Abstract
A novel method to control multistability of nonlinear oscillators by applying dual-frequency driving is presented. The test model is the Keller–Miksis equation describing the oscillation of a bubble in a liquid. It is solved by an in-house initial-value problem solver capable to exploit the high computational resources of professional graphics cards. During the simulations, the control parameters are the two amplitudes of the acoustic driving at fixed, commensurate frequency pairs. The high-resolution bi-parametric scans in the control parameter plane show that a period-2 attractor can be continuously transformed into a period-3 one (and vice versa) by proper selection of the frequency combination and by proper tuning of the driving amplitudes. This phenomenon has opened a new way to drive the system to a desired, pre-selected attractor directly via a non-feedback control technique without the need of the annihilation of other attractors. Moreover, the residence in transient chaotic regimes can also be avoided. The results are supplemented with simulations obtained by the boundary-value problem solver AUTO, which is capable to compute periodic orbits directly regardless of their stability, and trace them as a function of a control parameter with the pseudo-arclength continuation technique.
Highlights
One of the common features of nonlinear systems is multistability [1]; that is, two or more attractors can coexist at a fixed parameter combination
Even an infinite number of attracting periodic orbits called Gavrilov–Shilnikov–Newhouse sinks can coexist [15,16]. Another important mechanism leading to multistability is the coupling of a large number of identical systems, where the number of the attractors scales with the number of oscillators [17]. This results in a so-called attractor crowding that makes the system extremely sensitive to external noise leading to random hopping between many coexisting states
We focus on a very specific phenomenon, the branching mechanism, turning out to be a special feature of dual-frequency driven nonlinear oscillators with rational frequency combinations, which can be used for the control of multistability
Summary
One of the common features of nonlinear systems is multistability [1]; that is, two or more attractors can coexist at a fixed parameter combination. The appearance of homoclinic tangencies or the coupling of systems with a large number of unstable invariant manifolds can lead to a high level of multistability In these cases, even an infinite number of attracting periodic orbits called Gavrilov–Shilnikov–Newhouse sinks can coexist [15,16]. Another important mechanism leading to multistability is the coupling of a large number of identical systems, where the number of the attractors scales with the number of oscillators [17] This results in a so-called attractor crowding that makes the system extremely sensitive to external noise leading to random hopping between many coexisting states. Several other mechanisms such as delayed feedback [18], parametric forcing [19] or external noise [20] can induce multistability
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