Invariance Preserving Discretization Methods of Dynamical Systems

TL;DR

This paper develops invariance-preserving discretization methods for dynamical systems, establishing conditions under which certain sets remain invariant under numerical schemes like Euler methods.

Contribution

It introduces new invariance-preserving steplength thresholds for various discretization methods applied to linear and nonlinear systems, including general convex and conic sets.

Findings

Existence of local and uniform invariance thresholds for Euler methods on polyhedra, ellipsoids, and Lorenz cones.

Quantification of steplength thresholds for backward Euler on linear systems.

Main results on invariance thresholds for general discretization methods on convex and proper cone sets.

Abstract

In this paper, we consider local and uniform invariance preserving steplength thresholds on a set when a discretization method is applied to a linear or nonlinear dynamical system. For the forward or backward Euler method, the existence of local and uniform invariance preserving steplength thresholds is proved when the invariant sets are polyhedra, ellipsoids, or Lorenz cones. Further, we also quantify the steplength thresholds of the backward Euler methods on these sets for linear dynamical systems. Finally, we present our main results on the existence of uniform invariance preserving steplength threshold of general discretization methods on general convex sets, compact sets, and proper cones both for linear and nonlinear dynamical systems.