Chua's Attractor Research Articles

Chaotic Attractor Generated by Combining Chua Attractor with Another Circuit

Chaotic Attractor Generated by Combining Chua Attractor with Another Circuit

The Routh Hurwitz Criteria for Studying The Stability and Bifurcation in Multispiral Chua Chaotic Attractor

This article discusses the multispiral Chua Chaotic attractor’s hidden bifurcations that are generated by the sine function. The number of spirals (also known as a multiscroll attractor) that are controlled by the integer parameter c can be used to describe the basic shape of chaotic attractors. Since this parameter is an integer, increasing it by one does not allow the observation of bifurcations from n to n + 2 spirals. The method of hidden bifurcations, however, enables the observation of such bifurcations by adding a real parameter ε. Chaotic attractors with either an even or an odd number of spirals are visible along the marked paths of bifurcation. Moreover, this additional hidden parameter allows finding the bifurcation of the multispiral Chua attractor from a stable state to a chaotic state. Furthermore, the Routh-Hurwitz criteria are used to study the stability of the original equilibrium point of the Chua attractor.

The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system.

Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor. We derive a Poincaré return map to prove the existence of the double-scroll attractor and explicitly characterize its global dynamical properties. In particular, we reveal a hidden set of countably many saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets emerge from an ordered iterative process that yields sequential intersections between different horseshoes and their preimages. This novel distinctive feature differs from the classical Smale horseshoes, directly intersecting with their own preimages. Our global analysis suggests that the structure of the classical Chua attractor and other figure-eight attractors might be more complex than previously thought.

Studying the Chaotic Dynamics Using Rossler-Chua Systems Combined with A Semiconductor Laser

In this paper, two different chaotic dynamic systems are coupled using a semiconductor laser to produce a new chaotic system. These two chaotic systems are Rossler and Chua systems. X-dynamic of Rossler system was coupled optically using optical fiber as a carrier of signal with x, y, and z-dynamics of Chua system. The results were analyzed and the behavior of Chua system was found to be changing in time series which, in turn, changed the attractor. The Chua attractor was converted from double scroll to single scroll. The results obtained from connecting two different systems in chaotic behavior showed a remarkable increase in the bandwidth of Chua system. This increase in bandwidth opens up a wide field for many applications, the most important of which is in the field of secure communications.

New numerical approximation for Chua attractor with fractional and fractal-fractional operators

In this study, we present new numerical scheme for modified Chua attractor model with fractional operators. However we give numerical solution of the considered model with fractal-fractional operators. Also, we offer error analysis for a general Cauchy problem with fractional and fractal-fractional operators. For numerical solution of the considered equation, we use new numerical scheme which is established with an efficient polynomial known as Newton interpolation polynomial. The results are discussed with some examples and simulations.

Modified Chua chaotic attractor with differential operators with non-singular kernels

In the present paper we analysis the Modified Chua attractor using new concept of fractional differentiation with non-local and non-singular kernel. A new numerical scheme that was recently suggested was used for the Volterra equation with Atangana–Baleanu fractional integral, Caputo–Fabrizio integral and finally Riemann–Liouvile integral. The numerical solution obtained from the new numerical scheme let no doubt than to believe that the new numerical scheme is very efficient and converges toward exact solution very rapidly. Applicability and suitability of the scheme is justified when applied to solve some novel chaotic system with fractional order. Existence and uniqueness of the Volterra type is presented. We presented some numerical simulations for different values of fractional order.

Fractional stochastic modelling illustration with modified Chua attractor

Very recently a new concept to capture more complexities in nature was suggested. The concept combines two important concepts of modeling including fractional differentiation and stochastic approach. In this work, we aim to investigate new chaotic attractors using the modified Chuan models and the new approach. We use the log-normal distribution to convert constant parameters into distribution. Then we use 3 different types of differential operators including Caputo, Caputo-Fabrizio and Atangana-Baleanu derivatives. We solve the new equations by using the newly introduced numerical scheme. Our numerical simulations display very new attractors.

Dynamic reconstruction of chaotic system based on exponential weighted online sequential extreme learning machine with kernel

