School of Mathematical and Computational Sciences, Massey University, Palmerston North 4410, New Zealand.
Chaos. 2022 Apr;32(4):043120. doi: 10.1063/5.0079807.
The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on R 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.
R 上所有非退化的、连续的、两段的、分段线性映射的集合可以简化为一个四参数族，称为二维边碰撞正规形。我们证明，在参数空间的一个开区域内，这个族有一个吸引子，满足德瓦尼对混沌的定义。这加强了分段线性映射中混沌鲁棒性的现有结果。我们进一步表明，鞍点固定点的稳定流形尽管是一维的，但却密集地充满了包含吸引子的开区域。最后，我们确定了一个以前没有描述过的异宿分岔，在这个分岔处，吸引子经历了一次危机并可能被破坏。
