Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields
Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields
复制标题
DOI：
10.1016/j.jde.2015.07.027
复制
发表时间：
2015-12
影响因子：
2.4
通讯作者：
Haihua Liang;Joan Torregrosa
中科院分区：
文献类型：
--
作者：
Haihua Liang;Joan Torregrosa
Christopher在2006年证明了在某些假设下，Lyapunov常数关于参数的线性部分给出了初等中心的循环性。本文致力于建立一种新的方法，即并行化，来计算Lyapunov常数的线性部分。更具体地说，与其他传统机制相比，并行化可以在更短的时间内计算这些线性部分。为了证明这种方法的有效性，我们研究了一般n次多项式摄动下全纯中心z·=Iz+z2+z3+⋯+zn的循环性，对于n-≤13，从计算的角度来看，在哈密顿中心、时间可逆中心和Darboux中心中，全纯中心是获得任意阶高循环性例子的最佳候选者.对于n=4，5，…，13，证明了全纯中心的循环性至少为n2+n−2。这一结果给出了M(6)，M(7)，…的最大下界作为直接推论，我们还得到了Hilbert数H(6)≥40，H(8)≥70和H(10)≥10 8的最大下界，因为到目前为止最好的结果是H(6)≥39，H(8)≥67和H(10)≥10 0。
Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devoted to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is shown that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms. To show the power of this approach, we study the cyclicity of the holomorphic center z˙= i z+ z 2+ z 3+⋯+ z n under general polynomial perturbations of degree n, for n≤ 13. We also exhibit that, from the point of view of computation, among the Hamiltonian, time-reversible, and Darboux centers, the holomorphic center is the best candidate to obtain high cyclicity examples of any degree. For n= 4, 5,…, 13, we prove that the cyclicity of the holomorphic center is at least n 2+ n− 2. This result gives the highest lower bound for M (6), M (7),…, M (13) among the existing results, where M (n) is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n. As a direct corollary we also obtain the highest lower bound for the Hilbert numbers H (6)≥ 40, H (8)≥ 70, and H (10)≥ 108, because until now the best result was H (6)≥ 39, H (8)≥ 67, and H (10)≥ 100.