救助嵌入映射中的聚类、混沌与危机
Clustering, Chaos and Crisis in a Bailout Embedding Map
摘要：
本文研究二维不可压缩流中惯性质点的动力学。粒子动力学模型由四维耗散救助嵌入映射的基流，这是由二维面积保持映射表示。嵌入式地图的相图是丰富和有趣的气溶胶制度，其中的颗粒的密度大于基流，以及气泡制度，其中的颗粒密度小于基流。嵌入映射显示了三种类型的动力学行为，周期轨道，混沌结构和混合区域。因此，嵌入映射可以针对周期性轨道，以及在一定值的耗散参数的气溶胶和气泡制度的混沌结构。四维图的分叉图有助于识别这种结构的区域，在气溶胶的特殊区域，可以看到吸引子合并和扩大的危机。在危机时刻，两个周期为10的吸引子合并并同时加宽为一个混沌吸引子。在相图的某些点上可以看到危机引起的不稳定性。在危机爆发前的特征时间显示出作为耗散参数的函数的幂律行为。虽然气泡的分叉图看起来与气溶胶的分叉图相似，但气泡没有看到这样的危机状态。我们的研究结果可以在不同的应用环境中的杂质的动力学的影响。
英文摘要：
We study the dynamics of inertial particles in two dimensional incompressible flows. The particle dynamics is modelled by four dimensional dissipative bailout embedding maps of the base flow which is represented by 2-d area preserving maps. The phase diagram of the embedded map is rich and interesting both in the aerosol regime, where the density of the particle is larger than that of the base flow, as well as the bubble regime, where the particle density is less than that of the base flow. The embedding map shows three types of dynamic behaviour, periodic orbits, chaotic structures and mixed regions. Thus, the embedding map can target periodic orbits as well as chaotic structures in both the aerosol and bubble regimes at certain values of the dissipation parameter. The bifurcation diagram of the 4-d map is useful for the identification of regimes where such structures can be found. An attractor merging and widening crisis is seen for a special region for the aerosols. At the crisis, two period-10 attractors merge and widen simultaneously into a single chaotic attractor. Crisis induced intermittency is seen at some points in the phase diagram. The characteristic times before bursts at the crisis show power law behaviour as functions of the dissipation parameter. Although the bifurcation diagram for the bubbles looks similar to that of aerosols, no such crisis regime is seen for the bubbles. Our results can have implications for the dynamics of impurities in diverse application contexts.
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