生物学中分数阶系统的李雅普诺夫函数：方法与应用
Lyapunov functions for fractional-order systems in biology: methods and applications
摘要：
对某些特定的函数证明了$α\in (0,1]$阶Caputo导数的新估计.所示的估计是有用的构造Lyapunov函数的分数阶微分方程系统在生物学中，那些已知的常微分方程的基础上，因此，有用的确定的全局稳定性的平衡点的分数阶系统。为了说明我们的理论结果的有用性，分数HIV人口模型和分数细胞模型进行了研究。更准确地说，我们构造合适的李雅普诺夫泛函证明的自由和地方病平衡点的全局稳定性，分数模型，我们还进行了一些数值模拟，确认我们的选择。
英文摘要：
We prove new estimates of the Caputo derivative of order $α\in (0,1]$ for some specific functions. The estimations are shown useful to construct Lyapunov functions for systems of fractional differential equations in biology, based on those known for ordinary differential equations, and therefore useful to determine the global stability of the equilibrium points for fractional systems. To illustrate the usefulness of our theoretical results, a fractional HIV population model and a fractional cellular model are studied. More precisely, we construct suitable Lyapunov functionals to demonstrate the global stability of the free and endemic equilibriums, for both fractional models, and we also perform some numerical simulations that confirm our choices.
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