In this work, we consider a class of dynamical systems described by ordinary differential equations under the assumption that the global asymptotic stability (GAS) of equilibrium points is established based on the Lyapunov stability theory with the help of quadratic Lyapunov functions. We employ the Micken's methodology to construct a family of explicit nonstandard finite difference (NSFD) methods preserving any given quadratic Lyapunov function $V$, i.e. they admit $V$ as a discrete Lyapunov function. Here, the proposed NSFD methods are derived from a novel non-local approximation for the zero vector function. Through rigorous mathematical analysis, we show that the constructed NSFD methods have the ability to preserve any given quadratic Lyapunov functions regardless of the values of the step size. As an important consequence, they are dynamically consistent with respect to the GAS of continuous-time dynamical systems. On the other hand, the positivity of the proposed NSFD methods is investigated. It is proved that they can also preserve the positivity of solutions of continuous-time dynamical systems. Finally, the theoretical findings are supported by a series of illustrative numerical experiments, in which advantages of the NSFD methods are demonstrated.

翻译：本文考虑一类由常微分方程描述的动力系统，假设基于李雅普诺夫稳定性理论并借助二次李雅普诺夫函数建立平衡点的全局渐近稳定性。采用Micken方法构造一族显式非标准有限差分方法，此类方法能保持任意给定的二次李雅普诺夫函数$V$，即承认$V$为离散李雅普诺夫函数。此处所提出的非标准有限差分方法源于零向量函数的新型非局部逼近。通过严谨的数学分析，我们证明所构造的非标准有限差分方法无论步长取值如何，均能保持任意给定的二次李雅普诺夫函数。重要推论是，这些方法在连续时间动力系统的全局渐近稳定性方面具有动态一致性。另一方面，本文研究了所提非标准有限差分方法的正性，证明其亦能保持连续时间动力系统解的正性。最后，通过一系列数值实验验证理论结果，展示了非标准有限差分方法的优势。