Synchronization of chaos

Synchronization of chaos is a phenomenon in which the behavior of two or more coupled, dissipative , chaotic systems is synchronized . Due to the exponential divergence of two adjacent trajectories in a chaotic system, synchronization initially seems astonishing. Nevertheless, the possibility of synchronizing coupled or driven oscillators is relatively well established and understood experimentally and theoretically. It is a multifaceted phenomenon with many possible 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 research and physical cybernetics .

The possibility of synchronizing chaotic systems was discovered in 1990 through the work of Louis M. Pecora and Thomas L. Carroll and then further developed two years later by Kevin M. Cuomo and Alan V. Oppenheim. Before this discovery, many people would probably have doubted whether two chaotic systems could be synchronized, after all they have the property of being very sensitive to slight changes in the initial conditions .

Depending on the type of systems and couplings considered, it takes on different forms. All forms of synchronization have in common that they are asymptotically stable . This means that as soon as the synchronous oscillation has set in, small disturbances which would destroy the synchronization are quickly dampened so that the synchronous oscillation is restored. Mathematically, this asymptotic stability is shown by the fact that the positive Lyapunov exponent of the overall system, consisting of all oscillators, becomes negative when chaotic synchronization is achieved.

Forms of synchronization

Identical synchronization

This type of synchronization is also called complete or complete synchronization. It can take place with identical chaotic systems. The systems are called completely synchronized if there are initial conditions under which the systems develop identically in the further course. An example for one of the simplest cases are two diffusively coupled systems. This case can be described with the following differential equations :

${\displaystyle x^{\prime }=F(x)+\alpha (y-x)}$

${\displaystyle y^{\prime }=F(y)+\alpha (x-y)}$

here is the vector field that describes the dynamics of the individual chaotic systems and the coupling parameter . The equation defines an invariant subspace of the coupled system. If this subspace is locally attractive , then identical synchronization can be observed. ${\displaystyle F}$ ${\displaystyle \alpha }$ ${\displaystyle x(t)=y(t)}$

If the coupling of the two oscillators becomes too low, the chaotic behavior leads to the closely spaced trajectories diverging. However, if the coupling parameter is large enough, this behavior is suppressed by the coupling. To find the critical value of the coupling parameter at which this behavior change occurs, we consider the difference . We develop this vector field in a Taylor series . If the value is assumed to be small, the higher order terms can be neglected. This gives a linear differential equation that describes the behavior of the difference. ${\displaystyle v=x-y}$ ${\displaystyle v}$

${\displaystyle v^{\prime }=DF(x(t))v-2\alpha v}$

here is the Jacobian matrix of the vector field with respect to the solution direction. If is, we get ${\displaystyle DF(x(t))}$ ${\displaystyle \alpha =0}$

${\displaystyle u^{\prime }=DF(x(t))u,}$

The dynamics of a chaotic system are given by the equation , where is the largest Lyapunov exponent of the system. The approach leads from the equation for to an equation for . We thus get ${\displaystyle \|u(t)\|\leq \|u(0)\|e^{\lambda t}}$ ${\displaystyle \lambda }$ ${\displaystyle v=ue^{-2\alpha t}}$ ${\displaystyle v}$ ${\displaystyle u}$

${\displaystyle \|v(t)\|\leq \|u(0)\|e^{(-2\alpha +\lambda )t}.}$

So if the strength of the coupling for all is above the critical point , the system shows complete synchronization. The existence of this critical point depends on the properties of the individual chaotic systems. ${\displaystyle \alpha >\alpha _{c}}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha _{c}=\lambda /2}$

The above procedure usually gives the correct value of the critical coupling constant for the synchronization of the systems. In some cases, however, it can happen that the synchronization of the systems is lost even if the coupling strengths are greater than the calculated critical value. This behavior is due to the fact that the nonlinear terms that were neglected in the above linearization can play an important role by destroying the exponential bond for the behavior of the difference . However, it is always possible to use a more thorough method to solve this problem and calculate a critical coupling constant so that the stability is no longer influenced by the nonlinear terms. ${\displaystyle v}$

Generalized synchronization

This type of synchronization is usually observed when the coupled chaotic oscillators are different, but it has also been observed with identical oscillators. After an initial transition period, the states of the two systems are then linked via a function . ${\displaystyle \Phi }$

