Introduction

According to the New Oxford American Dictionary, a definition of to synchronize is to occur at the same time or rate. This is how synchronization is usually understood in the sciences; e.g. the synchronization of fireflies [Buck and Buck 1976, Buck 1988], timing synchronization in digital circuits [Rabaey et. al 2003] and phase lock loops [Best 1984], synchronization of pendulum clocks [Bennett et. al 2001], synchronization of nano-oscillators [Mohanty 2005], and synchronization in neural networks occurring in the visual cortex [König et al 1996, Castelo-Branco et. al 1998, Engel et. al 2001] and during epilepsy [Traub and Wong 1982]. Since there is an inherent rate in the definition, the systems that synchronize are generally periodic or quasi-periodic and this type of synchronization is referred to as phase synchronization. Some recent work on phase synchronization can be found in [Mirollo and Strogatz 1990, Strogatz 2000, Shilnikov et. al 2004]. One of the first to study synchronization is Christiaan Huygens in the seventeenth century where he noticed that two pendulum clocks mounted on the same frame will synchronize after some time [Bennett et. al 2001]. At the beginning of the twentieth century, the synchronization of clocks plays an important role in Einstein's derivation of the theory of relativity [Galison 2003].

In this discussion we are talking about a stronger kind of synchronization. This concept first came about when coupled chaotic systems are studied [Fujisaka and Yamada, 1983]. Systems operating in the chaotic regime are aperiodic and exhibit sensitive dependence on initial conditions, i.e. a small change in initial conditions or parameters leads to locally divergent or uncorrelated trajectories. There is no specific fixed period or frequency in the system and thus the traditional definition of synchronization is not applicable. In this case, synchronization is meant to indicate that the behavior between two systems approaches each other. This type of synchronization is called identical synchronization or complete synchronization. This is almost the antithesis of the sensitive dependence on initial conditions property in chaotic systems. Pecora and Carroll were the first to report that surprisingly, identical synchronization is possible in two chaotic systems coupled in a master-slave configuration [Pecora and Carroll 1990]. Identical synchronization is a stronger form of synchronization than phase synchronization and as a result is easier to study and analyze. As many natural, biological and man-made systems can exhibit chaotic behavior, identical synchronization has been observed in many physical systems [Elson et. al 1998, Mosekilde et. al 2002].

We give a concise presentation on identical synchronization and describe various applications.

Basic framework and notation

We assume that the coupled systems under consideration are either ordinary differential equations for the continuous-time case or maps for the discrete-time case.

For the continuous-time case, consider a collection of n systems. The i-th system is written as

where x_i is the state vector of the i-th system. We consider coupling among the n systems such that the state equation of the coupled ensemble can be written as:

For the discrete-time case, each system is defined as

and the state equations of the coupled systems are given by:

In this case t are natural numbers. We assume that for each set of initial conditions at t₀ there exists a unique trajectory for all time t≥t₀. For an initial condition x(t₀), we denote the corresponding trajectory as x(t).

Definition 1

The coupled system Eq. (1) or Eq. (2) is said to synchronize if ||x_i - x_j|| → 0 for t→∞, all i,j and all initial conditions.

Since the synchronization occurs independent of initial conditions, this is also referred to as global synchronization. A notion of local synchronization can be defined by restricting the set of initial conditions [Wu & Chua, 1995].

An equivalent way to define synchronization is that as t→∞, the states approach the linear synchronization manifold M, defined as the set M = {(x₁,..., x_n):x_i=x_j, for all i,j}. A third way to characterize synchronization is to say that the diameter of the convex hull of x₁,..., x_n vanishes as t→∞. This third characterization will be useful in some applications.

Throughout this article we will use

to denote the aggregate vector of all the state vectors x_i of the n systems, where x_i ∈ R^m for each i.

We say a (not necessarily symmetric) real square matrix A is positive semidefinite if x^TAx≥0 for all x. We denote this as A≽0. Similarly, we write A≻0 to say that A is positive definite (x^TAx>0 for all x≠0). The vector of all 1's is denoted 1. For a matrix A with real eigenvalues, we list them in increasing order as: λ₁(A) ≤ λ₂(A) ≤ ... ≤ λ_n(A). In this case, we also use λ_min(A) and λ_max(A) to denote λ₁(A) and λ_n(A) respectively.

Next section: synchronization of two coupled nonlinear dynamical systems

References

[Bennet et al. 2002] M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld. Huygens's clocks. Proceedings of the Royal Society A, 458(2019):563--579, 2002.

[Best 1984] R. Best. Phase-locked loops, theory, design and applications. McGraw-Hill, 1984.

[Buck 1988] J. Buck. Synchronous rhythmic flashing of fireflies II. The Quarterly Review of Biology, 63:265--287, 1988.

[Buck and Buck 1976] J. Buck and E. Buck. Synchronous fireflies. Scientific American, 234:74--85, 1976.

[Castelo-Branco et al. 1998] M. Castelo-Branco, S. Neuenschwander, and W. Singer. Synchronization of visual responses between the cortex, lateral geniculate nucleus, and retina in the anesthetized cat. The Journal of Neuroscience,, 18(16):6395--6410, 1998.

[Elson et. al 1998] R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel. Synchronous behavior of two coupled biological neurons. Physical Review Letters, 81:5692--5695, 1998.

[Engel et. al 2001] A. K. Engel, P. Fries, and W. Singer. Dynamic predictions: oscillations and synchrony in top-down processing. Nature Reviews Neuroscience, 2:704--716, 2001.

[Fujisaka and Yamada 1983] H. Fujisaka and T. Yamada. Stability theory of synchronized motion in coupled-oscillator systems, Progress of Theoretical Physics, 69(1):32-47, 1983.

[Galison 2003] P. L. Galison. Einstein's Clocks, Poincaré's Maps: Empires of Time. W. W. Norton & Company, 2003.

[König et. al 1996] P. König, P. R. Roelfsema, A. K. Engel, and W. Singer. Innate microstrabismus and synchronization of neuronal activity in primary visual cortex of cats. Society for Neuroscience Abstracts, 22:282, 1996.

[Mirollo and Strogatz 1990] R. E. Mirollo and S. H. Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM Journal of Applied Mathematics, 50(6):1645--1662, 1990.

[Mohanty 2005] P. Mohanty. Nano-oscillators get it together. Nature, 437:325--326, 2005.

[Mosekilde et. al 2002] E. Mosekilde, Y. Maistrenko, and D. Postnov. Chaotic Synchronization: Applications to Living Systems. World Scientific, 2002.

[Pecora and Carroll 1990] L. M. Pecora and T. L. Carroll. Synchronization in chaotic systems. Physical Review Letters, 64(8):821--824, feb 1990.

[Rabaey et al 2003] J. M. Rabaey, A. Chandrakasan, and B. Nikolic. Digital Integrated Circuits. Prentice Hall, 2nd edition, 2003.

[Shilnikov et. al 2004] A. L. Shilnikov, L. P. Shilnikov, and D. V. Turaev. On some mathematical topics in classical synchronization. a tutorial. International Journal of Bifurcation and Chaos, 14(7):2143--2160, 2004.

[Strogatz 2000] S. H. Strogatz. From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143:1--20, 2000.

[Traub and Wong 1982] R. D. Traub and R. K. Wong. Cellular mechanism of neuronal synchronization in epilepsy. Science, 216(4547):745--747, 1982.

Copyright 2006-2009 by Chai Wah Wu.