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Chaotic synchronization of coupled ergodic maps.

D. G. Sterling1

  • 1Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Boulder, Colorado 80303.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
Summary

Chaotic systems without dissipation can synchronize, though some orbits never synchronize. A modified coupling scheme ensures robust synchronization even with noise, useful for communication systems.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Statistical Mechanics

Background:

  • Most research on chaotic synchronization focuses on dissipative systems.
  • Chaotic systems lacking dissipation can also achieve synchronization.
  • Synchronization in non-dissipative systems has potential applications in communication.

Purpose of the Study:

  • Investigate the global stability of synchronization in coupled, non-dissipative ergodic maps.
  • Analyze the behavior of asynchronous orbits and synchronization times.
  • Develop a noise-robust synchronization method for communication systems.

Main Methods:

  • Coupling a family of ergodic maps on the N-torus.
  • Analyzing the stability of the synchronous state.
  • Deriving bounds for average synchronization time in 2D systems.
  • Numerical simulations in higher dimensions.
  • Proposing and testing a modified coupling scheme.

Main Results:

  • Identified a measure zero set of asynchronous orbits that never synchronize.
  • Demonstrated that typical trajectories converge to synchronization.
  • Derived bounds on average synchronization time for 2D random initial conditions.
  • Numerical evidence suggests similar bounds in higher dimensions.
  • Standard coupling is not robust to noise; a modified scheme provides robustness.

Conclusions:

  • Synchronization is possible in non-dissipative chaotic systems.
  • Asynchronous orbits exist but are rare.
  • Synchronization time can be bounded, especially in lower dimensions.
  • A modified coupling scheme enhances synchronization robustness against noise.

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