What is the lowest order of the fractional-order chaotic systems to behave chaotically?

Abstract

Considering the fractional-order simplified Lorenz system as an example, we analyze the lowest order under the effects of variations in the numerical methods employed to solve the system. We varied the commensurability of the system equations, the system parameter and iteration step to reveal how they affect the appearance of chaos. The results show that the lowest order is obtained using the Adomian decomposition method (ADM), and it is smaller than that obtained through the Adams-Bashforth-Moulton (ABM) algorithm. The fractional-order system in the case of incommensurate order always has a smaller lowest order for chaos than does the system with commensurate order. We found that the lowest order of the fractional-order system decreases as the system parameter increases or as the iteration step decreases. In addition, by selecting the optimal results from a previous analysis, we obtain the lowest order of the fractional-order simplified Lorenz system is 1.136, which is lower than all results previously reported. Finally, we propose a corollary: the lowest order of a fractional-order chaotic system returned by an exact solution condition is lower than the approximate solution derived by numerical simulations. This work facilitates us to find a lower lowest order of fractional-order nonlinear systems, which are relevant to the study of fractional-order nonlinear system for engineering application.