In this paper certain global properties of dynamical systems governed by nonlinear difference equations are studied. When an asymptotically stable equilibrium state or periodic solution exists, it is desirable to be able to determine a global region of asymptotic stability in the state space. In this paper an effective method is presented for the determination of such a region. It will be seen that once certain features of the backward mapping have been properly delineated, the development of the method becomes a rather simple one. The method is mainly presented for second-order systems but the basic ideas are also applicable to higher-order systems. Through the development of the theory and examples, one also sees that, in general, the region of asymptotic stability for a nonlinear difference system is of extremely complex shape.

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