The fractional-order hyperchaotic Lorenz system is solved as a discrete map by applying the Adomian decomposition method (ADM). Lyapunov Characteristic Exponents (LCEs) of this system are calculated according to this deduced discrete map. Complexity of this system versus parameters are analyzed by LCEs, bifurcation diagrams, phase portraits, complexity algorithms. Results show that this system has rich dynamical behaviors. Chaos and hyperchaos can be generated by decreasing fractional order q in this system. It also shows that the system is more complex when q takes smaller values. SE and C 0 complexity algorithms provide a parameter choice criteria for practice applications of fractional-order chaotic systems. The fractional-order system is implemented by digital signal processor (DSP), and a pseudo-random bit generator is designed based on the implemented system, which passes the NIST test successfully.

Researches on the fracmemristor have aroused increasing interest in the last several years, but there are no reports on design of the discrete fracmemristor. Based on the fractional-order difference and the mathematical model of the charge-controlled memristor, the discrete fracmemristor is designed where the amount of charge is determined by a fractional-order discrete system. In the numerical simulations, it shows that the pinched hysteresis loops are observed, which imply that the proposed memristor satisfies the definition of the memristor. As an application, the fracmemristor sine map is designed, and multistability is observed regarding the initial conditions of both the memristor and system. It provides a potential model for different applications such as cellular neural networks, modulators, sensors, chaotic systems, and programmable digital circuits.