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Complex networks have been extensively investigated in recent years. However, the dynamics, especially chaos and bifurcation, of the complex-valued complex network are rarely studied. In this paper, a star network of coupled complex-valued van der Pol oscillators is proposed to reveal the mechanism of star coupling. By the aid of bifurcation diagram, Lyapunov exponent spectrum and phase portrait in this study, chaos, hyper-chaos, and multi-existing chaotic attractors are observed from the star network, although there are only periodic states in a complex-valued van der Pol oscillator. Complexity versus coupling strength and nonlinear coefficient shows that the bigger the network size, the larger the parameter range within the chaotic (hyper-chaotic) region. It is revealed that the chaotic bifurcation path is highly robust against the size variation of the star network, and it always evolves to chaos directly from period-1 and quasi-periodic states, respectively. Moreover, the coexistence of chaotic phase synchronization and complete synchronization among the peripherals is also found from the star network, which is a symmetrybreaking phenomenon.

For the wide frequency spectrum of chaotic signals, it is difficult to realize chaotic signal conditioning. Therefore, researchers turn to the exploration of chaotic systems with independent non-bifurcation control for easy chaos modification. In this paper, a system with only one non-quadratic term is modified for providing multiscale amplitude/frequency control. By adjusting the feedback with an odd higher degree term, a switchable chaotic oscillator is obtained, which provides the different scales of amplitude/frequency control with the chaotic signal. Multisim-based circuit simulation shows the efficiency and convenience of chaotic signal control.

This paper proposes a novel sliding mode control (SMC) scheme to stabilize a class of fractional-order chaotic systems. Through constructing two sliding mode variables, the control problem of n-dimensional system can be transformed to the equivalent stabilizing problem of a reduced-order system. Subsequently, on the basis of second-order sliding mode (SOSM) technique, a robust control law is designed, which strongly attenuates the chattering phenomenon inherent in traditional sliding mode controller, and guarantees the existence of sliding motion in a finite time. The stability of two sliding mode variables to the origin is proved by conventional and fractional Lyapunov theories, respectively. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach. 246Copyright ⓒ 2014 SERSC to realize the control/synchronization of fractional-order chaotic systems in the past decades [13][14][15][16][17].However, the traditional SMC is of the first order, and there exists an inevitable drawback in applying such SMC, that is the so-called chattering phenomenon cannot be avoided. In order to improve the control accuracy and avoid, or at least strongly attenuate the undesired chattering effect, the second-order sliding mode (SOSM) method is proposed. The SOSM control approach was developed starting from the mid eighties, and its main aim is enhancing the control precision and eliminating the chattering phenomenon by removing the control discontinuity. To the best of our knowledge, so far, there are few works available in stabilizing/synchronizing fractional-order chaotic systems by SOSM control approach.Take the above discussions into account, this article considers the roust stabilization problem of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. The main contribution of this article is that using the novel SMC approach, the control task of n-dimensional system is transformed to the equivalent problem of stabilizing a reduced-order system. In other words, under the effect of the proposed method, if the reduced-order system is stable, then the n-dimensional system also is stable, so we only need to ensure the asymptotic stability of the reduced-order system. We use the traditional and fractional Lyapunov functions to demonstrate that the reaching phases of two sliding surfaces are stable. Besides, on the basis of SOSM control theory, a robust control law is designed, which can strongly attenuates the chattering phenomenon inherent in conventional sliding mode controller.The structure of this paper is organized as follows. In Section 2, relevant definitions and lemmas are presented. Section 3 gives the main results. In Section 4, two simulation examples are provided to verify the effectiveness of the proposed method. Conclusions are drawn in Section 5.