Parametric Resonant Control of Macroscopic Behaviors of Multiple Oscillators

Consider a finite collection of oscillators, which a user has limited means to perturb due to physical restrictions. We show that as long as the stiffness parameters of these oscillators can be harmonically perturbed, one can design a single shared perturbation, such that macroscopic trajectory tracking is achieved independently in each oscillator; that is, the oscillation amplitudes of all oscillators will approximate, respectively, an arbitrary collection of target functions. This control mechanism is based on the dynamical phenomenon of parametric resonance, which not only permits both increase and decrease of the oscillation amplitude by design, but also the simultaneous control of multiple oscillators with distinct intrinsic frequencies. A simulated animation of a remotely-powered-and-controlled array of circuits illustrates the efficacy of this control. Oscillators that can be controlled by this mechanism are not limited to harmonic ones, but those subject to additional weak damping, noise, and nonlinearity.