论文标题
爆炸的非自主矢量场：无限延迟方程和不变歧管
Blowing-up Nonautonomous Vector Fields: Infinite Delay Equations and Invariant Manifolds
论文作者
论文摘要
我们显示了平面，渐近自主微分方程的非自主不变流形的存在，它们具有零lyapunov光谱的平衡溶液。这些不变的歧管对应于我们使用爆破方法获得的降级方程的稳定和不稳定的歧管。更准确地说，爆炸方法扩展到非自主设置，并将原始的有限维常规微分方程转换为具有无限延迟但双曲线结构的无限差延迟方程。在用于延迟方程的不变歧管的技术构建中，我们必须仔细研究用于吹入空间中降低的时间的时间修复的效果，以确保足够的规律性。这使我们能够采用lyapunov-perron论点获得不变的歧管的存在。我们将最后一步与隐式函数参数结合在一起，以研究歧管的可不同性。最后，我们逆转了获得最初被考虑的方程式不变流形的时空变量的爆炸转换。
We show the existence of nonautonomous invariant manifolds for planar, asymptotically autonomous differential equations, that have equilibrium solutions with zero Lyapunov spectrum. These invariant manifolds correspond to the stable and unstable manifold of a desingularized equation, that we obtain by using the blow-up method. More precisely, the blow-up method is extended to the nonautonomous setting and transforms the original finite-dimensional ordinary differential equation into an infinite-dimensional delay equation with infinite delay, but hyperbolic structure. In the technical construction of the invariant manifolds for the delay equation, we have to carefully study the effect of the time reparametrization used for desingularization in the blown-up space to guarantee sufficient regularity. This allows us to employ a Lyapunov-Perron argument to obtain existence of an invariant manifold. We combine the last step with an implicit function argument to study differentiability of the manifold. Finally, we reverse the blow-up transformation of space and time variables obtaining invariant manifolds of the initially considered equation.
