具有零行列式的二维边界碰撞标准型
The two-dimensional border-collision normal form with a zero determinant
AI总结：
本文研究零行列式二维边界碰撞标准型随参数变化的丰富动力学，识别周期加级、锁模和混沌吸引子倍增等区域，并通过流行病与摩擦振子模型验证其分析价值。
摘要：
边界碰撞标准型是一个连续映射的分段线性族，用于描述边界碰撞分岔附近的动力学。以往大多数研究假设标准型的每一段都是可逆的，这从抽象观点来看是通有的，但在应用问题中，映射的一段常具有退化值域，对应于零行列式。这带来了简化，然而即使在二维情形下，其动力学也可能极其丰富。本文旨在广泛确定具有零行列式的二维边界碰撞标准型的动力学如何随其参数取值的不同而变化。我们识别出周期加级、周期递增、锁模以及混沌吸引子分量倍增的参数区域，并刻画了主要的分岔边界。其意图是让这些结果能够更容易、更有效地分析数学模型中的边界碰撞分岔；我们用一个流感流行病模型和两个黏滑摩擦振子模型对此进行了说明。我们还描述了三种尚待探索的新型分岔结构。
英文摘要：
The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an abstract viewpoint, but in applied problems one piece of the map often has degenerate range, corresponding to a zero determinant. This provides simplification, yet even in two dimensions the dynamics can be incredibly rich. The purpose of this paper is to determine broadly how the dynamics of the two-dimensional border-collision normal form with a zero determinant differs for different values of its parameters. We identify parameter regions of period-adding, period-incrementing, mode-locking, and component doubling of chaotic attractors, and characterise the dominant bifurcation boundaries. The intention is for the results to enable border-collision bifurcations in mathematical models to be analysed more easily and effectively, and we illustrate this with a flu epidemic model and two stick-slip friction oscillator models. We also describe three novel bifurcation structures that remain to be explored.
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