNR to conventional nonlinear resonance as relevant parameters are varied, the occurrence of dynamical chaos associated with ZDNR, and the influence of noise, are analysed and discussed for both Hamiltonian and dissipative systems.