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Related Experiment Videos

Dangerous bifurcation at border collision: when does it occur?

Anindita Ganguli1, Soumitro Banerjee

  • 1Center for Theoretical Studies, Indian Institute of Technology, Kharagpur-721302, India.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2005
PubMed
Summary

Dangerous bifurcations occur when a stable fixed point loses its basin of attraction at a border collision, causing unbounded orbits. This study explains this phenomenon and provides analytical conditions for its occurrence.

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Area of Science:

  • Dynamical Systems
  • Nonlinear Dynamics
  • Bifurcation Theory

Background:

  • Border collision bifurcations in piecewise smooth maps can lead to unexpected dynamics.
  • A stable fixed point may coexist with unbounded orbits due to shrinking basins of attraction.

Purpose of the Study:

  • To explain the phenomenon of

Main Methods:

  • Analysis of piecewise smooth maps.
  • Investigation of fixed point stability and basin of attraction dynamics.
  • Development of analytical conditions for bifurcation occurrence.

Main Results:

  • A detailed explanation of how stable fixed points can lead to unbounded orbits at bifurcation points.
  • Identification of specific parameter conditions that trigger these dangerous bifurcations.

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Conclusions:

  • Dangerous bifurcations represent a critical phenomenon in piecewise smooth dynamical systems.
  • Understanding these conditions is crucial for predicting and controlling system behavior.