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A large number of animal experiments show that there is irregular chaos in the biological nervous systems. An artificial chaotic neural network is a highly nonlinear dynamic system, which can realize a series of complex dynamic behaviors, optimize global search and neural computation, and generate pseudo-random sequences for information encryption. According to the superposition theory of sinusoidal signals with different frequencies of brain waves, a non-monotone activation function based on the multifrequency-frequency conversion sinusoidal function and a piecewise function is proposed to make a neural network more consistent with the biological characteristics. The analysis shows that by adjusting the parameters, the activation function can exhibit the EEG signals in its different states, which can simulate the rich and varying brain activities when the brain waves of different frequencies and types work at the same time. According to the activation function we design a new chaotic cellular neural network. The complexity of the chaotic neural network is analyzed by the structural complexity based SE algorithm and C0 algorithm. By means of Lyapunov exponential spectrum, bifurcation diagram and basin of attraction, the effects of the activation function’s parameters on its dynamic characteristics are analyzed in detail, and it is found that a series of complex phenomena appears in the chaotic neural network, such as many different types of chaotic attractors, coexistent chaotic attractors and coexistence limit cycles, which improves the performance of the chaotic neural network, and proves that the multi-frequency sinusoidal chaotic neural network has rich dynamic characteristics, so it has a good prospect in information processing, information encryption and other aspects.

Based on two modified Rosslor hyperchaotic systems, which are derived from the chaotic Rosslor system by introducing a state feedback controller, this paper proposes a new switched Rosslor hyperchaotic system. The switched system contains two different hyperchaotic systems and can change its behaviour continuously from one to another via a switching function. On the other hand, it presents a systematic method for designing the circuit of realizing the proposed hyperchaotic system. In this design, circuit state equations are written in normalized dimensionless form by rescaling the time variable. Furthermore, an analogous circuit is designed by using the proposed method and built for verifying the new hyperchaos and the design method. Experimental results show a good agreement between numerical simulations and experimental results.

In this paper, a new four-dimensional autonomous hyperchaotic system is designed for generating complex chaotic signals. In the design, its parameters are selected according to the requirements for chaos and hyperchaos. The hyperchaotic Nature is verified theoretically by using the bifurcation analysis and demonstrated experimentally by the implementation of an analogue electronic circuit. Moreover, the Field Programmable Gate Array (FPGA) technology is applied to implementing a continuous system in a digital form by using a chip of Altera Cyclone II EP2C35F484C8. The digital sequence generated from the FPGA device is observed in our experimental setup.

Based on the Lu system, a new chaotic system is constructed, which can generate a Lorenz-like attractor, Chen-like attractor, Lu-like attractor and new attractor when its parameters are chosen appropriately. The detailed dynamical behaviours of this system are also investigated, including equilibria and stability, bifurcations, and Lyapunov exponent spectrum. Moreover, a novel analogue circuit diagram is designed for the verification of various attractors.

A meminductor is a new type of memory device. It is of importance to study meminductor model and its application in nonlinear circuit prospectively. For this purpose, we present a novel mathematical model of meminductor, which considers the effects of internal state variable and therefore will be more consistent with future actual meminductor device. By using several operational amplifiers, multipliers, capacitors and resistors, the equivalent circuit of the model is designed for exploring its characteristics. This equivalent circuit can be employed to design meminductor-based application circuits as a meminductor emulator. By employing simulation experiment, we investigate the characteristics of this meminductor driven by sinusoidal excitation. The characteristic curves of current-flux (i-φ), voltage-flux (v-φ), v-ρ (internal variable of meminductor) and φ-ρ for the meminductor model are given by theoretical analyses and simulations. The curve of current-flux (i-φ) is a pinched hysteretic loop passing through the origin. The area bounding each sub-loop deforms as the frequency varies, and with the increase of frequency, the shape of the pinched hysteretic loop tends to be a straight line, indicating a dependence on frequency for the meminductor. Based on the meminductor model, a meminductive Wien-bridge chaotic oscillator is designed and analyzed. Some dynamical properties, including equilibrium points and the stability, bifurcation and Lyapunov exponent of the oscillator, are investigated in detail by theoretical analyses and simulations. By utilizing Lyapunov spectrum, bifurcation diagram and dynamical map, it is found that the system has periodic, quasi-periodic and chaotic states. Furthermore, there exist some complicated nonlinear phenomena for the system, such as constant Lyapunov exponent spectrum and nonlinear amplitude modulation of chaotic signals. Moreover, we also find the nonlinear phenomena of coexisting bifurcation and coexisting attractors, including coexistence of two different chaotic attractors and coexistence of two different periodic attractors. The phenomenon shows that the state of this oscilator is highly sensitive to its initial valuse, not only for chaotic state but also for periodic state, which is called coexistent oscillation in this paper. The basic mechanism and potential applications of the existing attractors are illustrated, which can be used to generate robust pseudo random sequence, or multiplexed pseudo random sequence. Finally, by using the equivalent circuit of the proposed meminducive model, we realize an analog electronic circuit of the meminductive Wien-bridge chaotic system. The results of circuit experiment are displayed by the oscilloscope, which can verify the chaotic characteristics of the oscillator. The oscillator, as a pseudo random signal source, can be used to generate chaotic signals for the applications in chaotic cryptography and secret communications.

