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.     

Nonlinear electronic circuit, Part II: synchronization in a chaotic MODEM scheme

In this work we present a thorough investigation of the effect of noise (internal or external) on the synchronization of a drive-response configuration system (unidirectional coupling between two identical systems). Moreover, since in every practical implementation of a communication system, the transmitter and receiver circuits (although identical) operate under slightly different conditions it is essential to consider the case of the mismatch between the parameters of the transmitter and the receiver. In our work we consider the non-autonomous 2nd order nonlinear oscillator system presented in [G. Mycolaitis, A. Tamasevicious, A. Cenys, A. Namajunas, K. Navionis, A. N. Anagnostopoulos, Globally synchronizable non-autonomous chaotic oscillator, in: Proc. of 7th International Workshop on Nonlinear Dynamics of Electronic Systems, Denmark, July 1999, pp. 277-280] which is particularly suitable for digital communications.Furthermore, we modified the previous chaotic communication system in order to exhibit enhanced security features. The enhancement in the security of the system is achieved by introducing a set of parameters used in the encoding and decoding of the message signal. We also introduce a time delay parameter in the dynamical system which on the one hand improves the chaotic behavior of the system and on the other hand, adds further security in the encoding-decoding scheme.     

Chaotic scrambling for wireless analog video

We report the implementation of a novel in-band chaotic scrambler for securing wireless analog video. In this demonstration system, an analog video signal is injected into a chaotic oscillator and the output is transmitted through a standard wireless radio link. At the receiver, a descrambler separates the video from the chaotic signal in real time. Experimental results show the scrambled signal effectively hides the original video image, yet the descrambler recovers the original color video with reasonable clarity and detail. Compared to digital encryption, chaotic scrambling offers an efficient, low-cost alternative for masking time-critical analog communications.     

Nonlinear electronic circuit,Part II: synchronization in a chaotic MODEM scheme

In this work we present a thorough investigation of the effect of noise (internal or external) on the synchronization of a drive–response configuration system (unidirectional coupling between two identical systems). Moreover, since in every practical implementation of a communication system, the transmitter and receiver circuits (although identical) operate under slightly different conditions it is essential to consider the case of the mismatch between the parameters of the transmitter and the receiver. In our work we consider the non-autonomous 2nd order nonlinear oscillator system presented in [G. Mycolaitis, A. Tamasevicious, A. Cenys, A. Namajunas, K. Navionis, A. N. Anagnostopoulos, Globally synchronizable non-autonomous chaotic oscillator, in: Proc. of 7th International Workshop on Nonlinear Dynamics of Electronic Systems, Denmark, July 1999, pp. 277–280] which is particularly suitable for digital communications.Furthermore, we modified the previous chaotic communication system in order to exhibit enhanced security features. The enhancement in the security of the system is achieved by introducing a set of parameters used in the encoding and decoding of the message signal. We also introduce a time delay parameter in the dynamical system which on the one hand improves the chaotic behavior of the system and on the other hand, adds further security in the encoding–decoding scheme.     

Periodic and chaotic synchronizations of two distinct dynamical systems under sinusoidal constraints

In this paper, periodic and chaotic synchronizations between two distinct dynamical systems under specific constraints are investigated from the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator were obtained, and the invariant domain of sinusoidal synchronization is achieved. From analytical conditions, the control parameter map is developed. Numerical illustrations for partial and full sinusoidal synchronizations of chaotic and periodic motions of the controlled pendulum with the Duffing oscillator are carried out. This paper presents how to apply the theory of discontinuous dynamical systems to dynamical system synchronization with specific constraints. The function synchronization of two distinct dynamical systems with specific constraints should be identified only by G-functions. The significance of function synchronization of distinct dynamical systems is to make the synchronicity behaviors hidden, which is very useful for telecommunication synchronization and network security.     

Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal

This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date.     

基于EMD方法的混沌信号中周期分量的提取

提出一种从Duffing振子产生的混沌信号中提取谐波分量的方法．依据任何信号由不同的固有简单振动模态组成的概念,利用经验模式分解（EMD）方法,将混沌信号分离为不同的内在模态函数(IMF),并在特定参数下从中分解出单一频率成分的谐波信号,从而成功地将混沌信号和谐波分量分离．仿真实验都表明该方法非常有效．     

Synchronization of chaotic solid-state Nd:YAG lasers: Application to secure communication

In this paper, the synchronization problem of coupled chaotic lasers in master–slave configuration is numerically studied. The approach used allows to give a simple design procedure for the slave laser. In particular, we consider a complex system composed by two chaotic Nd:YAG lasers coupled through the first state variable of the master laser. Synchronization of chaotic Nd:YAG lasers is achieved by injecting the chaotic signal from the master Nd:YAG laser into the slave Nd:YAG laser. The robustness of synchronization is discussed when a mismatch of parameters occurs, and the effects of the channel noise on recovered information are showed. A potential application of chaotic synchronization of Nd:YAG lasers to transmit encrypted digital information is also given.     

