Control of chaos
Encyclopedia
In chaos theory
, control of chaos is based on the fact that any chaotic attractor
contains an infinite number of unstable periodic orbits. Chaotic dynamics then consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different unstable periodic orbit where it remains for a limited time, and so forth. This results in a complicated and unpredictable wandering over longer periods of time.
Control of chaos is the stabilization, by means of small system perturbations, of one of these unstable periodic orbits. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage. The perturbation must be tiny, to avoid significant modification of the system's natural dynamics.
Several techniques have been devised for chaos control, but most are developments of two basic approaches: the OGY (Ott, Grebogi and Yorke) method, and Pyragas continuous control. Both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed.
In the OGY method, small, wisely chosen, swift kicks are applied to the system once per cycle, to maintain it near the desired unstable periodic orbit. In the Pyragas method, an appropriate continuous controlling signal is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit.
Experimental control of chaos by one or both of these methods has been achieved in a variety of systems, including turbulent fluids, oscillating chemical reactions, magneto-mechanical oscillators, and cardiac tissues. Sarnobat et al. (2000) attempt the control of chaotic bubbling with the OGY method and using electrostatic potential as the primary control variable.
The number of publications devoted to control of chaos is huge, see e.g. Chaos control bibliography (1997-2000)
Forcing two systems into the same state is not the only way to achieve synchronization of chaos
.
Both control of chaos
and synchronization
constitute parts of Cybernetical Physics
. Cybernetical physics
is a research area on the border between Physics
and Control Theory
.
González-Miranda, J. M. (2004). Synchronization and Control of Chaos: An Introduction for Scientists and Engineers. London: Imperial College Press.
Fradkov A.L., Pogromsky A.Yu. (1998). Introduction to Control of Oscillations and Chaos. Singapore: World Scientific Publishers.
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
, control of chaos is based on the fact that any chaotic attractor
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
contains an infinite number of unstable periodic orbits. Chaotic dynamics then consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different unstable periodic orbit where it remains for a limited time, and so forth. This results in a complicated and unpredictable wandering over longer periods of time.
Control of chaos is the stabilization, by means of small system perturbations, of one of these unstable periodic orbits. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage. The perturbation must be tiny, to avoid significant modification of the system's natural dynamics.
Several techniques have been devised for chaos control, but most are developments of two basic approaches: the OGY (Ott, Grebogi and Yorke) method, and Pyragas continuous control. Both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed.
In the OGY method, small, wisely chosen, swift kicks are applied to the system once per cycle, to maintain it near the desired unstable periodic orbit. In the Pyragas method, an appropriate continuous controlling signal is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit.
Experimental control of chaos by one or both of these methods has been achieved in a variety of systems, including turbulent fluids, oscillating chemical reactions, magneto-mechanical oscillators, and cardiac tissues. Sarnobat et al. (2000) attempt the control of chaotic bubbling with the OGY method and using electrostatic potential as the primary control variable.
The number of publications devoted to control of chaos is huge, see e.g. Chaos control bibliography (1997-2000)
Forcing two systems into the same state is not the only way to achieve synchronization of chaos
Synchronization of chaos
Synchronization of chaos is a phenomenon that may occur when two, or more, chaotic oscillators are coupled, or when a chaotic oscillator drives another chaotic oscillator...
.
Both control of chaos
Control of chaos
In chaos theory, control of chaos is based on the fact that any chaotic attractor contains an infinite number of unstable periodic orbits. Chaotic dynamics then consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different...
and synchronization
Synchronization
Synchronization is timekeeping which requires the coordination of events to operate a system in unison. The familiar conductor of an orchestra serves to keep the orchestra in time....
constitute parts of Cybernetical Physics
Cybernetical physics
Cybernetical physics is a scientific area on the border of Cybernetics and Physics which studies physical systems with cybernetics methods. Cybernetics methods are understood as methods developed within control theory, information theory, systems theory and related areas: control design,...
. Cybernetical physics
Cybernetical physics
Cybernetical physics is a scientific area on the border of Cybernetics and Physics which studies physical systems with cybernetics methods. Cybernetics methods are understood as methods developed within control theory, information theory, systems theory and related areas: control design,...
is a research area on the border between Physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and Control Theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
.
External links
Books
Eckehard Schöll and Heinz Georg Schuster (Eds). Handbook of Chaos Control Wiley-VCH; 2nd Revision, Enlarged edition (2007) Weinheim.González-Miranda, J. M. (2004). Synchronization and Control of Chaos: An Introduction for Scientists and Engineers. London: Imperial College Press.
Fradkov A.L., Pogromsky A.Yu. (1998). Introduction to Control of Oscillations and Chaos. Singapore: World Scientific Publishers.