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In this work, a novel memristive Chua's oscillator circuit is presented. In the proposed circuit, a linear negative resistor, which is parallel coupled with a first-order memristive diode bridge, is used instead of the well-known Chua's diode. Following this, an extensive theoretical and dynamical analysis of the circuit is conducted. This involves numerical computations of the system's phase portraits, bifurcation diagrams, Lyapunov exponents, and continuation diagrams. A comprehensive comparison is made between the numerical simulations and the circuit's simulations performed in Multisim. The analysis reveals a range of intriguing phenomena, including the route to chaos through a period-doubling sequence, antimonotonicity, and coexisting attractors, all of which are corroborated by the circuit's simulation in Multisim. [ABSTRACT FROM AUTHOR]

Since the fixed-time stability forms of nonlinear systems satisfy strict conditions, there are few general forms for nonlinear systems to achieve fixed-time stability. This work proposes a new class of more general fixed-time stability criteria. It is worth mentioning that, compared with the traditional method of estimating the convergence time, this paper obtains a more conservative stable time estimation formula through the integration method of the generalized integral mean theorem. In addition, given that the fixed-time stabilization of neural networks and chaotic oscillators have attracted extensive attention in recent years, and there are still many fixed-time stabilizations of nonlinear systems that have not been studied. Therefore, a discontinuous controller is designed in this paper. The above stability theory results are applied to the fixed-time stabilization of the Takagi-Sugeno (T-S) fuzzy competitive neural network and chaotic system (coupled Chua's oscillator). Finally, the validity and applicability of the theoretical results are verified by examples. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a new system called an alternate control system with double impulses. The present system is a cyclic control system, composed of four parts in a circle: to the first and last halves of each period of the system we add different continuous controls, and at the half-period time and the end of each period of the system we add different impulses. We then investigate the exponential stability of the considered system. An example based on Chua’s circuit is provided to confirm the effectiveness of the theoretical result.

In this paper, we propose a new system called an alternate control system with double impulses. The present system is a cyclic control system, composed of four parts in a circle: to the first and last halves of each period of the system we add different continuous controls, and at the half-period time and the end of each period of the system we add different impulses. We then investigate the exponential stability of the considered system. An example based on Chua's circuit is provided to confirm the effectiveness of the theoretical result. [ABSTRACT FROM AUTHOR]

This contribution uncovers numerical evidence of hysteric dynamical behaviors for the same set of the circuit parameters of the Chua’s circuit with traditional piecewise-linear nonlinearity. Stationary points and the symmetry property of the model first forecast the possible evidence of coexisting attractors. Then, well known nonlinear analysis approach based on the bifurcation diagrams, two-parameter diagrams, phase portraits, two parameter Lyapunov exponent diagrams, graph of maximum Lyapunov exponents, and attraction basins are exploited to characterize the dynamical behavior of the oscillator including coexisting orbits. Finally, the simultaneous existence of both periodic and chaotic orbits highlighted in the Chua’s oscillator is also annihilated based on linear controller. Numerical findings indicate control method ’s efficacy by combining two periodic routes and one chaotic route with another chaotic route.

The advancement in cryptographic algorithms is the primary need for modern secure communication systems. In secure communication system, the stream cipher serves the basis for cryptographic algorithms. The properties of cryptographic algorithms are often analyzed on the basis of randomness involved in implementing cipher texts while complexity of the hardware is also taken care of. Therefore, the main objective of this paper is to propose a new stream cipher scheme having higher complexity (high degree of randomness) while keeping the hardware as simple as possible. The proposed stream cipher algorithm gives the new direction to develop the highly secure and simple models in communication field. The proposed scheme offers the benefit of adjusting variable-length key(s) according to the variable length of plaintext using modular arithmetic and memristor—based chaotic behavioral Chua's circuit. To explain the encryption–decryption process at transmitter and receiver ends, the Devanagari script has been taken. Although, any other script or data type is supported by the proposed algorithm. In addition, the synchronization is also achieved without having any requirement of additional process. [ABSTRACT FROM AUTHOR]

A flux-controlled memristor emulator built with off-the-shelf electronic devices and based on a TiO2 model is presented in this article. The circuit proposed in this article uses the current mode approach based analog building blocks such as a second-generation current conveyor (CCII) and an operational transconductance amplifier (OTA) as an active element with few passive elements. The circuit shows a clear fingerprint of an ideal memristor. The offered emulator circuit can be made to operate in incremental and decremental modes and functions well up to 26.3 MHz. Nonvolatility, Monte Carlo sampling, and corner analysis simulations are executed to verify the robustness of the circuit. The functional verification of the presented circuit is performed using the 0.18- $\mu \text{m}$ CMOS parameter at a supply voltage of ±1.2 V. The experimental demonstration is carried out by making a prototype on a breadboard using ICs AD844AN and CA3080, which exhibits a good agreement with theoretical and simulation results. The layout of the circuit, which requires a total chip area of $75\times 70\,\,\mu \text{m}^{2}$ , is also created. Single/parallel combinations of a memristor, a high-pass filter, and a chaotic system are presented to demonstrate its application. [ABSTRACT FROM AUTHOR]

