Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions

We consider in this paper the dynamics of the self-sustained electromechanical system with multiple functions, consisting of an electrical Rayleigh–Duffing oscillator, magnetically coupled with linear mechanical oscillators. The averaging and the harmonic balance method are used to find the amplitudes of the oscillatory states respectively in the autonomous and nonautonomous cases, and analyze the condition in which the quenching of self-sustained oscillations appears. The influence of system parameters as well as the number of linear mechanical oscillators on the bifurcations in the response of this electromechanical system is investigated. Various bifurcation structures, the stability chart and the variation of the Lyapunov exponent are obtained, using numerical simulations of the equations of motion.     

A special full-state hybrid projective synchronization in symmetrical chaotic systems

A special full-state hybrid projective synchronization type is proposed in this paper. The anti-synchronization and complete synchronization can be achieved simultaneously in this new synchronization phenomenon. We point out how to realize this synchronization in chaotic systems: anti-synchronization in symmetrical coordinate subspace and complete synchronization in its normal coordinate subspace. Two illustrative examples, multi-scroll chaotic system by the partial Lyapunov stability theory, and a four-dimensional chaotic system by the invariance principle of differential equation are presented to exhibit this new synchronization.     

Estimation of unknown parameters and adaptive synchronization of hyperchaotic systems

This paper investigates the chaos synchronization of two hyperchaotic systems. Based on Lasalle invariance principle, adaptive schemes are derived to make two unidirectional coupling and mutual coupling hyperchaotic systems asymptotically synchronized whether the parameters are given or uncertain, and unknown parameters are identified simultaneously in the process of synchronization. Numerical simulations of hyperchaotic Chen systems are presented to show the effectiveness of the proposed chaos synchronization schemes.     

Multiswitching dual combination synchronization of time‐delay chaotic systems

This paper presents a novel synchronization scheme of multiswitching dual combination synchronization which is first of its kind. Multiswitching dual combination synchronization is achieved for 6 time‐delay chaotic systems. Asymptotically stable synchronization states are established by nonlinear control method and Lyapunov Krasovskii functional. To elaborate the proposed scheme, an example of time‐delay Rossler, Chen, and Shimizu Morioka systems is considered, where time‐delay Rossler system and Chen system are considered as drive systems and time‐delay Shimizu Morioka system is considered as response system. Theoretical analysis and computational results are in excellent agreement.     

Cluster synchronization in colored community network with different order node dynamics

In this paper, a colored community network model with different order nodes dynamics is introduced for the first time. The color of nodes or edges are assumed to be identical if they belong to the same community and nonidentical if they belong to different communities. The color of edges between any pair of communities are assumed to be identical if they connect the same pair of communities and nonidentical if they connect different pair of communities. Further, the order of the node dynamics in different communities can be totally different. Firstly, based on the Lyapunov stability theory, adaptive feedback controllers are designed for achieving cluster synchronization. Secondly, periodically intermittent controllers are designed for achieving cluster synchronization and the synchronization conditions are derived by using mathematical induction method and the analysis technique. Finally, several numerical examples are provided to illustrate the effectiveness of the derived theoretical results.     

Dynamics of a self-sustained electromechanical system with flexible arm and cubic coupling

This paper studies the dynamics of a self-sustained electromechanical system with nonlinear coupling. The mechanical part is a flexible beam. The Krylov–Bogoliubov averaging method is used to derive oscillatory solutions. Focus is made on the effects of the detuning parameter and nonlinear coupling. The largest Lyapunov exponent is used to determine chaotic domains of the system.     

Exponential synchronization for fractional‐order chaotic systems with mixed uncertainties

This article focuses on the problem of exponential synchronization for fractional‐order chaotic systems via a nonfragile controller. A criterion for α‐exponential stability of an error system is obtained using the drive‐response synchronization concept together with the Lyapunov stability theory and linear matrix inequalities approach. The uncertainty in system is considered with polytopic form together with structured form. The sufficient conditions are derived for two kinds of structured uncertainty, namely, (1) norm bounded one and (2) linear fractional transformation one. Finally, numerical examples are presented by taking the fractional‐order chaotic Lorenz system and fractional‐order chaotic Newton–Leipnik system to illustrate the applicability of the obtained theory. © 2014 Wiley Periodicals, Inc. Complexity 21: 114–125, 2015     

Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems

In this paper, a new type of anticipating synchronization, called time-varying anticipating synchronization, is defined firstly. Then novel adaptive schemes for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems are designed based on the Lyapunov function and invariance principle. The update gain of coupling strength can be automatically adapted to a suitable strength depending on the initial values and can be properly chosen to adjust the speed of achieving synchronization, so these schemes are analytical and simple to implement in practice. A classical chaotic dynamical system is used to demonstrate the effectiveness of the proposed adaptive schemes with or without parameter uncertainties.     

