Synchronization of chaos
Encyclopedia
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. Because of the butterfly effect
, which causes the exponential divergence of the trajectories of two identical chaotic system started with nearly the same initial conditions, having two chaotic systems evolving in synchrony might appear quite surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably well understood theoretically.
It has been found that chaos synchronization is quite a rich phenomenon that may present a variety of forms. When two chaotic oscillators are considered, these include:
All these forms of synchronization share the property of asymptotic stability. This means that once the synchronized state has been reached, the effect of a small perturbation that destroys synchronization is rapidly damped, and synchronization is recovered again. Mathematically, asymptotic stability is characterized by a positive Lyapunov exponent
of the system composed of the two oscillators, which becomes negative when chaotic synchronization is achieved.
Some chaotic systems allow even stronger control of chaos
.
Both synchronization of chaos
and control of chaos
constitute parts of Cybernetical Physics
.
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...
are coupled, or when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly effect
Butterfly effect
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions; where a small change at one place in a nonlinear system can result in large differences to a later state...
, which causes the exponential divergence of the trajectories of two identical chaotic system started with nearly the same initial conditions, having two chaotic systems evolving in synchrony might appear quite surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably well understood theoretically.
It has been found that chaos synchronization is quite a rich phenomenon that may present a variety of forms. When two chaotic oscillators are considered, these include:
- Identical synchronization. This is a straightforward form of synchronization that may occur when two identical chaotic oscillators are mutually coupled, or when one of them drives the other. If (x1,x2,,...,xn) and (x'1, x'2,...,x'n) denote the set of dynamical variables that describe the state of the first and second oscillator, respectively, it is said that identical synchronization occurs when there is a set of initial conditions [x1(0), x2(0),...,xn(0)], [x'1(0), x'2(0),...,x'n(0)] such that, denoting the time by t, |x'i(t)-xi((t)|→0, for i=1,2,...,n, when t→∞. That means that for time large enough the dynamics of the two oscillators verifies x'i(t)=xi(t), for i=1,2,...,n, in a good approximation. This is called the synchronized state in the sense of identical synchronization.
- Generalized synchronization. This type of synchronization occurs mainly when the coupled chaotic oscillators are different, although it has also been reported between identical oscillators. Given the dynamical variables (x1,x2,,...,xn) and (y1,y2,,...,ym) that determine the state of the oscillators, generalized synchronization occurs when there is a functional, Φ, such that, after a transitory evolution from appropriate initial conditions, it is [y1(t), y2(t),...,ym(t)]=Φ[x1(t), x2(t),...,xn(t)]. This means that the dynamical state of one of the oscillators is completely determined by the state of the other. When the oscillators are mutually coupled this functional has to be invertible, if there is a drive-response configuration the drive determines the evolution of the response, and Φ does not need to be invertible. Identical synchronization is the particular case of generalized synchronization when Φ is the identity.
- Phase synchronization. This form of synchronization, which occurs when the oscillators coupled are not identical, is partial in the sense that, in the synchronized state, the amplitudes of the oscillator remain unsynchronized, and only their phases evolve in synchrony. Observation of phase synchronization requires a previous definition of the phase of a chaotic oscillator. In many practical cases, it is possible to find a plane in phase space in which the projection of the trajectories of the oscillator follows a rotation around a well-defined center. If this is the case, the phase is defined by the angle, φ(t), described by the segment joining the center of rotation and the projection of the trajectory point onto the plane. In other cases it is still possible to define a phase by means of techniques provided by the theory of signal processingSignal processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, such as the Hilbert transformHilbert transformIn mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
. In any case, if φ1(t) and φ2(t) denote the phases of the two coupled oscillators, synchronization of the phase is given by the relation nφ1(t)=mφ2(t) with m and n whole numbers.
- Anticipated and lag synchronization. In these cases the synchronized state is characterized by a time interval τ such that the dynamical variables of the oscillators, (x1,x2,,...,xn) and (x'1, x'2,...,x'n), are related by x'i(t)=xi(t+τ); this means that the dynamics of one of the oscillators follows, or anticipates, the dynamics of the other. Anticipated synchronization may occur between chaotic oscillators whose dynamics is described by delay differential equationDelay differential equationIn mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
s, coupled in a drive-response configuration. In this case, the response anticipates the dynamics of the drive. Lag synchronization may occur when the strength of the coupling between phase-synchronized oscillators is increased.
- Amplitude envelope synchronization. This is a mild form of synchronization that may appear between two weakly coupled chaotic oscillators. In this case, there is no correlation between phases nor amplitudes; instead, the oscillations of the two systems develop a periodic envelope that has the same frequency in the two systems. This has the same order of magnitude than the difference between the average frequencies of oscillation of the two chaotic oscillator. Often, amplitude envelope synchronization precedes phase synchronization in the sense that when the strength of the coupling between two amplitude envelope synchronized oscillators is increased, phase synchronization develops.
All these forms of synchronization share the property of asymptotic stability. This means that once the synchronized state has been reached, the effect of a small perturbation that destroys synchronization is rapidly damped, and synchronization is recovered again. Mathematically, asymptotic stability is characterized by a positive Lyapunov exponent
Lyapunov exponent
In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
of the system composed of the two oscillators, which becomes negative when chaotic synchronization is achieved.
Some chaotic systems allow even stronger 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...
.
Both 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...
and 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...
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,...
.