Border-Collision bifurcations in 1D Piecewise-Linear Maps and Leonov's Approach
Border-Collision bifurcations in 1D Piecewise-Linear Maps and Leonov's Approach
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一维分段线性映射中的边界碰撞分叉和列昂诺夫方法
DOI：
10.1142/s021812741002757x
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
M. Schanz
中科院分区：
文献类型：
--
作者：
L. Gardini;F. Tramontana;V. Avrutin;M. Schanz
50年前(1959年)，在Leonov的一系列出版物中，对分段线性不连续一维映射中出现的嵌套加周期分叉结构进行了详细的分析研究。列昂诺夫获得的结果鲜为人知，尽管它们允许以一种优雅且比通常更有效的方式进行边界碰撞分支子空间的分析计算。在这项工作中，我们回顾了列奥诺夫的方法，并解释了它为什么有效。此外，我们略微改进了该方法，避免了不必要的坐标变换，并证明了该方法不仅适用于边界碰撞分叉曲线的计算。
50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonov's approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves.