On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations

TL;DR

This paper establishes a precise correspondence between symmetries of two-dimensional autonomous dynamical systems in their differential equation form and their phase plane representations, showing how symmetries can be transferred between these formulations.

Contribution

It proves that each symmetry in one formulation uniquely induces a symmetry in the other, via solving a linear PDE called the lifting condition, with detailed examples.

Findings

Every symmetry generator in the differential equation formulation induces a phase plane symmetry.

Every phase plane symmetry can be lifted to a symmetry of the original system.

The lifting process involves solving a linear partial differential equation.

Abstract

We consider the relationship between symmetries of two-dimensional autonomous dynamical system in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to the symmetries of their governing differential equations, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. Our main result is to show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original autonomous system, which is unique up to constant translations in time. The process of lifting requires the solution of a linear partial differential equation, which we refer to as the lifting condition. We discuss in detail the solution of this equation in general, and exemplify the lift of symmetries in two commonly occurring examples; a mass conserved linear model and a non-linear oscillator model.