Global dynamics of fractional-order systems with an extended generalized cell mapping method

Global dynamics of fractional-order systems is studied with an extended generalized cell mapping (EGCM) method. The one-step transition probability matrix of Markov chain of the EGCM is generated by means of the improved predictor-corrector approach for fractional-order systems. The one-step mapping time of the proposed method is evaluated with the help of the short memory principle for fractional derivatives to deal with its non-local property and to properly define a bound of the truncation error and a function M by considering the features of cell mapping. In this way, a method of generalized cell mapping for global dynamics of a fractional-order system is developed. Three examples of global analysis on fractional-order systems are given to demonstrate the validity and efficiency of the proposed method. And attractors, boundaries, basins of attraction, and saddles are obtained by the EGCM.