On Tensor-based Polynomial Hamiltonian Systems
Shaoxuan Cui, Guofeng Zhang, Hildeberto Jardon-Kojakhmetov, Ming Cao
TL;DR
The paper generalizes the linear Hamiltonian correspondence to tensor-based polynomial systems by introducing Hamiltonian cubical tensors and proving a necessary-and-sufficient condition: a tensor-based polynomial system is Hamiltonian with a polynomial Hamiltonian if and only if all system tensors are Hamiltonian cubical tensors. It provides a tractable equilibrium stability criterion grounded in tensor operations and validates the theory with numerical examples such as the anharmonic oscillator and related polynomial Hamiltonians. The results offer a practical framework for recognizing and analyzing Hamiltonian structure in nonlinear polynomial dynamics using tensor methods, with potential extensions to port-Hamiltonian formulations and links to optimal control.
Abstract
It is known that a linear system with a system matrix A constitutes a Hamiltonian system with a quadratic Hamiltonian if and only if A is a Hamiltonian matrix. This provides a straightforward method to verify whether a linear system is Hamiltonian or whether a given Hamiltonian function corresponds to a linear system. These techniques fundamentally rely on the properties of Hamiltonian matrices. Building on recent advances in tensor algebra, this paper generalizes such results to a broad class of polynomial systems. As the systems of interest can be naturally represented in tensor forms, we name them tensor-based polynomial systems. Our main contribution is that we formally define Hamiltonian cubical tensors and characterize their properties. Crucially, we demonstrate that a tensor-based polynomial system is a Hamiltonian system with a polynomial Hamiltonian if and only if all associated system tensors are Hamiltonian cubical tensors-a direct parallel to the linear case. Additionally, we establish a computationally tractable stability criterion for tensor-based polynomial Hamiltonian systems. Finally, we validate all theoretical results through numerical examples and provide a further intuitive discussion.