Abstract We propose a mathematical model that can control the stability of an unstable system. Periodicity is an important feature of the system. We add a continuous control to the first half of each period of the system and then add an impulse control J1 at the 1 / 2 $1/2$ period time. Again, we do not control the rest half of each period of the system. Finally, we add an impulse control J2 at the end of each period of the system. The system is called an alternate-continuous-control system with double-impulse. We study the stability of the current system by constructing the Lyapunov function. Using the proposed method, we can control the Chua oscillator. The system has two impulse inputs per period, which is more in line with natural law than the system that only has a single-impulse input. Therefore, the system proposed in this paper is more practical than current mature control systems.

In this paper, we formulate a new system, named by sandwich control system with impulse time windows. The present system is a cyclic control system, composed of three parts in a round: the first and last parts are continuous subsystems, while the middle one includes an impulsive operation. Different from the most existing results for impulsive systems, we assume that the impulse moments are unknown but limited to certain intervals (namely, impulse time windows). We then study the stability of the considered systems and obtain an exponential stability criterion in terms of a set of linear matrix inequalities. As a numerical example, Chua oscillator is stabilized by the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the problem of locally exponential stabilization (LES) of nonlinear systems under a class of hybrid control in the framework of actuator saturation, where the limitation of actuator saturation on both continuous state feedback control and impulsive control are fully considered. Based on impulsive control theory and differential inclusion approach, some sufficient conditions for LES are derived, where a novel set inclusion relation is proposed to handle the double saturation nonlinearities. Different from the existing results that each part of saturated hybrid control (SHC) is required to stabilize the system individually, our results relax the requirement by making full use of the correction effect of the impulse. Moreover, the maximum of estimation of domain of attraction is obtained by a convex optimal problem and corresponding algorithm. The result is applied to the robustness for a class of nonlinear systems. Finally, the validity of the results is shown by two examples and their simulations, where the synchronization problem of Chua's oscillator is illustrated in the framework of actuator saturation. [ABSTRACT FROM AUTHOR]

We propose a mathematical model that can control the stability of an unstable system. Periodicity is an important feature of the system. We add a continuous control to the first half of each period of the system and then add an impulse control J1 at the $1/2$ period time. Again, we do not control the rest half of each period of the system. Finally, we add an impulse control J2 at the end of each period of the system. The system is called an alternate-continuous-control system with double-impulse. We study the stability of the current system by constructing the Lyapunov function. Using the proposed method, we can control the Chua oscillator. The system has two impulse inputs per period, which is more in line with natural law than the system that only has a single-impulse input. Therefore, the system proposed in this paper is more practical than current mature control systems. [ABSTRACT FROM AUTHOR]

We study a van der Pol-like memristor oscillator, obtained by substituting a Chua's diode with an active controlled memristor in a van der Pol oscillator with Chua's diode. The mathematical model for the studied circuit is given by a three-dimensional piecewise linear system of ordinary differential equations, depending on five parameters. We show that this system has a line of equilibria given by the -axis and the phase space is foliated by invariant planes transverse to this line, which implies that the dynamics is essentially two-dimensional. We also show that in each of these invariant planes may occur limit cycles and relaxation oscillations (that is, nonsinusoidal repetitive (periodic) solutions), depending on the parameter values. Hence, the oscillator studied here, constructed with a memristor, is also a relaxation oscillator, as the original van der Pol oscillator, although with a main difference: in the case of the memristor oscillator, an infinity of oscillations are produced, one in each invariant plane, depending on the initial condition considered. We also give conditions for the nonexistence of oscillations, depending on the position of the invariant planes in the phase space. [ABSTRACT FROM AUTHOR]

Chua's circuit is one of the well-known nonlinear circuits which have been used to study a rich variety of nonlinear dynamic behaviors such as bifurcation, chaos, and routes to chaos. In this work, I consider the dynamics of Chua's circuit with a smooth cubic nonlinearity. The dynamics of the model is investigated using standard nonlinear analysis techniques including time series, bifurcation diagrams, phase space trajectories plots, Lyapunov exponents, and basins of attraction. Both period-doubling and crisis routes to chaos are reported. One of the major results of this work is the numerical finding of a parameter region in which Chua's circuit experiences multiple attractors' behavior (i.e., coexistence of four different periodic and chaotic attractors). This phenomenon was not reported previously in the Chua's circuit (despite the huge amount of related research works) and thus represents an enriching contribution to the understanding of the dynamics of Chua's oscillator. Basins of attraction of various coexisting attractors are depicted showing complex basin boundaries. The results obtained in this work let us conjecture that there are still some unknown and striking behaviors of Chua's oscillator (e.g., the phenomenon of extreme multistability, i.e., infinitely many attractors) that need to be uncovered. [ABSTRACT FROM AUTHOR]

