On characterizations of exponential stability of nonlinear discrete dynamical systems on bounded regions

In this paper, we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space. Through introducing two new concepts - the Lip constant of a nonlinear operator and strongly equivalent metrics of a norm - we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics, or if and only if the Lip constant of the nonlinear operator is less than one. In the latter case, we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the Lip constant. Based on the obtained results, we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the Lip constant and strongly equivalent metrics. The obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new, useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems.