Exterior dissipation, proportional decay, and integrals of motion

TL;DR

This paper introduces a method to modify dynamical systems with conserved quantities, creating new systems where these quantities evolve monotonically and proportionally, enabling analysis of extremal states and new integrable systems.

Contribution

It develops a multi-parameter family of systems that generalize existing dissipation techniques using exterior products, allowing systematic reduction of integrability and discovery of special states.

Findings

Applied to the three-body Toda lattice, demonstrating transition from aperiodic orbit to a limit cycle.

Provides a framework for constructing new integrable and extremal states in complex systems.

Generalizes existing dissipation methods with a novel exterior product approach.

Abstract

Given a dynamical system with $m$ independent conserved quantities, we construct a multi-parameter family of new systems in which these quantities evolve monotonically and proportionally, and are replaced by $m-1$ conserved linear combinations of themselves, with any of the original quantities as limiting cases. The modification of the dynamics employs an exterior product of gradients of the original quantities, and often evolves the system towards asymptotic linear dependence of these gradients in a nontrivial state. The process both generalizes and provides additional structure to existing techniques for selective dissipation in the literature on fluids and plasmas, nonequilibrium thermodynamics, and nonlinear controls. It may be iterated or adapted to obtain any reduction in the degree of integrability. It may enable discovery of extremal states, limit cycles, or solitons, and the construction of new integrable systems from superintegrable systems. We briefly illustrate the approach by its application to the cyclic three-body Toda lattice, driven from an aperiodic orbit towards a limit cycle.