Chua System Research Articles

Anti‐Synchronization and Intermittent Anti‐Synchronization of Two Identical Delay Hyperchaotic Chua Systems Via Linear Control

AbstractThis work is concerned with anti‐synchronization and intermittent anti‐synchronization of two identical delay hyperchaotic Chua systems with linear control approaches. In the first two schemes, based on Lyapunov stability theory, some sufficient conditions for achieving anti‐synchronization of two identical delay hyperchaotic Chua systems are derived, numerical simulation results are presented to demonstrate the effectiveness of the proposed anti‐synchronization schemes. In the third scheme, the anti‐synchronization conditions are obtained by numerical method. In the fourth scheme, only one state of the response system is controlled via linear control, we report an interesting phenomenon of intermittent anti‐synchronization.

Generation of 2N + 1-scroll existence in new three-dimensional chaos systems.

We propose a systematic methodology for creating 2N + 1-scroll chaotic attractors from a simple three-dimensional system, which is named as the translation chaotic system. It satisfies the condition a12a21 = 0, while the Chua system satisfies a12a21 > 0. In this paper, we also propose a successful (an effective) design and an analytical approach for constructing 2N + 1-scrolls, the translation transformation principle. Also, the dynamics properties of the system are studied in detail. MATLAB simulation results show very sophisticated dynamical behaviors and unique chaotic behaviors of the system. It provides a new approach for 2N + 1-scroll attractors. Finally, to explore the potential use in technological applications, a novel block circuit diagram is also designed for the hardware implementation of 1-, 3-, 5-, and 7-scroll attractors via switching the switches. Translation chaotic system has the merit of convenience and high sensitivity to initial values, emerging potentials in future engineering chaos design.

Localization Method of Compact Invariant Sets with Application to the Chua System

In this paper, we consider the problem of compact invariant sets localization for the Chua system. To obtain our results we develop and apply a localization method. This method allows us to find two types of subsets in the phase space of a nonlinear system. The first type consists of Poincaré sections having a nonempty intersection with any compact invariant set of the system. The second type consists of localizing sets containing all compact invariant sets of the system. The considered localization method produces systems of inequalities describing the localizing sets and specifies the equations of the appropriate global sections. These inequalities and equations depend on parameters of the system and, therefore, the obtained localization results can be used in the bifurcation analysis. We find one-parametric families of both compact global sections and nontrivial localizing sets for the Chua system. These localizing sets are compact or unbounded. The intersection of unbounded localizing sets in some cases is a compact localizing set. We indicate the domains where trajectories of the Chua system go to infinity.

Study of bifurcations and chaos in the Muthuswamy–Chua system

The dynamical behavior of an autonomous simplest electronic circuit that consists of three elements connected in series governed by three ordinary differential equations was investigated. The circuit elements are a linear passive inductor, a linear passive capacitor, and a nonlinear active memristor. Under variation of two parameters we observed a rich variety of bifurcation phenomena, including periodic, quasi-periodic, intermittent, and chaotic behaviors associated with this very simple nonlinear system. One and two-parameter bifurcation diagrams were studied. Route of period-doubling to chaos through different structures was observed. Our theoretical results are compared with the available experimental results and are found to be in good agreement with these results.

Multistability in Chua's circuit with two stable node-foci.

Only using one-stage op-amp based negative impedance converter realization, a simplified Chua's diode with positive outer segment slope is introduced, based on which an improved Chua's circuit realization with more simpler circuit structure is designed. The improved Chua's circuit has identical mathematical model but completely different nonlinearity to the classical Chua's circuit, from which multiple attractors including coexisting point attractors, limit cycle, double-scroll chaotic attractor, or coexisting chaotic spiral attractors are numerically simulated and experimentally captured. Furthermore, with dimensionless Chua's equations, the dynamical properties of the Chua's system are studied including equilibrium and stability, phase portrait, bifurcation diagram, Lyapunov exponent spectrum, and attraction basin. The results indicate that the system has two symmetric stable nonzero node-foci in global adjusting parameter regions and exhibits the unusual and striking dynamical behavior of multiple attractors with multistability.

Vibrational resonance and implementation of dynamic logic gate in a piecewise-linear Murali–Lakshmanan–Chua circuit

We report the occurrence of vibrational resonance in piecewise-linear non-autonomous system. Especially, we show that an optimal amplitude of the high frequency second harmonic driving enhances the response of a piece-wise linear non-autonomous Murali–Lakshmanan–Chua (MLC) system to a low frequency first harmonic signal. This phenomenon is illustrated with the analytical solutions of circuit equations characterising the system and finally compared with the numerical method. Further, it has been enunciated explicitly, the implementation of the fundamental NOR/NAND gate via vibrational resonance, both by numerical and analytical solutions. In addition, these logical behaviours (AND/NAND/OR/NOR) can be decided by the amplitude of the input square waves without altering the system parameters.

