Effects of system parameter and fractional order on dynamic behavior evolution in fractional-order Genesio-Tesi system

The object of this paper is to reveal the relationship between dynamics of the fractional-order Genesio-Tesi system and its parameter, especially the order of fractional derivatives. At first, the transcritical bifurcation is carried out based on the stability analysis of equilibrium points. Then, effects of system parameter and fractional order on the dynamics have been investigated deeply and systematically. For commensurate fractional-order system, period-doubling bifurcation, reverse period-doubling bifurcation, period window, transient chaos and steady chaos are found with parameter and fractional order varying simultaneously. For incommensurate fractional-order system, there also exists the route to chaos by period-doubling bifurcation. Furthermore, the effect of different fractional orders on dynamical behavior evolution has been compared. These new findings contribute to successful selecting the most efficient control function, which has been validated by some numerical tools.