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Control of multistability. (English) Zbl 1357.34105

Summary: Multistability or coexistence of different attractors for a given set of parameters is one of the most exciting phenomena in dynamical systems. It can be found in different areas of science, such as physics, chemistry, biology, economy, and in nature. The final state of a multistable system depends crucially on the initial conditions. From the viewpoint of applications, there are two major issues related to the emergence of multistability. On one hand, this phenomenon often can create inconvenience, as for instance, in the design of a commercial device with specific characteristics, where multistability needs to be avoided or the desired state has to be stabilized against a noisy environment, and on the other hand, the coexistence of different stable states offers a great flexibility in the system performance without major parameter changes, that can be used with the right control strategies to induce a definite switching between different coexisting states. These two examples alone illustrate the importance of multistability control in applied nonlinear science. For the last decade a lot of research has been devoted to the development of control techniques of multistable systems. These methods cover several strategies, going from feedback control methods to nonfeedback, such as periodic or stochastic perturbations capable of changing the coexisting states stability and driving the system from multistability to monostability. We review the most representative control strategies, discuss their theoretical background and experimental realization.

MSC:

34H10 Chaos control for problems involving ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G35 Dynamical aspects of attractors and their bifurcations
78A60 Lasers, masers, optical bistability, nonlinear optics
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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