A Hamiltonian with periodic orbits having several delays
A Hamiltonian with periodic orbits having several delays
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DOI：
10.1016/j.jde.2005.08.013
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发表时间：
2006-03
影响因子：
2.4
通讯作者：
Solomon Jekel;Christopher Johnston
中科院分区：
文献类型：
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作者：
Solomon Jekel;Christopher Johnston
1974年Kaplan和Yorke引入了具有任意数量延迟的某延迟方程，x ' (t)=-f(x(t-1))-f(x(t-2))-⋯-f(x(t-n))，推测当f是实数的奇同胚且在原点和无穷远处可微时，它具有周期解。通过将延迟方程与Rn+1上的向量场耦合，证明了当f在0和∞处的导数满足一定条件时，存在一个延迟和两个延迟版本。我们发现耦合向量场的闭合轨道出现在两个超平面场的交点上，这两个超平面场在流动和变形下都是不变的。通过分析线性向量场的性质，在满足Kaplan和Yorke的自然推广条件时，给出了广义时滞方程[n+12]周期解的初等构造。
In 1974 Kaplan and Yorke introduced a certain delay equation with an arbitrary number of delays, x′(t)=-f(x(t-1))-f(x(t-2))-⋯-f(x(t-n)), conjecturing that it has periodic solutions when f is an odd homeomorphism of the reals which is differentiable at the origin and infinity. By coupling the delay equation to a vector field on Rn+1they were able to prove, when certain conditions on the derivative of f at 0 and ∞ are satisfied, one delay and two delay versions. We find that the closed orbits of the coupled vector field occur at the points of intersection of two hyperplane fields which are invariant under the flow and invariant under deformation to a linear vector field. By analyzing properties of the linear vector field we are able to give an elementary construction of [n+12] periodic solutions to the general delay equation when conditions naturally extending those of Kaplan and Yorke are satisfied.