论文标题
关于kolmogorov polycycles的循环性
On the cyclicity of Kolmogorov polycycles
论文作者
论文摘要
在本文中，我们研究了平面多项式kolmogorov的差异系统\ [ X_μ\ Quad \ sist {Xf（X，Y;μ），} {Yg（X，Y;μ），} \]，参数$μ$在开放子集$λ\ subset \ r^n $中变化。将$x_μ$ comx_μ$交给庞加莱碟片，第一个象限的边界是一个不变的三角形$γ$，我们认为它是一个夸张的一层多环，正好在所有$μ\inλ$的顶点上有三个鞍点。也就是说，从$γ$分叉的限制周期的数量作为我们扰动$μ。在我们的主要结果中，我们定义了三个函数，这些功能对于多循环的循环性起着相同的作用，因为多环的循环性是前三个lyapunov量，用于焦点的环境。作为一个应用程序，我们研究了两个立方kolmogorov家族，其中$ n = 3 $和$ n = 5 $，在这两种情况下，我们都能确定所有$μ\inλcle的多环的循环性，包括$γ$的返回图的参数，包括标识。
In this paper we study planar polynomial Kolmogorov's differential systems \[ X_μ\quad\sist{xf(x,y;μ),}{yg(x,y;μ),} \] with the parameter $μ$ varying in an open subset $Λ\subset\R^N$. Compactifying $X_μ$ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle $Γ$, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all $μ\inΛ.$ We are interested in the cyclicity of $Γ$ inside the family $\{X_μ\}_{μ\inΛ},$ i.e., the number of limit cycles that bifurcate from $Γ$ as we perturb $μ.$ In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with $N=3$ and $N=5$, and in both cases we are able to determine the cyclicity of the polycycle for all $μ\inΛ,$ including those parameters for which the return map along $Γ$ is the identity.