This paper addresses the problems of robust stability analysis and stabilization for continuous-time systems with state-dependent uncertainties via constructing a new parameter-dependent Lyapunov function. On the basis of such a parameter-dependent Lyapunov function, the stability conditions for open-loop uncertain systems are first presented. Then a relaxed stability analysis approach that utilizes a property of the time derivatives of uncertain parameters is obtained. To take full advantage of the parameter-dependent Lyapunov function, a model-dependent state-feedback stabilization scheme is proposed, which can provide more flexibilities in controller synthesis. The static state-feedback controller can be seen as a special case of the model-dependent state-feedback controller. A numerical example is provided to illustrate the effectiveness of the proposed approaches. Finally, the developed controller design methodology is applied to stabilization and synchronization of Chua's oscillator, which has wide applications in secure communication systems and power electronic systems. [ABSTRACT FROM AUTHOR]

This paper investigates the global lagged finite-time synchronization of the master-slave Lur'e systems subject to time delay of signal transmission. By designing a variable-substitution and feedback controller, a master-slave finite-time synchronization scheme for the Lur'e systems with time delay is built up. Two delay-independent global lagged finite-time synchronization criteria are proved in the forms of linear matrix inequalities (LMIs), and the corresponding settling time of synchronization is analytically estimated. The obtained LMI criteria are applied to Chua's oscillators, obtaining some easily implemented algebraic criteria under various single-variable-substitution and feedback controller, which are then optimized to improve their conservative property. Finally, several numerical examples are illustrated to verify the effectiveness of the optimized criteria. [ABSTRACT FROM AUTHOR]

In this paper, we develop a parameter matching technique using adaptive synchronization with a chaotic attractor. Two Chua's oscillators, identical in every parameter except for one, are set-up in a master/slave configuration where the slave's mismatched parameter is adaptable. Using Lyapunov functions and incorporating the presence of chaotic signals, adaptive control laws are designed to ensure exact parameter matching. One of the derived adaptive controllers is experimentally validated by using an adaptive inductor-gyrator composed of current feedback op-amps (CFOAs). The experimental results are compared to high-fidelity SPICE simulations, and performance of adaptive controllers are compared over a wide range of parameter values using MATLAB. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents the universality of the coexistence of complete synchronization with the recently new proposed inverse π-lag synchronization, in the case of mutually coupled systems of Chua's circuit family. The phenomenon of multistability and the nature of this circuit family systems are the key-points which lead to the explanation of this coexistence. For the need of this work, the most representative circuit of this circuits' family, the Chua oscillator, is used. Simulation results confirm this universality in the class of Chua's circuit family. [ABSTRACT FROM AUTHOR]

The modeling and numerical investigation of the dynamical behavior of a semiconductor ring laser (SRL) driven by Chua’s oscillator are reported. By increasing the coupling strength between the SRL and Chua’s oscillator at a fixed bias current, the SRL exhibits an intermittency route to anti-phase chaos. However, for a fixed value of the coupling strength, we report a period-doubling route to out-of-phase and anti-phase chaos when varying one of the parameters of the Chua’s oscillator are reported. We also demonstrate that a SRL driven by the chaotic output of Chua’s oscillator generates a more complex chaos compared to the one found in a SRL subject to a sinusoidally modulated current. This new way of modulation of semiconductor lasers would not only bring a general benefit in the physical equipment and reduce their cost but could have an impact for some relevant engineering applications. [ABSTRACT FROM AUTHOR]

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions. [ABSTRACT FROM AUTHOR]

This work shows the experimental implementation of a chaotic communication system based on two Chua's oscillators which are synchronized by Hamiltonian forms and observer approach. The chaotic communication scheme is realized by using the commercially available positive-type second generation current conveyor (CCII+), which is included into the AD844 device. As a result, experimental measurements are provided to demonstrate the suitability of the CCII+ to implement chaotic communication systems. [ABSTRACT FROM AUTHOR]

