拟线性奇摄动系统的闭轨的分歧

Bifurcation of Closed Path of Quasi-linear Singularly Perturbed Systems

本文作为文〔3〕的推广。考虑了一般的拟线性奇摄动系统,其退化系统有一族闭轨。讨论了在何种条件下,相应的完全系统才能产生孤立的闭轨。

The conditions for the general quasilinear singularly perturbed system (1.1)to have isolated closed paths have been found when the corresponding degenerate system (1.2) has a family of closed paths (2.5).The main result here is the following theorem.Theorem: Under the hypotheses(1),(2),(3)and(4), if a~*∈R and M~*∈R^(m-1) satisfy the equation(2.8)and the inequality(2.9), then there exists such a function α(ε) continuous on [0,δ](δ>0, small enough) with a(0)=a~*,for any ε∈[0,δ] and near the closed path Γ_0: x=x_0(t)Φ(t)M~*,the system (1.1) has a unique closed path Γ_e:x=x(t,ε),y=y(t,ε)of period (1+εα(ε)) with (2.10) being valid uniformlY in (-∞,∞).Conversely, if the system (1.1) has such a closed path Γ_e:x=x(t,ε), y=y(t,ε)of period (1+εα(ε))with α(ε) being a function continuons on [0,δ],(2.10) holds uniformly in (-∞,∞), then α~*α(0) and M~* satisfy the equation(2.8)with P(α,M) defined by(2.7).