Dynamical properties and chaos synchronization of improved Colpitts oscillators

In this paper, the dynamics and synchronization of improved Colpitts oscillators designed to operate in ultrahigh frequency range are considered. The model is described by a continuous time four-dimensional autonomous system with an exponential nonlinearity. The system is integrated numerically and various bifurcation diagrams and corresponding graphs of largest 1D Lyapunov exponent are plotted to summarize different scenarios leading to chaos. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling, intermittency and interior crisis routes when monitoring the bias (i.e. power supply) in tiny ranges. In order to promote chaos-based synchronization designs of this type of oscillators, a synchronization strategy based upon the design of a nonlinear state observer is successfully adapted. The suggested approach enables synchronization to be achieved via a scalar transmitted signal which represents a suitable feature for communication applications. Numerical simulations are performed to demonstrate the effectiveness and feasibility of the proposed technique.     

On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals

In this paper, we have examined effects of forcing a periodic Colpitts oscillator with periodic and chaotic signals for different values of coupling factors. The forcing signal is generated in a master bias-tuned Colpitts oscillator having identical structure as that of the slave periodic oscillator. Numerically solving the system equations, it is observed that the slave oscillator goes to chaotic state through a period-doubling route for increasing strengths of the forcing periodic signal. For forcing with chaotic signal, the transition to chaos is observed but the route to chaos is not clearly detectable due to random variations of the forcing signal strength. The chaos produced in the slave Colpitts oscillator for a chaotic forcing is found to be in a phase-synchronized state with the forced chaos for some values of the coupling factor. We also perform a hardware experiment in the radio frequency range with prototype Colpitts oscillator circuits and the experimental observations are in agreement with the numerical simulation 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.     

A novel chaos synchronization of uncertain mechanical systems with parameter mismatchings, external excitations, and chaotic vibrations

In this paper, a new concept of chaos synchronization, which is superior to generalized exponential synchronization, generalized virtual synchronization, and generalized complete synchronization, is firstly introduced and the chaos synchronization of a pair of Duffing-Holmes oscillators with parameter mismatchings, external excitations, and chaotic vibrations is investigated. Based on the time-domain approach with differential inequality, a feedback control is proposed to realize generalized synchronization (generalized exponential synchronization, respectively) for a pair of Duffing-Holmes oscillators without uncertainties (with uncertainties, respectively). In addition, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but also the critical time can be correctly estimated. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the obtained result.     

Chaos time-domain reflectometry for fault location on live wires

We propose a chaos time-domain reflectometry (CTDR) for locating faults on live wires. This method uses a chaotic output of an improved Colpitts oscillator as probe signal, and detects wire faults by correlating a duplicate with the echo of the probe signal. Benefiting from the anti-jamming of the correlation function of the wideband chaos, fault location on live wires can be achieved. We experimentally demonstrate the detection for live wires in a digital communication system, in which a type of digital signal named high density bipolar of order 3 (HDB3) is transmitted. The effects of the chaotic probe signal on the bit error rate (BER) of the transmitted HDB3 at different rates are analyzed. Meanwhile, the influences of the backward HDB3 reflected by wiring faults on the signal-noise-ratio (SNR) of CTDR measurement are examined experimentally. The results show that fault detection on live wires is achieved when the power of the chaotic probe signal is about from -24.8 dB to -13.5 dB lower than that of the transmitted digital signal. In this case, the BER is kept less than 3E-10, and the SNR of CTDR is higher than 3 dB. Besides, the auto-correlation properties of the improved Colpitts oscillator at different states are investigated experimentally to explore the suitable chaotic states for the CTDR.     

An adaptive feedback control for chaos synchronization of nonlinear systems with different order

This paper deals with the chaotic synchronization of fourth-order system and second driven oscillators. Such a problem is related to the synchronization of strictly different chaotic systems. We show that the dynamical evolution of second-order driven oscillators can be synchronized with the projection on canonical planes of a fourth-order chaotic system. In this sense, it is said that the synchronization is achieved in reduced-order. Based on the Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. An application to secure chaotic communication is also discussed. Numerical simulations are presented to demonstrate the efficiency of the proposed synchronization and secure communication schemes.     

