Complete description of the Lyapunov spectra of continuous families of linear differential systems with unbounded coefficients,Izvestiya: Mathematics

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Complete description of the Lyapunov spectra of continuous families of linear differential systems with unbounded coefficients
Izvestiya: Mathematics ( IF 0.8 ) Pub Date : 2020-12-30 , DOI: 10.1070/im8976
V. V. Bykov ₁

Affiliation

For every positive integer $n$ and every metric space $M$ we consider the class $\widetilde{\mathcal{U}}^n(M)$ of all parametric families $\dot x = A(t, \mu)x$ , where $x\in\mathbb{R}^n$ , $t\ge 0$ , $\mu\in M$ , of linear differential systems whose coefficients are piecewise continuous and, generally speaking, unbounded on the time semi-axis for every fixed value of the parameter $\mu$ such that if a sequence $(\mu_k)$ converges to $\mu_0$ in the space of parameters, then the sequence $(A(\,{\cdot}\,,\mu_k))$ converges uniformly on the semi-axis to the matrix $A(\,{\cdot}\,,\mu_0)$ . For the families in $\widetilde{\mathcal{U}}^n(M)$ , we obtain a complete description of individual Lyapunov exponents and their spectra as functions of the parameter.

中文翻译：

无界系数线性微分系统连续族的李雅普诺夫谱的完整描述

对于每个正整数 $n$ 和每个度量空间 $M$ ，我们考虑 $\widetilde{\mathcal{U}}^n(M)$ 所有参数族的类 $\dot x = A(t, \mu)x$ ，其中 $x\in\mathbb{R}^n$ , $t\ge 0$ , $\mu\in M$ , 线性微分系统的系数是分段连续的，一般来说，对于每个固定的参数值，在时间半轴上是无界的， $\亩$ 例如如果一个序列在参数空间中 $(\mu_k)$ 收敛到 $\亩_0$ ，那么该序列 $(A(\,{\cdot}\,,\mu_k))$ 在半轴上均匀地收敛到矩阵 $A(\,{\cdot}\,,\mu_0)$ 。对于中的族 $\widetilde{\mathcal{U}}^n(M)$ ，我们获得了对单个 Lyapunov 指数及其光谱作为参数函数的完整描述。

更新日期：2020-12-30

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