A new method is offered here for global analysis of nonlinear dynamical systems. It is based upon the idea of constructing the associated cell-to-cell mappings for dynamical systems governed by point mappings or governed by ordinary differential equations. The method uses an algorithm which allows us to determine in a very effective manner the equilibrium states, periodic motions and their domains of attraction when they are asymptotically stable. The theoretic base and the detail of the method are discussed in the paper and the great potential of the method is demonstrated by several examples of application.

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