On the stability of discrete-time homogeneous polynomial dynamical systems

Every homogeneous polynomial dynamical system (HPDS) can be uniquely represented by a tensor. In our recent article (Chen, IEEE Trans Autom Control), we established necessary and sufficient stability criteria for certain continuous-time HPDSs by exploiting tensor spectral theory. In this article, we extend these results to discrete-time HPDSs. In particular, if the state transition tensor of a discrete-time HPDS is orthogonally decomposable (odeco), we can derive its explicit solution. We refer to such HPDSs as odeco HPDSs. Building upon the form of the explicit solution, we demonstrate that the Z-eigenvalues of the state transition tensor offer necessary and sufficient stability conditions, analogous to the continuous-time case. The region of attraction can also be obtained for the odeco HPDS. Additionally, by employing the upper bounds of Z-spectral radii, we can efficiently determine the asymptotic stability of odeco HPDSs. Finally, we leverage tensor singular values to analyze the stability properties of general discrete-time HPDSs, where the state transition tensors are not odeco. We illustrate our framework with numerical examples.