生物相互作用网络非振荡行为的鲁棒李雅普诺夫判据
A robust Lyapunov criterion for non-oscillatory behaviors in biological interaction networks
摘要：
我们介绍了非振荡的概念，提出了一个建设性的方法，其强大的验证，并研究其应用于生物相互作用网络（也称为，化学反应网络）。我们首先在研究非线性系统的雅可比矩阵的第二加性复合变分系统的基础上，重新讨论Muldowney关于周期解不存在的结果。我们表明，指数稳定的后者排除了极限环，拟周期解，和广泛的类的振荡行为。然后，我们专注于生物相互作用网络与一般动力学中产生的非线性方程，我们表明，上述变分系统的动态可以嵌入在一个线性微分包含。然后，我们提出了算法构造分段线性李雅普诺夫函数，以证明全球强大的非振荡行为。最后，我们应用我们的技术来研究几种受调节的酶循环，其中可用的方法无法提供有关其定性全局行为的任何信息。
英文摘要：
We introduce the notion of non-oscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowney's result on non-existence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust non-oscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles where available methods are not able to provide any information about their qualitative global behavior.
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