逆积分因子与Poincaré映射
The inverse integrating factor and the Poincaré map
摘要：
研究了具有定义在正则轨道邻域内的解析逆积分因子的平面实解析微分系统。我们发现，逆积分因子定义了一个常微分方程的过渡映射沿轨道。当正则轨道为极限环时，我们可以用逆积分因子确定其对应的Poincaré返回映射。特别地，我们证明了极限环的重数与其上的逆积分因子的重数为零是一致的，并将这一结果应用于同宿环分支的研究。我们只考虑临界点是双曲鞍点且Poincaré返回映射不是恒等式的同宿环。通过对鞍点邻域内逆积分因子的局部分析，我们可以用逆积分因子在其上的消失重数来确定多圈的循环性，我们的结果也适用于同宿圈的鞍点可线性化的特殊情况，即，在这种情况下，该图形的循环性的界限不能通过代数方法确定。
英文摘要：
This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the inverse integrating factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincaré return map is not the identity. A local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.
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