For the dynamic reconstruction of the chaotic dynamical system, a method of identifying an exponential weighted online sequential extreme learning machine with kernel(EW-KOSELM) is proposed. The kernel recursive least square (KRLS) algorithm is directly extended to an online sequential ELM framework, and weakens the effect of old data by introducing a forgetting factor. Meanwhile, the proposed algorithm can deal with the ever-increasing computational difficulties inherent in online kernel learning algorithms based on the ‘fixed-budget’ memory technique. The employed EW-KOSELM identification method is firstly applied to the numerical example of Duffing-Ueda oscillator for chaotic dynamical system based on simulated data, the qualitative and quantitative analysis for various validation tests of the dynamical properties of the original system as well as the identification model are carried out. A set of qualitative validation criteria is implemented by comparing chaotic attractors i.e. embedding trajectories, computing the corresponding Poincare mapping, plotting the bifurcation diagram, and plotting the steady-state trajectory i.e. the limit cycle between the original system and the identification model. Simultaneously, the quantitative validation criterion which includes computing the largest positive Lyapunov exponent and the correlation dimension of the chaotic attractors is also calculated to measure the closeness i.e. the approximation error between the original system and the identification model. The employed method is further applied to a practical implementation example of Chua's circuit based on the experimental data which are generated by sampling and recording the measured voltage across a capacitor, the inductor current from the double-scroll attractor, the measured voltage across a capacitor from the Chua's spiral attractor and an experimental time series from a chaotic circuit. The digital filtering technique is then used as a preprocessing approach, on the basis of wavelet denoising the measured data with lower signal-noise ratio (SNR) which can produce the double-scroll attractor or the spiral attractor, the reconstruction attractor of the identification model is compared with the reconstruction attractor from the experimental data for original system. The above experimental results confirm that the EW-FB-KOSELM identification method has a better performance of dynamic reconstruction, which can produce an accurate nonlinear model of process exhibiting chaotic dynamics. The identification model is dynamically equivalent or system approximation to the original system.

Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor

In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.

Hidden Bifurcations in the Multispiral Chua Attractor

We introduce a novel method revealing hidden bifurcations in the multispiral Chua attractor in the case where the parameter of bifurcation [Formula: see text] which determines the number of spiral is discrete. This method is based on the core idea of the genuine Leonov and Kuznetsov method for searching hidden attractors (i.e. applying homotopy and numerical continuation) but used in a very different way. Such hidden bifurcations are governed by a homotopy parameter [Formula: see text] whereas [Formula: see text] is maintained constant. This additional parameter which is absent from the initial problem is perfectly fitted to unfold the actual structure of the multispiral attractor. We study completely the multispiral Chua attractor, generated via sine function, and check numerically our method for odd and even values of [Formula: see text] from 1 to 12. In addition, we compare the shape of the attractors obtained for the same value of parameter [Formula: see text] while varying the parameter [Formula: see text].

Hidden attractor and its dynamical characteristic in memristive self-oscillating system

The classical attractors, defined as self-excited attractors, such as Lorenz attractor, Rssler attractor, Chua's attractor and many other well-known attractors, are all excited from unstable index-2 saddle-foci, namely, an attractor with an attraction basin corresponds to an unstable equilibrium. A new type of attractors, defined as hidden attractors, was first found and reported in 2011, whose attraction basin does not intersect with small neighborhoods of the equilibria of the system. Due to the existences of hidden attractors, some particular dynamical systems associated with line equilibrium, or no equilibrium, or stable equilibrium have attracted much attention recently. Additionally, by introducing memristors into existing oscillating circuits or substituting nonlinear resistors in classical chaotic circuits with memristors, a variety of memristor based chaotic and hyperchaotic circuits are simply established and has been broadly investigated in recent years. Motivated by these two considerations, in this paper, we present a novel memristive system with no equilibrium, from which an interesting and striking phenomenon of coexistence of the behaviors of hidden multiple attractors and the corresponding multistability is perfectly demonstrated by numerical simulations and experimental measurements. According to a newly proposed circuit realization scheme, a new type of four-dimensional memristive self-oscillated system is easily implemented by directly replacing a linear coupling resistor in an existing three-dimensional self-oscillated system circuit with a voltage-controlled memristor. The proposed system has no equilibrium, but can generate various hidden attractors including periodic limit cycle, quasi-periodic limit cycle, chaotic attractor, and coexisting attractors and so on. Based on bifurcation diagram, Lyapunov exponent spectra, and phase portraits, complex hidden dynamics with respect to a system parameter of the memristive self-oscillated system are studied. Specially, when different initial conditions are used, the system displays the coexistence phenomenon of chaotic attractors with different topological structures or quasi-periodic limit cycle and chaotic attractor, as well as the phenomenon of multiple attractors of quasi-periodic limit cycle and chaotic attractors with multiple topological structures. The results imply that some coexisting hidden multiple attractors reflecting the emergences of multistability can be observed in the proposed memristive self-oscillated system, which are well illustrated by several conventional dynamical analysis tools. Based on PSIM circuit simulation model, the memristive self-oscillated system is easily made in at a hardware level on a breadboard and two kinds of dynamical behaviors of coexisting hidden multiple attractors are captured in hardware experiments. Hardware experimental measurements are consistent with numerical simulations, which demonstrates that the proposed memristive self-oscillated system has very abundant and complex hidden dynamical characteristics.

Chaotic attractors with separated scrolls.

