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Catastrophe of riddling

Lai1

  • 1Department of Mathematics, Department of Electrical Engineering, and Department of Physics, Center for Systems Science and Engineering Research, Arizona State University, Tempe, Arizona 85287, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|November 23, 2000
PubMed
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Deterministic perturbations destroy riddled basins, replacing them with fractal ones. This study analyzes unstable periodic orbits to reveal the dynamical mechanism behind this catastrophe and identifies novel transient chaotic behavior.

Area of Science:

  • Dynamical systems theory
  • Chaos theory
  • Nonlinear dynamics

Background:

  • Riddling bifurcations are a phenomenon in dynamical systems where basins of attraction become intricately intertwined.
  • Existing research often assumes the presence of invariant subspaces for riddling to occur.

Purpose of the Study:

  • To investigate the impact of small deterministic perturbations on riddled basins.
  • To elucidate the dynamical mechanism responsible for the destruction of riddling.
  • To identify and characterize novel transient chaotic behaviors.

Main Methods:

  • Analysis of unstable periodic orbits within the dynamical system.
  • Investigation of critical behaviors near bifurcations.
  • Numerical simulations to observe the effects of perturbations.

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Main Results:

  • Arbitrarily small deterministic perturbations typically destroy riddled basins.
  • Riddled basins are replaced by fractal basins, a phenomenon termed 'catastrophe of riddling'.
  • A novel form of transient chaotic behavior was identified during critical transitions.

Conclusions:

  • The assumption of invariant subspaces is not robust against small perturbations.
  • Unstable periodic orbits play a crucial role in the mechanism of the catastrophe of riddling.
  • The identified transient chaos offers new insights into the complex dynamics of perturbed systems.