Abstract
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
Highlights
Because of its positive Lyapunov exponents, an isolated chaotic system synchronizes with nothing else
We review some of the history and early work in the area of synchronization in chaotic systems
For two separated chaotic systems, trajectories starting at close initial condition will diverge at first
Summary
Because of its positive Lyapunov exponents, an isolated chaotic system synchronizes with nothing else. The synchronization of two or more chaotic systems is another version of the usual thought experiment where one imagines two initial conditions starting close to each other in phase space. For two separated chaotic systems, trajectories starting at close initial condition will diverge at first. We realize that two identical uncoupled chaotic systems have twice as many positive Lyapunov exponents as either of the systems by themselves. In our original synchronizing system, we went from a total of two positive Lyapunov exponents in the combination of two uncoupled systems to one positive Lyapunov exponent in the pair of coupled systems. As we go through this article, we will show how this approach matured and became general with a geometric way to view the synchronization in large networks of coupled oscillators
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