Continuous Chaotic System Research Articles

Dynamic analysis and synchronisation control of a novel chaotic system with coexisting attractors

In this paper, a novel four-dimensional continuous chaotic system is constructed from the simplified Lorenz-like system. The novel system has three equilibria, strange attractors, coexisting attractors, and performs Hopf bifurcation with the variation of system parameters. The coexisting attractors, which are the most remarkable dynamic features of the system, are numerically studied. The coexisting attractors show that the system coexists as a pair of point, periodic, and chaotic attractors. Some basic dynamic behaviours are studied as well. The synchronisation control problem of the system is analysed. The theoretical and numerical analyses demonstrate that the system can easily achieve synchronisation by using the passive control technique.

Detecting chaos in fractional-order nonlinear systems using the smaller alignment index

The identification between chaos and ordered states in fractional-order chaotic systems is a challenge as well as a hot topic due to the complex of fractional calculus. In this paper, the smaller alignment index (SALI) is developed to detect chaos in the fractional-order chaotic systems by introducing the fractional-order tangent systems. Numerical simulations are carried out based on the fractional-order simplified Lorenz system and the fractional-order Hénon map, which are continuous chaotic system and discrete chaotic system, respectively. It shows that the proposed method is effective for distinguishing chaos and order in different kinds of fractional-order chaotic systems.

Adaptive Synchronization Strategy between Two Autonomous Dissipative Chaotic Systems Using Fractional-Order Mittag-Leffler Stability.

Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos.

A new computer-controlled platform for ADC-based true random number generator and its applications

The basis of encryption techniques is random number generators (RNGs). The application areas of cryptology are increasing in number due to continuously developing technology, so the need for RNGs is increasing rapidly, too. RNGs can be divided into two categories as pseudorandom number generator (PRNGs) and true random number generator (TRNGs). TRNGs are systems that use unpredictable and uncontrollable entropy sources and generate random numbers. During the design of TRNGs, while analog signals belonging to the used entropy sources are being converted to digital data, generally comparators, flip-flops, Schmitt triggers, and ADCs are used. In this study, a computer-controlled new and flexible platform to find the most appropriate system parameters in ADC-based TRNG designs is designed and realized. As a sample application with this new platform, six different TRNGs that use three different outputs of Zhongtang, which is a continuous time chaotic system, as an entropy source are designed. Random number series generated with the six designed TRNGs are put through the NIST800–22 test, which has the internationally highest standards, and they pass all tests. With the help of the new platform designed, ADC-based high-quality TRNGs can be developed fast and also without the need for expertise. The platform has been designed to decide which entropy source and parameter are better by comparing them before complex embedded TRNG designs. In addition, this platform can be used for educational purposes to explain how to work an ADC-based TRNG. That is why it can be utilized as an experiment set in engineering education, as well.

Comparative Controlling of the Lorenz Chaotic System Using the SMC and APP Methods

The Lorenz chaotic system is based on a nonlinear behavior and this causes the system to be unstable. Therefore, two different controller models were developed and named as the adaptive pole placement and sliding mode control (SMC) methods for the establishment of continuous time nonlinear Lorenz chaotic system. In order to achieve this, an improved controller structure was developed first theoretically for both the controller methods and then tested practically using the numerical samples. During the establishment of adaptive pole placement method for the Lorenz chaotic system, various stages were applied. The nonlinear chaotic system was also linearized by means of Taylor Series expansion processes. In addition, the feedback matrix of the adaptive pole placement method was determined using linear Jacobian matrix. The chaotic system reached an equilibrium point by using both the SMC and adaptive pole placement methods; however the simulation results of the SMC had better success than adaptive pole placement control technique.

A strange novel chaotic system with fully golden proportion equilibria and its mobile microcomputer-based RNG application

A strange novel three-dimensional quadratic continuous autonomous chaotic system with fully golden proportion equilibria is proposed. It has only seven terms, three quadratic nonlinearities and one parameter ‘a’. The system equations have four equilibrium points and very interestingly all the equilibrium points have fully golden proportion values. Besides, this chaotic system has hidden amplitude control properties. The dynamic analyses of the system are presented such as equilibrium points, dissipativity, Lyapunov exponents, bifurcation diagrams, phase portraits and hidden amplitude control properties. Later, electronic circuit of the system is simulated in software and implemented in real environment. Finally, microcomputer-based random number generator (RNG) application and its NIST-800-22 tests are executed as another real-time application.

