Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Phys Rev E. 2017 Oct;96(4-1):042213. doi: 10.1103/PhysRevE.96.042213. Epub 2017 Oct 19.
While structures and bifurcations controlling tracer particle transport and mixing have been studied extensively for systems with only stretching-and-folding, and to a lesser extent for systems with only cutting-and-shuffling, few studies have considered systems with a combination of both. We demonstrate two bifurcations for nonmixing islands associated with elliptic periodic points that only occur in systems with combined cutting-and-shuffling and stretching-and-folding, using as an example a map approximating biaxial rotation of a less-than-half-full spherical granular tumbler. First, we characterize a bifurcation of elliptic island containment, from containment by manifolds associated with hyperbolic periodic points to containment by cutting line tangency. As a result, the maximum size of the nonmixing region occurs when the island is at the bifurcation point. We also demonstrate a bifurcation where periodic points are annihilated by the cutting-and-shuffling action. Chains of elliptic and hyperbolic periodic points that arise when invariant tori surrounding an elliptic point break up [according to Kolmogorov-Arnold-Moser (KAM) theory] can annihilate when they meet a cutting line. Consequently, the Poincaré index (a topological invariant of smooth systems) is not preserved.
虽然仅具有拉伸-折叠的系统以及在较小程度上仅具有切割-混合的系统中，已经广泛研究了控制示踪粒子输运和混合的结构和分支，但很少有研究考虑到同时具有这两种情况的系统。我们使用一个近似于半满球形颗粒摇床的双轴旋转的映射作为示例，展示了与椭圆周期点相关的非混合岛屿的两个分支，这些分支仅在同时具有切割-混合和拉伸-折叠的系统中出现。首先，我们描述了椭圆岛包含的分支，从与双曲周期点相关的流形包含到切割线相切的包含。因此，当岛位于分支点时，非混合区域的最大尺寸发生。我们还展示了一个分支，其中周期性点被切割-混合作用消除。当围绕椭圆点的不变环破裂时出现的椭圆和双曲周期点链[根据Kolmogorov-Arnold-Moser (KAM)理论]在遇到切割线时可能会被消除。因此，Poincaré 指标（光滑系统的拓扑不变量）不被保留。
