Controlling Multistability by Small Periodic Perturbation

Abstract

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.

All-optical controlled switching between time-periodic square waves in diode lasers with delayed feedback

“…Multistate switching devices are needed for the next generation of all-optical computing and communication [13]. The multistability phenomenon is observed for many nonlinear systems and the transition of complex multistability to controlled monostability has raised a lot of attention [14,15]. For Vertical-Cavity Surface-Emitting Lasers (VCSELs) with the PROF setup, it was recently shown that a second weak feedback allows generation of robust SWs with respect to parameter changes [10].…”

Extreme multistability in a chemical model system

“…The exception, of course, is a perturbation in the values of the variables that preserves the value of the conserved quantity, for example, a perturbation that changes y 1 and y 2 equally in Eqs. (3), so that the value of the conserved quantity C remains the same. However, this argument does not apply to the system described by Eqs.…”

“…This occurs in two subsystems with an intermediate from each subsystem involved in the other subsystem, comparable to the model (R 7 )-(R 18 ) described by Eqs. (3). In the course of the conversion, two products, C and D, are formed that do not participate in the dynamics, as well as the main product E. We therefore do not include the time evolution of these species in deriving system (A1), below.…”

“…Most of the work devoted to dynamical systems theory deals with systems having one or only a few attractors for a given set of parameters. However, many physical and biological systems are known in which there are a multitude of coexisting attractors [3][4][5]. Examples include systems from laser physics [6,7], semiconductor physics [8,9], chemistry [10,11], neuroscience [12], and population dynamics [13].…”

Controlling Multistability in Chaotic Systems

“…Recently, controlling multistability [8] related nonlinear phenomena have become an important direction of applied research in Nonlinear Science [9]. In particular, suitable small-periodic perturbation has been found to be very efficient in order to reduce the number of coexisting attractors.…”