Applications of symbolic dynamics in counteracting the dynamical degradation of digital chaos

Chaotic systems which are realized on the finite precision devices suffer from dynamical degradation. In the literature, there does not seem to be good basis for designing schemes to reduce its negative influence on digital chaotic systems. Meanwhile, symbolic dynamics is often used to study the behaviors of complicated dynamical systems. In this paper, a new mechanism based on symbolic dynamics is proposed for designing effective schemes, in order to counteract the dynamical degradation of digital chaotic systems. A concrete scheme with hybrid structure is discussed to show significance of the mechanism. Symbolic dynamics are used to rigorously prove that a class of chaos-based digital systems can be perturbed to be chaotic again by a continuous chaotic system. Numerical experiments demonstrate that this scheme can recover the dynamical properties of original system, which is different from the existing remedies.

Dynamical analysis and FPGA implementation of a large range chaotic system with coexisting attractors

Abstract Based on the generalized 3-D Lu chaotic system, this paper presents a new 3-D autonomous continuous chaotic system with a large range parameter b ∈ [ 7500 ] . The system has coexisting chaotic attractors depending on the distances from unstable equilibrium points to a given initial point. The topological horseshoes and entropy are obtained by the aid of the topological horseshoes theory and numerical computations. The results show that the new system is chaotic. Moreover, the system’s chaotic characteristics are verified by Multisim simulation and FPGA implementation.

Fractional-order PWC systems without zero Lyapunov exponents

In this paper, it is shown numerically that a class of fractional-order piece-wise continuous systems, which depend on a single real bifurcation parameter, have no zero Lyapunov exponents but can be chaotic or hyperchaotic with hidden attractors. Although not analytically proved, this conjecture is verified on several systems including a fractional-order piece-wise continuous hyperchaotic system, a piece-wise continuous chaotic Chen system, a piece-wise continuous variant of the chaotic Shimizu-Morioka system and a piece-wise continuous chaotic Sprott system. These systems are continuously approximated based on results of differential inclusions and selection theory, and numerically integrated with the Adams-Bashforth-Moulton method for fractional-order differential equations. It is believed that the obtained results are valid for many, if not most, fractional-order PWC systems.

Generating a Double-Scroll Attractor by Connecting a Pair of Mutual Mirror-Image Attractors via Planar Switching Control

An original three-dimensional (3D) smooth continuous chaotic system and its mirror-image system with eight common parameters are constructed and a pair of symmetric chaotic attractors can be generated simultaneously. Basic dynamical behaviors of two 3D chaotic systems are investigated respectively. A double-scroll chaotic attractor by connecting the pair of mutual mirror-image attractors is generated via a novel planar switching control approach. Chaos can also be controlled to a fixed point, a periodic orbit and a divergent orbit respectively by switching between two chaotic systems. Finally, an equivalent 3D chaotic system by combining two 3D chaotic systems with a switching law is designed by utilizing a sign function. Two circuit diagrams for realizing the double-scroll attractor are depicted by employing an improved module-based design approach.

Complex dynamics of a new 3D Lorenz-type autonomous chaotic system

This paper investigates a new three-dimensional continuous quadratic autonomous chaotic system which is not topologically equivalent to the Lorenz system. The dynamical behaviours of this system are further investigated in detail, including the ultimate boundedness, the invariant sets and the global attraction domain according to Lyapunov stability theory of dynamical systems. The innovation of the paper lies in the fact that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets with respect to the parameters of this system. To validate the ultimate bound estimation, numerical simulations are also investigated. Numerical simulations verify the effectiveness and feasibility of the theoretical scheme.

A new two-level data hiding algorithm for high security based on a nonlinear system

In the recent years, interdisciplinary studies on and applications of nonlinear systems have been increased significantly. Some of these studies are chaos-based applications. Chaos-based encryption, one of the most popular type of applications, has some security drawbacks. In this article, to overcome those drawbacks a secure steganography application with chaos-based encryption is realized by employing a four-dimensional continuous chaotic system. As a first step, the chaotic system that is used in the study is described and analyzed, and then random number generator (RNG) is designed for encryption application. In the next step, the data is encrypted with the designed RNG and then embedded in a carrier in chaos-based manner. As the final step, the security analysis of encryption and data hiding processes is carried out to evaluate their performances. The most important feature of the article contains both encryption and data hiding processes with successful security performances, and these types of studies are rare in the literature.

Dynamical behaviors, control and synchronization of a new chaotic model with complex variables and cubic nonlinear terms

A novel chaotic model with complex variables and cubic non-linear terms was proposed. The new system is a six dimensional continuous real autonomous chaotic system. The characteristics of this system containing invariance, dissipation, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams and chaotic achievement are studied. Converting and turning the system chaotic behavior to its unstable trivial fixed point via the Lyapunov stability theorem. An approach proposed to analyze the system chaos synchronization. Analytical expressions are derived for control functions. The chaos synchronization results were employed to develop a simple application in secure communication. Numerical effects computed to experiment the control forces scientific expressions gravity and to show the chaos synchronization of a chaotic system.

