论文标题
较高维度的关键点，I：Lozi家族中周期性轨道创建的反向顺序
Critical points in higher dimensions, I: Reverse order of periodic orbit creations in the Lozi family
论文作者
论文摘要
我们介绍了一个重新归一化模型，该模型解释了离散时间连续动力系统的行为如何随着系统的尺寸而变化。该模型适用于一些二维系统，包括Hénon和Lozi地图。在这里，我们专注于维护Lozi家族的定向，这是一个连续的分段仿射图的两参数家族，并将家族视为帐篷家族的扰动。 首先，我们给出一个新的证明，可以通过使用符号动力来对所有周期轨道进行分类。对于每个编码，相关的周期性轨道在分析上取决于参数的存在。当发生边界碰撞时，会发生或歼灭周期性轨道。接下来，我们证明某些类型的周期轨道的分叉参数在参数空间中形成分析曲线。这改善了Ishii定理（1997）。最后，我们使用模型和分析曲线来证明，当Lozi家族与帐篷家族的任意亲近时，定期轨道创建的顺序会逆转。这表明在轨道创作上的强迫关系（Guckenheimer 1979和Collett and Eckmann 1980）在二维上崩溃了。实际上，即使家庭任意接近一维，强迫关系也没有延续到二维。
We introduce a renormalization model which explains how the behavior of a discrete-time continuous dynamical system changes as the dimension of the system varies. The model applies to some two-dimensional systems, including Hénon and Lozi maps. Here, we focus on the orientation preserving Lozi family, a two-parameter family of continuous piecewise affine maps, and treat the family as a perturbation of the tent family from one to two dimensions. First, we give a new prove that all periodic orbits can be classified by using symbolic dynamics. For each coding, the associated periodic orbit depends on the parameters analytically on the domain of existence. The creation or annihilation of periodic orbits happens when there is a border collision bifurcation. Next, we prove that the bifurcation parameters of some types of periodic orbits form analytic curves in the parameter space. This improves a theorem of Ishii (1997). Finally, we use the model and the analytic curves to prove that, when the Lozi family is arbitrary close to the tent family, the order of periodic orbit creation reverses. This shows that a forcing relation (Guckenheimer 1979 and Collett and Eckmann 1980) on orbit creations breaks down in two dimensions. In fact, the forcing relation does not have a continuation to two dimensions even when the family is arbitrary close to one dimension.
