Three-dimensional Autonomous Chaotic System Research Articles

A memristor-based chaotic system and its field programmable gate array implementation

A nanoscale memristor can replace the nonlinear part of a chaotic system, which can greatly reduce the physical size of the chaotic system. More importantly, it can enhance the complexity of the chaotic system and the randomness of signals. In this paper, a new memristor-based chaotic system is designed based on a new three-dimensional autonomous chaotic system. In order to study the complex dynamic characteristics of the memristive system, the chaotic system is investigated by the theoretical derivation, numerical simulation, stabilization of equilibrium points, and Lyapunov exponent spectrum. The influences of different parameters on the phase diagram and the stability of equilibrium point of this system are also discussed in detail. It is interesting that when system parameters a and c take different values, the location and stability of the equilibrium point of the system will be changed, then two scrolls of the system will be overturned at a different angle, and it will produce a different degree of aliasing between the two scrolls. Parameter b has a large variable range, when it is changed, and the system will transform into three kinds of classical chaotic systems defined by Vaněček and Celikovsk. These indicate that the memristor-based chaotic system has a lot of valuable dynamic behaviors, so it has applications in the field of secure communication, information processing etc. Field programmable gate array (FPGA) technology has a large capacity and high reliability, which is widely used in modern digital signal processing. And with the development of FPGA technology, applying FPGA technology to realizing the chaotic systems has gradually become a hot topic. Moreover, the improved Newton iteration method is used to design a square root operator of memristor in this paper by using verilog hardware description language (verilog HDL) which only needs three times iteration to reach the required accuracy. The results of FPGA hardware are consistent with the numerical simulation results. It breaks through the previous bottleneck that the chaotic system based on titanium dioxide memristor can only be simulated in computer, which is of great significance for further studing of memristor, and provides a reference for further research on the memristor-based chaotic system and applications in secure communication and information processing.

Dynamic analysis of a chaotic system

A three-dimensional (3D) autonomous chaotic system is reported in this paper. Some basic properties are analyzed carefully by means of phase portraits, bifurcation diagrams, Lyapunov exponent spectrums and equilibrium points. The remarkable particularity of the system is that it possesses the dynamical properties as constant Lyapunov exponent spectrum and adjustable signal amplitude versus parameters b, d, and f. Further, based on the topological horseshoe theory, the horseshoe chaos in this system is investigated.

A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium

A new hyper-chaotic system is presented in this paper by adding a smooth flux-controlled memristor and a cross-product item into a three-dimensional autonomous chaotic system. It is exciting that this new memristive system can show a four-wing hyper-chaotic attractor with a line equilibrium. The dynamical behaviors of the proposed system are analyzed by Lyapunov exponents, bifurcation diagram and Poincare maps. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of hyperchaos in the system is verified theoretically. Finally, an electronic circuit is designed to implement the hyper-chaotic memristive system.

The chaotic Dadras–Momeni system: control and hyperchaotification

In this paper a novel three-dimensional autonomous chaotic system, the so called Dadras-Momeni system, is considered and two different control techniques are employed to realize chaos control and chaos synchronization. Firstly, the optimal control of the chaotic system is discussed and an open loop feedback controller is proposed to stabilize the system states to one of the system equilibria, minimizing the cost function by virtue of the Pontryagin’s minimum principle. Then, an adaptive control law and an update rule for uncertain parameters, based on Lyapunov stability theory, are designed both to drive the system trajectories to an equilibrium or to realize a complete synchronization of the system in periodic and chaotic regimes. Numerical simulations are included to show the effectiveness of the designed controllers in realizing the stabilization and the synchronization of the chaotic system. Finally, a new hyperchaotic system is introduced by adding a controller to the second equation of the three-dimensional autonomous Dadras-Momeni chaotic system and the corresponding hyperchaotic attractor is shown via computer simulations.

Analysis of Dynamical Behaviors in a Continuous Chaotic System Without Šil'nikov Orbits

This paper discusses a simple yet interesting three-dimensional autonomous quadratic chaotic system, which has two fixed points of saddle-focus type but does not belong to the class of Šil'nikov type since there are neither homoclinic nor heteroclinic orbits. Dynamical behaviors of this chaotic system are investigated in detail through analyzing bifurcation routes, Lyapunov exponents and Poincaré mappings. These investigations partially reveal the mechanism of how chaos in non-Šil'nikov type of systems is generated.

