Controlling Multistability by Small Periodic Perturbation

A small perturbation of any system parameters may not in general create any significant qualitative change in dynamics of a multistable system. However, a slow-periodic modulation with properly adjusted amplitude and frequency can do so. In particular, it can control the number of coexisting attractors. The basic idea in this controlling mechanism is to introduce a collision between an attractor with its basin boundary. As a consequence, the attractor is destroyed via boundary crisis, and the chaotic transients settle down to an adjacent attractor. These features have been observed first theoretically with the Hénon map and laser rate equations, and then confirmed experimentally with a cavity-loss modulated CO 2 laser and a pump-modulated fiber laser. The number of coexisting attractors increases as the dissipativity of the system reduces. In the low-dissipative limit, the creation of attractors obeys the predictions of Gavrilov, Shilnikov and Newhouse, when the attractors, referred to as Gavrilov–Shilnikov–Newhouse (GSN) sinks, are created in various period n-tupling processes and remain organized in phase and parameter spaces in a self-similar order. We demonstrate that slow small-amplitude periodic modulation of a system parameter can even destroy these GSN sinks and the system is suitably converted again to a controllable monostable system. Such a control is robust against small noise as well. We also show the applicability of the method to control multistability in coupled oscillators and multistability induced by delayed feedback. In the latter case, it is possible to annihilate coexisting states by modulating either the feedback variable or a system parameter or the feedback strength.