A survey of Wien bridge-based chaotic oscillators: Design and experimental issues

This paper presents a comparative study on design and implementation of Wien type chaotic oscillators. By making a collection of almost all Wien bridge-based chaotic circuits, we have investigated these oscillators in terms of chaotic dynamics, circuit structures, active building blocks, nonlinear element structures and operating frequency by using PSpice simulations and laboratory experiments. In addition to this comparative investigation, we present our two basic experimental contributions to referred implementations. While the first of our experimental contributions consists of the experimentally implementation of CFOA-based Chua’s circuit modified for very high chaotic oscillations, the scope of the second is to experimentally implement a Wien type high frequency chaos generator, which has the diode–inductor composite, in the inductorless form by using CFOA-based synthetic inductor.     

Synchronous states in time-delay coupled periodic oscillators: A stability criterion

There are several works showing that nonzero time delay between nodes in an oscillator network can be responsible for several kinds of behavior as synchronization and chaos. Here, by using the Lyapunov linearizing method, in a system of two coupled oscillators derived as a particular case of the full connected network, it is shown that the time delay parameter has two sets of values: one that destabilizes the whole system and other that implies stability. Besides, there is a set of time delay values responsible for chaotic behaviors, even in a simple coupled oscillators system.     

Adaptive synchronization to a general non-autonomous chaotic system and its applications

In this paper, new adaptive synchronous criteria for a general class of n-dimensional non-autonomous chaotic systems with linear and nonlinear feedback controllers are derived. By suitable separation between linear and nonlinear terms of the chaotic system, the phenomenon of stable chaotic synchronization can be achieved using an appropriate adaptive controller of feedback signals. This method can also be generalized to a form for chaotic synchronization or hyper-chaotic synchronization. Based on stability theory on non-autonomous chaotic systems, some simple yet less conservative criteria for global asymptotic synchronization of the autonomous and non-autonomous chaotic systems are derived analytically. Furthermore, the results are applied to some typical chaotic systems such as the Duffing oscillators and the unified chaotic systems, and the numerical simulations are given to verify and also visualize the theoretical results.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

Synchronization of hyperchaotic oscillators via single unidirectional chaotic-coupling

In this paper, synchronization of two hyperchaotic oscillators via a single variable’s unidirectional coupling is studied. First, the synchronizability of the coupled hyperchaotic oscillators is proved mathematically. Then, the convergence speed of this synchronization scheme is analyzed. In order to speed up the response with a relatively large coupling strength, two kinds of chaotic coupling synchronization schemes are proposed. In terms of numerical simulations and the numerical calculation of the largest conditional Lyapunov exponent, it is shown that in a given range of coupling strengths, chaotic-coupling synchronization is quicker than the typical continuous-coupling synchronization. Furthermore, A circuit realization based on the chaotic synchronization scheme is designed and Pspice circuit simulation validates the simulated hyperchaos synchronization mechanism.     

Synchronization of Colpitts oscillators with different orders

In this paper, we consider chaos synchronization between chaotic Colpitts oscillators with different orders, consisting of standard and improved version of Colpitts oscillators. Firstly, the normalized state equation of the improved version of the Colpitts oscillator designed to operate in the ultrahigh frequency range are presented. It is found that this version is described by fourth-order nonlinear differential equations. The equations of motion are solved numerically using the Runge–Kutta algorithm and simulations demonstrate chaos in the microwave frequencies range. Secondly, the problem of synchronization dynamics of third and fourth orders systems in the chaotic states is also investigated, and a controller is proposed based on stability theory by constructing the Lyapunov function, to ensure synchronization between both oscillators. Computer experiments demonstrate the effectiveness and feasibility of the proposed technique for these oscillators.     

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.     

A simple adaptive control for full and reduced-order synchronization of uncertain time-varying chaotic systems

In this paper, a simple adaptive feedback control is proposed for full and reduced-order synchronization of time-varying and strictly uncertain chaotic systems. Our method uses only one feedback gain with parameter adaptation law and converges very fast even in the presence of noise. For full synchronization, a drive-response system consisting of two second-order identical parametrically excited oscillators achieve global synchronization; while for reduced-order synchronization, the dynamical evolution of a second-order parametrically driven oscillator is synchronized with the projection of a third-order time-varying chaotic system. The effectiveness of our approach is demonstrated using numerical simulations.     

Synchronizing chaotic systems using control based on tridiagonal structure

The direct design approach based on tridiagonal structure combines the structure analysis with the design of stabilizing controller and the original nonlinear affine systems is transformed into a stable system with special tridiagonal structure using the method. In this study, the direct method is proposed for synchronizing chaotic systems. There are several advantages in this method for synchronizing chaotic systems: (a) it presents an easy procedure for selecting proper controllers in chaos synchronization; (b) it constructs simple controllers easy to implement. Examples of Lorenz system, Chua’s circuit and Duffing system are presented.     

A simple global synchronization criterion for coupled chaotic systems

Based on the Lyapunov stabilization theory and Gerschgorin theorem, a simple generic criterion is derived for global synchronization of two coupled chaotic systems with a unidirectional linear error feedback coupling. This simple criterion is applicable to a large class of chaotic systems, where only a few algebraic inequalities are involved. To demonstrate the efficiency of design, the suggested approach is applied to some typical chaotic systems with different types of nonlinearities, such as the original Chua’s circuit, the modified Chua’s circuit with a sine function, and the Rössler chaotic system. It is proved that these synchronizations are ensured by suitably designing the coupling parameters.