Three-dimensional Autonomous Chaotic System Research Articles

On a New Three-Dimensional Chaotic System with Adaptive Control and Chaos Synchronization

A three-dimensional autonomous deterministic chaotic system having six parameters is explored within this article. The dynamical characteristics of the proposed system are investigated through eigenvalues structure, bifurcation diagrams, Kaplan–Yorke dimension, Lyapunov exponents, time response, and phase plane trajectories. For the suitable design of the system parameters, it is found that the system can exhibit periodic, period-n, or chaotic oscillations. Accordingly, the system’s dynamical behavior to the variation of its coefficients has been explored. The obtained results revealed that the proposed dynamical system does not lose its chaotic oscillations for the small fluctuations of one or more of the values of its parameters. In addition, chaos control and chaos synchronization have been studied by means of the adaptive control strategy relying on Lyapunov’s second method of stability. The numerical simulation revealed that superior chaos control and master-slave synchronization have been achieved by the applied control laws. Finally, the obtained results have been simulated via a nonlinear electronic circuit that demonstrated the feasibility of the purposed chaotic system for different engineering applications such as secure communications, cryptosystems, image encryption, and image processing.

Dynamical Analysis of a One- and Two-Scroll Chaotic System

In this paper, a three-dimensional (3D) autonomous chaotic system is introduced and analyzed. In the system, each equation contains a quadratic crossproduct. The system possesses a chaotic attractor with a large chaotic region. Importantly, the system can generate both one- and two-scroll chaotic attractors by choosing appropriate parameters. Some of its basic dynamical properties, such as the Lyapunov exponents, Lyapunov dimension, Poincaré maps, bifurcation diagram, and the chaotic dynamical behavior are studied by adjusting different parameters. Further, an equivalent electronic circuit for the proposed chaotic system is designed according to Kirchhoff’s Law, and a corresponding response electronic circuit is also designed for identifying the unknown parameters or monitoring the changes in the system parameters. Moreover, numerical simulations are presented to perform and complement the theoretical results.

A new autonomous memristive megastable oscillator and its Hamiltonian-energy-dependent megastability.

Multistability is a special issue in nonlinear dynamics. In this paper, a three-dimensional autonomous memristive chaotic system is presented, with interesting multiple coexisting attractors in a nested structure observed, which indicates the megastability. Furthermore, the extreme event is investigated by local riddled basins. Based on Helmholtz's theorem, the average Hamiltonian energy with respect to initial-dependent dynamics is calculated and the energy transition explains the occurrence mechanisms of the megastability and the extreme event. Finally, by configuring initial conditions, multiple coexisting megastable attractors are captured in PSIM simulations and FPGA circuits, which validate the numerical results.

Experimental Validation of Multiple Attractors in a Simple Chaotic Circuit

In this paper, a simple jerk circuit that allows studying the dynamical behavior of a three-dimensional autonomous chaotic system with only one nonlinear term is further investigated by numerical simulations and experimental validation. Depending on a single tuning parameter, the chaotic system is theoretically studied using standard techniques such as equilibrium analysis, bifurcation diagram, and Lyapunov exponents. Subsequently, the circuit that models the chaotic system is implemented to validate theoretical prediction experimentally. Despite the simple structure of the jerk circuit, experimental study of Fourier spectra has shown that the jerk circuit displays complex dynamics characterized by periodic limit cycles and aperiodic strange attractors. In addition, the jerk circuit has exhibited a wide tuning range and experimental results have shown good agreement with theoretical prediction except for few cases where numerical simulation has failed to accurately match experimental results due to sensitivity to initial conditions which is a signature of chaotic nonlinear systems.

Coexisting asymmetric behavior and free control in a simple 3-D chaotic system

The design and circuit implementation of flexible chaotic systems with desired properties have been of great interest due to their promising applications in various chaos-based information systems. In this paper, a simple three-dimensional autonomous chaotic system with only one quadratic nonlinearity and six linear terms is proposed. The system is asymmetric, which has two equilibria and can generate a single-scroll attractor. This system exhibits asymmetric bi-stability behaviors of chaotic attractors coexisting with periodic limit cycles and other chaotic attractors. By introducing control functions to coefficients of specific linear term or quadratic term, the system can be extended as an amplitude-controllable, equilibrium-shiftable chaotic system. Consequently, attractor rotation is implemented via polarity symmetry, and pseudo-multi-scroll attractors are further generated via nonautonomous approach. The theoretical analyses are confirmed by both the numerical simulations and circuit verifications.