Adaptive‐impulsive function projective synchronization for a class of time‐delay chaotic systems

This article investigates the function projective synchronization (FPS) for a class of time‐delay chaotic system via nonlinear adaptive‐impulsive control. To achieve the FPS, suitable nonlinear continuous and impulsive controllers are designed based on adaptive control theory and impulsive control theory. Using the generalized Babarlat's lemma, a general condition is given to ensure the FPS. Here, the time‐delay chaotic system is assumed to satisfy the Lipschitz condition while the Lipschitz constants are estimated by augmented adaptation equations. Numerical simulation results are also presented to verify the effectiveness of the proposed synchronization scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 333–341, 2015     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Numerical study of flow asymmetry and self-sustained jet oscillations in geometrically symmetric cavities

In this paper we present the results of numerical investigation of self-sustained oscillations of a jet confined in a symmetric cavity. This work represents an attempt to reproduce empirical observations of asymmetric flows in geometrically symmetric systems and to extend the jet flow investigations to more complex possible scenarios. A well-known example of such two-dimensional flow has been reported experimentally and reproduced numerically for simple flow [E. Schreck, M. Schaefer, Numerical study or bifurcation in three-dimensional sudden channel expansions, Comput. Fluids 29 (2000) 583–593]. It has been found that for some particular control parameter, above its critical value (bifurcation point), the jet can be deflected to either of the two sides of the cavity. In this paper we report a similar behaviour which is, however, characterized by a more complicated flow pattern. While simple flow appears only within small cavity lengths the complex flows develops with increased cavity lengths. Unlike stationary asymmetric solutions accompanied by cavity jet oscillations, as experimentally reported in e.g., [A. Maurel, P. Ern, B.J.A. Zielinska, J.E. Wesfreid, Experimental study of self-sustained oscillations in a confined jet, Phys Rev. E 54 (1996) 3643–3651], in our investigations of both simple and complex asymmetric flow we observed the slow periodical drift of the jet from one to another side of the cavity. The essential control parameters were Reynolds number Re and the ratio length to inlet width L/d. According to experiments of Maurel et al. (1996), the jet is stable and symmetric, when both L/d and Re are below certain critical values, otherwise jet oscillations appear in both experiment and our simulation (cavity oscillations regime). However, further increase of either (or both) L/d and Re leads of so called free jet type oscillations regime. This paper describes complex jet behaviour within the later oscillations regime. We believe that both simple “classical” and “our” complex stationary asymmetric solutions (as well as superimposed cavity-type and free-jet oscillations) can be explained based on physical arguments as already done in previous works. However, the origin of slow drift motion remains still to be resolved. This might be of high importance for clear distinguishing between relevant physical and numerical features in future codes developments.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Generalized synchronization of chaotic systems by pure error dynamics and elaborate Lyapunov function

The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this paper. Generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation, instead of current mixed error dynamics in which master state variables and slave state variables are presented. The elaborate Lyapunov function is applied rather than the current plain square sum Lyapunov function, deeply weakening the power of Lyapunov direct method. The scheme is successfully applied to both autonomous and nonautonomous double Mathieu systems with numerical simulations.     

Synchronizing chaotic dynamics with uncertainties using a predictable synchronization delay design

This paper deals with synchronization and optimization problems of second-order chaotic oscillators by applying a novel control scheme. The approach developed considers incomplete state measurements and no detailed model of the systems to guarantee robust stability. This approach includes an uncertainty estimator and leads to a robust predictable feedback control scheme. The synchronization of the ⁶-Duffing and ⁶-Van der Pol oscillators was used as an illustrative example. A fairly good agreement is obtained between the analytical and numerical results.     

