The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove that throughout an open region of parameter space, this family has an attractor satisfying Devaney’s definition of chaos. This strengthens the existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one-dimensional object, densely fills an open region containing the attractor. Finally, we identify a heteroclinic bifurcation, not described previously, at which the attractor undergoes a crisis and may be destroyed.
Robust Devaney chaos in the two-dimensional border-collision normal form
Note: This article is part of the Focus Issue, Non-smooth Dynamics.
I. Ghosh, D. J. W. Simpson; Robust Devaney chaos in the two-dimensional border-collision normal form. Chaos 1 April 2022; 32 (4): 043120. https://doi.org/10.1063/5.0079807
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