Periodic and chaotic dynamics in an asymmetric elastoplastic oscillator

We consider the dynamics of a harmonically forced oscillator with an asymmetric elastic–perfectly plastic stiffness function. The computed bifurcation diagrams for the oscillator show regions of periodic motion, hysteresis and large regions of chaotic motion. These different regions of dynamical behaviour are plotted in a two-dimensional parameter space consisting of forcing amplitude and forcing frequency. Examples of the chaotic motion encountered are shown using a discontinuity crossing map. Comparisons are made with the symmetric oscillator by computing a typical bifurcation diagram and considering previously published results for the symmetric system. From this we conclude that the asymmetric system is dominated by a large region of chaotic motion whereas in the symmetric oscillator period one motion and coexisting period three motion predominates.     

Fast synchronization of non-identical chaotic modulation-based secure systems using a modified sliding mode controller

In this paper, a secure communication scheme based on chaotic modulation is proposed using a reversible process and a robust controller with efficient cost and complexity to synchronize two different chaotic systems. In the controller design, a sliding mode control with an adaptive rule is used for non-linear inputs. The adaptive rule is applied to ensure the synchronization when uncertainties, non-modeled dynamics or external distortions are at work. The message signal is recovered at the receiver using a recursive process at the end. The effectiveness of the proposed algorithm is confirmed via the simulation results for the synchronization of the transmitted signal modulated by Chen chaotic system at the transmitter and Genesio chaotic system at the receiver, and those for the information recovery process.     

Implementation of chaotic secure communication systems based on OPA circuits

In this paper, we proposed a novel three-order autonomous circuit to construct a chaotic circuit with double scroll characteristic. The design idea is to use RLC elements and a nonlinear resistor. The one of salient features of the chaotic circuit is that the circuit with two flexible breakpoints of nonlinear element, and the advantage of the flexible breakpoint is that it increased complexity of the dynamical performance. Here, if we take a large and suitable breakpoint value, then the chaotic state can masking a large input signal in the circuit. Furthermore, we proposed a secure communication hyperchaotic system based on the proposed chaotic circuits, where the chaotic communication system is constituted by a chaotic transmitter and a chaotic receiver. To achieve the synchronization between the transmitter and the receiver, we are using a suitable Lyapunov function and Lyapunov theorem to design the feedback control gain. Thus, the transmitting message masked by chaotic state in the transmitter can be guaranteed to perfectly recover in the receiver. To achieve the systems performance, some basic components containing OPA, resistor and capacitor elements are used to implement the proposed communication scheme. From the viewpoints of circuit implementation, this proposed chaotic circuit is superior to the Chua chaotic circuits. Finally, the test results containing simulation and the circuit measurement are shown to demonstrate that the proposed method is correct and feasible.     

A chaos secure communication scheme based on multiplication modulation

A secure spread spectrum communication scheme using multiplication modulation is proposed. The proposed system multiplies the message by chaotic signal. The scheme does not need to know the initial condition of the chaotic signals and the receiver is based on an extended Kalman filter (EKF). This signal encryption scheme lends itself to cheap implementation and can therefore be used effectively for ensuring security and privacy in commercial consumer electronics products. To illustrate the effectiveness of the proposed scheme, a numerical example based on Genesio-Tesi system and also Chen dynamical system is presented and the results are compared.     

Encryption and decryption of information in chaotic communication systems governed by delay-differential equations

We consider different ways for encryption and decryption of information in communication systems using chaotic signal of a time-delay system as a carrier. A method is proposed for extracting a hidden message in the case when the parameters of the chaotic transmitter are a priori unknown. For different configurations of the transmitter the procedure of information signal extraction from the transmitted signal is demonstrated using numerical data produced by nonlinear mixing of the chaotic signal of the Mackey–Glass system and frequency-modulated harmonic signal.     

