Chua's Chaotic Circuit Research Articles

Physical implementation of cobalt ferrite memristor in Chua's circuit for chaotic encryption

Memory resistor, or memristor, has been realized as a discrete electronic device and has a perspective application in the field of cryptography. The physical implementation of the memristor in chaotic circuits has been scarcely explored. In this paper, a memristor is fabricated by spin-coating a cobalt ferrite precursor on a processed silicon and is then electro-sputtered with silver to act as the anode with the base silicon as the cathode. This fabrication process has a scalability potential in conjunction with integrated circuit fabrication techniques and complementary metal oxide semiconductor (CMOS) technologies. The fabricated cobalt ferrite memristor has shown a ratio between the on and off resistance of >1000 and has been implemented in a chaotic Chua's circuit, making it one of few physical implementations of a physical memristor in a physical circuit. The analysis and characterization of this circuit using bifurcation diagrams and Lyapunov exponent prove the chaotic behavior of a real Chua's circuit. This chaotic behavior can be useful in chaotic cryptography as nonperiodic oscillations can be leveraged to make sensitive information more difficult to interpret by bad actors.

Reconstructing bifurcation diagrams of chaotic circuits with reservoir computing.

Model-free reconstruction of bifurcation diagrams of Chua's circuits using the technique of parameter-aware reservoir computing is investigated. We demonstrate that (1) reservoir computer can be utilized as a noise filter to restore the system dynamics from noisy signals; (2) for a single Chua circuit, a machine trained by the noisy time series measured at several sampling states is capable of reconstructing the whole bifurcation diagram of the circuit with a high precision; and (3) for two coupled chaotic Chua circuits with mismatched parameters, the machine trained by the noisy time series measured at several coupling strengths is able to anticipate the variation of the synchronization degree of the coupled circuits with respect to the coupling strength over a wide range. Our studies verify the capability of the technique of parameter-aware reservoir computing in learning the dynamics of chaotic circuits from noisy signals, signifying the potential application of this technique in reconstructing the bifurcation diagram of real-world chaotic systems.

Addressing asymmetric saturation in chaotic Chua's circuit switched system: A combined adaptive sliding mode strategy

SummaryThe purpose of this research is to develop a combined adaptive nonlinear feedback integral sliding mode (CANFISM) for the Chua's switched nonlinear circuit systems. The suggested adaptive composite nonlinear technique is improved to provide a robust controller with enhanced performance in the face of uncertainty and asymmetric saturation. The proposed technique has been theoretically validated, and its closed‐loop exponentially boundedly stable has been demonstrated. Meanwhile, CNF (composite nonlinear feedback) parameters are then determined by employing the linear matrix inequality technique, and the notion of robust tracking is performed and compared in Chua's switched nonlinear circuit systems. This method guarantees robustness against the effects of uncertainty and input asymmetric saturation and decreases chattering. The effectiveness and merits of the suggested technique are then achieved by utilizing two simulation examples of Chua's circuit system.

Electronic Implementation of a Deterministic Small-World Network: Synchronization and Communication.

In this paper, synchronization and encrypted communication transmissions of analog and digital messages in a deterministic small-world network (DSWN) are presented. In the first instance, we use a network with 3 coupled nodes in a nearest-neighbor (NN) topology, then the amount of nodes is increased until reaching a DSWN with 24 nodes. The synchronization and encrypted communication transmissions using a DSWN are presented experimentally by using Chua's chaotic circuit as node, in both analog and digital electronic implementations, where for the continuous version (CV) we use operational amplifiers (OA), and in the discretized version (DV) we use Euler's numerical algorithm implemented in an embedded system by using an Altera/Intel FPGA and external digital-to-analog converters.

