The model of the consecutive memcapacitor has been widely used in chaotic circuits. However, the model of discrete memcapacitor and its application in chaotic systems have not been further studied. In this paper, a model of discrete memcapacitor is proposed. And the dynamical characteristics of the discrete memcapacitor model are analyzed. The memristive Chebyshev map is obtained by coupling the discrete memcapacitor with the Chebyshev map. Since memristive Chebyshev map has linear fixed points, the memristive Chebyshev map is unstable or critically stable, depending on the internal parameters and the initial condition of the chaotic map. The dynamical behavior of control parameter dependence of memristive Chebyshev map is studied by using several analysis methods, and its hyperchaotic attractor is found. The special phenomenon of coexistence of attractors is also found. Finally, the memristive Chebyshev map is realized by DSP. And the results of simulation are further verified. The results of this study supply a theoretical basis for the application of discrete memcapacitor in the design of discrete chaotic systems.

Research on discrete memristor models applied to discrete maps deserves more in-depth discussion. In this paper, a continuous memristor is introduced and the discrete memristor model is obtained by the forward Eulerian difference algorithmic discretization. This model is coupled to a cosine map to further obtain a two-dimensional memristor cosine map. The dynamical characteristics of the memristor cosine map are investigated through numerical simulations and other analytical methods. For example, the phase diagram, the bifurcation diagram, the Lyapunov exponential spectrum and the Spectral Entropy complexity with parameters, etc., In addition, multi-stability phenomena of the system are identified. The results show that the cosine map coupled with a discrete memristor has more complex dynamical behaviors and is more suitable for applications in cryptography.

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In this paper, a hyperchaotic circuit consisting of a series memristor, meminductor, and memcapacitor is proposed. The dimensionless mathematical model of the system is established by the state equation of the circuit. The stability of equilibrium point of the system is analyzed by using the traditional dynamic analysis method. Then, the dynamical characteristics of the chaotic system with parameters are analyzed in detail. In addition, the system also has some particular phenomena such as attractor coexistence and state transition. Finally, the circuit is realized by DSP, and the result is consistent with that of numerical simulation. This proves the accuracy of the theoretical analysis. Numerical simulation result shows which hyperchaotic system has very abundant dynamical characteristics.