Smooth transformations and ruling out closed orbits in planar systems

TL;DR

This paper explores the geometric properties of planar dynamical systems under smooth transformations and introduces criteria to exclude closed orbits, enhancing understanding and computational detection of periodic solutions.

Contribution

It provides a unified geometric framework for analyzing planar systems and reformulates Bendixson's criterion using coordinate-independent methods.

Findings

Unified geometric perspective on planar systems

Reformulation of Bendixson's criterion

Automated exclusion of closed orbits in phase space

Abstract

This work deals with planar dynamical systems with and without noise. In the first part, we seek to gain a refined understanding of such systems by studying their differential-geometric transformation properties under an arbitrary smooth mapping. Using elementary techniques, we obtain a unified picture of different classes of dynamical systems, some of which are classically viewed as distinct. We specifically give two examples of Hamiltonian systems with first integrals, which are simultaneously gradient systems. Potential applications of this apparent duality are discussed. The second part of this study is concerned with ruling out closed orbits in steady planar systems. We reformulate Bendixson's criterion using the coordinate-independent Helmholtz decomposition derived in the first part, and we derive another, similar criterion. Our results allow for automated ruling out of closed orbits in certain regions of phase space, and could be used in the future for efficient seeding of initial conditions in numerical algorithms to detect periodic solutions.