On novel Hamiltonian descriptions of some three-dimensional non-conservative systems
Aritra Ghosh, Anindya Ghose-Choudhury, Partha Guha
TL;DR
The paper develops a resistive-Hamiltonian framework for describing non-conservative dynamics in three-dimensional systems by coupling a Poisson part $\mathcal{J}$ with a symmetric resistance part $\mathcal{R}$, yielding dynamics $\dot{\mathbf{x}}=(\mathcal{J}-\mathcal{R})\nabla H$ and nonzero divergence. It demonstrates that the reduced three-wave-interaction model, as well as the Chen, Lü, and Qi chaotic systems (and the Rabinovich model), admit such formulations, with explicit expressions for $H$, $\mathcal{J}$, and $\mathcal{R}$; it further develops higher-degree Poisson matrices via the Jordan-like product $\mathcal{N}=\mathcal{J}\mathcal{R}+\mathcal{R}\mathcal{J}$ to obtain bi-Hamiltonian descriptions when $\mathcal{N}$ is Poisson. The authors also establish Jordan-like transformations to generate new bi-Hamiltonian systems and show that conformal Hamiltonian dynamics on Poisson manifolds can reproduce several of the same models by introducing a constant Euler dilatation, offering an alternative viewpoint on non-conservative evolution. Overall, the work broadens Hamiltonian formulations of non-conservative 3D dynamics and links resistance-based, Jordan-product, and conformal perspectives to well-known chaotic and wave-interaction systems.
Abstract
We present 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 $\mathcal{J}$ is supplemented by a resistance matrix $\mathcal{R}$. While such resistive-Hamiltonian systems are manifestly non-conservative, we construct higher-degree Poisson matrices via the Jordan product as $\mathcal{N} = \mathcal{J} \mathcal{R} + \mathcal{R} \mathcal{J}$, 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 non-conservative part of the dynamics is described with the aid of the Euler vector field.