${\displaystyle [y_{1}(t),\dotsc ,y_{n}(t)]=\Phi [x_{1}(t),\dotsc ,x_{n}(t)]}$

The vectors and describe the respective states of the system. The equation then states that the state of one system can be completely determined by the state of the other. If the systems are mutually coupled, the function must be invertible . If it's just a drive-response ratio, it doesn't have to be. The identical synchronization is a special case of the generalized synchronization. With her, the function is the identity . ${\displaystyle [y_{1}(t),\dotsc ,y_{n}(t)]}$ ${\displaystyle [x_{1}(t),\dotsc ,x_{n}(t)]}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi }$

Phase synchronization

A coupling in which only the phase shift of the coupled chaotic oscillators remains constant, while the amplitudes are independent of one another, is called phase synchronization. Such synchronization is also possible with systems that are not identical. In order to be able to define a phase for the oscillation, one must first find a hyperplane in phase space, similar to a Poincaré mapping , on which the projection of the movement of the oscillator can be represented as a rotation around a well-defined center. In this case, the phase is defined by the angle that results from the segment if the position of the oscillator and the projection of the same onto the hyperplane are each connected to the center. ${\displaystyle \phi (t)}$

In the event that such a center cannot be found, a phase can be defined using other signal processing techniques, such as a Hilbert transform . In any case, the phase synchronization can then be achieved through the relation

${\displaystyle n\phi _{1}(t)=m\phi _{2}(t)}$

express, where and are respectively the phases of the system and and integers. ${\displaystyle \phi _{1}}$ ${\displaystyle \phi _{2}}$ ${\displaystyle m}$ ${\displaystyle n}$

Pre-run and post-run synchronization

In this case the states of the chaotic systems are connected by a time interval . ${\displaystyle \tau }$

${\displaystyle [y_{1}(t),\dotsc ,y_{n}(t)]=[x_{1}(t+\tau ),\dotsc ,x_{n}(t+\tau )]}$

This means that one oscillator follows the movement of the other with a time delay or runs ahead of the other. Such a lead can be observed in a system of retarded differential equations that are coupled in a drive-reaction configuration. Tracking synchronization can occur with phase-synchronously coupled oscillators if the strength of the coupling is increased.

Amplitude envelope synchronization

This weak form of synchronization can occur between two weakly coupled chaotic oscillators. In this case there is neither a match between the amplitudes nor the phases, as is the case with phase-locked oscillators. Instead, a periodic envelope function develops , which has the same frequency in both systems. Similar to coupled pendulums , the frequency has the same order of magnitude as the difference in the mean oscillation frequencies of the two chaotic oscillators. Such an amplitude envelope synchronization often precedes a phase synchronization, which is to say that when the strength of the coupling is increased, a phase synchronization is established.

Example of synchronization with a Lorenz attractor

The synchronization of chaotic systems can be used for the encrypted transmission of messages. One possibility is chaotic lasers or electrical circuits . For example, an electrical implementation of the Lorenz attractor can be constructed for this purpose. One encryption method is chaos masking , in which the actual signal is superimposed with a much stronger chaotic oscillation, so that an outsider only receives noise. With an appropriate receiving circuit, however, the chaotic oscillation can be reconstructed and thus subtracted from the signal. The receiver must have a chaotic oscillator similar to the transmitter, which can be synchronized with it. As an example we take a Lorenz attractor, which can be described by the following differential equations:

${\displaystyle {\begin{aligned}{\dot {u}}&=\sigma (v-u)\\{\dot {v}}&=ru-v-20uw\\{\dot {w}}&=5uv-bw\\\end{aligned}}}$

With a corresponding receiver circuit that is operated by the signal . ${\displaystyle u(t)}$

${\displaystyle {\begin{aligned}{\dot {u_{r}}}&=\sigma (v_{r}-u(t))\\{\dot {v_{r}}}&=ru(t)-v_{r}-20u(t)w_{r}\\{\dot {w_{r}}}&=5u(t)v_{r}-bw_{r}\\\end{aligned}}}$

We define the status of the sender , that of the receiver and the error . It can now be shown that the error tends towards zero. To do this, we first subtract the equations of the receiver from the equations of the sender and get ${\displaystyle {\vec {d}}=(u,v,w)}$ ${\displaystyle {\vec {r}}=(u_{r},v_{r},w_{r})}$ ${\displaystyle {\vec {e}}={\vec {d}}-{\vec {r}}}$ ${\displaystyle t\to \infty }$

${\displaystyle {\begin{aligned}{\dot {e_{1}}}&=\sigma (e_{2}-e_{1})\\{\dot {e_{2}}}&=-e_{2}-20u(t)e_{3}\\{\dot {e_{3}}}&=5u(t)e_{2}-be_{3}\\\end{aligned}}}$

