Measure-preserving symmetries and reversibilities of ordinary differential systems

TL;DR

This paper investigates measure-preserving symmetries and reversibilities in differential systems, showing how they affect divergence and its derivatives, with applications to specific biological and physical models.

Contribution

It establishes new theoretical results on how measure-preserving symmetries and reversibilities influence divergence properties in differential systems.

Findings

Measure-preserving symmetries preserve divergence and its derivatives.

Reversibilities affect odd and even divergence derivatives differently.

All area-preserving symmetries and reversibilities of certain biological systems are characterized.

Abstract

We prove that measure-preserving symmetries of an $n$-dimensional differential system preserve its divergence and the divergence derivatives along the solutions. Also, we prove that measure-preserving reversibilities preserve odd-order divergence derivatives along the solutions, and that even-order derivatives are multiplied by $-1$. We apply such results to find all the area-preserving symmetries and reversibilities of planar Lotka-Volterra and Li\'enard systems.