Multistability and Hidden Attractors in a Three-Dimensional Chaotic System

This paper proposes a novel three-dimensional autonomous chaotic system. Interestingly, when the system has infinitely many stable equilibria, it is found that the system also has infinitely many hidden chaotic attractors. We show that the period-doubling bifurcations are the routes to chaos. Moreover, the Hopf bifurcations at all equilibria are investigated and it is also found that all the Hopf bifurcations simultaneously occur. Furthermore, the approximate expressions and stabilities of bifurcating limit cycles are obtained by using normal form theory and bifurcation theory.

Jacobi analysis for an unusual 3D autonomous system

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

Analysis and Control of Fractional Order Generalized Lorenz Chaotic System by Using Finite Time Synchronization

In this paper, we present a corresponding fractional order three-dimensional autonomous chaotic system based on a new class of integer order chaotic systems. We found that the fractional order chaotic system belongs to the generalized Lorenz system family by analyzing its linear term and topological structure. We also found that the equilibrium point generated by the fractional order system belongs to the unstable saddle point through the prediction correction method and the fractional order stability theory. The complexity of fractional order chaotic system is given by spectral entropy algorithm andC0algorithm. We concluded that the fractional order chaotic system has a higher complexity. The fractional order system can generate rich dynamic behavior phenomenon with the values of the parameters and the order changed. We applied the finite time stability theory to design the finite time synchronous controller between drive system and corresponding system. The numerical simulations demonstrate that the controller provides fast and efficient method in the synchronization process.

Dynamics of a new composite four–Scroll chaotic system

A new three-dimensional autonomous chaotic system is proposed. This new system can generate single-scroll, double-scroll, three-scroll and four-scroll attractors under different system parameters. Particularly, it can generate a four-scroll chaotic attractor composed of a large Chua-like attractor and a small Lorenz-like attractor. And the system can also generate a nested three-scroll attractor and the multi-double-scroll chaotic attractor. In addition, the system possesses the chaotic state transition, and the number of scrolls will change in the state transition process. The formation mechanism of the composite four-scroll chaotic attractor is analyzed in detail. The dynamic analysis methods include time series, 0–1 test chart, phase diagram, bifurcation diagram and Lyapunov exponents are used to describe some basic dynamics behaviors of the proposed system.

Dynamics, implementation and stability of a chaotic system with coexistence of hyperbolic and non-hyperbolic equilibria

Hyperbolic-type equilibrium requires that all the real parts of the corresponding eigenvalues are nonzero. In this paper, a three-dimensional autonomous chaotic system is introduced, and interestingly we find that one non-hyperbolic equilibrium point and two hyperbolic equilibrium points coexist in this system, which, according to the information we know, has not been previously reported. We first reveal the basic dynamics of the system through analyzing phase portrait, frequency spectrum, Poincaré map, bifurcation diagram and Lyapunov exponent. Then, based on the idea of the improved modular technology, we build an analog circuit to realize the chaotic system, which further verifies the theoretical results. Finally, we design a simple feedback controller on account of Lyapunov asymptotic stability theory, to globally suppress the system to its equilibrium points.

Dynamics, circuit implementation and synchronization of a new three-dimensional fractional-order chaotic system

In this paper, a new three-dimensional autonomous fractional-order chaotic system is presented. There are four different nonlinear terms in the governed equations. By means of theoretical analysis and numerical simulation, some basic dynamical properties, such as phase diagrams, bifurcation diagram, Lyapunov exponents spectrum and chaos diagram of the new fractional-order chaotic system are investigated. Then, we completed the circuit design of the fractional-order chaotic system, and realized the analog hardware circuit. Both the hardware experiment results and circuit simulation results are agreed well with those of numerical simulation. Finally, based on the theory of sliding-mode control and the fractional stability theory, a suitable sliding mode controller is designed for the novel three-dimensional fractional-order chaotic system. Numerical simulations are provided to verify the effectiveness of the proposed synchronization scheme.

Bistable Hidden Attractors in a Novel Chaotic System with Hyperbolic Sine Equilibrium

For the past 4 years, there has been a rapid rise in the study of chaotic systems with curves of equilibria which are categorized as systems with hidden attractors. There is still significant controversy surrounding the shapes of equilibrium points. This paper presents a new three-dimensional autonomous chaotic system with hyperbolic sine equilibrium. Fundamental dynamical properties and complex dynamics of the system have been discovered by using equilibrium analysis, phase portrait, Poincare map, bifurcation diagram and Lyapunov spectrum. It is crucial to note that there are bistable hidden chaotic attractors in the introduced system. Furthermore, in order to show the feasibility of the new system with hyperbolic sine equilibrium, its electronic circuit has been implemented.

