SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1356-1376, January 2020.
Motivated by a problem in the management of ecological populations, we study the bifurcation structure known as period adding structure for a family of one-dimensional bimodal piecewise linear maps. This structure is rather degenerate compared to the general case usually addressed in the literature. The degeneracy affects both the type of border collision bifurcations constituting the bifurcation structure and the number and location of the bifurcation points in the parameter space. We provide rigorous theoretical results that yield a complete description of the degenerate border collision bifurcations and a full determination of the bifurcation structure. This allows us to extend partial results previously reported about a similar problem. From an ecological point of view, we provide numerical simulations showing potential risks and opportunities associated with the bifurcation structure studied here. Moreover, we provide examples of applications of our results to some well-known population models, showing that the period adding structure ranges from very simple to very intricate.

中文翻译：

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一维双峰分段线性映射的简并周期加分叉结构

SIAM应用数学杂志，第80卷，第3期，第1356-1376页，2020年1月。
受生态种群管理问题的影响，我们研究了一维双峰分段线性图族的分叉结构，称为周期相加结构。与文献中通常提到的一般情况相比，这种结构简直是简陋的。简并性既影响构成分叉结构的边界碰撞分叉的类型，也影响参数空间中分叉点的数量和位置。我们提供了严格的理论结果，可完整描述退化的边界碰撞分叉以及对分叉结构的完整确定。这使我们能够扩展先前报告的有关类似问题的部分结果。从生态的角度来看，我们提供了数值模拟，显示了与此处研究的分叉结构相关的潜在风险和机会。此外，我们提供了将结果应用到一些著名的人口模型的示例，显示出周期增加结构的范围从非常简单到非常复杂。