Synchronization of chaos

Synchronization of chaos is a phenomenon in which the behavior of two or more coupled dissipative, chaotic systems synchronized. Due to the exponential divergence of two neighboring trajectories in a chaotic system synchronization initially seems amazing. Nevertheless, the possibility of synchronization of coupled oscillators driven or is experimentally and theoretically relatively well established and understood. It is a versatile phenomenon with many potential applications in different fields.

Properties and History

Chaotic synchronization is closely related to the even stronger controlled chaos. Synchronization and controlled chaos are components of chaos theory and the physical cybernetics.

The possibility of synchronization of chaotic systems was discovered in 1990 by the work of Louis M. Pecora and Thomas L. Carroll and then two years later further developed by Kevin M. Cuomo and Alan V. Oppenheim. Prior to this discovery, many people would probably have doubts that you can synchronize two chaotic systems, finally, the latter have the property of being very sensitive to slight changes in the initial conditions.

Depending on the nature of the systems and couplings it takes different forms. All forms of synchronization have in common that they are asymptotically stable. This means that once the synchronous oscillation has ceased to small errors, which would destroy the synchronization can be quickly attenuated, so that the synchronous oscillation will be restored. Mathematically, this shows asymptotic stability by the fact that the positive Lyapunov exponent of the whole system, consisting of all oscillators is negative when a chaotic synchronization is achieved.

Forms of synchronization

Identical synchronization

This type of synchronization is also called complete or full synchronization. They can occur at identical chaotic systems. The system is called completely synchronized if there are initial conditions for which the systems evolve identically in the further course. An example of one of the simplest cases, two diffusion- coupled systems. This case can be described by the following differential equations:

Here, the vector field, which describes the dynamics of the different chaotic systems and the coupling parameter. The equation defines an invariant subspace of the coupled system. If this subspace is locally attractive, then we can observe identical synchronization.

If the coupling of the two oscillators becomes too low leads to the chaotic behavior, the trajectories diverge closely spaced apart. However, if the feedback parameter is large enough, this behavior is suppressed by the coupling. To find the critical value of the coupling parameter at which this change in behavior occurs, we look at the difference. We develop this vector field in a Taylor series. If one assumes that the value is small, the higher order terms can be neglected. This gives a linear differential equation which describes the behavior of the difference.

Here, the Jacobian matrix of the vector field with respect to the direction of solution. If it is, we get

The dynamics of a chaotic system provides the equation where the largest Lyapunov exponent of the system. With the approach to get from the equation for an equation for. We thus obtain

If the strength of the coupling for all is therefore above the critical point, the system is full synchronization. The existence of this critical point depends on the properties of the individual chaotic systems.

With the above method usually yields the correct value of the critical coupling constant for the synchronization of the systems. However, in some cases, it may happen that the synchronization of the systems is lost is greater than the calculated critical value and for coupling strengths. This behavior arises from the fact that the nonlinear terms which have been neglected in the above linearization can play an important role by destroying the exponential binding for the behavior of the difference. However, it is always possible with a more detailed method to solve this problem and to compute a critical coupling constant, so that the stability is not affected by the non-linear terms.

Generalized synchronization

This type of synchronization is usually observed if the coupled chaotic oscillator are different, but it is also already been observed with identical oscillators. After an initial transition period, the states of the two systems then are related by a function.

The vectors and thereby describe the respective states of the system. Then the equation indicates that the state of a system can be fully determined by the state of the other. If the systems are mutually coupled, the function must be invertible. If it is merely a drive - response relationship, this need not be the case. The identical synchronization is a special case of the generalized synchronization. In her function is the identity.

Phase synchronization

A coupling in which only the phase shift of the coupled chaotic oscillator remains constant while the amplitudes are independent of each other is called synchronization phase. Such synchronization is possible even with non-identical systems. In order to define a phase of the oscillation, one must first similar to a Poincaré mapping find a hyperplane in phase space on which the projection of the motion of the oscillator can be represented as a rotation around a well-defined center. In this case, the phase is defined by the angle which results from the segment when one connects the position of the oscillator and the projection thereof on the hyperplane with respective center.

In the event that can not be found such a center, you can phase through other techniques of signal processing, such as defining a Hilbert transform. In any case, the phase synchronization can be defined by the relation

Express where and are respectively the phases of the system and and are integers.

Pre- and post- synchronization

In this case, the states of the chaotic systems are connected by a time interval.

This means that an oscillator followed by a time delay of the movement of the other or running ahead of the other. Such advance running can be observed retarded differential equations, which are coupled in a drive -response configuration in a system. Tracking synchronization may occur in phase synchronism coupled oscillators, when the strength of the coupling is increased.

Amplitude Einhüllendensynchronisation

This weak form of synchronization can occur between two weakly coupled chaotic oscillators. In this case, there is no coincidence of the amplitudes or the phases as phase-locked oscillators. Instead, a periodic envelope function, which has the same frequency in both systems developed. Similarly coupled oscillation, the frequency has the same order of magnitude as the difference in the mean frequencies of two oscillators chaotic. Frequently such amplitude Einhüllendensynchronisation phase synchronization precedes, which is to say that if the strength of the coupling is increased, adjusts a phase synchronization.

Example of synchronization in a Lorenz attractor

The synchronization of chaotic systems can be used for encrypted transmission of messages. One possibility is chaotic laser or electrical circuits. These can be for example an electrical implementation of the Lorenz attractor construct. An encryption method is the chaos masking, in which the actual signal is superimposed on a much stronger chaotic oscillation, so that an outsider receives only a noise. With a corresponding receiver circuit can, however, reconstruct the chaotic oscillation and thus subtract from the signal. The recipient must have similar to the transmitter, which can be synchronized with this chaotic oscillator. As an example we take a Lorenz attractor, which can be described by the following differential equations:

With a corresponding receiver circuit, which is operated by the signal.

We defininieren the status of the sender, the recipient's and the error. It can be shown now that strives for the error to zero. Accordingly, we first subtract the equations of the recipient of the equations of the transmitter and receive

The resulting system is indeed linearly, but has a time-dependence from the chaotic signal received. We now construct a Lyapunov function, so that cancels out the dependence. The addition of the second and third equations, the second and the third equation is multiplied to result in:

We define the Lyapunov function, therefore, as

Is positive definite and it can be shown that is a Lyapunov function and so therefore is a globally stable fixed point is simple, and the decreases exponentially.