ABSTRACT

This paper investigates the limit cycles for planar differential system $\dot{x}=\lambda x-y+P_n(x,y),\dot{y}=x+\lambda y+Q_n(x,y),$where $P_n(x,y),Q_n(x,y)$ are homogeneous polynomials of degree n. Using the method of upper and lower solutions we show that this system has a unique limit cycle under some suitable conditions. Moreover, by employing the monotone iterative technique, we provide a computational method to obtain the expression, or the asymptotical one, of this limit cycle in the polar form. The error estimation of the asymptotical solutions is also provided.

中文翻译：

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基于单调迭代技术的平面多项式微分系统极限环的唯一性及表达

摘要

本文研究了平面微分系统的极限环$\dot{X}=\lambda X-\mathrm{是的}+{磷}_n(X,\mathrm{是的}),\dot{\mathrm{是的}}=X+\lambda \mathrm{是的}+{问}_n(X,\mathrm{是的}),$在哪里${磷}_n(X,\mathrm{是的}),{问}_n(X,\mathrm{是的})$是n次齐次多项式。使用上下解法我们证明了该系统在某些合适的条件下具有唯一的极限环。此外，通过采用单调迭代技术，我们提供了一种计算方法来获得极坐标形式的这个极限环的表达式或渐近表达式。还提供了渐近解的误差估计。