Quadratic Systems with Center and Their Perturbations
Quadratic Systems with Center and Their Perturbations
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DOI：
10.1006/jdeq.1994.1049
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发表时间：
1994-04
影响因子：
2.4
通讯作者：
H. Zoladek
中科院分区：
文献类型：
--
作者：
H. Zoladek
本文研究平面z =（i + λ）z + Az 2 + B上的二次系统|z| 2 + Cz 2，z = x + iy。给出了中心条件λ = 0且B = 0或A = − 1 2，B = 1或A = 1，B = 1，C = C或A = 2，B = 1，|C| = 1，以及Bautin定理，即从中心或焦点分叉的小振幅极限环的数量不大于3。我们给出了有中心系统的分叉图和相图，并精确描述了每种情况下的周期性。本文指出了Bautin工作中的一个错误，并回答了有关焦点数的一些问题。我们还研究了全球（不小）极限环的中心的情况下，两个不变的线（Lotka-Volterra系统）使用阿贝尔积分的小扰动。我们证明了相应的阿贝尔积分的零点个数是0，1或2。本文研究了具有二重中心的系统在小扰动下的极限环问题（当两个中心条件成立时）。该问题等价于某些多项式系数积分的零点问题。
Abstract We study the quadratic systems on a plane z = (i + λ)z + Az2 + B |z|2 + Cz2, z = x + iy. We give a simple algebraic proof of the center conditions, λ = 0 and B = 0 or A = − 1 2 , B = 1 or A = Ā, B = 1, C = C or A = 2, B = 1, |C| = 1, and of the theorem of Bautin that the number of small-amplitude limit cycles bifurcating from a center or a focus is not greater than 3. We present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case. Here one mistake in Bautin′s work is revealed and some questions concerning focus numbers are answered. We study also global (not small) limit cycles for a small perturbation of the center case with two invariant lines (the Lotka-Volterra system) using Abelian integrals. We show that the number of zeroes of the corresponding Abelian integral is 0, 1, or 2. Some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two-fold center (when two center conditions hold). The problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients.