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In this paper, a modified degree-based strategy is proposed for pinning control of complex networks under constrained control capacity. Limited by the total control strength, the pinning strategy design problem is reconsidered as a constrained optimization problem. To provide a fast and effective solution, a decrease-and-conquer approach has been designed to assign the best control strengths to the pinned highest-degree nodes. Simulation results have demonstrated that the obtained strategy provides an outperformed control effect on various kinds of complex networks, as compared with strategies derived from degree, betweenness or closeness using evenly-distributed control strength.

Lorenz system, as the first classical chaotic system, has been intensively investigated over the last four decades. Based on the sawtooth wave function, this paper initiates a novel approach for generating multi-wing butterfly chaotic attractors from the generalized first and second kinds of Lorenz-type systems. Compared with the traditional ring-shaped multi-scroll Lorenz chaotic attractors, the proposed multi-wing butterfly chaotic attractors are much easier to be designed and implemented by analog circuits. The dynamical behaviors of these multi-wing butterfly chaotic systems are further studied. Theoretical analysis shows that every index-2 saddle-focus equilibrium corresponds to a unique wing in the butterfly attractors. Finally, a module-based unified circuit diagram is constructed for realizing various multi-wing butterfly attractors. It should be especially pointed out that this is the first time in the literature that a maximal 10-wing butterfly chaotic attractor is experimentally verified by analog circuits.

In this paper, a modulated optical injection system is proposed. It mainly consists of a semiconductor laser subject to a modulated optical injection. It is known that optical injection will force a laser intensity into high-speed periodic oscillation without modulation, while the oscillation will be locked with weak external modulation. However, the route to chaos in such a system is observed and reported for the first time. It is shown that the generated oscillation frequency shifts away from the external frequency when the modulation amplitude becomes sufficiently large. Due to the nonlinearities of the laser, the frequencies are mixed inside the laser cavity, which causes the dynamics to follow a quasi-periodic route to chaos. In addition, the resultant chaotic signal possesses a larger bandwidth than that attainable with constant optical injection alone. Since the system is capable of generating both narrowband photonic microwave signals with low phase variances and chaotic signals with large bandwidths, it is practically important for Doppler and chaotic lidar applications.

This paper proposes a new three-dimensional chaotic generator, for which the nonlinearity is based on sinusoidal function. Based on a practical circuit design for realizing a wide angular range of sinusoidal functions, chaotic attractors with large number of scrolls have been obtained. Different types of attractors are available, governed by the choices of variables used in the sinusoidal function. The design is illustrated with both simulations and experiments.

Synchronization of chaotic systems with unknown parameters, or namely auto-synchronization, has aroused a lot of interest in the past few years. In this paper, we are interested in solving one of the critical cases in Lorenz system. Based on the past researches, it was concluded that the auto-synchronization of Lorenz system was inachievable if the unknown parameters resided in all of its three dynamical equations while only the first state was measurable. However, as demonstrated in this paper, this problem can be resolved by an adaptive observer following simple design rules. The design is justified with the conditional Lyapunov exponents, and its effectiveness is clearly illustrated in simulations.

Random number generators are widely used in different applications. However, it is difficult to obtain a good random number generator in low precision and resource constrained system, such as an eight-bit micro-controller system which is still commonly used in industrial and consumer markets. This paper provides a practical solution for this problem based on chaotic systems. By the use of a modified Chua's circuit, it is demonstrated that the sampled state, after post-processing by a high-dimensional chaotic map, can be used as a random source even in an eight-bit environment. The randomness of the generated sequence is testified and confirmed by different statistical tests and the up-to-date statistical suite.

In this letter, a three-dimensional continuous-time smooth autonomous system with quadratic nonlinear terms is proposed for generating multiscroll chaotic attractors. Observation of 2 × 2-scroll attractors generated from this kind of system is reported for the first time. The result is confirmed by both numerical simulations and electronic circuit experiments.

In this paper, the generation of n × m-scroll attractors under a Chua-circuit framework is presented. By using a sawtooth function, f1(x), and a staircase function, f2(y), n × m-scroll attractors can be generated and observed from a third-order circuit. Its dynamical behaviors are investigated by means of theoretical analysis as well as numerical simulation. Moreover, two electronic circuits are designed for its realization, and experimental observations of n × m-scroll attractors based on Chua's circuit are reported, for the first time in the literature.

In this paper, a fast chaos-based pseudo-random number generator (PRNG) is proposed for secured communications. In order to achieve fast throughput and facilitate hardware realization, 32-bit fixed point representation and arithmetic are used. Even under such configuration with quantization errors which will make the normal chaos-based PRNG impractical, our scheme can pass all the statistical tests in the up-to-date National Institute of Standards and Technology (NIST) test suite with the output bit rate up to 134 Mbps in a 2.6 GHz Pentium-4 machine. With such a fast PRNG, a stream cipher is hence designed for the application of online secure voice communication system with User Datagram Protocol (UDP).