This paper proposes a new behavior of chaotic attractors with separated scrolls while combining Julia's process with Chua's attractor and Lorenz's attractor. The main motivation of this work is the ability to generate a set of separated scrolls with different behaviors, which in turn allows us to choose one or many scrolls combined with modulation (amplitude and frequency) for secure communication or synchronization. This set seems a new class of hyperchaos because each element of this set looks like a simple chaotic attractor with one positive Lyapunov exponent, so the cardinal of this set is greater than one. This new approach could be used to generate more general higher-dimensional hyperchaotic attractor for more potential application. Numerical simulations are given to show the effectiveness of the proposed theoretical results.

Simplified Chua's attractor via bridging a diode pair

In this paper, a simplified Chua's circuit is realised by bridging a diode pair between a passive LC (inductance and capacitance in parallel connection - LC) oscillator and an active RC (resistance and capacitance in parallel connection - RC) filter. The dynamical behaviours of the circuit are investigated by numerical simulations and verified by experimental measurements. It is found that the simplified Chua's circuit generates Chua's attractors similarly and demonstrates complex non-linear phenomena including coexisting bifurcation modes and coexisting attractors in particular.

Nonlinear Dynamics of Chaotic Attractor of Chua Circuit and Its Application for Secure Communication

The Chua circuit is among the simplest non-linear circuits that shows most complex dynamical behavior, including chaos which exhibits a variety of bifurcation phenomena and attractors. In this paper, Chua attractor's chaotic oscillator, synchronization and masking communication circuits were designed and simulated. The electronic circuit oscilloscope outputs of the realized Chua system is also presented. Simulation and oscilloscope outputs are used to illustrate the accuracy of the designed and realized Chua chaotic oscillator circuits. The Chua system is addressed suitable for chaotic synchronization circuits and chaotic masking communication circuits using Matlab® and MultiSIM® software. Simulation results are used to visualize and illustrate the effectiveness of Chua chaotic system in synchronization and application of secure communication. Keywords: chua nonlinear circuit, chaotic attractor, chaotic synchronization, secure communication.

Localization of hidden attractors of the generalized Chua system based on the method of harmonic balance

Chua’s circuits, which were introduced by Leon Ong Chua in 1983, are simplest electric circuits operating in the mode of chaotic oscillations. Systems of differential equations describing the behavior of Chua’s circuits are three-dimensional autonomous dynamical systems with scalar nonlinearity. In the standard Chua system, chaotic oscillations are excited in the classical manner, namely, starting from the neighborhood of the unstable zero equilibrium, after the transient process, the system trajectory tends to a Chua attractor.

PERIOD-DOUBLING AND SCROLL-DOUBLING BIFURCATIONS IN A CLASS OF NONSYMMETRIC CELLULAR NEURAL NETWORKS

Two kinds of bifurcation behaviors in a simple nonsymmetric CNN system are discovered with the help of computer simulation. The first kind of bifurcation exhibits a cascade of period doubling bifurcations which lead to the birth of two coexisting Chua's single-scroll attractors. The second one presents a scroll doubling bifurcation which gives rise from Chua's single-scroll attractor to a Chua's double-scroll attractor.

Stabilizing Unstable Periodic Orbits of the Multi-Scroll Chua's Attractor

This paper addresses the control of the n-scroll Chua’s circuit. It will be shown how chaotic systems with multiple unstable periodic orbits (UPOs) detected in the Poincar´e section can be stabilized as well as taking the system dynamics from one UPO to another.

Fault diagnosis for analog circuits based on chaotic signal excitation

A fault diagnosis method for analog circuit is proposed in this paper. An all-purpose amplifier is taken as the typical circuit under test (CUT). The chaotic signal, which is generated by an improved Chua's circuit, is employed as the excitation signal of CUT. The algorithm for phase-space reconstruction of chaotic time series is a combination algorithm of multiple autocorrelation and Γ-test. The circuit state is estimated based on detecting the geometric change of Chua's attractor with a data-mining approach. For the purpose of information fusion, another eight features are extracted from the testing data to comprehensively determine the CUT states. A discussion is made for comparing the effectiveness of each feature according to the testing results.

Another Chemical Chaotic Attractor for Chua's Equation

A new chemical kinetic system for Chua's equation is presented. In contrast to the double-scroll Chua's attractor, the chaotic attractor obtained from this system has only one scroll.

NONLINEAR IDENTIFICATION USING PRIOR KNOWLEDGE OF FIXED POINTS: A MULTIOBJECTIVE APPROACH

This paper is devoted to the problem of model building from data produced by a nonlinear dynamical system. Unlike most published works that address the problem from a black-box perspective, in the present paper a procedure is developed that permits the use of prior knowledge about the location of fixed-points in addition to the data thus resulting in a gray-box approach. Numerical results using Chua's double-scroll attractor and the sine map are presented. As discussed, the suggested procedure is useful as a means to partially compensate for the loss of information due to noise and to improve dynamical performance in the presence of model structure mismatches. Preliminary results have indicated that the procedure outlined in this paper is a systematic way of searching for models in the vicinity of the black-box solution. This could have important consequences not only in model building but also in model validation.