A fractional map with hidden attractors: chaos and control

This paper studies the dynamics of a new fractional-order map with no fixed points. Through phase plots, bifurcation diagrams, largest Lyapunov exponent, it is shown that the proposed fractional map exhibit chaotic and periodic behavior. New Hidden chaotic attractors are observed, and transient state is found to exist. Complexity of the new map is also analyzed by employing approximate entropy. Results, show that the fractional map without fixed point have high complexity for certain fractional order. In addition, a control scheme is introduced. The controllers stabilize the states of the fractional map and ensure their convergence to zero asymptotically. Numerical results are used to verify the findings.