论文标题
有条件的Lipschitz为普通微分方程阴影
Conditional Lipschitz shadowing for ordinary differential equations
论文作者
论文摘要
我们介绍了有条件的Lipschitz阴影的概念，该概念不是旨在遮蔽每个伪轨，而是属于某个规定的集合。我们建立了两种类型的足够条件，在某些非\ -auto \ - 非态的普通微分方程具有这样的属性。第一个标准适用于半线性微分方程，只要其线性部分是双曲线，并且在规定集合的邻域中的非线性很小。第二个标准要求在规定集合的附近，相对于状态变量，右侧的衍生物的对数规范在状态变量方面是统一的。结果适用于重要的模型方程类别，包括逻辑方程，该方程最近已研究其条件阴影。构建了几个示例，表明所获得的条件是最佳的。
We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain non\-auto\-nomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to the state variable is uniformly negative in a neighborhood of the prescribed set. The results are applicable to important classes of model equations including the logistic equation, whose conditional shadowing has recently been studied. Several examples are constructed showing that the obtained conditions are optimal.
