Unstable dimension variability and heterodimensional cycles in the border-collision normal form

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Bifurcation Theory

Background:

• Chaotic attractors often exhibit periodic solutions with unstable manifolds of varying dimensions.
• These complex dynamics are not observed in hyperbolic attractors.
• Border-collision bifurcations generate unique dynamical phenomena in diverse applications.

Purpose of the Study:

• To highlight the existence of complex dynamical phenomena in the border-collision normal form.
• To analyze the properties of continuous, piecewise-linear maps relevant to border-collision bifurcations.
• To investigate phenomena arising from differing unstable manifold dimensions.

Main Methods:

• Explicitly identifying parameter values for heterodimensional cycles.
• Analyzing a one-parameter subfamily to study unstable dimension variability.
• Utilizing fast and accurate computations of periodic solutions for analysis.

Main Results:

• Demonstrated the existence of heterodimensional cycles in the border-collision normal form.
• Argued that heterodimensional cycles can be dense in parameter space.
• Identified key bifurcations associated with unstable dimension variability.

Conclusions:

• The border-collision normal form provides a valuable testbed for studying complex dynamics.
• Piecewise-linear maps allow for relatively exact analysis of chaotic phenomena.
• Unstable dimension variability leads to a rich zoo of dynamical behaviors.