论文标题
非线性动力学非自主阴影的一般方法
A general approach to nonautonomous shadowing for nonlinear dynamics
论文作者
论文摘要
给定一个非自主和非线性微分方程\ begin {equation} \ label {de} x'= a（t）x+f（t）x+f（t，t，x）\ quad t \ geq 0，\ end en \ end \ end {equation}在任意的banach Space $ x $上，我们为相关的线性方程组建了非常普遍的条件，以$ x $ $ x'= a = a = a（t） $ f：[0，+\ infty）\ times x \ to x $在上面的系统下满足阴影属性的适当版本。更确切地说，我们要求$ x'= a（t）x $承认一种非常一般的二分法类型，其中包括经典的双曲线行为是非常特殊的情况。此外，我们要求$ f $是第二个变量中的Lipschitz，并具有足够小的Lipschitz常数。我们的一般框架使我们能够处理以前没有获得阴影结果的各种设置。此外，我们能够恢复并完善几个已知结果。 我们还展示了如何将我们的主要结果应用于高阶微分方程的阴影属性的研究。最后，我们通过提供结果的离散时间版本来结束论文。
Given a nonautonomous and nonlinear differential equation \begin{equation}\label{DE} x'=A(t)x+f(t,x) \quad t\geq 0, \end{equation} on an arbitrary Banach space $X$, we formulate very general conditions for the associated linear equation $x'=A(t)x$ and for the nonlinear term $f:[0,+\infty)\times X\to X$ under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that $x'=A(t)x$ admits a very general type of dichotomy, which includes the classical hyperbolic behaviour as a very particular case. In addition, we require that $f$ is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting a discrete time versions of our results.
