Super-slow nonlinear hysteresis loop “tracking” for managing energy intake in the forced Duffing oscillator

Hysteresis loops are ubiquitous in harmonically (and not only) forced nonlinear oscillators. These loops result due to the well-known nonlinear bi-stability phenomenon, i.e., the co-existence of steady state responses, with the initial conditions dictating the attraction of a specific orbit by either of these steady states. This introduces an element of uncertainty in the forced dynamics. Hence, if one targets the excitation of a higher-amplitude co-existing steady state solution, e.g., to maximize the energy input into the forced oscillator, this sensitivity on initial conditions introduces uncertainty. Moreover, the role of such hysteresis loops in terms of nonlinear physics is not entirely clear; this contrasts with similar hysteresis loops appearing in, e.g., in cyclically loaded viscoelastic materials, which denote the energy dissipated per excitation cycle. Here a methodology is presented for removing the uncertainty of the forced dynamics on initial conditions, and, in the process, for better understanding and exploiting nonlinear hysteresis loops. Considering the forced Duffing oscillator, as an example, harmonic excitations with super-slowly modulated amplitudes are considered as a means of “tracking” the hysteresis loop during a super-slow cycle of the applied modulated force. For either single or repetitive cycles of super-slow force modulations one may tune the modulations to predictively maximize or minimize the energy intake into the forced oscillator depending on which regions of the hysteresis loop are tracked. Importantly, computational studies indicate that the forced dynamics become independent of the specific initial conditions, thus removing the uncertainty in the response due to bi-stability. These findings elucidate the physical significance of nonlinear hysteresis in terms of energy transfer in forced oscillators, and are applicable to a broad class of forced dynamical systems exhibiting nonlinear hysteresis.