Resonances of nonlinear oscillators

We show that nonlinear oscillators have a large response to special aperiodic driving forces. If these forces are selected to minimize the driving effort for a given terminal energy, these forces are given by the time-reflected transient of the unperturbed dynamics (the ‘‘principle of the dynamical key’’). We provide a proof of this principle. We find that these optimal forcing functions have very similar dynamics for several different norms. We present a quantitative comparison of the energy transfer for sinusoidal and optimal driving forces. We find that aperiodic driving forces are most effective for large nonlinearity and small friction. We show that this optimal control is stable for several important systems.