A New Chaotic Attractor and Its Synchronization Implementation

A new three-dimensional autonomous chaotic system is proposed in this paper. It analyzes dynamic behaviors of this new system, including the stability of equilibria, the system dissipativity, poincare maps, Lyapunov exponent, bifurcation, wave forms, and spectrum. Numeral simulation is performed in Matlab. A chaotic circuit is designed, and the corresponding circuit simulation is conducted. The results show that the chaotic attractor exists in the new system. The synchronization problem of the chaotic system is solved with a synchronization method, and its synchronization circuit is designed. Numeral simulation and circuit simulation show that the synchronization problem of this chaotic system proposed is solved.

Estimating ultimate bound and finding topological horseshoe for a new chaotic system

This paper presents a new three-dimensional autonomous chaotic system with only one positive term. Basic dynamical properties of the new attractor are demonstrated in terms of phase portraits, equilibria, Lyapunov exponents, Poincare mapping, bifurcation diagram. Furthermore, we derive a three-dimensional spheriform ultimate bound and positively invariant set for all the positive values of its parameters a, b, c. At last, the horseshoe chaos in this system is investigated based on the topological theory.

Numerical analysis and circuit realization of the modified LÜ chaotic system

A novel three-dimensional autonomous chaotic system from the LÜ chaotic system is given. By using the theoretical analysis and numerical simulation, we provide an insight into the dynamic properties and characterizations of this system, such as Hopf bifurcation. In particular, we are interested in focusing on the dependence of varying parameters on chaos with the help of some chaos indicators including the fast Lyapunov indicator, small alignment indexes and Lyapunov exponent. It is shown that growing the parameter c leads to the extent of chaos. Finally, a chaotic electronic circuit is designed for the realization of the chaotic attractor with aim of Multisim software, and it gives almost the same rules of types of orbits as numerical ones by an alternating value of a circuital resistor.

Circuit Simulation of the Modified Lorenz System

We propose a novel three-dimensional autonomous chaotic system originating from the Lorenz chaotic system. With help of the theoretical analysis and the numerical simulation, the dynamic properties and characterization of this system are studied such as Hopf bifurcation etc.. In particular, fast Lyapunov indicator, small alignment indexes and Lyapunov exponent are used to estimate the effects on the dynamics, of varying parameter that correspond to ordered or chaotic orbits. It is found that increasing value of the parameter r always leads to the extent of chaos. Finally, by Multisim software, a chaotic electronic circuit is designed for the realization of the chaotic attractor, and along with the change of circuital resistor value, it gives almost the same rules of types of orbits as numerical ones.

Stability Analysis and Control of A 3-D Autonomous Ai-Yuan-Zhi-Hao Hyperchaotic System

The stability analysis and control of a three-dimensional autonomous chaotic system that exhibits hyperchaotic properties is presented in this paper. The system's algebraic structure consistsof six variable control parameters, three nonlinear terms and two cross-products which triggers hypersensitivity in its dynamic state evolutions. Stability analysis and control via fuzzy modelling confirms the controllability of its unstable orbits in well-tuned regions of stability using Lyapunov principles.

On Dynamics Analysis of a Novel Three-Dimensional Autonomous Chaotic System

This paper reports a novel three-dimensional autonomous chaotic system. By choosing an appropriate bifurcation parameter, we prove that a Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value, and some basic dynamical properties, such as Lyapunov exponents, fractal dimension, bifurcation diagram, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied. Furthermore, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis.

A GALLERY OF LORENZ-LIKE AND CHEN-LIKE ATTRACTORS

In this article, three-dimensional autonomous chaotic systems with two quadratic terms, similar to the Lorenz system in their algebraic forms, are studied. An attractor with two clearly distinguishable scrolls similar to the Lorenz attractor is referred to as a Lorenz-like attractor, while an attractor with more intertwine between the two scrolls similar to the Chen attractor is referred to as a Chen-like attractor. A gallery of Lorenz-like attractors and Chen-like attractors are presented. For several different families of such systems, through tuning only one real parameter gradually, each of them can generate a spectrum of chaotic attractors continuously changing from a Lorenz-like attractor to a Chen-like attractor. Some intrinsic relationships between the Lorenz system and the Chen system are revealed and discussed. Some common patterns of the Lorenz-like and Chen-like attractors are found and analyzed, which suggest that the instability of the two saddle-foci of such a system somehow determines the shape of its chaotic attractor. These interesting observations on the general dynamic patterns hopefully could shed some light for a better understanding of the intrinsic relationships between the algebraic structures and the geometric attractors of these kinds of chaotic systems.

Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation

A three-dimensional autonomous chaotic system is presented and physically implemented. Some basic dynamical properties and behaviors of this system are described in terms of symmetry, dissipative system, equilibria, eigenvalue structures, bifurcations, and phase portraits. By tuning the parameters, the system displays chaotic attractors of different shapes. For specific parameters, the system exhibits periodic and chaotic bursting oscillations which resemble the conventional heart sound signals. The existence of Shilnikov type of heteroclinic orbit in the three-dimensional system is proven using the undetermined coefficients method. As a result, Shilnikov criterion guarantees that the three-dimensional system has the horseshoe chaos. The corresponding electronic circuit is designed and implemented, exhibiting experimental chaotic attractors in accord with numerical simulations.

Complete switched modified function projective synchronization of a five-term chaotic system with uncertain parameters and disturbances

A five-term three-dimensional (3D) autonomous chaotic system with an exponential nonlinear term is reported in this paper. Basic dynamical behaviours of the chaotic system are further investigated. Then a new synchronization phenomenon, complete switched modified function projective synchronization (CSMFPS), for this novel five-term chaotic system with uncertain parameters and disturbances is investigated. This paper extends previous work, where CSMFPS of chaotic systems means that all the state variables of the drive system synchronize with different state variables of the response system. As the synchronization scheme has many combined forms, it is a promising type of synchronization and can provide greater security in secure communication. Based on Lyapunov stability theory, a robust adaptive controller is contrived to acquire CSMFPS, parameter identification and suppress disturbances simultaneously. Finally, the Lorenz system and the new five-term chaotic system are taken as examples and the corresponding numerical simulations are presented to verify the effectiveness and feasibility of the proposed control scheme.

A New Chaotic Attractor and its Digital Implementation Based on LabVIEW

In this paper, a novel three-dimensional autonomous chaotic system with six terms and two quadratic nonlinearities is presented. Some basic dynamical properties of the new chaotic system are analyzed by means of equilibrium points, eigenvalue structures, Lyapunov exponent and Lyapunov dimension. In order to overcome the external conditions affected by the analog circuit’s chaotic system, digital implementation of the new chaotic system based on LabVIEW is also proposed. The results show that the experimental results by LabVIEW are consistent with the theoretical simulation results by Matlab, and the method is an effective digital implementation method.

The Related Extension and Application of the Ši'lnikov Theorem

The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.

Adaptive Sliding Mode Control for Synchronization of a Fractional-Order Chaotic System

This paper proposes a three-dimensional autonomous chaotic system which displays some interest dynamical behaviors such as invariable Lyapunov exponent spectrums and controllable signal amplitude. The corresponding fractional version of the proposed system is obtained. A single state controller for synchronization of this fractional-order chaotic system is developed based on the techniques of sliding mode control and adaptive control. Numerical simulations are provided to demonstrate the feasibility of the presented synchronization method.

Modeling and Simulation of a Switchable Chaotic System with Fractional Order

A new three-dimensional autonomous chaotic system is constructed. Some basic dynamical characteristics of this system with integer order are studied. The switchable chaotic system is constitutive of two subsystems. The dynamic characteristics of the fractional-order switchable system varying with the differential operator orders with fixed parameter are analyzed. Numerical simulations on the dynamics of new system with fractional-order have been carried out.

Analysis and adaptive synchronization for a new chaotic system

In this paper, a new three-dimensional autonomous chaotic system is presented. There are three control parameters and three different nonlinear terms in the governed equations. The new chaotic system has six equilibrium points. Basic dynamic properties of the new system are investigated via theoretical analysis and numerical simulation. The nonlinear characteristic of the new chaotic system are demonstrated in terms of equilibria, Jacobian matrices, Lyapunov exponents, a dissipative system, Poincare maps and bifurcations. Then, an adaptive control law is derived to make the states of two identical chaotic systems asymptotically synchronized based on the Lyapunov stability theory. Finally, a numerical simulation is presented to verify the effectiveness of the proposed synchronization scheme.

Analysis of a novel three-dimensional chaotic system

This paper proposes a novel three-dimensional (3D) autonomous chaotic system, which displays complicated dynamical behaviors. Basic dynamical properties are analyzed by means of phase portraits, Poincare mapping and equilibria. The different dynamic behaviors of the system are investigated especially when changing each system parameter. It is found that the Lyapunov exponent spectrums keep invariable when the parameters d, e and f vary; the signal amplitude can be adjusted by d, e and f; and the phase of signals x1 and x3 in the system can be controlled by parameter e. Finally, the amplitude modulation factor of this 3D chaotic system is studied carefully.