Regularity of invariant graphs for forced systems
Regularity of invariant graphs for forced systems
复制标题
受迫系统不变图的正则性
DOI：
10.1017/s0143385799126555
复制
发表时间：
1999
影响因子：
0.9
通讯作者：
J. Stark
中科院分区：
文献类型：
--
作者：
J. Stark
非线性动力学的许多应用涉及受迫系统。我们考虑的情况下，对于一个固定的输入驱动系统的收缩，这是例如在某些类别的过滤器的情况下，并在同步的研究。当这种收缩是均匀的，它可以很容易地表明，存在一个全局吸引不变集，这是一个从驱动状态空间到驱动状态空间的函数的图形;这是一个更一般的系统惯性流形的众所周知的概念的特殊情况。如果驱动状态空间是一个流形，并且收缩足够强，则这个不变集是一个正常的双曲流形，因此是光滑的。本文的目的是在两个方向上扩展这个结果：第一，我们只有一致的压缩为一个紧凑的不变集的输入状态，第二，其中的收缩率是不一致的（因此定义的李雅普诺夫指数和类似的数量）。在这两种情况下，不变图只定义在输入空间的闭子集上，因此我们需要为这样的函数定义一个适当的光滑度概念。这是根据惠特尼扩张定理完成的：如果一个函数满足这个定理的条件，那么它被认为是惠特尼光滑的，因此可以扩展到整个输入空间的光滑函数。
Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space; this is a special case of the well known concept of an inertial manifold for more general systems. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set is a normally hyperbolic manifold, and hence smooth. The aim of this paper is to extend this result in two directions: firstly, where we only have uniform contraction for a compact invariant set of input states, and secondly where the contraction rates are non-uniform (and hence defined by Liapunov exponents and analogous quantities). In both cases the invariant graph is only defined over closed subsets of the input space, and hence we need to define an appropriate notion of smoothness for such functions. This is done in terms of the Whitney extension theorem: a function is considered Whitney smooth if it satisfies the conditions of this theorem and hence can be extended to a smooth function of the whole input space.