Bifurcation and synchronization of a new fractional-order system

In this paper, bifurcation and adaptive synchronization of a new fractional-order system are investigated. Firstly, bifurcations for the system with the variation of a system parameter and a derivative order are studied and presented in a three-dimensional space. The dynamics of the system changes from regular to chaos with an increase of derivative orders. It is found that three derivative orders have different contributions to a critical total order for the system to change from regular to chaos, meaning that the nonlinearity in the system downgrades the critical order for transitions to chaos. By the analysis of bifurcations in a two-parameter space, we obtain that the minimal order is 2.62 for the system to remain chaos. Secondly, the entropy is analyzed to measure the level of chaos present in the fractional-order system. Finally, based on the stability theory of fractional-order systems, synchronization of the fractional-order system with fully uncertain parameters is realized by designing appropriate adaptive controllers and estimation laws. Numerical simulations are implemented to demonstrate the effectiveness and flexibility of the synchronization controllers and the estimation laws for the unknown parameters.