Numerical study of a nonlinear fractional chaotic Chua's circuit

<abstract> <p>As an exponentially growing sensitivity to modest perturbations, chaos is pervasive in nature. Chaos is expected to provide a variety of functional purposes in both technological and biological systems. This work applies the time-fractional Caputo and Caputo-Fabrizio fractional derivatives to the Chua type nonlinear chaotic systems. A numerical analysis of the mathematical models is used to compare the chaotic behavior of systems with differential operators of integer order versus systems with fractional differential operators. Even though the chaotic behavior of the classical Chua's circuit has been extensively investigated, our generalization can highlight new aspects of system behavior and the effects of memory on the evolution of the chaotic generalized circuit.</p> </abstract>

Successive lag synchronization of nonlinear stochastic networks with arbitrary structure via pinning control

AbstractIn this article, a new nonlinear stochastic network with arbitrary structure and noise is proposed for realizing the successive lag synchronization (SLS). Both the constant and adaptive pinning control laws are designed respectively to regulate the SLS to desired chaotic states, where the network structure can be directed, weighted, time‐varying and even not strongly connected. Several sufficient conditions for guaranteeing the stochastic asymptotic stability of SLS are derived by means of the Lyapunov stability theory. First, we demonstrate that the pinned nodes can be fixed via 1‐dimensional inequalities, which explicitly provide how many nodes and which nodes should be controlled. Besides, those connections from a node with a given number to other nodes with smaller numbers will lead to the increase of pinned nodes. Second, our results indicate that, in order to make the SLS stable the coupling strength must belong to a bounded interval for both the pinning control schemes. Finally, we conclude that strong noise perturbation will lead to the increase of pinned nodes and even break the SLS. The theoretical results are validated with different numerical examples by embedding the chaotic Chua's circuit to each node.

Mode selection and stability of attractors in Chua circuit driven by piezoelectric sources

The piezoelectric ceramic can be activated to produce continuous voltage and it is often used to excite nonlinear circuits, which is tamed as functional circuit to detect the acoustic wave and mechanic forcing. When two piezoelectric devices are combined to couple a nonlinear circuit, any switch between two channels embeds with piezoelectric ceramic will induce distinct mode transition in the output voltages. In addition, synchronous inputs from two piezoelectric ceramics will induce competition on exciting the nonlinear circuit and then the mode selection becomes competitive. In this paper, two piezoelectric devices are incorporated in two different branch circuits of the Chua circuit, and each piezoelectric ceramic is connected with a resistor in series in the branch circuit, the Chua circuit will be driven by two different external stimuli generated from the piezoelectric devices along the two branch circuits in parallel. The two branch circuits embedded with piezoelectric devices are controlled by two switchers S1, S2, and the induced piezoelectric voltages are described by periodic from without losing generality. Three cases are investigated for detecting the stability of attractors and mode selection in the chaotic Chua circuit driven by piezoelectric voltage sources. It is confirmed that the involvement of piezoelectric ceramic into the nonlinear circuit can control the dynamics and firing patterns effectively.

Nonlinear system identification in Sobolev spaces

We consider the problem of approximating an unknown function from experimental data, while approximating at the same time its derivatives. Solving this problem is useful, for instance, in the context of nonlinear system identification, for obtaining models that are more accurate and reliable than the traditional ones based on plain function approximation. Indeed, models identified by accounting for the derivatives can provide improved performance in several endeavours, such as in multi-step prediction, simulation, Nonlinear Model Predictive Control, and control design in general. In this paper, we propose a novel approach based on convex optimisation, allowing us to solve the aforementioned identification problem. We develop an optimality analysis, showing that models derived using this approach enjoy suitable optimality properties in Sobolev spaces. The optimality analysis also leads to the derivation of tight uncertainty bounds on the unknown function and its derivatives. We demonstrate the effectiveness of the approach with three numerical examples, concerned with univariate function identification, multi-step prediction of the Chua chaotic circuit, and control of the inverted pendulum.

A Comprehensive Study and Analysis of the Chaotic Chua Circuit

was studied by taking several different values for the constant α and fixing the other three variables β, c and d with the values 25.58, -0.7142857, and -1.142, respectively. The purpose of this paper is to know the values by which the system transforms from a steady state to a chaotic state under the initial conditions x, y, and z that equal -1.6, 0 and 1.6 respectively. It was found that when the value of α is equal to 0, the Chua system is in a steady state, and when the value of α is equal to 9.5 and the wave is sinusoidal, the system is in oscillation, and when α is equal 13.4 the system is in a Quasi-chaotic state, and finally the system turns to the chaotic state when the value of α equals 15.05.

