Lorenz-like Chaotic System Research Articles

Design of a new multi-wing chaotic system and its application in color image encryption

Due to its chaotic carrier function and pseudorandom nature, chaotic systems are frequently applied in the area of secure communication. To render chaotic signals more complicated, a novel multi-wing chaotic system is developed in this study. Initially, a double-wing attractor generated by a novel 3D Lorenz-like chaotic system that comprises two quadratic terms and four linear terms. By inserting a nonlinear feedback controller, the complex system gets transformed into a multi-wing chaotic system. It is noteworthy that the topological shape of the attractor and the number of equilibrium points can both be altered in the particular multi-wing chaotic system. Secondly, the dynamic features of the multi-wing chaotic system are discussed via employing the bifurcation diagram and the Lyapunov exponential spectrum. According to results from simulation, the system displays widely dispersed chaotic characteristics over a large range of parameters. In addition, the complexity analysis findings of chaotic sequences testify that the chaotic sequences have strong complexity. Therefore, the multi-wing chaotic system will have an excellent effect in image encryption. Finally, a scrambling-diffusion encryption scheme is built using the rich chaotic sequences discussed above. The scheme includes two times scrambling of pixel position and once transform of pixel value by the virtue of Galois field (GF). The proposed algorithm has superior efficiency and excellent security according to the results of the feasibility and security analysis. Moreover, the investigation of the multi-wing chaotic system property and its usage in image encryption additionally broadens the applicability of chaos systems and offers some standards for confidential communication.

A fast quantum image encryption algorithm based on affine transform and fractional-order Lorenz-like chaotic dynamical system

A fast quantum image encryption algorithm based on affine transform and fractional-order Lorenz-like chaotic dynamical system

Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

Dynamical Analysis and Simulation of a New Lorenz-Like Chaotic System

This work presents and investigates a new chaotic system with eight terms. By numerical simulation, the two-scroll chaotic attractor is found for some certain parameters. And, by theoretical analysis, we discuss the dynamical behavior of the new-type Lorenz-like chaotic system. Firstly, the local dynamical properties, such as the distribution and the local stability of all equilibrium points, the local stable and unstable manifolds, and the Hopf bifurcations, are all revealed as the parameters varying in the space of parameters. Secondly, by applying the way of Poincaré compactification inℝ3, the dynamics at infinity are clearly analyzed. Thirdly, combining the dynamics at finity and those at infinity, the global dynamical behaviors are formulated. Especially, we have proved the existence of the infinite heteroclinic orbits. Furthermore, all obtained theoretical results in this paper are further verified by numerical simulations.

DYNAMICS OF A GENERALIZED LORENZ-LIKE CHAOS DYNAMICAL SYSTEMS

<abstract> In this work, a new seven-parameter Lorenz-like chaotic system is presented and discussed by combining nonlinear dynamical systems theory with computer simulation. The existence of the ultimate bound set and global exponential attractive set of this chaotic system is proved by using Lyapunov's direct method. A family of analytic mathematical expression of the ultimate bound sets and global exponential attractive sets involving two parameters are obtained, respectively. Meanwhile, the volumes of the ultimate bound set and global exponential attractive set are obtained, respectively. Numerical simulations are conducted which validates the correctness of the proposed theoretical analysis. </abstract>

A Symmetric Controllable Hyperchaotic Hidden Attractor

By introducing a simple feedback, a hyperchaotic hidden attractor is found in the newly proposed Lorenz-like chaotic system. Some variables of the equilibria-free system can be controlled in amplitude and offset by an independent knob. A circuit experiment based on Multisim is consistent with the theoretic analysis and numerical simulation.

Hyperchaotic Circuit Based on Memristor Feedback with Multistability and Symmetries

A novel hyperchaotic circuit is proposed by introducing a memristor feedback in a simple Lorenz-like chaotic system. Dynamic analysis shows that it has infinite equilibrium points and multistability. Additionally, the symmetrical coexistent attractors are investigated. Further, the hyperchaotic system is implemented by analogue circuits. Corresponding experimental results are completely consistent with the theoretical analysis.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

