Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields
Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields
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DOI：
10.1142/s0218127415501503
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发表时间：
2015-10
期刊：
影响因子：
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通讯作者：
O. Diaconescu;D. Schlomiuk;N. Vulpe
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文献类型：
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作者：
O. Diaconescu;D. Schlomiuk;N. Vulpe
本文考虑一类gcd（p，q）= 1的真实的二次微分系统，其不变线的总重数为4和2，且有一个真实的无穷奇点.我们首先构造紧标准型的类，以便包括极限点在12维参数空间的这个类。接下来我们构造这些紧化标准型的分歧图。这些图包含许多重复的相图，我们表明，这些是由于许多对称性下的集团行动。为了保留动力学的本质，我们最终在仿射变换和时间同构的群G = Aff（2，λ）× λ * 的作用下构造商空间，并将相图放置在这些商空间中。最后的图只保留了必要的信息，以捕捉参数空间中的运动下的动力学，以及在这个群体的行动。本文还给出了二次系统仿射线重数k不变的充要条件。
In this article, we consider the class of all real quadratic differential systems , with gcd(p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.