Multiple attractors in a non-ideal active voltage-controlled memristor based Chua's circuit

Multiple attractors can be found in many nonlinear dynamical system with multistability. Recently, experimental attractors with two stable saddle-foci were reported to find in a non-ideal active voltage-controlled memristor based Chua's circuit. In this paper we focus on the multiple attractors found in the proposed memristive Chua's circuit. Concretely, by numerical simulations of mathematical model, hardware circuit experiments and PSIM circuit simulations, multiple attractors with different initial states are revealed and with the dimensionless system equations, complex dynamics with different initial conditions are further discussed. Theoretical derivation results indicate that the normalized memristive Chua's system has two stable nonzero saddle-foci in globally adjusting normalized parameter region and exhibits the unusual and striking dynamical behavior of multiple attractors with multistability.

Method of “transition into space of derivatives”: 40 years of evolution

In 1975, the “method of transition into space of derivatives” was proposed. It is an efficiently verifiable frequency criterion for the existence of a nontrivial periodic solution in multidimensional models of automatic control systems with one differentiable nonlinear term. The method used the classical torus principle and refrained from any constructions in the phase space of the system under study. Moreover, the method allowed researchers to broaden the class of systems to which it could be applied. In this work, we give a survey of the results presenting generalization and expansion of the method. We also show the connection between the method of transition into space of derivatives, the well-known generalized Poincare–Bendixson principle proposed by R. A. Smith, and the results of contemporary authors who are active in the theory of oscillations in multidimensional systems. In the recent years, the author obtained frequency criteria for the existence of orbitally stable cycles in multiinput multioutput (MIMO) control systems and methods for the construction of multidimensional systems having a unique equilibrium and an arbitrarily prescribed number of orbitally stable cycles, which are described in the paper. The extension of the generalized Poincare–Bendixson principle to multidimensional systems with angular coordinate is presented. We show the application of described methods of investigation of oscillation processes in multidimensional dynamical systems to solving S. Smale’s problem in the chemical kinetics theory of biological cells and also to finding hidden attractors of the generalized Chua system and the minimal global attractor of a system with a polynomial nonlinear term. The publication is illustrated by numerous examples.

Synchronization of Chaotic Systems with Uncertain Time-Varying Parameters

In this paper, a method for digitally redesign of observer-based synchronization for chaotic systems with uncertain time-varying parameters is presented. The proposed design methodology in this work can be considered as an observer design problem where two systems states synchronize. The proposed method starts by having the system in a form suitable for the design; hence, the optimal linearization method (OLM) of the nominal system, around each operating point of the system trajectory, is used. The uncertain time-varying parameters terms are decomposed such that the nonlinear system can be presented as a set of local linear model with disturbed inputs. Then, H∞ robust observer is designed in continuous-time domain based on the linearized model of the system to force the system to synchronize. Finally, a robust digital observer is obtained as a result of applying digital redesign of the designed continuous-time H∞ robust observer using the state matching technique. The effectiveness of the proposed design method is illustrated through numerical examples solving the Chua and Chen chaotic systems. The examples are numerically simulated with Matlab® using a Runge Kutta fourth-order algorithm. The results are provided at the end and then compared with others to show the effectiveness of this method. The results show a robust performance of the proposed method where the trajectory of the states for the drive and response systems synchronizes.

FPGA implementation of a UPT chaotic signal generator for image encryption

In this study, the validity and feasibility of UPT chaos for image encryption is demonstrated. This paper describes the design and implementation of a UPT chaotic signal generator for image encryption based on FPGA and Chua's system. In order to demonstrate the validity of the design, the signal generated by the chaotic signal generator is compared with Matlab simulations and used in an image encryption experiment. All of the experimental results indicate that using a UPT chaotic signal generator for image encryption is both practical and effective.

Control problems of a smooth Chua system by a single input controller

This paper investigates the control problems of a smooth Chua chaotic system. Three single input controllers are obtained to solve the stabilization, synchronization and anti-synchronization problems of such system, respectively. It should be pointed out that the obtained controllers are not only simple, but also physical in applications. Numerical simulation verifies the effectiveness and the validity of the obtained results.

Design of Grid Multiscroll Chaotic Attractors via Transformations

Three transformation approaches for generating grid multiscroll chaotic attractors are presented through theoretical analysis and numerical simulation. Three kinds of grid multiscroll chaotic attractors are generated based on one-dimensional multiscroll Chua system. The dynamics of the multiscroll chaotic attractors are analyzed by means of equilibrium points, eigenvalues, the largest Lyapunov exponent and complexity. As the experimental verification, we implemented the circular grid multiscroll attractor on DSP platform. The simulation and experimental results are consistent well with that of theoretical analysis, and it shows that the design approaches are effective.

Global Synchronization of a New Chua's System

The problem of global synchronization control for a class of new Chua's system is studied in this paper. Several linear controllers are proposed to realize the global exponential synchronization of two Chua's systems. Decoupling feedback control method is used to make Chua's system global exponential synchronized. Global exponential synchronization with respect to (w.r.t) partial state variables is studied when one of the error variables is zero. Finally, an adaptive synchronization controller is designed to make the response and the driven systems synchronized. Moreover, the estimate errors of the uncertain parameters converge to zero since the persistent excitation condition holds. Additionally, numerical simulations show the effectiveness of the proposed controllers.

