Local Stability and Lyapunov Functionals for n-Dimensional Quasipolynomial Conservative Systems

TL;DR

The paper introduces a method to analyze local stability of equilibrium points in high-dimensional, nonlinear conservative systems using Poisson structures and generalized Lyapunov functionals, extending classical approaches.

Contribution

It develops a novel approach combining Poisson reformulation and energy-Casimir Lyapunov functionals for stability analysis of complex n-dimensional quasipolynomial systems.

Findings

The method applies to systems with arbitrary species number, including odd dimensions.

Constructs generalized Lyapunov functionals for these systems.

Provides examples demonstrating the effectiveness of the approach.

Abstract

We present a method for determining the local stability of equilibrium points of conservative generalizations of the Lotka-Volterra equations. These generalizations incorporate both an arbitrary number of species -including odd-dimensional systems- and nonlinearities of arbitrarily high order in the interspecific interaction terms. The method combines a reformulation of the equations in terms of a Poisson structure and the construction of their Lyapunov functionals via the energy-Casimir method. These Lyapunov functionals are a generalization of those traditionally known for Lotka-Volterra systems. Examples are given.