A new three-dimensional chaotic system: Dynamical properties and simulation

This paper recomends a new three-dimensional autonomous chaotic system with six terms including three multipliers, which is different from the Lorenz system and other existing systems. Basic dynamical characteristics of this new system are analyzed via equilibria, Lyapunov exponent spectrum, a dissipative system, phase portraits, the Poincaré map, bifurcation diagrams and Hopf bifurcation. The compound structures of this new system are also investigated. Analysis of the results show that this chaotic attractor has complex dynamics with some interesting characteristics.

A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form

A three-dimensional autonomous chaotic system with an infinite number of equilibrium points located on a line and a hyperbola is proposed in this paper. To analyze the dynamical behaviors of the proposed system, mathematical tools such as Routh-Hurwitz criteria, Lyapunov exponents and bifurcation diagram are exploited. For a suitable choice of the parameters, the proposed system can generate periodic oscillations and chaotic attractors of different shapes such as bistable and monostable chaotic attractors. In addition, an electronic circuit is designed and implemented to verify the feasibility of the proposed system. A good qualitative agreement is shown between the numerical simulations and the Orcard-PSpice results. Moreover, the fractional-order form of the proposed system is studied using analog and numerical simulations. It is found that chaos, periodic oscillations and periodic spiking exist in this proposed system with order less than three. Then an electronic circuit is designed for the commensurate fractional order α = 0.98, from which we can observe that a chaotic attractor exists in the fractional-order form of the proposed system. Finally, the problem of drive-response generalized projective synchronization of the fractional-order form of the chaotic proposed autonomous system is considered.

A New Simple Chaotic Lorenz-Type System and Its Digital Realization Using a TFT Touch-Screen Display Embedded System

This paper presents a new three-dimensional autonomous chaotic system. The proposed system generates a chaotic attractor with the variation of two parameters. Analytical and numerical studies of the dynamic properties to generate chaos, for continuous version (CV) and discretized version (DV), for the new chaotic system (NCS) were conducted. The CV of the NCS was implemented by using an electronic circuit with operational amplifiers (OAs). In addition, the presence of chaos for DV of the NCS was proved by using the analytical and numerical degradation tests; the time series was calculated to determine the behavior of Lyapunov exponents (LEs). Finally, the DV of NCS was implemented, in real-time, by using a novel embedded system (ES) Mikromedia Plus for PIC32MX7 that includes one microcontroller PIC32 and one thin film transistor touch-screen display (TFTTSD), together with external digital-to-analog converters (DACs).

Image Encryption Algorithm Based on a Novel Improper Fractional-Order Attractor and a Wavelet Function Map

This paper presents a three-dimensional autonomous chaotic system with high fraction dimension. It is noted that the nonlinear characteristic of the improper fractional-order chaos is interesting. Based on the continuous chaos and the discrete wavelet function map, an image encryption algorithm is put forward. The key space is formed by the initial state variables, parameters, and orders of the system. Every pixel value is included in secret key, so as to improve antiattack capability of the algorithm. The obtained simulation results and extensive security analyses demonstrate the high level of security of the algorithm and show its robustness against various types of attacks.

A simple chaotic system with non-hyperbolic equilibria

This paper constructs a three-dimensional autonomous chaotic system with simple algebraic structure of only five terms. Interestingly, this system is found to possess two non-hyperbolic equilibria, thus it is not under the class of Šil’nikov sense chaos for owning neither heteroclinic orbit nor homoclinic orbit. Dynamical properties have been revealed with the help of bifurcation route, Poincaré map, frequency spectrum, amplitude modulation and topological horseshoe. We hope that our work can motivate a stimulus for the further study on non-Šil’nikov chaotic system with non-hyperbolic equilibrium.

Amplitude-Phase Modulation, Topological Horseshoe and Scaling Attractor of a Dynamical System**Supported by Hunan Provincial Natural Science Foundation of China under Grant No. 2016JJ4036, University Natural Science Foundation of Jiangsu Province under Grant No. 14KJB120007 and the National Natural Science Foundation of China under Grant Nos. 11504176 and 11602084

A three-dimensional autonomous chaotic system is discussed in this paper. Some basic dynamical properties of the system, including phase portrait, Poincaré map, power spectrum, Kaplan–Yorke dimension, Lyapunov exponent spectra, signal amplitude and topological horseshoe are studied theoretically and numerically. The main finding by analysis is that the signal amplitude can be modulated via controlling the coefficients of the linear term, cross-product term and squared term simultaneously or respectively, and the phase of x3 can be modulated by the product of the coefficients of the linear term and cross-product term. Furthermore, scaling chaotic attractors of this system are achieved by modified projective synchronization with an optimization-based linear coupling method, which is safer for secure communications than the existed synchronization scheme since the scaling factors can be regarded as the security encoding key.

A Chaotic System with Different Families of Hidden Attractors

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form

A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.