论文标题
二维慢速系统中的nilpotent接触点的分形编成
Fractal codimension of nilpotent contact points in two-dimensional slow-fast systems
论文作者
论文摘要
在本文中，我们介绍了nilpotent接触点$ p $的分形编织概念，以$λ=λ_0$，平滑的平面慢$ - $ fast Systems $ x_ {ε，λ} $时，当接触订单$ n_ {λ_0}（λ_0}（λ_0}（p）$ p $ $ p $）$ s $ $ pug $ s_ v $ s_ / $ p $具有有限的慢速发散，即，$ s_ {λ_0}（p）\ leq 2（n_ {λ_0}（p）（p）-1）$。 $ p $的分形成是对（Dumortier and Roussarie（2009））的liénard类型的慢速hopf点的传统的概括的概括，并且它是本质上定义的，即可以直接计算它而无需首先将系统带入正常形式。分形成词的概念的内在性质源于点的分形序列的Minkowski维度，这些序列使用so $ - $称为输入$ - $ $退出关系定义在$ p $接近$ p $，而慢速差异差异积分。我们将我们的方法应用于缓慢的$ - $ fast Hopf点，并阅读其退化（即第一个非零lyapunov数量）以及从其分形代码直接的hopf点附近的极限循环的数量。我们通过使用一个简单的公式来计算分形成曲线，以数字上的一些有趣的示例来证明我们的结果。我们通过使用一个简单的公式来计算分形成曲线，以数字上的一些有趣的示例来证明我们的结果。
In this paper we introduce the notion of fractal codimension of a nilpotent contact point $p$, for $λ=λ_0$, in smooth planar slow$-$fast systems $X_{ε,λ}$ when the contact order $n_{λ_0}(p)$ of $p$ is even, the singularity order $s_{λ_0}(p)$ of $p$ is odd and $p$ has finite slow divergence, i.e., $s_{λ_0}(p)\leq 2(n_{λ_0}(p)-1)$. The fractal codimension of $p$ is a generalization of the traditional codimension of a slow-fast Hopf point of Liénard type, introduced in (Dumortier and Roussarie (2009)), and it is intrinsically defined, i.e., it can be directly computed without the need to first bring the system into its normal form. The intrinsic nature of the notion of fractal codimension stems from the Minkowski dimension of fractal sequences of points, defined near $p$ using the so$-$called entry$-$exit relation, and slow divergence integral. We apply our method to a slow$-$fast Hopf point and read its degeneracy (i.e., the first nonzero Lyapunov quantity) as well as the number of limit cycles near such a Hopf point directly from its fractal codimension. We demonstrate our results numerically on some interesting examples by using a simple formula for computation of the fractal codimension. We demonstrate our results numerically on some interesting examples by using a simple formula for computation of the fractal codimension.