A Novel Encryption Method for Dorsal Hand Vein Images on a Microcomputer

In this paper, a Lorenz-like chaotic system was developed to encrypt the dorsal hand patterns on a microcomputer. First, the dorsal hand vein images were taken from the subjects via an infrared camera. These were subjected to two different processes called contrast enhancement and segmentation of vein regions. Second, the pre- and post-processed images were encrypted with a new encryption algorithm in the microcomputer environment. For the encryption process, random numbers were generated by the chaotic system. These random numbers were subjected to NIST-800-22 test which is the most widely accepted statistical test suite. The speeded up robust feature (SURF) matching algorithm was utilized in the initial condition sensitivity analysis of the encrypted images. The results of the analysis have shown that the proposed encryption algorithm can be used in identification and verification systems. The encrypted images were analyzed with histogram, correlation, entropy, pixel change rate (NPCR), initial condition sensitivity, data loss, and noise attacks which are frequently used for security analyses in the literature. In addition, the images were analyzed after noise attacks by means of peak signal-to-noise ratio (PSNR), mean square error (MSE), and the structural similarity index (SSIM) tests. It has been shown that the dorsal hand vein images can be used in identification systems safely with the help of the proposed method on microcomputers.

Efficient image encryption method based on improved Lorenz chaotic system

A new image encryption method based on permutation–diffusion architecture is proposed. Permutation and diffusion operations are performed row-wise and column-wise on input image. To permute and diffuse the pixels, Lorenz-like chaotic system with varying bifurcation parameter is used to generate six random sequences. Initially, three sequences help in change the pixel’s position and next three used to diffuse the pixels. The main benefit of this method is that it is more sensitive to input image. Since input parameters required for secure key generation depend upon the input image.

Analysis, circuit implementation and applications of a novel chaotic system

PurposeLack of optimization and improvement on experimental circuits precludes comprehensive statements. It is a deficiency of the existing chaotic circuit technology. One of the aims of this paper is to solve the above mentioned problems. Another purpose of this paper is to construct a 10 + 4-type chaotic secure communication circuit based on the proposed third-order 4 + 2-type circuit which can output chaotic phase portraits with high accuracy and high stability.Design/methodology/approachIn Section 2 of this paper, a novel third-order 4 + 2 chaotic circuit is constructed and a new third-order Lorenz-like chaotic system is proposed based on the 4 + 2 circuit. Then some simulations are presented to verify that the proposed system is chaotic by using Multisim software. In Section 3, a fourth-order chaotic circuit is proposed on the basis of the third-order 4 + 2 chaotic circuit. In Section 4, the circuit design method of this paper is applied to chaotic synchronization and secure communication. A new 10 + 4-type chaotic secure communication circuit is proposed based on the novel third-order 4 + 2 circuit. In Section 5, the proposed third-order 4 + 2 chaotic circuit and the fourth-order chaotic circuit are implemented in an analog electronic circuit. The analog circuit implementation results match the Multisim results.FindingsThe simulation results show that the proposed fourth-order chaotic circuit can output six phase portraits, and it can output a stable fourth-order double-vortex chaotic signal. A new 10 + 4-type chaotic secure communication circuit is proposed based on the novel third-order 4 + 2 circuit. The scheme has the advantages of clear thinking, efficient and high practicability. The experimental results show that the precision is improved by 2-3 orders of magnitude. Signal-to-noise ratio meets the requirements of engineering design. It provides certain theoretical and technical bases for the realization of a large-scale integrated circuit with a memristor. The proposed circuit design method can also be used in other chaotic systems.Originality/valueIn this paper, a novel third-order 4 + 2 chaotic circuit is constructed and a new chaotic system is proposed on the basis of the 4 + 2 chaotic circuit for the first time. Some simulations are presented to verify its chaotic characteristics by Multisim. Then the novel third-order 4 + 2 chaotic circuit is applied to construct a fourth-order chaotic circuit. Simulation results verify the existence of the new fourth-order chaotic system. Moreover, a new 10 + 4-type chaotic secure communication circuit is proposed based on chaotic synchronization of the novel third-order 4 + 2 circuit. To illustrate the effectiveness of the proposed scheme, the intensity limit and stability of the transmitted signal, the characteristic of broadband and the requirements for accuracy of electronic components are presented by Multisim simulation. Finally, the proposed third-order 4 + 2 chaotic circuit and the fourth-order chaotic circuit are implemented through an analog electronic circuit, which are characterized by their high accuracy and good robustness. The analog circuit implementation results match the Multisim results.

Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma

In this paper, a new Lorenz-like chaotic system describing the interaction of three resonantly coupled waves in plasma is studied. Explicit ultimate boundedness and global attraction domain are derived according to stability theory of dynamical systems. The innovation of the paper is 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 for this system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained. Finally, numerical localization of attractor is presented.

Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable whenα∈(0,α0)and unstable whenα∈(α0,∞); Hopf bifurcation occurs whenαcrosses a critical valueα0by choosingαas a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regardingαas a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter valueα.

The Design and Its Application in Secure Communication and Image Encryption of a New Lorenz-Like System with Varying Parameter

A new Lorenz-like chaotic system with varying parameter is proposed by adding a state feedback function. The structure of the new designed system is simple and has more complex dynamic behaviors. The chaos behavior of the new system is studied by theoretical analysis and numerical simulation. And the bifurcation diagram shows a chaos-cycle-chaos evolution when the new parameter changes. Then a new synchronization scheme by a single state variable drive is given based on the new system and a chaotic parameter modulation digital secure communication system is also constructed. The results of simulation demonstrate that the new proposed system could be well applied in secure communication. Otherwise, based on the new system, the encryption and decryption of image could be achieved also.

Simplified hyper-chaotic systems generating multi-wing non-equilibrium attractors

Abstract By a simple feedback control, a new hyper-chaotic system was constructed based on a three-dimensional Lorenz-like chaotic system. This hyper-chaotic system can show butterfly shape (two-wing) strange attractor. The remarkable feature of this system is that it has no equilibrium. By Lyapunov exponents, bifurcation diagram and Poincare maps, the dynamical behaviors of the proposed system were analyzed; and this system was also realized by an electronic circuit. To extend this two-wing non-equilibrium hyper-chaotic system to multi-wing chaotic systems, a piecewise-linear (PWL) function was designed and applied to the proposed hyper-chaotic system, which can produce four-wing, six-wing, eight-wing and ten-wing chaotic attractors. Several experimental circuits were also designed to show these multi-wing chaotic attractors.

Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system

We present a global dynamical analysis of the following quadratic differential system $$\dot{x}{=}a(y{-}x), \dot{y}\!=\!dy-xz, \dot{z}\!=\!-bz+fx^2+gxy$$ , where $$(x,y,z)\in {\mathbb {R}}^3$$ are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in $${\mathbb {R}}^3$$ , alike Lorenz, Rossler, Chen and other. By using the Poincare compactification for a polynomial vector field in $${\mathbb {R}}^3$$ , we study the dynamics of this system on the Poincare ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.

Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos

This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively.

SINGLE BACKSTEPPING CONTROL INPUT OF A NOVEL 3D AUTONOMOUS CHAOTIC SYSTEM

In this paper, we have proposed a novel three-dimensional Lorenz-like chaotic system. Some basic properties of the system, such as dynamical behaviors, bifurcation diagram. Lyapunov exponents and Poincare mapping are investigated either analytically or numerically. Furthermore, the control problem of the new chaotic system was studied via nonlinear backstepping method. The single backstepping control input was designed according to Lyapunov stability criterion. Numerical simulations are carried out in order to demonstrate the effectiveness of the proposed control design.

On the boundedness of solutions to the Lorenz-like family of chaotic systems

This paper deals with a class of three-dimensional autonomous nonlinear systems which have potential applications in secure communications, and investigates the localization problem of compact invariant sets of a class of Lorenz-like chaotic systems which contain T system with the help of iterative theorem and Lyapunov function theorem. Since the Lorenz-like chaotic system does not have y in the second equation, the approach used to the Lorenz system cannot be applied to the Lorenz-like chaotic system. We overcome this difficulty by introducing a cross term and get an interesting result, which includes the most interesting case of the chaotic attractor of the Lorenz-like systems. Furthermore, the results obtained in this paper are applied to study complete chaos synchronization. Finally, numerical simulations show the effectiveness of the proposed scheme.

Dynamical properties and simulation of a new Lorenz-like chaotic system

This paper formulates a new three-dimensional chaotic system that originates from the Lorenz system, which is different from the known Lorenz system, Rossler system, Chen system, and includes Lu systems as its special case. By using the center manifold theorem, the stability character of its non-hyperbolic equilibria is obtained. The Hopf bifurcation and the degenerate pitchfork bifurcation, the local character of stable manifold and unstable manifold, are also in detail shown when the parameters of this system vary in the space of parameters. Corresponding bifurcation cases are illustrated by numerical simulations, too. The existence or non-existence of homoclinic and heteroclinic orbits of this system is also studied by both rigorous theoretical analysis and numerical simulation.