Global Dynamics of Fractional-Order Discrete Maps

Abstract

In this paper, global dynamics of fractional-order discrete maps is analyzed by an extended generalized cell mapping (EGCM) method. Considering the lack of valid global analysis methods, the EGCM method is used to explore the global dynamics for fractional-order discrete maps. Firstly, considering the slowly convergence speed of solution of fractional-order discrete maps, the one-step mapping time of the EGCM method should be sufficient long to guarantee the precision of the results. Secondly, global dynamics of three typical fractional-order discrete maps is analyzed by the EGCM method. The stable and the unstable invariant sets can be obtained by the method. The results confirm their previous results, and furthermore obtain the global dynamics in the interesting region which includes attractors, saddles, basin boundaries and domains of attraction. These indicate that the EGCM method is also valid and efficient for fractional-order discrete maps.