Abstract:
We present two results on the analysis of discrete dynamical systems and finite
difference discretizations of continuous dynamical systems, which preserve their
dynamics and essential properties. The first result provides a sufficient condition for
forward invariance of a set under discrete dynamical systems of specific type, namely
time-reversible ones. The condition involves only the boundary of the set. It is a
discrete analog of the widely used tangent condition for continuous systems (viz. the
vector field points either inwards or is tangent to the boundary of the set). The
second result is nonstandard finite difference (NSFD) scheme for dynamical systems
defined by systems of ordinary differential equations. The NSFD scheme preserves the
hyperbolic equilibria of the continuous system as well as their stability. Further, the
scheme is time reversible and, through the first result, inherits from the continuous
model the forward invariance of the domain. We show that the scheme is of second
order, thereby solving a pending problem on the construction of higher-order
nonstandard schemes without spurious solutions. It is shown that the new scheme
applies directly for mass action-based models of biological and chemical processes.
The application of these results, including some numerical simulations for invariant
sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type
epidemiological model, which may have arbitrary large number of infective or
recovered/removed compartments.
