  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.