Adaptive Tracking for Uncertain MIMO Nonlinear Systems With Time-Varying Parameters and Bounded Disturbance

In this article, tracking control is considered for a class of uncertain multi-input-multi-output (MIMO) nonlinear systems, where the time-varying parameters, the time-varying control coefficient and the time-varying disturbance are assumed to be unknown but to be bounded. Three stable adaptive tracking schemes for a given reference signal are proposed by devising different control algorithms. In the first scheme, bounded-error tracking is achieved in the sense that the tracking error converges exponentially to an adjustable region around the origin, where the σ-modification adaptive laws are used to ensure the boundedness of all closed-loop signals. In the second scheme, asymptotic tracking is obtained in the sense that the tracking error converges to zero asymptotically, where the strictly positive and integral functions are employed in the control law to ensure the signal boundedness and zero-error tracking. In the third scheme, exponential tracking is gotten in the sense that the tracking error exponentially converges to zero with a given convergence speed, where exponential functions are incorporated into control law and adaptive laws to ensure system stability and the faster convergence. Three adaptive tracking schemes are, respectively, applied to nonlinear chaotic Chua's circuit with control inputs. The parametric model is developed for Chua's circuit with uncertain parameters and external disturbances. The effectiveness of the proposed control algorithms is demonstrated by comparative simulation studies.

Phase synchronization between nonlinear circuits by capturing electromagnetic field energy

Smart nonlinear circuits can be tamed to reproduce the main dynamical properties in neural activities and thus neural circuits are built to estimate the occurrence of multiple modes in electric activities. In the presence of electromagnetic radiation, the cardiac tissue, brain and neural circuits are influenced because field energy is injected and captured when induction field and current are generated in the media and system. In this paper, an isolated Chua circuit is exposed to external electromagnetic field and energy capturing is estimated for nonlinear analysis from physical viewpoint. Furthermore, two Chua circuits without direct variable coupling are exposed to the same electromagnetic field for energy capturing. Periodical and noise-like radiations are imposed on the Chua circuits which can capture the magnetic field energy via the induction coil. It is found that the two Chua circuits (periodical or chaotic) can reach phase synchronization and phase lock in the presence of periodical radiation. On the other hand, noise-like radiation can realize complete synchronization between two chaotic Chua circuits while phase lock occurs between two Chua circuits in periodical oscillation. It gives some important clues to control the collective behaviors of neural activities under external field.

Control and synchronization in nonlinear circuits by using a thermistor

The survival and occurrence of chaos are much dependent on the intrinsic nonlinearity and parameters region for deterministic nonlinear systems, which are often represented by ordinary differential equations and maps. When nonlinear circuits are mapped into dimensional dynamical systems for further nonlinear analysis, the physical parameters of electric components, e.g. capacitor, inductor, resistance, memristor, can also be replaced by dynamical parameters for possible adjustment. Slight change for some bifurcation parameters can induce distinct mode transition and dynamics change in the chaotic systems only when the parameter is adjustable and controllable. In this paper, a thermistor is included into the chaotic Chua circuit and the temperature effect is considered by investigating the mode transition in oscillation and the dependence of Hamilton energy on parameters setting in thermistor. Furthermore, the temperature of thermistor is adjusted for finding possible synchronization between two chaotic Chua circuits connected by a thermistor. When the coupling channel via thermistor connection is activated, two identical Chua circuits (periodical or chaotic oscillation) can reach complete synchronization. In particular, two periodical Chua circuits can be coupled to present chaotic synchronization by taming parameters in thermistor of coupling channel. However, phase synchronization is reached while complete synchronization becomes difficult when the coupling channel is activated to coupling a periodical Chua circuit and a chaotic Chua circuit. It can give guidance for further control of firing behaviors in neural circuits when the thermistor can capture the heat effectively.

Capturing and shunting energy in chaotic Chua circuit

Abstract Nonlinear electric devices are critical for building chaotic circuits and the outputs voltage from the capacitor are often detected for further analyzing the dynamics of the nonlinear circuits. Continuous exchange between the electric field energy in the capacitor and magnetic field energy in the induction coil is effective to keep continuous oscillation in the circuit. That is, energy encoding and transmission can regulate the dynamical behaviors in chaotic circuits. In this paper, a branch circuit, which is built by using a capacitor and induction coil, is paralleled with one output end of chaotic Chua circuit, and this external branch circuit is activated to control chaos by pumping energy from Chua circuit and capturing external electromagnetic radiation. From dynamical control view, it explains the mechanism for differential control via capacitor and integral control via induction coil. While in the view of energy encoding, the control branch of circuit built by using a capacitor connected with induction coil in series can capture some external field energy and thus the nonlinear behaviors are controlled by generating equivalent current in the branch of control circuit. The circuit equation and also the dimensionless dynamical system under energy control are obtained, and numerical studies are supplied to confirm the control mechanism from physical view. In the end, it gives possible suggestions to enhance the control effectiveness in experimental way.

