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Robust Devaney chaos in the two-dimensional border-collision normal form.

I Ghosh1, D J W Simpson1

  • 1School of Mathematical and Computational Sciences, Massey University, Palmerston North 4410, New Zealand.

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|April 30, 2022
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Summary
This summary is machine-generated.

This study reveals that a specific family of maps exhibits chaotic attractors, enhancing our understanding of chaos robustness. A novel bifurcation is identified where these attractors can be destroyed.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Piecewise-linear maps are fundamental in dynamical systems.
  • Border-collision bifurcations are critical phenomena in these maps.
  • Understanding the conditions for chaos is essential in nonlinear dynamics.

Purpose of the Study:

  • To investigate the chaotic dynamics of the two-dimensional border-collision normal form.
  • To analyze the properties of attractors within this parameter space.
  • To identify novel bifurcations and their impact on chaotic behavior.

Main Methods:

  • Reduction of map families to a normal form.
  • Analysis of attractors using Devaney's definition of chaos.
  • Investigation of stable manifold properties.
  • Identification of heteroclinic bifurcations.

Main Results:

  • The two-dimensional border-collision normal form exhibits chaotic attractors in an open parameter region.
  • The stable manifold of a saddle fixed point densely fills the region containing the attractor.
  • A new heteroclinic bifurcation is identified, leading to attractor crisis.

Conclusions:

  • Chaos is robust in this family of piecewise-linear maps.
  • The geometric properties of stable manifolds play a crucial role in chaotic dynamics.
  • Attractor destruction via crisis is a significant phenomenon in this system.