Inverse Jacobi multiplier as a link between conservative systems and Poisson structures

TL;DR

This paper explores the connection between conservative dynamical systems and Poisson structures, analyzing local singularities and global properties, and introduces weak conservativeness via inverse Jacobi multipliers with applications to various systems.

Contribution

It characterizes Poisson structures near zero-Hopf singularities and links conservativeness with Poisson structures, extending inverse Jacobi multipliers to weak solutions.

Findings

Poisson structures are characterized around zero-Hopf singularities.

Strict conservativeness is linked to the existence of Poisson structures depending on phase space dimension.

Weak conservativeness is introduced via inverse Jacobi multipliers as weak solutions.

Abstract

Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the Flow-Box Theorem we restrict ourselves to neighborhoods of singularities. In this sense, we characterize Poisson structures around the typical zero-Hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non-vanishing first jet by connecting with the classical Poincar\'e center problem. From the global point of view, we connect the property of being strictly conservative (the invariant measure must be positive) with the existence of a Poisson structure depending on the phase space dimension. Finally, weak conservativeness in dimension two is introduced by the extension of inverse Jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed. Examples including Lotka-Volterra systems, quadratic isochronous centers, and non-smooth oscillators are provided.