Steplength Thresholds for Invariance Preserving of Discretization Methods of Dynamical Systems on a Polyhedron

TL;DR

This paper establishes methods to compute steplength thresholds ensuring invariance preservation in discretization methods for dynamical systems on polyhedra, including Taylor, rational, and Euler methods, with efficient algorithms and theoretical insights.

Contribution

It introduces new algorithms and theoretical results for calculating steplength thresholds for invariance preservation across various discretization methods.

Findings

Steplength thresholds can be found via polynomial zeroes for Taylor methods.

A simple algorithm computes thresholds for rational function methods.

The largest threshold for Euler method is obtained through linear optimization.

Abstract

Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algorithm is proposed to numerically compute the steplength threshold. For rational function type discretization methods we derive a valid steplength threshold for invariance preserving, which can be computed by using an analogous algorithm as in the first case. The relationship between the previous two types of discretization methods and the forward Euler method is studied. Finally, we show that, for the forward Euler method, the largest steplength threshold for invariance preserving can be computed by solving a finite number of linear optimization problems.