An elementary result in the stability theory of time-invariant nonlinear discrete dynamical systems

Abstract

The stability of the equilibria of time-invariant nonlinear dynamical systems with discrete time scale is investigated. We present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle, then the equilibrium maybe unstable, or marginally stable, or even asymptotically stable, which show that the necessary condition cannot be further restricted in general. In addition, the necessary condition is given in terms of spectral radius and matrix norms.