Exterior dissipation, proportional decay, and integrals of motion

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.