Piecewise-linear maps with heterogeneous chaos
Piecewise-linear maps with heterogeneous chaos
复制标题
具有异构混沌的分段线性映射
DOI：
10.1088/1361-6544/ac0d45
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发表时间：
2021
期刊：
影响因子：
1.7
通讯作者：
H. Takahasi and J. A. Yorke
中科院分区：
文献类型：
--
作者：
Y. Saiki;H. Takahasi and J. A. Yorke
混沌动力学可以是非常异质的，在这个意义上，在某些区域中的动力学在更多的方向上比在其他区域中不稳定。当轨迹在这些区域之间徘徊时，动力学是复杂的。我们说一个混沌不变集是异质的，当任意靠近集合的每个点都有不同的周期点，具有不同数量的不稳定维数。我们称这种动力学为异质混沌（hetero-chaos）。虽然我们相信物理系统是异混沌的，但很少有明确的例子被证明是异混沌的。在这里，我们提出了两个显式动力系统，特别是简单和易于处理的计算机。它将提供更多的直觉，以了解即使是简单的系统也可以有多复杂。我们的地图有一个密集的周期点的轨道是一维不稳定的，另一个密集的周期点的轨道是二维不稳定的。此外，它们相对于勒贝格测度是遍历的。
Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos). While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.