Secure communication by chaotic synchronization: Robustness under noisy conditions

In this work we present a thorough investigation of the effect of noise (internal or external) on the synchronization of a drive-response configuration system (unidirectional coupling between two identical systems). Moreover, since in every practical implementation of a communication system, the transmitter and receiver circuits (although identical) operate under slightly different conditions it is essential to consider the case of the mismatch between the parameters of the transmitter and the receiver. In our work we consider the non-autonomous second order non-linear oscillator system presented by G. Mycolaitis et al. in Proceedings of Seventh International Workshop on Nonlinear Dynamics of Electronic Systems [Globally synchronizable non-autonomous chaotic oscillator, Denmark, July 1999, pp. 277–280], which is particularly suitable for digital communications. Binary information is encoded by combining square pulses of two different frequencies selected so that the system is always in the chaotic regime independent of the encoded message.     

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.     

Sinusoidal modulation control method in a chaotic neural network

Chaotic neural networks (CNNs) have chaotic dynamic associative memory properties: The memory states appear non-periodically, and cannot be converged to a stored pattern. Thus, it is necessary to control chaos in a CNN in order to recognize associative memory. In this paper, a novel control method, the sinusoidal modulation control method, has been proposed to control chaos in a CNN. In this method, a sinusoidal wave simplified from brain waves is used as a control signal to modulate a parameter of the CNN. The simulation results demonstrate the effectiveness of this control method. The controlled CNN can be applied to information processing. Moreover, the method provides a way to associate brain waves by controlling CNNs.     

Universal prediction-based adaptive fault estimator applied to secure communication

In this paper, a communication scheme that could use a nonlinear dynamical system to create encrypted keys with an additional dimension is proposed, and the scheme could keep encrypted keys not to diverge. Since the divergence of encrypted keys (nonlinear signals) easily happens in non-linear systems coupled with other systems, the adaptive control approach, proposed in this paper, uses the universal state-space adaptive observer-based fault diagnosis/estimator and the high-performance tracker to eliminate the divergence of encrypted keys. At the same time, the receiver of communication retrieves informal messages by the universal state-space adaptive observer-based fault diagnosis/estimator and the high-performance tracker. Thus, this paper takes advantage of the merit of digital redesign methodology for a practical implementation of secure-communication, and the estimator solves the problem of secure communication. Thus developed a new approach could add more dimensions into nonlinear secure-communication systems without having the problem of divergence of encrypted keys.     

Reduced-order synchronization of uncertain chaotic systems via adaptive control

We consider the coupling of two uncertain dynamical systems with different orders using an adaptive feedback linearization controller to achieve reduced-order synchronization between the two systems. Reduced-order synchronization is the problem of synchronization of a slave system with projection of a master system. The synchronization scheme is an exponential linearizing-like controller and a state/uncertainty estimator. As an illustrative example, we show that the dynamical evolution of a second-order driven oscillator can be synchronized with the canonical projection of a fourth-order chaotic system. Simulation results indicated that the proposed control scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed output feedback controller in the application of secure communication.     

Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing

We examine the Melnikov criterion for a global homoclinic bifurcation and a possible transition to chaos in case of a single degree of freedom nonlinear oscillator with a symmetric double well nonlinear potential. The system was subjected simultaneously to parametric periodic forcing and self-excitation via negative damping term. Detailed numerical studies confirm the analytical predictions and show that transitions from regular to chaotic types of motion are often associated with increasing the energy of an oscillator and its escape from a single well.     

An improved chaos-based secure communication technique using a novel encryption function with an embedded cipher key

In this paper, a secure communication technique, using a chaotic system with a single adjustable parameter and a single observable time series, is proposed. The chosen chaotic system, which is a variant of the famous Rikitake model, has a special structure for which the adjustable parameter appears in the dynamic equation of the observable time series. This particular structure is used to build a synchronization-based state observer that is decoupled from the adaptive parameter identifier. A local Lyapunov function is used to design the parameter identifier, with an adjustable convergence rate that guarantees the stability of the overall system. A two-channel transmission method is used to exemplify the suggested technique where the secret message is encoded using a nonlinear function of both the chaotic states and the adjustable parameter of the chaotic system that acts as a secret key. Simulations show that, at the receiver, the signal can be efficiently retrieved only if the secret key is known, even when both the receiver and the transmitter are in perfect synchronization. The proposed technique is demonstrated to have improved security and privacy against intruders, when compared to other techniques reported in the literature, while being simple to implement using both analog and digital hardware. In addition, the chosen chaotic system is shown to be flexible in accommodating the transmission of signals with variable bandwidths, which promotes the superiority and versatility of the suggested secure communication technique.