Finite-time synchronization for coupled systems with time delay and stochastic disturbance under feedback control

This paper proposes a framework for finite-time synchronization of coupled systems with time delay and stochastic disturbance under feedback control. Combining Kirchhoff"s Matrix Tree Theorem with Lyapunov method as well as stochastic analysis techniques, several sufficient conditions are derived. Differing from previous references, the finite time provided by us is related to topological structure of networks. In addition, two concrete applications about stochastic coupled oscillators with time delay and stochastic Lorenz chaotic coupled systems with time delay are presented, respectively. Besides, two synchronization criteria are provided. Ultimately, two numerical examples are given to illustrate the effectiveness and feasibility of the obtained results.     

Stability of synchronized dynamics and pattern formation in coupled systems: Review of some recent results

In arbitrarily coupled dynamical systems (maps or ordinary differential equations), the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) and the formation of patterns from loss of stability of the synchronized states are two problems of current research interest. These two problems are often treated separately in the literature. Here, we present a unified framework in which we show that the eigenvalues of the coupling matrix determine the stability of the synchronized state, while the eigenvectors correspond to patterns emerging from desynchronization. Based on this simple framework three results are derived: First, general approaches are developed that yield constraints directly on the coupling strengths which ensure the stability of synchronized dynamics. Second, when the synchronized state becomes unstable spatial patterns can be selectively realized by varying the coupling strengths. Distinct temporal evolution of the spatial pattern can be obtained depending on the bifurcating synchronized state. Third, given a desired spatiotemporal pattern, one is able to design coupling schemes which give rise to that pattern as the coupled system evolves. Systems with specific coupling schemes are used as examples to illustrate the general methods.     

Modified function projective lag synchronization of chaotic systems with disturbance estimations

This paper addresses the modified function projective lag synchronization (MFPLS) for a class of chaotic systems with unknown external disturbances. The disturbances are supposed to be generated by the exogenous systems. By using the disturbance-observer-based control and the linear matrix inequality approach, the disturbance observers are developed to ensure the boundedness of the disturbance error dynamics. Then by employing the sliding mode control (SMC) technique, an active SMC law is established to guarantee the disturbance rejection and realize MFPLS between the master and slave systems. And the corresponding numerical simulation is provided to illustrate the effectiveness of the proposed method.     

Adaptive cluster synchronization in directed networks with nonidentical nonlinear dynamics

In this article, the problem of cluster synchronization in the complex networks with nonidentical nonlinear dynamics is considered. By Lyapunov functional and M‐matrix theory, some sufficient conditions for cluster synchronization are obtained. Moreover, the least number of nodes which should be pinned is given. It is shown that when the root nodes of all the clusters are pinning‐controlled, cluster synchronization with adaptive coupling strength can be achieved. Different from the constraints of many literatures, the assumption is that each row sum for all diagonal submatrices of the Laplacian matrix is equal to zero. Finally, a numerical simulation in the network with three scale‐free subnetwork is provided to demonstrate the effectiveness of the theoretical results. © 2016 Wiley Periodicals, Inc. Complexity 21: 380–387, 2016     

Dead-beat full state hybrid projective synchronization for chaotic maps using a scalar synchronizing signal

This paper provides a contribution to the topic of full state hybrid projective synchronization (FSHPS) by introducing an observer-based approach that enables synchronization to be achieved via a scalar synchronizing signal. The method is based on a theorem that assures dead-beat synchronization (i.e., exact synchronization in finite time) to a wide class of discrete-time chaotic (hyperchaotic) systems. Two examples, involving the hyperchaotic Grassi-Miller map and the hyperchaotic double scroll map, show that FSHPS can be effectively achieved in finite time using a scalar synchronizing signal only.     

Asymptotic pinning synchronization of nonlinear multi-agent systems: Its application to tunnel diode circuit

This work proposes a pinning state-feedback control technique for synchronizing non-linear multi-agent systems (MASs) with time delays. A collection of switching-directed graphs describes the communication exchanges between all of the agents. The challenge of asymptotic stability analysis for some error systems is translated into the construction of a leader-following synchronization of the relevant MASs. The closed-loop system could be acquired by building a convenient Lyapunov–Krasovskii functional (LKF) that has two integral terms, and by using Kronecker product qualities combined with matrix inequality techniques. When these conditions are met, a state-feedback pinning controller can be built with linear matrix inequalities (LMIs), which can be derived easily from a number of efficient optimization algorithms. Further, the performance of the proposed control design system is verified based on a tunnel diode circuit (TDC) by numerical simulations.