Hiding of Speech based on Chaotic Steganography and Cryptography Techniques

The technique of embedding secret information into cover media, like image, video, audio and text called Steganography, so that only the sender and the authorized recipient who have a key can detect the presence of secret information. In this paper Steganography and Cryptography techniques of speech present with Chaos. Fractional order Lorenz and Chua systems that provides an expanded in key space are used to encrypt speech message. The large key space addition to all properties of randomness and nonlinearity which are possessed these chaotic systems ensure a highly robustness and security for cryptography process. As well as Modified Android Cat Map (MACM) offers additional space and security for steganography process. The irregular outputs of the MACM are used in this paper to embed a secret message in a digital cover image. The results show a large key sensitivity to a small change in the secret key or parameters of MACM. Therefore, highly security hiding for speech will be

Anticipating chaotic synchronization via act-and-wait coupling

Act-and-wait concept is utilized to design a coupling law for anticipating synchronization of chaotic systems. The time-delay coupling term in this algorithm is periodically switched on and off such that the phase space of the whole system remains finite-dimensional despite the presence of the delay. Consequently, the stability of the anticipated synchronization manifold is easily achieved, since it is definite by a finite number of Lyapunov exponents. We show that the stable synchronization regime with considerably large anticipation time can be attained even for single-input single-output systems. The results are demonstrated with the Rossler, Chua and Lorenz chaotic systems.

Passification-based decentralized adaptive synchronization of dynamical networks with time-varying delays

This paper is aimed at application of the passification based adaptive control to decentralized synchronization of dynamical networks. We consider Lurie type systems with hyper-minimum-phase linear parts and two types of nonlinearities: Lipschitz and matched. The network is assumed to have both instant and delayed time-varying interconnections. Agent model may also include delays. Based on the speed-gradient method decentralized adaptive controllers are derived, i.e. each controller measures only the output of the node it controls. Synchronization conditions for disturbance free networks and ultimate boundedness conditions for networks with disturbances are formulated. The proofs are based on Passification lemma in combination with Lyapunov–Krasovskii functionals technique. Numerical examples for the networks of 4 and 100 interconnected Chua systems are presented to demonstrate the efficiency of the proposed approach.

Generation and circuit implementation of multi-block multidirectional grid multi-scroll chaotic attractors

Due to the dynamic characteristics of the Chua's system, multi-scroll chaotic attractors are still confined in a single block and fail to break the limit. This paper proposes an approach for generating novel multi-block multidirectional grid multi-scroll chaotic attractors that can break the limit via novel nonlinear modulating functions. According to this method, the recursion rules used to generate multi-block multidirectional grid multi-scroll attractors are mathematically obtained. The new system is autonomous; the effectiveness of this method has been verified by theoretical analysis, numerical simulation, and circuit implementation.

Anti-synchronization and intermittent anti-synchronization of two identical hyperchaotic Chua systems via impulsive control

This work is concerned with anti-synchronization and intermittent anti-synchronization of two identical hyperchaotic Chua systems using an impulsive control technique. Sufficient condition of anti-synchronization is derived from Lyapunov stability theory, and numerical simulation results are presented to demonstrate the effectiveness of the proposed anti-synchronization scheme. In addition, we report an interesting phenomenon: when the second state of the response system is controlled via impulse, there exist narrow regions of the impulsive control gain, where intermittent anti-synchronization can occur.

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

Although Chua's circuit is one of the most studied nonlinear dynamical systems, its version with negative parameters remains practically untouched. This work reports an interesting and rich dynamic scenery that was hidden in this almost unexplored region. The study is focused on 2D parameter space and presents an analysis of stability based on describing functions. Numerical investigations present a gallery of period-adding cascades and a strong presence of basin boundary metamorphoses. The key to this new scenario is that for negative parameters, Chua's system does not satisfy the Shilnikov condition and it is shown that the homoclinic orbit organizes the parameter space completely different from as known. The obtained experimental results corroborate with the numerical and theoretical investigations.

Grid multi-wing butterfly chaotic attractors generated from a new 3-D quadratic autonomous system

Due to the dynamic characteristics of the Lorenz system, multi-wing chaotic systems are still confined in the positive half-space and fail to break the threshold limit. In this paper, a new approach for generating complex grid multi-wing attractors that can break the threshold limit via a novel nonlinear modulating function is proposed from the firstly proposed double-wing chaotic system. The proposed method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. The new system is autonomous and can generate various grid multi-wing butterfly chaotic attractors without requiring any external forcing, it also can produce grid multi-wing both on the xz-plane and yz-plane. Basic properties of the new system such as dissipation property, equilibrium, stability, the Lyapunov exponent spectrum and bifurcation diagram are introduced by numerical simulation, theoretical analysis and circuit experiment, which confirm that the multi-wing attractors chaotic system has more rich and complicated chaotic dynamics. Finally, a novel module-based unified circuit is designed which provides some principles and guidelines for future circuitry design and engineering application. The circuit experimental results are consistent with the numerical simulation results.