Lyapunov Functions for General Nonuniform Dichotomies
Lyapunov Functions for General Nonuniform Dichotomies
复制标题
DOI：
10.1007/s00032-013-0198-y
复制
发表时间：
2013-03
影响因子：
1.7
通讯作者：
L. Barreira;Jifeng Chu;C. Valls
中科院分区：
文献类型：
--
作者：
L. Barreira;Jifeng Chu;C. Valls
对于非自治线性方程x ' = A(t)x，我们给出了用Lyapunov函数表示的指数行为存在性的完整刻画。特别地，我们得到了一个反定理，给出了每个指数二分法的显式李雅普诺夫函数。我们工作的主要新颖之处在于我们考虑了一种非常一般的非均匀指数二分法。这包括均匀指数二分类，非均匀指数二分类和多项式二分类。我们还考虑了二分类的均匀部分和非均匀部分的不同增长率的情况。作为我们工作的一个应用，我们以一种非常直接的方式建立了非均匀指数二分类在足够小的线性扰动下的鲁棒性。
For nonautonomous linear equationsx′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.