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.

In this paper, we investigate the mathematical models of discrete memristors based on Caputo fractional difference and G–L fractional difference. Specifically, the integer-order discrete memristor is a special model of those two cases. The “∞”-type hysteresis loop curves are observed when input is the bipolar periodic signal. Meanwhile, numerical analysis results show that the area of hysteresis decreases with the increase of frequency of input signal and the decrease of derivative order. Moreover, the memory effect, characteristics and physical realization of the discrete memristors are discussed, and a discrete memristor with short memory effects is designed. Furthermore, discrete memristive systems are designed by introducing the fractional-order discrete memristor and integer-order discrete memristor to the Sine map. Chaos is found in the systems, and complexity of the systems is controlled by the parameter of the memristor. Finally, FPGA digital circuit implementation is carried out for the integer-order and fractional-order discrete memristor and discrete memristive systems, which shows the potential application value of the discrete memristor in the engineering application field.

A universal modular plus parallel (MPP) method is proposed to construct enhanced chaotic maps, including one-dimensional MPP chaotic map (1D-MPPCM) and high-dimensional MPPCM (HD-MPPCM). It is theoretically proved that 1D-MPPCM model can significantly increase the Lyapunov exponent (LE) and parameter range of seed chaotic maps. To further increase the system dimension, the HD-MPPCM model is established through the close-loop parallel coupling mechanism. Based on several typical seed chaotic maps, some new parallel chaotic maps are obtained by self-parallel and hybrid-parallel, and their dynamics are analyzed by phase diagram, LEs, permutation entropy (PE) complexity and statistic $\chi^2$ . The simulation results show that the proposed maps have large maximum Lyapunov exponent (MLE), PE complexity, and uniform distribution. In particular, HD-MPPCMs have some interesting characteristics, such as full positive LEs, hyperchaotic behavior, global chaos, and full attractor distribution, which are the potential model for engineering applications. To further verify the practicability, the proposed maps are implemented on DSP platform, and applied to pseudo-random number generator (PRNG).

In this article, an improved memristive Hénon map is proposed by using the state variable difference instead of the state variable itself as the input of a discrete memristor. As a result, the cumulative module in the discrete memristor can avoid divergence. The bifurcation diagrams, Lyapunov exponent spectrums, and sample entropy complexity analysis results show that the system has rich dynamical behaviors. In addition, analog circuits of the discrete memristor and the memristive chaotic map are designed for the first time. PSIM circuit simulations and breadboard experiments verify the effectiveness of the proposed models. The analog circuits indicate the physical realizability of the discrete memristor. It lays the foundation for the applications of discrete memristor and helps to explain their physics significance.

To solve the ubiquitous problems of frail chaos and dynamical degradation, a delayed feedback method is proposed in this article to construct hyperchaotic maps. In this method, the delayed feedback control increases the complexity, dimension, and period length of a chaotic map, and the use of modular operation for boundary control further improves its chaotic range and ergodicity. Taking three 1-D chaotic maps as seeds, some new high-dimensional chaotic maps are proposed based on this method. The chaotic properties of the proposed maps are analyzed by the definition of Lyapunov exponent (LE) and stability analysis. Performance evaluations show that the proposed maps have large chaotic ranges, positive LEs, hyperchaotic behavior, high complexity, uniform distribution, multiple coexistence attractor, and strong resistance to parameter estimation and dynamical degradation. To further verify its practicability, the proposed maps are implemented on a field-programmable gate array platform, and applied to the pseudorandom number generator and image compression.