Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the chaos control and the generalized projective synchronization methods for heavy symmetric gyroscope systems via Gaussian radial basis adaptive variable structure control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise like, synchronized chaotic systems can be used as cipher generators for secure communication. This paper presents chaos synchronization of two identical chaotic motions of symmetric gyroscopes. In this paper, the switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. Using the neural variable structure control technique, control laws are established which guarantees the chaos control and the generalized projective synchronization of unknown gyroscope systems. In the neural variable structure control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimator are derived in the sense of Lyapunov function. Thus, the unknown gyro systems can be guaranteed to be asymptotically stable. Also, the proposed method can achieve the control objectives. Numerical simulations are presented to verify the proposed control and synchronization methods. Finally, the effectiveness of the proposed methods is discussed.     

Adaptive generalized function projective lag synchronization of different chaotic systems with fully uncertain parameters

In this paper, a novel projective synchronization scheme called adaptive generalized function projective lag synchronization (AGFPLS) is proposed. In the AGFPLS method, the states of two different chaotic systems with fully uncertain parameters are asymptotically lag synchronized up to a desired scaling function matrix. By means of the Lyapunov stability theory, an adaptive controller with corresponding parameter update rule is designed for achieving AGFPLS between two diverse chaotic systems and estimating the unknown parameters. This technique is employed to realize AGFPLS between uncertain Lü chaotic system and uncertain Liu chaotic system, and between Chen hyperchaotic system and Lorenz hyperchaotic system with fully uncertain parameters, respectively. Furthermore, AGFPLS between two different uncertain chaotic systems can still be achieved effectively with the existence of noise perturbation. The corresponding numerical simulations are performed to demonstrate the validity and robustness of the presented synchronization method.     

Time and resonance patterns in chaotic piece-wise linear systems

The aim of this paper is to characterize time and resonant behavior in a class of piece-wise linear systems evolving chaotically. To this end, statistical methods are used to reveal interactions of different parts of the system. The system under study comprises a continuous time subsystem and a switching rule that induces an oscillatory path by switching alternately between stable and unstable conditions. Since the system is not continuous, the principal oscillation frequency depends on the switching regime and linear subsystem parameters; therefore, many time and resonant patterns can be observed. It is shown that the system may display resonance produced by the action of a external signal (switching law), as well as internal and combinational resonance. The effect of system parameters on time evolution and resonance is studied. It is shown that the nature of subsystems eigenvalues plays a crucial role in the type of resonance observed, producing in some cases complex interaction of resonance modes.     

Robust controlling of chaotic behavior in supply chain networks

The supply chain network is a complex nonlinear system that may have a chaotic behavior. This network involves multiple entities that cooperate to meet customers demand and control network inventory. Although there is a large body of research on measurement of chaos in the supply chain, no proper method has been proposed to control its chaotic behavior. Moreover, the dynamic equations used in the supply chain ignore many factors that affect this chaotic behavior. This paper offers a more comprehensive modeling, analysis, and control of chaotic behavior in the supply chain. A supply chain network with a centralized decision-making structure is modeled. This model has a control center that determines the order of entities and controls their inventories based on customer demand. There is a time-varying delay in the supply chain network, which is equal to the maximum delay between entities. Robust control method with linear matrix inequality technique is used to control the chaotic behavior. Using this technique, decision parameters are determined in such a way as to stabilize network behavior.     

Finite-time chaos control via nonsingular terminal sliding mode control

This paper considers the nonsingular terminal sliding mode control for chaotic systems with uncertain parameters or disturbances. The switching surface is designed technically to realize fast convergence. The controller derived from such switching surface is nonsingular and it can stabilize the chaotic systems in a finite time. Besides the second-order system and the triangular system, the proposed method can also be applied to a general class of uncertain nonlinear system. Finally, simulation results are presented to illustrate the effectiveness of the design.     

Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system

In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.     

不同超混沌系统的最优同步

在参数未知的情况下,通过设计最优控制器和参数自适应律实现了新的四维混沌系统与超混沌吕系统的同步.接着根据Lyapunov稳定性原理和Hamilton-Jacobi-Bellman方程,选取Lyapunov函数和合适的性能指标函数从理论上证明这种方法的有效性.理论证明结果表明所设计的控制器能使性能指标函数取得最小值,是最优的.最后又通过matlab软件对同步系统进行数值仿真,仿真结果显示驱动系统与响应系统能够很好地达到了同步,表明方法是可行有效的.     

