Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique


A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency–amplitudetextlessbrtextgreaterparameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. Totextlessbrtextgreaterillustrate this approach, it is applied to the Keller–Miksis equation describing the radial dynamics of a single spherical gas bubble placed intextlessbrtextgreateran infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a propertextlessbrtextgreaterselection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, thetextlessbrtextgreatertrajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. Thetextlessbrtextgreatertransformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibriumtextlessbrtextgreaterstate (absence of external driving) of the system.

Chaos: An Interdisciplinary Journal of Nonlinear Science 30: 073123