Heteroclinic Orbits for Retarded Functional Differential Equations
Heteroclinic Orbits for Retarded Functional Differential Equations
复制标题
DOI：
10.1016/0022-0396(86)90032-x
复制
发表时间：
1986-11
影响因子：
2.4
通讯作者：
J. Hale;Xiaoyan Lin
中科院分区：
文献类型：
--
作者：
J. Hale;Xiaoyan Lin
假设 Г 是 C k 泛函微分方程 x ̇ (t)= f (xi) 的异宿轨道，其中 α 极限集 α (Г) 和 ω 极限集 ω (Г) 为双曲平衡点或周期轨道。给出了C k 中存在接近f 的f̃tf 的充要条件，且x ̇ (t)=\̃ tf (x t) 具有接近于Γ 的异宿轨道\̃ gG。轨道\̃ gG 由有限数量的分岔函数 G (β, f ̃)ε R d∗, βε R d+ 1 的零点获得。 Γ 的横向性由导数 D β G 的非简并性来表征。还表明，\̃ tf 具有接近 Γ 的异宿轨道，位于有限余维= max {0,− indΓ} 的 C k 子流形上或者在它的闭包上，其中 ind Г 是 Г 的索引。
Suppose Γ is a heteroclinic orbit of a C k functional differential equation x ̇ (t)= ƒ (x i) with α-limit set α (Γ) and ω-limit set ω (Γ) being either hyperbolic equilibrium points or periodic orbits. Necessary and sufficient conditions are given for the existence of an f̃tf close to ƒ in C k with the property that x ̇ (t)=\̃ tf (x t) has a heteroclinic orbit\̃ gG close to Γ. The orbits\̃ gG are obtained from the zeros of a finite number of bifurcation functions G (β, f ̃)∈ R d∗, β∈ R d+ 1. Transversality of Γ is characterized by the nondegeneracy of the derivative D β G. It is also shown that the\̃ tf which have heteroclinic orbits near Γ are on a C k submanifold of finite codimension= max {0,− indΓ} or on the closure of it, where ind Γ is the index of Γ.