The resulting system is linear in , but has a time dependency on the chaotic received signal . We now construct a Lyapunov function so that the dependency cancels out. The addition of the second and third equations, where the second equation is multiplied by and the third by, results in: ${\displaystyle {\vec {e}}}$ ${\displaystyle u(t)}$ ${\displaystyle e_{2}}$ ${\displaystyle 4e_{3}}$

${\displaystyle {\begin{aligned}e_{2}{\dot {e}}_{2}+4e_{3}{\dot {e}}_{3}&=-e_{2}^{2}-20u(t)e_{2}e_{3}+20u(t)e_{2}e_{3}-4be_{3}^{2}\\={\frac {1}{2}}{\frac {\mathrm {d} }{\mathrm {d} t}}\left(e_{2}^{2}+4e_{3}^{2}\right)&=-e_{2}^{2}-4be_{3}^{2}\\\end{aligned}}}$

We therefore define the Lyapunov function as

${\displaystyle E({\vec {e}},t)={\frac {1}{2}}\left({\frac {1}{\sigma }}e_{1}^{2}+e_{2}^{2}+4e_{3}^{2}\right)}$.

${\displaystyle E}$is positive definite and it can be shown that such that is a Lyapunov function, and is therefore a globally stable fixed point, and that it decreases exponentially. ${\displaystyle {\dot {E}}\leq 0}$ ${\displaystyle E}$ ${\displaystyle {\vec {e}}=0}$ ${\displaystyle {\vec {e}}(t)}$

literature

• A. Pikovsky, M. Rosemblum, J. Kurths : Synchronization: A Universal Concept in Nonlinear Sciences . Ed .: Cambridge University Press. 1st edition. tape 12 , 2001, ISBN 0-521-53352-X , pp. 432 (English).
• JM González-Miranda: Synchronization and Control of Chaos. An introduction for scientists and engineers . Imperial College Press, 2004, ISBN 1-86094-488-4 , pp. 224 (English).
• AL Fradkov: Cybernetical physics: from control of chaos to quantum control . Springer, 2007 (preliminary Russian version: St.Petersburg, Nauka, 2003).
• E. Schöll, HG Schuster: Handbook of Chaos Control . Wiley-VCH, 2008, ISBN 978-3-527-40605-0 .

Web links

• Scholarpedia article on synchronization

Individual evidence

1. ↑ Alex Arenas, Albert Díaz-Guilera, Jürgen Kurths , Yamir Moreno, Changsong Zhou: Synchronization in complex networks . In: Physics Reports . tape 469 , no. 3 , December 2008, p. 93-153 , doi : 10.1016 / j.physrep.2008.09.002 .
2. ^ CW Wu: Synchronization in complex networks of nonlinear dynamical systems. World Scientific Publishing, Singapore 2007.
3. ^ Steven H. Strogatz : Nonlinear Dynamics and Chaos. Perseus Books Group, 2001; P. 335: “ Kevin Cuomo and Alan Oppenheim (1992, 1993) have implemented a new approach to this problem, building on Percora and Carroll's (1990) discovery of synchronized chaos ”; P. 338: “ Before their work, many people would have doubted that two chaotic systems could be made to synchronize. ”
4. ^ Peter Ashwin: Bubbling transition . In: Scholarpedia . tape 1 , no. 8 , 2006, p. 1725 , doi : 10.4249 / scholarpedia.1725 .
5. ^ Tiago Pereira: Stability of Synchronized Motion in Complex Networks . In: Adaptation and Self-Organizing Systems . December 10, 2011, arxiv : 1112.2297 .
6. ↑ Wolfgang Kinel, Ido Kanter: Secure Communication with Chaos Synchronization . In: Handbook of Chaos Control . Wiley-VCH, 2008, ISBN 978-3-527-40605-0 , pp. 303 (English): “Communication with synchronized chaotic lasers has been demonstrated by Van Wiggeren and Roy in 1998”
7. ↑ Kevin M. Cuomo, Alan V. Oppenheim: Circuit implementation of synchronized chaos with applications to communications . In: Physical Review Letters . tape 71 , no. 1 , July 5, 1993, p. 65-68 , doi : 10.1103 / PhysRevLett.71.65 .
8. ^ Proof in Steven H. Strogatz: Nonlinear Dynamics and Chaos . Perseus Books Group, 2001, p. 340