Chaos in a new fractional-order system without equilibrium points

Chaotic systems without equilibrium points represent an almost unexplored field of research, since they can have neither homoclinic nor heteroclinic orbits and the Shilnikov method cannot be used to demonstrate the presence of chaos. In this paper a new fractional-order chaotic system with no equilibrium points is presented. The proposed system can be considered “elegant” in the sense given by Sprott, since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. When the system order is as low as 2.94, the dynamic behavior is analyzed using the predictor–corrector algorithm and the presence of chaos in the absence of equilibria is validated by applying three different methods. Finally, an example of observer-based synchronization applied to the proposed chaotic fractional-order system is illustrated.     

Lag projective synchronization of a class of complex network constituted nodes with chaotic behavior

In this paper, a method of the lag projective synchronization of a class of complex network constituted nodes with chaotic behavior is proposed. Discrete chaotic systems are taken as nodes to constitute a complex network and the topological structure of the network can be arbitrary. Considering that the lag effect between network node and chaos signal of target system, the control input of the network and the identification law of adjustment parameters are designed based on Lyapunov theorem. The synchronization criteria are easily verified.     

Chaotic attractor generation and critical value analysis via switching approach

The generation of novel chaotic funnel-shaped attractors is introduced and the analysis of related critical values is given with a proposed switching method in this paper. The underlying mechanism involves a simple three-dimensional switched system and a hysteretically switching signal. Moreover, theoretic analysis is carried out to study the attractor generation and the corresponding critical values by fully utilizing the specific structure of the non-smooth system. Based on carefully derivation, the critical values and related stability regions of the created attractors are estimated explicitly, which is usually impossible for general non-smooth dynamics. In addition, it is demonstrated by simulation that various attractor patterns are generated conveniently by adjusting suitable system parameters.     

Adaptive anti control of chaos for robot manipulators with experimental evaluations

Roughly speaking, anti control of chaos consists in injecting a chaotic behavior to a system by means of a control scheme. This note introduces a new scheme to solve the anti control of chaos for robot manipulators. The proposed controller uses an adaption law to estimate the robot parameters on line. Thus, the controller does not require any knowledge of the physical parameters of the manipulator, such as masses, lengths of the links, moments of inertia, etc. The new scheme is based in the velocity field control paradigm, hence the specification of a chaotic system to define a desired velocity field is required. Experimental results in a two degrees-of-freedom direct-drive robot illustrate the practical feasibility of the introduced theory. In order to achieve anti control of chaos of our experimental system, two different chaotic attractors are used: the Genesio-Tesi system and a Jerk-type system. Results showed that the controller is able to inject the chaotic behavior to the robot while the robot parameters are estimated on line.     

Unknown nonlinear chaotic gyros synchronization using adaptive fuzzy sliding mode control with unknown dead-zone input

In this paper, the problem of synchronizing two chaotic gyros in the presence of uncertainties, external disturbances and dead-zone nonlinearity in the control input is studied while the structure of the gyros, parameters of the dead-zone and the bounds of uncertainties and external disturbances are unknown. The dead-zone nonlinearity in the control input might cause the perturbed chaotic system to show unpredictable behavior. This is due to the high sensitivity of these systems to small changes in their parameters. Thereby, the effect of these issues should not be ignored in the control design for these systems. In order to eliminate the effects from the dead-zone nonlinearity, in this paper, a robust adaptive fuzzy sliding mode control scheme is proposed to overcome the synchronization problem for a class of unknown nonlinear chaotic gyros. The main contribution of our paper in comparison with other works that attempt to solve the problem of dead-zone in the synchronization of chaotic gyros is that we assume that the structure of the system, uncertainties, external disturbances, and dead-zone are fully unknown. Simulation results are provided to illustrate the effectiveness of the proposed method.     

Finite-time synchronization and identification of complex delayed networks with Markovian jumping parameters and stochastic perturbations

In this paper, the finite-time synchronization and identification for the uncertain system parameters and topological structure of complex delayed networks with Markovian jumping parameters and stochastic perturbations is studied. On the strength of finite time stability theorem and appropriate stochastic Lyapunov–Krasovskii functional under the Itô’s formula, some sufficient conditions are obtained to assurance that the complex delayed networks with Markovian switching dynamic behavior can be identified the uncertain parameters and topological structure matrix in finite time under stochastic perturbations. In addition, three numerical simulations of different situation and dimension are presented to illustrate the effectiveness and feasibility of the theoretical results.     

