Nonstandard finite difference methods preserving general quadratic Lyapunov functions

TL;DR

This paper introduces a new class of explicit nonstandard finite difference methods that preserve quadratic Lyapunov functions and positivity, ensuring dynamic consistency with the stability properties of continuous dynamical systems.

Contribution

The paper develops a novel NSFD framework that guarantees preservation of quadratic Lyapunov functions and positivity, regardless of step size, for stable dynamical systems.

Findings

NSFD methods preserve quadratic Lyapunov functions

Methods maintain positivity of solutions

Numerical experiments demonstrate advantages

Abstract

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.