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

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.