Kolmogorov多圈的圈性
On the cyclicity of Kolmogorov polycycles
摘要：
本文研究了参数$μ$在开子集$Λ\subset\R^N$中变化的平面多项式Kolmogorov微分系统\[ X_μ\quad\sist{xf(x,y;μ),}{yg(x,y;μ),} \].将$X_μ$紧化为Poincaré圆盘，第一象限的边界是不变三角形$Γ$，我们假设它是一个双曲多环，对于所有$μ\inΛ.$，它的顶点正好有三个鞍点。我们感兴趣的是$Γ$在家族$\{X_μ\}_{μ\inΛ},$中的循环性，即，从$Γ$分叉的极限环的数量，因为我们扰动$μ.$在我们的主要结果中，我们定义了三个功能，发挥相同的作用，作为第一个三个李雅普诺夫量的周期性的焦点的多环的周期性。作为一个应用程序，我们研究了两个三次Kolmogorov家庭，$N=3$和$N=5$，在这两种情况下，我们能够确定所有的$μ\inΛ,$，包括那些参数的返回地图沿$Γ$的身份的多圈的循环性。
英文摘要：
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.
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