Energy translation symmetries and dynamics of separable autonomous two-dimensional ODEs

TL;DR

This paper investigates energy translation symmetries in separable autonomous 2D ODEs, proving their existence and extending their application to biological models, revealing insights into their phase plane dynamics and temporal behavior.

Contribution

It establishes the existence of orthogonal energy translation symmetries in separable autonomous 2D ODEs and extends these symmetries to include temporal dynamics in biological models.

Findings

Energy translation symmetries correspond to internal energy shifts.

Symmetries extend to temporal dynamics in biological models.

Analytic expressions for symmetry actions on models are derived.

Abstract

We study symmetries in the phase plane for separable, autonomous two-state systems of ordinary differential equations (ODEs). We prove two main theoretical results concerning the existence and non-triviality of two orthogonal symmetries for such systems. In particular, we show that these symmetries correspond to translations in the internal energy of the system, and describe their action on solution trajectories in the phase plane. In addition, we apply recent results establishing how phase plane symmetries can be extended to incorporate temporal dynamics to these energy translation symmetries. Subsequently, we apply our theoretical results to the analysis of three models from the field of mathematical biology: a canonical biological oscillator model, the Lotka--Volterra (LV) model describing predator-prey dynamics, and the SIR model describing the spread of a disease in a population. We describe the energy translation symmetries in detail, including their action on biological observables of the models, derive analytic expressions for the extensions to the time domain, and discuss their action on solution trajectories.