Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems via Tensor Orthogonal Decomposition.

In this paper, we investigate the explicit solutions and stability properties of certain continuous-time homogeneous polynomial dynamical systems via tensor algebra. In particular, if a system of homogeneous polynomial differential equations can be represented by an orthogonal decomposable tensor, we can write down its explicit solution in a simple fashion by exploiting tensor Z-eigenvalues and Z-eigenvectors. In addition, according to the form of the explicit solution, we explore the stability properties of the homogeneous polynomial dynamical system. We discover that the Z-eigenvalues from the orthogonal decomposition of the corresponding dynamic tensor can offer necessary and sufficient stability conditions, similar to these from linear systems theory. Finally, we demonstrate our results with numerical examples.