In this work, a class of continuous-time autonomous dynamical systems describing many important phenomena and processes arising in real-world applications is considered. We apply the nonstandard finite difference (NSFD) methodology proposed by Mickens to design a generalized NSFD method for the dynamical system models under consideration. This method is constructed based on a novel non-local approximation for the right-side functions of the dynamical systems. It is proved by rigorous mathematical analyses that the NSFD method is dynamically consistent with respect to positivity, asymptotic stability and three classes of conservation laws, including direct conservation, generalized conservation and sub-conservation laws. Furthermore, the NSFD method is easy to be implemented and can be applied to solve a broad range of mathematical models arising in real-life. Finally, a set of numerical experiments is performed to illustrate the theoretical findings and to show advantages of the proposed NSFD method.

翻译：本文研究了一类描述现实应用中诸多重要现象与过程的连续时间自治动力系统。我们采用Mickens提出的非标准有限差分（NSFD）方法论，为所考虑的动力系统模型设计了一种广义NSFD方法。该方法基于对动力系统右端函数的一种新型非局部逼近而构建。通过严谨的数学分析证明，该NSFD方法在正性、渐近稳定性以及三类守恒律（包括直接守恒律、广义守恒律和亚守恒律）方面具有动态一致性。此外，该NSFD方法易于实现，可应用于求解现实中涌现的广泛数学模型。最后，通过一组数值实验验证了理论结果，并展示了所提NSFD方法的优势。