一类参数非线性系统的穴盆与井喷分岔
The occurrence of riddled basins and blowout bifurcations in a parametric nonlinear system
摘要：
本文研究了平面上具有两个不同子空间不变的双参数映射族$F_{β_1,β_2}$。我们观察到，我们的模型具有两个混沌吸引子$A_i$，$i=0,1$，躺在这些不变的子空间，并确定的参数，在$A_i$有一个局部的吸引域或成为一个混沌鞍。在此基础上，在$β_1β_2$平面的开放区域内，研究了全球意义上的裂谷盆地的存在性。我们将系统半共轭到一个随机行走模型，并定义了一个分形边界来分隔两个混沌吸引子的吸引域，然后详细描述了这个混沌吸引域。我们发现，该模型经历了一系列的分支：“井喷分支”，“正常排斥的分支”和“通过创建一个新的混沌吸引子与混合盆地的分支”。数值模拟图形，以确认我们的结果的有效性。
英文摘要：
In this paper, a two parameters family $F_{β_1,β_2}$ of maps of the plane living two different subspaces invariant is studied. We observe that, our model exhibits two chaotic attractors $A_i$, $i=0,1$, lying in these invariant subspaces and identify the parameters at which $A_i$ has a locally riddled basin of attraction or becomes a chaotic saddle. Then, the occurrence of riddled basin in the global sense is investigated in an open region of $β_1β_2$-plane. We semi-conjugate our system to a random walk model and define a fractal boundary which separates the basins of attraction of the two chaotic attractors, then we describe riddled basin in detail. We show that the model undergos a sequence of bifurcations: "a blowout bifurcation", "a bifurcation to normal repulsion" and "a bifurcation by creating a new chaotic attractor with an intermingled basin". Numerical simulations are presented graphically to confirm the validity of our results.
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