Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

In this article, we provide a system-theoretic treatment of certain continuous-time homogeneous polynomial dynamical systems (HPDS) via tensor algebra. In particular, if a system of homogeneous polynomial differential equations can be represented by an orthogonally decomposable (odeco) tensor, we can construct its explicit solution by exploiting tensor Z-eigenvalues and Z-eigenvectors. We refer to such HPDS as odeco HPDS. By utilizing the form of the explicit solution, we are able to discuss the stability properties of an odeco HPDS. We illustrate that the Z-eigenvalues of the corresponding dynamic tensor can be used to establish necessary and sufficient stability conditions, similar to these from linear systems theory. In addition, we are able to obtain the complete solution to an odeco HPDS with constant control. Finally, we establish results that enable one to determine if a general HPDS can be transformed to or approximated by an odeco HPDS, where the previous results can be applied. We demonstrate our framework with simulated and real-world examples.

Automated summary

In this article, we use tensor algebra to provide a system-theoretic treatment of certain continuous-time homogeneous polynomial dynamical systems (HPDS). By representing the system of homogeneous polynomial differential equations with an orthogonally decomposable (odeco) tensor, we can construct its explicit solution using tensor Z-eigenvalues and Z-eigenvectors. We discuss the stability properties of an odeco HPDS by utilizing the form of the explicit solution and establish results for determining if a general HPDS can be transformed or approximated by an odeco HPDS.