Small amplitude limit cycles and the distribution of zeros of families of analytic functions

• 成果类型: Article
• 署名作者: Brudnyi, A
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3062096
• 发表日期: 2001
• 页码: 227-243
• 关键词: differential-equations; inequalities; real

摘要

We estimate the expected number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin provided that the field in a larger neighbourhood is close enough to a linear center. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter. We obtain this inequality by methods of pluripotential theory. This inequality also implies versions of a strong law of large numbers and the central limit theorem for a probabilistic scheme associated with the distribution of zeros.