A memcapacitor is a new type of memory capacitor. Before the advent of practical memcapacitor, the prospective studies on its models and potential applications are of importance. For this purpose, we establish a mathematical memcapacitor model and a corresponding circuit model. As a potential application, based on the model, a memcapacitor oscillator is designed, with its basic dynamic characteristics analyzed theoretically and experimentally. Some circuit variables such as charge, flux, and integral of charge, which are difficult to measure, are observed and measured via simulations and experiments. Analysis results show that besides the typical period-doubling bifurcations and period-3 windows, sustained chaos with constant Lyapunov exponents occurs. Moreover, this oscillator also exhibits abrupt chaos and some novel bifurcations. In addition, based on the digital signal processing (DSP) technology, a scheme of digitally realizing this memcapacitor oscillator is provided. Then the statistical properties of the chaotic sequences generated from the oscillator are tested by using the test suit of the National Institute of Standards and Technology (NIST). The tested randomness definitely reaches the standards of NIST, and is better than that of the well-known Lorenz system.

To develop real world memristor application circuits, an equivalent circuit model which imitates memductance (memory conductance) of the HP memristor is presented. The equivalent circuit can be used for breadboard experiments for various application circuit designs of memristor. Based on memductance of the realistic HP memristor and Chua’s circuit a new chaotic oscillator is designed. Some basic dynamical behaviors of the oscillator, including equilibrium set, Lyapunov exponent spectrum, and bifurcations with various circuit parameters are investigated theoretically and numerically. To confirm the correction of the proposed oscillator an analog circuit is designed using the proposed equivalent circuit model of an HP memristor, and the circuit simulations and the experimental results are given.

The dependence of sensitivity on initial conditions is the essence of chaos. And the randomness of chaos originates from the high sensitivity to initial values, which is measured by the Lyapunov exponents. It is found in this paper that the cascade of chaotic systems can considerably improve the Lyapunov exponents of cascade chaos and other dynamic properties. Therefore, in this paper, we study the cascade of chaotic systems and the influence on dynamic performances of the cascade chaos, and we present the definition and conditions of chaotic system cascade. It is proved in theory that the Lyapunov exponent of cascade chaos system is a sum of Lyapunov exponents of cascade subsystems. Appropriate cascade for chaotic systems can increase system parameters and expand parameter regions of chaos mapping and full mapping, thereby enhancing initial condition sensitivity of chaotic map and security of chaotic pseudo-random sequences. For logistic map, cubic map and tent map, the dynamic characteristics of logistic-logistic, logistic-cubic and logistic-tent cascade are investigated in detail, verifying the improvements on dynamic characteristics of cascade chaos systems. The proposed chaotic cascade system can be used to generate better pseudo-random sequences for initial condition sensitivity and security.

In this paper, a new hyperchaotic system is constructed by introducing an additional state variable into the third-order Lorenz system. Some basic properties, including dissipativity, equlibria, stability and Hopf bifurcation, of this hyperchaotic system are analyzed in detail, and the bifurcation routes to hyperchaos from periodic, chaotic evolutions are observed. The existence of hyperchaos is verified with Lyapunov exponent spectrum. Moreover, an analog electronic circuit is designed, and various hyperchaotic attractors of this system are observed from the circuit experiments.

In order to generate complex chaotic attractors, we construct a new three-dimensional quadratic autonomous chaotic system, in which each equation contains a single quadratic cross-product term and a system parameter. Basic dynamic properties of the new system are investigated via theoretical analysis and numerical simulation using the Lyapunov exponent spectrum and bifurcation diagram. Our results show that this system has five equilibria, therefore is not topologically equivalent to the Lorenz, Rsslor or the Chen and Lü systems, and the new system is chaotic when its parameters satisfy certain conditions. Compared with the systems mentioned above, the proposed system has larger positive Lyapunov exponent，displays a complex attractor and some other interesting properties. An electronic circuit was designed to realize the new chaotic system. Experimental chaotic behaviors of the system were found to be identical to the dynamic properties predicted by theoretical analysis and numerical simulations.