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Stability and attractivity in discrete dynamical systems

M Martelli1, D Marshall

  • 1Mathematics Department, California State University, Fullerton 92634, USA.

Mathematical Biosciences
|July 1, 1995
PubMed
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One-dimensional discrete dynamical systems cannot have unstable attractors. However, this study constructs a two-dimensional discrete system exhibiting an unstable global attractor, challenging previous assumptions in dynamical systems research.

Area of Science:

  • Dynamical Systems Theory
  • Mathematical Physics

Background:

  • Discrete dynamical systems are fundamental models in mathematics and physics.
  • The stability of attractors is a crucial concept in understanding system behavior.
  • Previous research suggested one-dimensional systems lack unstable attractors.

Purpose of the Study:

  • To investigate the existence of unstable attractors in discrete dynamical systems.
  • To challenge the established understanding of attractor stability in one-dimensional systems.
  • To construct a novel two-dimensional discrete system with specific attractor properties.

Main Methods:

  • Theoretical analysis of one-dimensional discrete dynamical systems.
  • Construction of a C1-smooth two-dimensional discrete dynamical system.

Related Experiment Videos

  • Analysis of the stability properties of the global attractor in the constructed system.

    • Main Results:

    • Proof that one-dimensional discrete dynamical systems cannot possess unstable attractors.
    • Successful construction of a C1 two-dimensional discrete system.
    • Demonstration that the constructed two-dimensional system possesses an unstable global attractor.

    Conclusions:

    • The findings confirm the theoretical impossibility of unstable attractors in one-dimensional discrete systems.
    • The constructed two-dimensional system serves as a counterexample, demonstrating that unstable global attractors can exist in higher dimensions.
    • This work opens new avenues for exploring complex behaviors in higher-dimensional discrete dynamical systems.