A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents
A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents
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DOI：
10.1016/j.jde.2012.02.002
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发表时间：
2012-05
影响因子：
2.4
通讯作者：
T. S. Doan;D. Karrasch;T. Y. Nguyen;S. Siegmund
中科院分区：
文献类型：
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作者：
T. S. Doan;D. Karrasch;T. Y. Nguyen;S. Siegmund
线性微分方程 x˙=A(t)x, t∈[t−,t+] 的双曲性概念被定义为统一不同的现有概念，如有限时间 Lyapunov 指数（Haller, 2001, [13], Shadden et al., 2005, [24]）、一致或 M-双曲性（Haller, 2001, [13], Berger et al., 2009，[6]）和（t−，（t+−t−））-二分法（Rasmussen，2010，[21]）。描述了它与二分谱（Sacker and Sell，1978，[23]，Siegmund，2002，[26]）、D-双曲性（Berger et al.，2009，[6]）和特征值实部（在 A 为常数的情况下）的关系。我们证明了谱定理并提供了谱区间的近似结果。
A hyperbolicity notion for linear differential equations x˙=A(t)x, t∈[t−,t+], is defined which unifies different existing notions like finite-time Lyapunov exponents (Haller, 2001, [13], Shadden et al., 2005, [24]), uniform or M-hyperbolicity (Haller, 2001, [13], Berger et al., 2009, [6]) and (t−,(t+−t−))-dichotomy (Rasmussen, 2010, [21]). Its relation to the dichotomy spectrum (Sacker and Sell, 1978, [23], Siegmund, 2002, [26]), D-hyperbolicity (Berger et al., 2009, [6]) and real parts of the eigenvalues (in case A is constant) is described. We prove a spectral theorem and provide an approximation result for the spectral intervals.