四分区分段线性映射中的不可逆性和分岔现象
Noninvertibility and Bifurcation Phenomena in a Four-Partitions Piecewise Linear Map
AI总结：
研究由四个分区定义的二维分段线性映射的全局动力学与分岔结构，分析不动点稳定性，揭示周期2动力学由翻转分岔引发，更高周期周期通过折叠边界碰撞分岔出现，还研究了退化参数区域，证明吸收不变区间存在及确定混沌动力学参数区。
AI中文摘要：
本文研究了一个由四个分区定义的二维分段线性映射的全局动力学和分岔结构，两个状态变量都涉及绝对值项。我们分析了不动点的稳定性，证明了2周期动力学的出现是由翻转分岔引发的。对这些周期的详细分析揭示了一个独特的分支，它以两种拓扑不同的构型出现，通过边界碰撞分岔相连。此外，我们表明更高周期的周期（3和4）通过折叠边界碰撞分岔出现在稳定-鞍点对中，导致由鞍轨稳定流形组织的多稳态区域。最后，我们研究了系统简化为一维不连续映射的退化参数区域。在这种情况下，我们证明了吸收不变区间的存在，并确定了表现出鲁棒混沌动力学的参数区域。
英文摘要：
This paper investigates the global dynamics and bifurcation structures of a two-dimensional piecewise linear map defined by four partitions, involving absolute value terms for both state variables. We analyze the stability of fixed points and demonstrate that the emergence of period-2 dynamics is initiated by a Flip bifurcation. A detailed analysis of these cycles reveals a unique branch manifesting in two topologically distinct configurations, connected via a Border Collision Bifurcation. Furthermore, we show that higher-period cycles (period 3 and 4) emerge in stable-saddle pairs through Fold Border Collision Bifurcations, leading to regions of multistability organized by the stable manifolds of saddle orbits. Finally, we examine the degenerate parameter regime where the system reduces to a one-dimensional discontinuous map. In this case, we prove the existence of an absorbing invariant interval and identify parameter regions exhibiting robust chaotic dynamics.