A New Nonlinear Chaotic Complex Model and Its Complex Antilag Synchronization

Another chaotic nonlinear Lü model with complex factors is covered here. We can build this riotous complex system when we add a complex nonlinear term to the third condition of the complex Lü system and think of it as if every one of the factors is mind boggling or complex. This system in real adaptation is a 6-dimensional continuous autonomous chaotic system. Different types of chaotic complex Lü system are developed. Also, another sort of synchronization is presented by us which is simple for anybody to ponder for the chaotic complex nonlinear system. This sort might be called a complex antilag synchronization (CALS). There are irregular properties for CALS and they do not exist in the literature; for example, (i) the CALS contains or fused two sorts of synchronizations (antilag synchronization ALS and lag synchronization LS); (ii) in CALS the attractors of the main and slave systems are moving opposite or similar to each other with time lag; (iii) the state variable of the main system synchronizes with a different state variable of the slave system. A scheme is intended to accomplish CALS of chaotic complex systems in light of Lyapunov function. The acquired outcomes and effectiveness can be represented by a simulation case for our new model.

A New 3D Autonomous Continuous System with Two Isolated Chaotic Attractors and Its Topological Horseshoes

Based on the 3D autonomous continuous Lü chaotic system, a new 3D autonomous continuous chaotic system is proposed in this paper, and there are coexisting chaotic attractors in the 3D autonomous continuous chaotic system. Moreover, there are no overlaps between the coexisting chaotic attractors; that is, there are two isolated chaotic attractors (in this paper, named “positive attractor” and “negative attractor,” resp.). The “positive attractor” and “negative attractor” depend on the distance between the initial points (initial conditions) and the unstable equilibrium points. Furthermore, by means of topological horseshoes theory and numerical computation, the topological horseshoes in this 3D autonomous continuous system is found, and the topological entropy is obtained. These results indicate that the chaotic attractor emerges in the new 3D autonomous continuous system.

A new 3D chaotic system: Dynamical analysis, electronic circuit design, active control synchronization and chaotic masking communication application

This study presents a new three dimensional continuous time chaotic system. Dynamical analysis and numerical simulation of the new chaotic system were first carried out. Then electronic circuit design of the new chaotic system was implemented considering practical applications. The numerical and electronic circuit simulation results were found to be in good accordance. Besides, synchronization by active control was realized for secure communication applications, followed by designing a communication system with chaotic masking method. Finally, a secure signal masking application was implemented on Matlab platform. In the developed secure communication system, information signal sent over noisy channel was successfully retrieved at the receiver.

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.

Potential Function in a Continuous Dissipative Chaotic System: Decomposition Scheme and Role of Strange Attractor

We demonstrate, first in literature, that potential functions can be constructed in a continuous dissipative chaotic system and can be used to reveal its dynamical properties. To attain this aim, a Lorenz-like system is proposed and rigorously proved chaotic for exemplified analysis. We explicitly construct a potential function monotonically decreasing along the system's dynamics, revealing the structure of the chaotic strange attractor. The potential function is not unique for a deterministic system. We also decompose the dynamical system corresponding to a curl-free structure and a divergence-free structure, explaining for the different origins of chaotic attractor and strange attractor. Consequently, reasons for the existence of both chaotic nonstrange attractors and nonchaotic strange attractors are discussed within current decomposition framework.

Control of Rabinovich chaotic system using sliding mode control

SUMMARYThis paper investigates the control of the continuous time Rabinovich chaotic system with the sliding mode control method. Based on the properties of the sliding mode theory, the controllers are designed and added to the nonlinear Rabinovich system. Numerical simulations show that the Rabinovich chaotic system can be regulated to its equilibrium points in the state space by using the sliding mode controllers, which verifies all the theoretical analyses. Simulation results of the proposed sliding mode control strategy have been also compared with the passive control method, and their performances are discussed. Copyright © 2013 John Wiley & Sons, Ltd.

Counteracting the dynamical degradation of digital chaos via hybrid control

Chaotic systems would degrade owing to finite computing precisions, and such degradation often seriously affects the performance of digital chaos-based applications. In this paper, a chaotification method is proposed to solve the dynamical degradation of digital chaotic systems based on a hybrid structure, where a continuous chaotic system is applied to control the digital chaotic system, and a unidirectional coupling controller that combines a linear external state control with a modular function is designed. Moreover, we proof rigorously that a class of digital chaotic systems can be driven to be chaotic in the sense that the system is sensitive to initial conditions. Different from the existing remedies, this method can recover the dynamical properties of system, and even make some properties better than those of the original chaotic system. Thus, this new approach can be applied to the fields of chaotic cryptography and secure communication.