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This technical note studies quantitatively asymptotic growth behaviors of trajectories (AGBT) of nonlinear autonomous discrete dynamical system that has unbounded domain, non-Lipschitz continuous nonlinear operator, and stable or unstable equilibrium point. We explain how trajectory motion speed is quantitatively determined in the system, and study exact computation and sharp estimation of the smallest exponential bound of trajectories. We characterize exponential stability and asymptotic stability of the system from a new point of view, and provide a simple condition to distinguish them from each other. These results extend existing results that were obtained in some special cases of the system, and are helpful for quantitative analysis and understanding of AGBT of the system. [ABSTRACT FROM AUTHOR]

Stability is arguably one of the core concepts upon which our understanding of dynamical and control systems has been built. The related notion of incremental stability, however, has received much less attention until recently, when it was successfully used as a tool for the analysis and design of intrinsic observers, output regulation of nonlinear systems, frequency estimators, synchronization of coupled identical dynamical systems, symbolic models for nonlinear control systems, and bio-molecular systems. However, most of the existing controller design techniques provide controllers enforcing stability rather than incremental stability. Hence, there is a growing need to extend existing methods or develop new ones for the purpose of designing incrementally stabilizing controllers. In this technical note, we develop a backstepping design approach for incremental stability. [ABSTRACT FROM AUTHOR]