A new random diffusion algorithm based on the multi-scroll Chua's chaotic circuit system

In this paper, a new image encryption algorithm was proposed based on the Chua's multi-scroll chaotic circuit system. In comparison with other classical discrete systems, the new multi-scroll system has higher complexity, thus it was more suitable for the design of the image encryption algorithm. The correlation of the adjacent pixels was greatly eliminated by using the Arnold cat transformation. According to the scroll number of the multi-scroll chaotic system on x-y direction, parameter p and q of the Arnold transformation can be determined. Especially, the pixel value of the ciphered image of the new random diffusion algorithm depends on two random non-adjacent pixels and the interference value of the multi-scroll chaotic system, it significantly reduced the correlation between the adjacent pixels, and it greatly increases the ability to resist deciphering. The experimental results show that the encryption effect of the ciphered image were significantly higher than that of other encryption algorithms. It provided a data support and theoretical direction for the research of the related cryptography fields.

Effects of Variable-Order Passive Circuit Element in Chua Circuit

In this paper, the behaviour of a variable-order passive circuit element which is used in Chua chaotic circuit is analysed. Firstly, the behaviour of a circuit with a variable-order memristor is presented. In the generalized Ohm’s law for a memory element, the order of a passive circuit element is defined as a kind of the element, so the circuit shows unusual behaviour. Finally, the chaotic circuit is designed with a variable-order memristor, and the effect of the variable-order element is shown in the Chua circuit. The chaotic circuit model with the novel memristor shows limit-cycle behaviour.

Wavelet Fuzzy Brain Emotional Learning Control System Design for MIMO Uncertain Nonlinear Systems.

This paper aims to present a novel efficient scheme in order to more effectively control the multiple input and multiple output (MIMO) uncertain nonlinear systems. A wavelet fuzzy brain emotional learning controller (WFBELC) model is proposed, which is comprises the benefit of wavelet function, fuzzy theory and brain emotional neural network. When it is used as the main tracking controller for a MIMO uncertain nonlinear systems, the performances of the system, such as the approximation ability, the learning performance and the convergence rate, will be effectively improved. Meanwhile, the gradient descent method is used to adjust the parameters online of WFBELC and the Lyapunov function is employed to guarantee the rapid convergence of the control systems. For the sake of the further illustrating the superiority of this model, two examples of uncertain nonlinear systems, a Duffing-Holmes chaotic system and a Chua's chaotic circuit, are studied. After compared with other models, the test results show that the proposed model can be applied to obtain more satisfactory control performance and be more suitable to deal with the influence of the uncertainty of the MIMO nonlinear systems.

Control of the modified chaotic Chua's circuit using threshold method

Introduction. General scientific fields where can be used circuits that realize chaotic behavior and generate chaotic oscillations are presented.Methods for control of chaotic oscillations are also presented. For modelling, analysis and demonstrate results was selected MultiSim software environment. Modelling and Analysis of Non-Linear Element. This modified Chua’s circuit has a simple non-linear element, designed to have a piecewise-linear characteristic, that is, a combination of an opamp with two diodes that are mutually inline. For realization of nonlinearity, for two diodes do not need a separate power source, only one bipolar power source for the opamp is enough. The scheme for modelling of the nonlinear element and the results of computer simulation, i.e. the volt-ampere characteristic (VAC) at certain values of the components of the scheme's nominal values, is presented. This modified Chua's circuit, which generates a chaotic and controlled attractor with a fixed period, can be used in modern transmission and reception systems of information. Modeling and Analysis of the Modified Chaotic Chua’s Generator. System’s behavior is investigated through numerical simulations, by using well known tools of nonlinear theory, such as chaotic attractor and time distributions of the chaotic coordinates. Threshold Method for Control of Chaotic Oscillations. System of equations that realize chaotic oscillations of Chua's circuit is presented. Using threshold method was practical realization of the control of chaotic attractor. This modified Chua’scircuit that generate a chaotic and controlled attractor with a fixed period can be used in modern systemstransmitting and receiving information. Number of periodic (controlled) attractor can be used as a keys formasking of information carrier. Conclusions. For the first time was used threshold method forcontrol of chaotic oscillations for modified Chua’schaotic generator. This modified Chua’s circuit thatgenerate a chaotic and controlled attractor with afixed period can be used in modern systems transmittingand receiving information. Number of periodic(controlled) attractor can be used as a keys for maskingof information carrier.

