Hidden Extreme Multistability and Initial Offset-boosting of Fract ional Order Hyperchaotic Map

At present, hidden dynamics and extreme multistability of fractional order systems are around much interest from researchers. In this paper, by utilizing the fractional $\mu$-Caputo operator a simple fractional order hyperchaotic map is introduced. What interest us is that the newly presented map has no equilibrium points and a line of equilibrium depending on the value of bifurcation parameters, which indicate that it can produce hidden attractors. For the different fractional order values and system parameters, the dynamics of the fractional order map is studied. It is verified that the new model undergoes doubling route to chaos. Furthermore, by varying the initial condition of the second state and selecting appropriate system parameters, the initial offset boosting phenomena is first discovered. This dynamic behaviour indicate that the system also has hidden multistability. The dynamic of this model is experimentally investigated via bifurcation diagrams, phase portraits and Lyapunov exponent.