Automatic Generation of Bounds for Polynomial Systems with Application to the Lorenz System

TL;DR

This paper presents an automated method combining Lyapunov techniques and quantifier elimination to compute bounds for polynomial dynamical systems, demonstrated on the Lorenz system, reducing reliance on expert insight.

Contribution

It introduces an automatic procedure for estimating attractor bounds in polynomial systems using software tools, improving efficiency over traditional manual methods.

Findings

Successfully applied to the Lorenz system

Provides less conservative attractor bounds

Automates the process of invariant set estimation

Abstract

This study covers an analytical approach to calculate positively invariant sets of dynamical systems. Using Lyapunov techniques and quantifier elimination methods, an automatic procedure for determining bounds in the state space as an enclosure of attractors is proposed. The available software tools permit an algorithmizable process, which normally requires a good insight into the systems dynamics and experience. As a result we get an estimation of the attractor, whose conservatism only results from the initial choice of the Lyapunov candidate function. The proposed approach is illustrated on the well-known Lorenz system.