Further Results on Quasi‐Synchronization of Delayed Chaotic Systems with Parameter Mismatches Via Intermittent Control

AbstractThis paper focuses on the problem of quasi‐synchronization of delayed chaotic systems with parameter mismatches via aperiodically intermittent control. First, the intermittently controlled systems are modeled as switched systems with two modes. Then, piecewise switching‐time‐dependent Lyapunov functional/function are introduced for stability analysis of the switched systems. As a result, two novel sufficient conditions for quasi‐synchronization are established under a small error bound. Based on the obtained quasi‐synchronization criteria, the existence of the aperiodically intermittent controllers is presented in the form of Linear Matrix Inequalities (LMIs). Finally, chaotic neural networks and Chua's circuit are provided to verify the validity of the proposed method.

Fault estimation and controller compensation in Lure systems by LPV-embedding

ABSTRACTPrevious works have considered the use of set invariance theory for fault detection and isolation in nonlinear Lure systems. This paper extends those results and proposes a new actuator fault-tolerant control approach. The fault-tolerant control scheme is designed based on linear parameter-varying (LPV) models of Lure systems. The actuator fault situation is diagnosed by an invariant set-based fault detection and isolation algorithm. Faults are compensated by adapting the controller gain based on estimates of the fault magnitude. Conditions for correct fault detection and isolation, and closed-loop stability are derived. The proposed fault-tolerant control scheme is compared with a linearised model approach and the performance of both, LPV-embedding and linearised, approaches are analysed for scalar and second-order systems. An example of a chaotic Chua circuit is also provided to illustrate the proposed fault-tolerant control scheme in higher-order systems.

Bi-stability in a fifth-order voltage-controlled memristor-based Chua's chaotic circuit

Generally, the occurrence of multiple attractors indicates that the multi-stability existing in a nonlinear dynamical system and the long-time motion behavior are essentially different, depending on which basin of attraction the initial condition belongs to. Up to now, due to the emergence of multi-stability, some particular memristor-based nonlinear circuits whose dynamical behaviors are extremely related to memristor initial conditions or other initial conditions have attracted considerable attention. By replacing linear or nonlinear resistors with memristor emulators in some already-existing oscillating circuits or introducing memristor emulators with different nonlinearities into these oscillating circuits, various memristor-based nonlinear dynamical circuits have been constructed and broadly investigated. Motivated by these considerations, we present a novel fifth-order voltage-controlled memristor-based Chua's chaotic circuit in this paper, from which a wonderful phenomenon of bi-stability is well demonstrated by numerical simulations and PSIM circuit simulations. Note that the bi-stability is a special kind of multi-stability, which is rarely reported in the memristor-based chaotic circuits.<br/>The proposed memristor-based Chua's chaotic circuit is constructed by inserting an inductor into the coupled resistor branch in series and substituting the Chua's diode with a voltage-controlled memristor in the classical Chua's circuit. Five-dimensional system model is established, of which the equilibrium point and its stability are investigated. Theoretical derivation results indicate that the proposed circuit owns one or three equilibrium points related to the circuit parameters. Especially, unlike the newly reported memristive circuit with bi-stability, the proposed memristor-based Chua's chaotic circuit has only one zero equilibrium point under the given parameters, but it can generate coexistent chaotic and periodic behaviors, and the bi-stability occurs in such a memristive Chua's circuit. By theoretical analyses, numerical simulations and PSIM circuit simulations, the bi-stability phenomenon of coexistent chaotic attractors and periodic limit cycles with different initial conditions and their formation mechanism are revealed and expounded. Besides, with the dimensionless system equations, the corresponding initial condition-dependent dynamical behaviors are further numerically explored through bifurcation diagram, Lyapunov exponents, phased portraits and attraction basin. Numerical simulation results demonstrate that the proposed memristive Chua's system can generate bi-stability under different initial conditions. The PSIM circuit simulations and numerical simulations are consistent well with each other, which perfectly verifies the theoretical analyses.