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

The stability of the equilibria of time-invariant nonlinear dynamical syste ms with discrete time scale is investigated, We present an elementary proof showing that in the case of a stable equilibrium and continuously differen tiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrat e via numerical examples that if some eigenvalues are on the unit circle an d all other eigenvalues are inside the unit circle, then the equilibrium ma ybe unstable, or marginally stable, or even asymptotically stable, which sh ow 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. (C) 1999 Elsevier Science Inc. All rights reserved.