Typical complex behaviors induced by numerical algorithm in dynamical analysis of fractional order nonlinear systems

Nowadays, identification, dynamical analysis, control and synchronization of fractional dynamical systems have become a focus topic in the nonlinear research fields. Researchers always us a linear time invariant transfer function (LTI) to approximate the fractional transfer function in related numerical investigation and then put their results into circuits designing, signal processing, etc. In this paper, we further carry out the investigation on the reliability of existed methods on this topic. Firstly, we discussed the serious topic via case study on the newly proposed Liu chaotic system and its commensurate fractional system. Our research reveals that, the most widely used Charef and Oustaloup methods may cause fake chaotic or fake periodic phenomena under some wrong conditions. That´s to say, the traditional frequency method may be invalid under some situation. Then, it is suggested that the modified Oustaloup method can improve the reliability of the traditional frequency LTI method, since the modified method behaves much better on the left and right boundary of a related interval of fitting. By using such a new method, possible cases of fake complex phenomena caused by traditional LTI methods might be avoided successfully. Lastly, it is also addressed that the ADM predictor-corrector scheme may also cause fake complex behaviors while using unsuitable length of iteration step by taking the hyper-chaotic system as illustrations. So, it is suggested that, the modified frequency approximate method is more suitable for numerical analysis for the case which is far from the transient state boundary between order and chaos. While using the ADM predictor-corrector scheme, smaller iteration step should be used for avoiding fake chaos. Numerical analysis further confirmed our analysis.