Entrainment by Chaos

A new phenomenon, the entrainment of limit cycles by chaos, which results from the appearance of cyclic irregular behavior, is discussed. In this study, sensitivity is considered as the main ingredient of chaos to be captured, and the period-doubling cascade is chosen for extension. Theoretical results are supported by simulations and discussions regarding Chua’s oscillators, entrainment of toroidal attractors by chaos, synchronization, and controlling problems. It is demonstrated that the entrainment cannot be considered as generalized synchronization of chaotic systems.     

Chaotic synchronization of two complex nonlinear oscillators

Synchronization is an important phenomenon commonly observed in nature. It is also often artificially induced because it is desirable for a variety of applications in physics, applied sciences and engineering. In a recent paper [Mahmoud GM, Mohamed AA, Aly SA. Strange attractors and chaos control in periodically forced complex Duffing’s oscillators. Physica A 2001;292:193–206], a system of periodically forced complex Duffing’s oscillators was introduced and shown to display chaotic behavior and possess strange attractors. Such complex oscillators appear in many problems of physics and engineering, as, for example, nonlinear optics, deep-water wave theory, plasma physics and bimolecular dynamics. Their connection to solutions of the nonlinear Schrödinger equation has also been pointed out.In this paper, we study the remarkable phenomenon of chaotic synchronization on these oscillator systems, using active control and global synchronization techniques. We derive analytical expressions for control functions and show that the dynamics of error evolution is globally stable, by constructing appropriate Lyapunov functions. This means that, for a relatively large set initial conditions, the differences between the drive and response systems vanish exponentially and synchronization is achieved. Numerical results are obtained to test the validity of the analytical expressions and illustrate the efficiency of these techniques for inducing chaos synchronization in our nonlinear oscillators.     

Adaptive control and synchronization of chaotic systems consisting of Van der Pol oscillators coupled to linear oscillators

This paper deals with the problem of control and synchronization of coupled second-order oscillators showing a chaotic behavior. A classical feedback controller is first used to stabilize the system at its equilibrium. An adaptive observer is then designed to synchronize the states of the master and slave oscillators using a single scalar signal corresponding to an observable state variable of the driving oscillator. An interesting feature of the proposed approach is that it can be used for chaos control as well as synchronization purposes. Numerical simulations results confirming the analytical predictions are shown and pspice simulations are also performed to confirm the efficiency of the proposed control scheme.     

Chaos synchronization for a class of nonlinear oscillators with fractional order

In this paper, a special kind of nonlinear chaotic oscillator, the Qi oscillator, is studied in detail. Since such systems are shown to possess a relatively wide spectral bandwidth, it is considerably beneficial to practical engineering in the secure communication field. The chaos synchronization problem of the fractional-order Qi oscillators coupled in a master-slave pattern is examined by applying three different kinds of methods: the nonlinear feedback method, the one-way coupling method and the method based on the state observer. Suitable synchronization conditions are derived by using the Lyapunov stability theory, and most importantly, a sufficient and necessary synchronization condition for the case with fractional order between 1 and 2 is presented. Results of numerical simulations validate the effectiveness and applicability of the proposed schemes.     

Synchronization of chaos in non-identical parametrically excited systems

In this paper, we investigate the synchronization of chaotic systems consisting of non-identical parametrically excited oscillators. The active control technique is employed to design control functions based on Lyapunov stability theory and Routh–Hurwitz criteria so as to achieve global chaos synchronization between a parametrically excited gyroscope and each of the parametrically excited pendulum and Duffing oscillator. Numerical simulations are implemented to verify the results.     

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.     

