User:Epaminondas Rosa Jr./Proposed/Synchronization of chaotic oscillators

• Dr. Epaminondas Rosa Jr., Deptarment of Physics, Illinois State University

This article is work in progress. Synchronization of Chaotic Oscillators

Chaotic synchronization can be defined as the adjustment of varying rhythms of coupled chaotic oscillators. From this viewpoint synchronization is a dynamical process rather than a state.

Examples

There are several particular cases of synchrony between two chaotic oscillators.

• Generalized synchronization
  • Two coupled chaotic oscillators are said be in generalized synchrony when there is a functional relationship connecting their dynamical variables.

• Complete (or total) synchronization
  • Consider two coupled chaotic systems with trajectories coinciding exactly with each other as time evolves. This realization, somewhat unexpected when first observed (Fujisaka and Yamada 1983, Afraimovich et al. 1986, Pecora and Carroll 1990), was then called synchronization in chaotic systems.

• Phase synchronization
  • This is the situation where two coupled chaotic oscillators keep their phases² in step with each other while their amplitudes remain uncorrelated (Rosenblum et al. 1996).

• Lag synchronization
  • If two chaotic oscillators are moving together but the phase of one of the oscillators is constantly lagging behind the phase of the other oscillator by a fixed angle less than 2\(\pi\ ,\) they are moving in a lag synchronous state.

Notes

(1) For chaotic attractors where a two-dimensional projection yields a well-defined center of rotation, the phase can be readily defined as the rotational angle with respect to a fixed reference axis. Sometimes, however, the phase of an oscillator cannot be readily defined and techniques like the Hilbert transform might be needed for studying the phase synchronous phenomenon.

References

• Afraimovich V.S., Verichev N.N., and Rabinovich M.I. (1986) Radiophys. Quantum Electron. 29, 795-802.
• Fujisaka H., and Yamada T. (1983) Prog. Theor. Phys. 69, 32-39.
• Pecora L.M. and Carroll T.L. (1990) Phys. Rev. Let. 64, 821-824.
• Rosenblum M.G., Pikvsky A.S., and Kurths J. (1996) Phys. Rev. Lett. 76, 1804-1807.