On Novel Hamiltonian Descriptions of Some Three-Dimensional Nonconservative Systems

Abstract

We present the novel Hamiltonian descriptions of some three-dimensional systems including two well-known systems describing the three-wave interaction problem and some well-known chaotic systems, namely, the Chen, Lü, and Qi systems. We show that all of these systems can be described in a Hamiltonian framework in which the Poisson matrix (Formula presented.) is supplemented by a resistance matrix (Formula presented.). While such resistive-Hamiltonian systems are manifestly nonconservative, we construct higher degree Poisson matrices via the Jordan product as (Formula presented.), thereby leading to new bi-Hamiltonian systems. Finally, we discuss conformal Hamiltonian dynamics on Poisson manifolds and demonstrate that by appropriately choosing the underlying parameters, the reduced three-wave interaction model as well as the Chen and Lü systems can be described in this manner where the concomitant nonconservative part of the dynamics is described with the aid of the Euler vector field.