Global consensus for discrete-time competitive systems

Grossberg established a remarkable convergence theorem for a class of competitive systems without knowing and using Lyapunov function for the systems. We present the parallel investigations for the discrete-time version of the Grossberg's model. Through developing an extended component-competing analysis for the coupled system, without knowing a Lyapunov function and applying the LaSalle's invariance principle, the global pattern formation or the so-called global consensus for the system can be achieved. A numerical simulation is performed to illustrate the present theory.