In this paper, a chaos-based cryptographic hash function is proposed for message authentication where a fixed length message digest can be generated from any messages with arbitrary length. The compression function in-use is developed according to the diffusion and confusion properties of a novel high-dimensional discrete chaotic map. A symmetric key structure can also be easily embedded based on the high sensitivity to initial conditions and system parameters of the map. As demonstrated in experiments, the throughput of the proposed scheme is about 1.5 times of that with MD5, and the hash value appears randomly without any correlation with the message or the key.

In this letter, a simple nonlinear state feedback controller is designed for generating hyperchaos from a three-dimensional autonomous chaotic system. The hyperchaotic system is not only demonstrated by computer simulations but also verified with bifurcation analysis, and is implemented experimentally via an electronic circuit.

In this paper, a neuron is introduced for chaotifying nonchaotic discrete-time systems with arbitrary dimensions. By modeling the neuron with a hyperbolic tangent activation function, a scalar feedback signal expressed in a linear combination of the neuron outputs is used. Chaos can then be generated from the controlled discrete-time system. The existence of chaos is verified by both theoretical proof and computer simulations.

In this paper, an online chatting system with an embedded real-time chaotic encryption/decryption method is designed for the Internet. Such system not only provides a real-time communication platform, but also ensures a secure channel for communication. By the use of cipher feedback and the skew tent map, the input text can be real-time encoded and the cipher text is sent via TCP/IP. With the properties of randomness of the map, and its sensitivity on system parameters and the initial conditions, the encrypted transmitting messages are difficult to be eavesdropped. The implemented method is simple and can be easily embedded in any existing systems.

This paper studies chaos synchronization between two coupled chaotic systems using linear state feedback control. A new, simple and yet easily applicable criterion is derived for chaos synchronization based on the Lyapunov stability theory and the Linear Quadratic Optimal Control theory. This criterion is given in terms of some simple algebraic inequalities established by the Gerschgorin theorem, and it is easily applicable to a large class of chaotic systems. As an example, the familiar Chua's circuit is simulated for demonstration.

Based on Lyapunov stabilization theory, this paper proposes a new generic criterion of global chaos synchronization between two coupled chaotic systems from a unidirectional linear error feedback coupling approach. The criterion is successfully applied to some typical chaotic systems with different types of nonlinearity, such as the classic Chua's circuit, the modified Chua's circuit with a sine function, and the Rössler and Lorenz chaotic systems. The coupling parameters are determined according to the new criterion so as to ensure the coupled systems' global chaos synchronization.

Chen's attractor was recently reported and proved to be topologically different from the Lorenz butterfly attractor. In this paper, Chen's attractor is realized with an electronic circuit. Although the circuitry is straight-forward, it is the first design reported. In addition, the synchronization scheme between two Chen's attractors is also considered and realized.

Based on the delayed feedback chaotic anticontrol systems, a new chaotic phase shift keying (CPSK) scheme is proposed for secure communications in this paper. The chaotic transmitter is a linear system with nonlinear delayed feedback in which a trigonometric function cos(·) is used. Such system can exhibit rich chaotic behavior with the choice of appropriate parameters. For an M-ary communication system where M=2n, each of these M possible symbols (n-bits) is firstly mapped to 2(m-1)π/M (with m=1, 2, …, M) which is used as the phase argument for the cos(·) function in the nonlinear feedback. Two different kinds of signals can be transmitted. In the first one, an appropriate linear combination of state variables is chosen as the transmitting signal based on the observer theory. In another one, a nonlinear component in the transmitter state equation is chosen. In both schemes, only a scalar chaotic signal is transmitted through the channel. Demodulation is based on the synchronization of the transmitter and the receiver, and different decoded phases correspond to different information signals.

In this paper, the effect of the parameter switching (PS) algorithm in a fractional order chaotic circuit is investigated both in simulation and experiment. The Chen system of fractional order is focused and realized in an electronic circuit. By designing a switching circuit, the PS algorithm is implemented and it is the first time, the paradoxical “Chaos [Formula: see text] Chaos [Formula: see text] Order” is presented in an electronic circuit. Both the simulation and experimental results confirm that the obtained attractor under switching approximates the attractor of the time-averaged model. Some important design issues for the circuitry realization of the PS scheme are pointed out. Finally, our work confirms the practical usage of PS algorithm in potential applications such as attractor synthesis and chaos control.

In this paper, we aim to identify a directed complex network with optimal controllability, for which pinning control is to be applied. Since the controllability of a network can be reflected by the smallest nonzero eigenvalue of a matrix related to its topology, an optimal network design problem is formulated based on the maximization of this eigenvalue. To better consider the practical reality, constraints on node degree sequence are specified. Based on the derived bounds of the eigenvalue, an effective rewiring scheme is designed and solutions close to or equal to the upper bound are obtained. Finally, the relationship between network characteristics and controllability is studied. Through complexity analysis, it is concluded that the network with high controllability should possess two properties, i.e. nodes with high out-degree for pinning and other nodes with uniform degree distribution.