On fractional lyapunov exponent for solutions of linear fractional differential equations
On fractional lyapunov exponent for solutions of linear fractional differential equations
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DOI：
10.2478/s13540-014-0169-1
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发表时间：
2014-03
影响因子：
3
通讯作者：
N. D. Cong;Doan Thai Son;H. T. Tuan
中科院分区：
文献类型：
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作者：
N. D. Cong;Doan Thai Son;H. T. Tuan
本文主要研究线性分数阶微分方程解的渐近性态。首先，我们证明了有界线性分数阶微分方程任意非平凡解的经典李雅普诺夫指数总是非负的。接下来，使用Mittag-Leffler函数，我们引入了一个适当的概念，分数李雅普诺夫指数的任意函数。我们证明了对于一个线性分数阶微分方程，由其解的所有可能的分数阶李雅普诺夫指数组成的分数阶李雅普诺夫谱可以很好地描述该方程的渐近行为.因此，分数阶线性微分方程的稳定性可以用分数阶李雅普诺夫谱来刻画。最后，为了说明理论结果，我们显式计算的分数阶微分方程的任意解的分数阶李雅普诺夫指数。
Our aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.