The iterative digraph cell mapping method of non-smooth dynamical systems

Digraph cell mapping method (DCMM) is an effectivetechnique to analyze the global behavior of dynamical systems. Recently, ithas been applied to the study of crises and stochastic bifurcation, whichachieved a series of good results. However, many researches by DCMM arebased on smooth dynamical systems, and it is unknown whether this method canbe applied to non-smooth systems. In this paper, the key problem namedexpansion of cell flow is discussed in detail when the digraph cell mappingmethod is applied to analyze non-smooth dynamical systems. It is found thatbecause of non-smooth boundary the expansion of cell flow is usuallyserious, which may result in analysis distortion of DCMM. For the aboveproblem, the notion of the artificial node set is introduced to record whichstate cells may cause the expansion of cell flow. Taking the artificial nodeset as the redivided object, an iterative version of DCMM together with itseffective algorithm is designed to decrease the expansion of cell flow andimprove the computing accuracy. Furthermore some remarks are suggested forreaders to analyze complex non-smooth systems. In the iterative processes,the present method can not only keep the integrality of all computingresults but also significantly enhance the computing efficiency. As anillustrative example, the Duffing-van der Pol vibro-impact system withcomplex nonlinear structures is taken to demonstrate the validity of theproposed method.