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Border-collision period-doubling scenario.

Viktor Avrutin1, Michael Schanz

  • 1Institute of Parallel and Distributed Systems (IPVS), University of Stuttgart, Universitätstrasse 38, D-70569 Stuttgart, Germany. Viktor.Avrutin@informatik.uni-stuttgart.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 28, 2004
PubMed
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This study explores a unique border-collision period-doubling bifurcation in Lorenz-type systems. It reveals a novel scenario involving pairs of border-collision and pitchfork bifurcations, leading to complex dynamical behaviors.

Area of Science:

  • Dynamical Systems Theory
  • Nonlinear Dynamics
  • Chaos Theory

Background:

  • The classical period-doubling route to chaos is well-understood.
  • Lorenz-type systems exhibit complex dynamics, often studied via Poincaré maps.
  • Border-collision bifurcations are a distinct class of bifurcations in piecewise-smooth systems.

Purpose of the Study:

  • To investigate the border-collision period-doubling bifurcation scenario in a one-dimensional dynamical system.
  • To analyze the sequence of bifurcations forming this scenario, specifically pairs of border-collision and pitchfork bifurcations.
  • To characterize emergent phenomena such as symmetry breaking/recovery and coexisting attractors.

Main Methods:

  • Utilizing a one-dimensional dynamical system model.

Related Experiment Videos

  • Representing the system using a Poincaré return map.
  • Analyzing the properties of bifurcations and attractors within the system.

    • Main Results:

    • Identified a novel bifurcation scenario: border-collision period-doubling.
    • This scenario consists of sequential pairs of border-collision and pitchfork bifurcations.
    • Observed characteristic properties including symmetry-breaking, symmetry-recovering, coexisting attractors, and noninvariant attractive sets.

    Conclusions:

    • The border-collision period-doubling scenario offers a distinct pathway to complex dynamics in Lorenz-type systems.
    • This scenario generates rich dynamical behaviors not typically seen in classical period-doubling routes.
    • Further research into piecewise-smooth systems and their bifurcations is warranted.