The memristor has attracted phenomenal worldwide attention since its debut on 1 May 2008 issue of Nature in view of its many potential applications, e.g. super-dense nonvolatile computer memory and neural synapses. The Hewlett–Packard memristor is a passive nonlinear two-terminal circuit element that maintains a functional relationship between the time integrals of current and voltage, respectively, viz. charge and flux. In this paper, we derive several nonlinear oscillators from Chua's oscillators by replacing Chua's diodes with memristors. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical study of chaotic dynamics via the Adomian decomposition method. The approach, which provides series solutions of the system equations, is first applied to Chua's circuit and Chua's oscillator, both with cubic nonlinearity. Successively, the method is utilized for obtaining hyperchaotic multiscroll attractors in a ring of three Chua's circuits, where the smooth nonlinearities are Hermite interpolating polynomials. The reported examples show that the approach presents two main features, i.e. the system nonlinearity is preserved and the chaotic solution is provided in a closed form. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present some of the problems we have faced in visualizing patterns coming from Chua''s circuit, with the main aim of sharing these investigations with researchers in Computer Graphics. Furthermore, the paper deals about advanced Computers Graphics techniques for managing complexity in representing chaotic and hyperchaotic systems. The problems we have faced deal about the nature of these configurations, their shapes and dimensions, the granularity of these objects, since the output could be made of different elements such as cubes, cylinders and so on. Instead, cognitive issues related to perception of forms in 3D spaces and their recognition are still unexplored. [Copyright &y& Elsevier]

This work presents a linguistic approach to the understanding of chaos. The idea comes from our work on translating into sounds and music the complexity of chaotic systems, as with Chua's attractors. Therefore, working with sounds, we have used a standardization criterion in order to detect only some of the features of the richness of chaos. This method involves the selection of a limited number of sounds, which, in turn, allow for the creation of other components of the system at different levels of organization, as with natural language. Thus, phonetic, morphological, syntactic and semantic levels, linked by a grammar of an "artificial chaotic language" are defined. Each linguistic unit of the artificial language presents networks of interrelated phenomena since for each of them, it is possible to detect its path length, tracing a graph of the mutual relationships of the system's components. We found that there is interaction among the different levels of the chaotic language. Furthermore, some traits of the dynamics of the evolution of language in human infants are found in the main routes to chaos. In this emergent dynamics, mutations, struggle for the fittest, natural selection, and the relative distribution of linguistic entities in genetic landscapes are observed. On the basis of this, a possible bridge of connection between the physical and the mental worlds may be developed. [ABSTRACT FROM AUTHOR]

The nonlinearity in Chua's oscillator is commonly implemented as a three-segment piecewise-linear resistor. The piecewise-linear nature of the element means that the implementation requires a significant amount of circuitry and the speed of operation is limited. The qualitative behavior of Chua's oscillator has also been captured using a smooth cubic nonlinearity; the implementation of the latter also requires a significant amount of circuitry and suffers from limited speed of operation. This work describes a novel implementation of Chua's oscillator using just four transistors and a battery to produce a cubic-like nonlinearity. The circuit is simple, robust, and capable of operating at frequencies over one thousand times greater than the original Chua's oscillator. [ABSTRACT FROM AUTHOR]

Due to strong contraction in the outer regions of Chua's circuit with piecewise-linear dynamics, the system can be described by an approximate one-dimensional map. We confirm by simulation and experiment that the initial slope of this map is defined by $e^{\frac{\sigma_1 2 \pi}{\omega_1}}$. [ABSTRACT FROM AUTHOR]

Modern Science is finding new methods of looking at biological, physical or social phenomena. Traditional methods of quantification are no longer sufficient and new approaches are emerging. These approaches make it apparent that the phenomena the observer is looking at are not classifiable by conventional methods. These phenomena are complex. A complex system, as Chua's oscillator, is a nonlinear configuration whose dynamical behavior is chaotic. Chua's oscillator equations allow to define the basic behavior of a dynamical system and to detect the changes in the qualitative behavior of a system when bifurcation occurs, as parameters are varied. The typical set of behavior of a dynamical system can be detailed as equilibrium points, limit cycles, strange attractors. The concepts, methods and paradigms of Dynamical Systems Theory can be applied to understand human behavior. Human behavior is emergent and behavior patterns emerge thanks to the way the parts or the processes are coordinated among themselves. In fact, the listening process in humans is complex and it develops over time as well. Sound and music can be both inside and outside humans. This tutorial concerns the translation of Chua's oscillators into music, in order to find a new way of understanding complexity by using music. By building up many computational models which allow the translation of some quantitative features of Chua's oscillator into sound and music, we have created many acoustical and musical compositions, which in turn present the characteristics of dynamical systems from a perceptual point of view. We have found interesting relationships between dynamical systems behavior and their musical translation since, in the process of listening, human subjects perceive many of the structures as possible to perceive in the behavior of Chua's oscillator. In other words, human cognitive abilities can analyze the large and complicated patterns produced by Chua's systems translated into music, achieving the cognitive economy and the coordination and synthesis of countless data at our disposal that occur in the perception of dynamic events in the real world. Music can be considered the semantics of dynamical systems, which gives us a powerful method for interpreting complexity. [ABSTRACT FROM AUTHOR]