A model-based characterization of the long-term asymptotic behavior of nonlinear discrete-time processes using map invariance

The present research work proposes a new approach to the problem of quantitatively characterizing the long-term dynamic behavior of nonlinear discrete-time processes. The formulation of the problem of interest can be naturally realized through a system of nonlinear functional equations (NFEs), for which a rather general set of conditions for the existence and uniqueness of a locally analytic solution is derived. The solution to the system of NFEs is then proven to represent a locally analytic invariant manifold for the nonlinear discrete-time process of interest. The local analyticity property of the invariant manifold map enables the development of a series solution method for the above system of NFEs, which can be easily implemented using MAPLE. Under a certain set of conditions, it is shown that the invariant manifold attracts all system trajectories, and therefore, the long-term dynamic behavior is determined through the restriction of the discrete-time process dynamics on the invariant manifold.