A memristive chaotic system with heart-shaped attractors and its implementation

As a controllable nonlinear element, memristor is easy to produce the chaotic signal. Most of the current researchers focus on the nonlinear characteristics of the memristor, however, its ability to control and adjust chaotic systems is often neglected. Therefore, a memristive chaotic system is introduced to generate a kind of heart-shaped attractors in this paper. To further understand the complex dynamics of the system, several basic dynamical behavior of the new chaotic system, such as dissipation and the stability of the equilibrium point is investigated. Some basic properties such as Poincaré-map, Lyapunov index and bifurcation diagram are presented, either analytically or numerically. In addition, the influence of parameters on the system's dynamic behavior is analyzed. Finally, an analog implementation based on PSPICE simulation is also designed. The obtained results clearly show this chaotic system has rich nonlinear characteristics. Some interesting conclusions can be drawn that memristors bring the following effects on chaotic systems: (a) when the polarity of the memristor is changed, a mirror image of the chaotic attractors will appeared in the system; (b) along with the proper choose of the memristor parameters, the chaotic motion of system will be suppressed and enhanced, which makes the system can be applied to the practice on either generating chaos signal or suppressing chaotic interference.     

Multi‐switching combination–combination synchronization of non‐identical fractional‐order chaotic systems

In this paper, multi‐switching combination–combination synchronization scheme has been investigated between a class of four non‐identical fractional‐order chaotic systems. The fractional‐order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional‐order Lü and Rössler chaotic systems. In multi‐switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional‐order chaotic systems, the multi‐switching combination–combination synchronization of four fractional‐order non‐identical systems has been investigated. For the synchronization of four non‐identical fractional‐order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.     

Chaos in periodically forced discrete-time ecosystem models

Natural populations whose generations are non-overlapping can be modelled by difference equations that describe how the populations evolve in discrete time-steps. These ecosystem models are, in general, nonlinear and contain system parameters that relate to such properties as the intrinsic growth-rate of a species. Typically, the parameters are kept constant. In this study, in order to simulate cyclic effects due to changes in environmental conditions, periodic forcing is applied to system parameters in four specific models, comprising three well-known, single-species models due to May, Moran–Ricker, and Hassell, and also a Maynard Smith predator–prey model. It is found that, in each case, a system that has simple (e.g., periodic) behavior in its unforced state can take on extremely complicated behavior, including chaos, when periodic forcing is applied, dependent on the values of the forcing amplitudes and frequencies. For each model, the application of forcing is found to produce an effective increase in the parameter space over which the system can behave chaotically. Bifurcation diagrams are constructed with the forcing amplitude as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, and attractor crises.     

A note on the fractional-order Volta’s system

This paper deals with the fractional-order Volta’s system. It is based on the concept of chaotic system, where the mathematical model of system contains fractional order derivatives. This system has simple structure and can display a double-scroll attractor. The behavior and stability analysis of the integer-order and the fractional commensurate and non-commensurate order Volta’s system with total order less than 3 which exhibits chaos are presented as well.     

Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses

In this paper, the chaotic behavior of a simplest autonomous memristor-based circuit of fractional order is suppressed by periodic impulses applied to one or several state variables. The circuit consists of two passive linear elements, a capacitor and an inductor, as well as a nonlinear memristive element. It is shown that by applying a sequence of adequate (identical or different) periodic impulses to one or several variables, the chaotic behavior can be suppressed. Impulse values and control timing are determined numerically, based on the bifurcation diagram with impulses as bifurcation parameters. Empirically, the probability to have a reasonably wide range of impulses to suppress chaos is quite large, ensuring that chaos suppression can be implemented, as demonstrated by several examples presented.     

A dynamic parameter estimator to control chaos with distinct feedback schemes

A dynamic strategy is proposed to estimate parameters of chaotic systems. The dynamic estimator of parameters can be used with diverse control functions; for example, those based on: (i) Lie algebra, (ii) backstepping, or (iii) variable feedback structure (sliding-mode). The proposal has adaptive structure because of interaction between dynamic estimation of parameters and a feedback control function. Without lost of generality, a class of dynamical systems with chaotic behavior is considered as benchmark. The proposed scheme is compared with a previous low-parameterized robust adaptive feedback in terms of execution and performance. The comparison is motivated to ask: What is the suitable adaptive scheme to suppress chaos in an specific implementation? Experimental results of proposed scheme are discussed in terms of control execution and performance and are relevant in specific implementations; for example, in order to induce synchrony in complex networks.     

Nonlinear dynamics and circuit realization of a new chaotic flow: A variant of Lorenz,Chen and Lü

In this paper, a new three-dimensional autonomous chaotic system is presented, and the range of the parameters which can induce the system to be unstable is analyzed. The dynamical behavior of this system is further investigated in some detail, including equilibria and stability, various attractors, together with the maximally complex attractor, Poincaré maps, bifurcations, and Lyapunov-exponent spectrum. The oscillator circuit of the new chaotic system is afterwards designed by using EWB software and a typical chaotic attractor is experimentally demonstrated.