Adaptive synchronization between two different chaotic dynamical systems

This work presents the synchronization between two different chaotic systems by using an adaptive feedback control scheme. The adaptive synchronization problem between an electrostatic system and electromechanical transducer has been investigated. An adaptive linear feedback law with two controllers is proposed to ensure the global chaos synchronization of the nonlinear electrostatic and electromechanical systems. Numerical simulations results are presented to demonstrate the effectiveness of the proposed method.     

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.     

Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller

In this paper, a robust adaptive sliding mode controller (RASMC) is proposed to realize chaos synchronization between two different chaotic systems with uncertainties, external disturbances and fully unknown parameters. It is assumed that both master and slave chaotic systems are perturbed by uncertainties, external disturbances and unknown parameters. The bounds of the uncertainties and external disturbances are assumed to be unknown in advance. Suitable update laws are designed to tackle the uncertainties, external disturbances and unknown parameters. For constructing the RASMC a simple sliding surface is first designed. Then, the RASMC is derived to guarantee the occurrence of the sliding motion. The robustness and stability of the proposed RASMC is proved using Lyapunov stability theory. Finally, the introduced RASMC is applied to achieve chaos synchronization between three different pairs of the chaotic systems (Lorenz–Chen, Chen–Lorenz, and Liu–Lorenz) in the presence of the uncertainties, external disturbances and unknown parameters. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed RASMC.     

Bifurcation analysis of a simple 3D oscillator and chaos synchronization of its coupled systems

Tama?evi?ius et al. proposed a simple 3D chaotic oscillator for educational purpose. In fact the oscillator can be implemented very easily and it shows typical bifurcation scenario so that it is a suitable training object for introductory education for students. However, as far as we know, no concrete studies on bifurcations or applications on this oscillator have been investigated. In this paper, we make a thorough investigation on local bifurcations of periodic solutions in this oscillator by using a shooting method. Based on results of the analysis, we study chaos synchronization phenomena in diffusively coupled oscillators. Both bifurcation sets of periodic solutions and parameter regions of in-phase synchronized solutions are revealed. An experimental laboratory of chaos synchronization is also demonstrated.     

Output tracking and synchronization of chaotic Chua’s circuit with disturbances via model predictive regulator

In this paper, we consider the output tracking and synchronization problem for a piecewise linear chaotic Chua’s circuit, which is expected to follow the reference signal generated by another autonomous chaotic Chua’s circuit. A novel model predictive regulator is proposed to guarantee the chaos synchronization in the presence of external disturbance. Moreover with this regulator, the optimal transient tracking performance for synchronization is ensured even when the piecewise autonomous and controlled Chua’s circuits switch asynchronously. The proposed approach is highly efficient in the sense that the regulation conditions can be satisfied by solving a set of linear matrix inequalities and several Sylvester equations instead of many partial differential regulator equations in the well-known nonlinear output regulation theory.     

Scaling features of multimode motions in coupled chaotic oscillators

Two different methods (the WTMM- and DFA-approaches) are applied to investigate the scaling properties in the return-time sequences generated by a system of two coupled chaotic oscillators. Transitions from twomode asynchronous dynamics (torus or torus–chaos) to different states of chaotic phase synchronization are found to significantly reduce the degree of multiscality. The influence of external noise on the possibility of distinguishing the various chaotic states is considered.     

Function projective synchronization for fractional-order chaotic systems

This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between the fractional-order Lorenz systems with different orders, and achieve synchronization between the fractional-order Lorenz system and fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

The function cascade synchronization method and applications

In this paper, a new function cascade synchronization method of chaos system is proposed to achieve generalized projective synchronization for chaotic systems. Based on Laypunov stability, the proposed synchronization technique is applied to three famous chaotic systems: the unified chaotic system, Liu system and Rössler system, which can make the states of two identical chaotic systems asymptotically synchronized by choosing different special suitable error functions. Numerical simulations are presented to show the effectiveness.