Hyperchaotic Lorenz System Research Articles

Synchronisation of different order chaotic systems with delay and parametric uncertainty

The paper addresses synchronisation of two different order chaotic systems with time delay in response system using modified projective synchronisation. The uncertainty in the parameters of both drive and response systems is considered while designing adaptive controller and adaptive parameter updating laws, based on Lyapunov stability theory. Numerical simulations for two cases, i.e., synchronisation between hyper-chaotic Lorenz and time delayed chaotic Lu system and hyper-chaotic Lu and advanced Lorenz system with delayed dynamics, are presented to show the feasibility and effectiveness of the proposed method.

Memristor-based Lorenz hyper-chaotic system and its circuit implementation

To study the application of memristor in chaotic system, we employ the smooth continuous nonlinear flux-controlled memristor model and feedback control technique to design a hyperchaotic system based on the simplified Lorenz system. By using memristor as a positive feedback of the simplified Lorenz system, the dimensionless mathematical model is derived. The differences between the memristor-based chaotic system and ordinary chaotic system are then further studied. Firstly, the stable equilibrium and unstable equilibrium point sets of the system are analyzed theoretically, and it is found that the system has infinite equilibrium points including stable and unstable equilibrium points. The stable and unstable ranges of the system with different parameters are also determined. Theoretical analysis shows that the system has the same symmetry as the simplified Lorenz system. Thus the system has rich dynamical behaviors, such as limit cycle, chaotic attractor, and hyper-chaotic attractor. Secondly, by the methods of bifurcation diagram, Lyapunov exponent spectrum, Poincar section, and Spectral Entropy algorithm, the dynamical behaviors of the system are analyzed in detail. By calculating the Lyapunov exponent spectrum, the dynamical behaviors are studied and they change with system parameters and the initial conditions of memristor respectively. The maximum positive Lyapunov exponent of the memristor-based Lorenz hyperchaotic system is higher than that of the simplified Lorenz system, which indicates the memristor-based Lorenz hyperchaotic system is more complex. Further, we find all the complex dynamical behaviors to be coexisting with the infinite equilibrium sets, which is quite different from those of many ordinary hyper-chaotic systems. Meanwhile, we observe the attractors coexisting and state transition phenomenon in this system, caused by changing the initial conditions of the memristor. State transition phenomenon is then further studied by means of phase portraits and spectral entropy algorithm for the first time. Finally, by using operational amplifiers, diodes and other discrete components, we design an equivalent circuit of the smooth continuous nonlinear flux-controlled memristor model, and the equivalent circuit is used to design and realize the analog electronic circuit of the memristor-based Lorenz hyper-chaotic system. By using an analog oscilloscope, the phase portraits of hyper-chaotic attractor are observed clearly. The state transition phenomenon can also be seen using the oscilloscope. It is found that the circuit experimental results are in agreement with those of the theoretical analysis and numerical simulation. It verifies that the system is physically realizable, and lays a strong foundation for its applications in engineering. Next, we will try to investigate the chaotic secure communication based on this hyper-chaotic system.

Cooperative output regulation for a class of nonlinear multi-agent systems with arbitrarily large uncertainty

This paper studies the cooperative global robust output regulation problem for the nonlinear multi-agent systems in output feedback form with relative degree $r=2$ subject to arbitrarily large uncertainty. The robust control method alone cannot handle arbitrarily large uncertainty. By combining the robust and adaptive high gain technique, we develop a recursive procedure to construct a distributed output feedback control law to deal with the problem. The theoretical result of this paper is applied to the leader-following consensus problem of a family of hyper-chaotic Lorenz systems.

Finite-Time Synchronization of Chaotic Systems with Different Dimension and Secure Communication

Finite-time synchronization of chaotic systems with different dimension and secure communication is investigated. It is rigorously proven that global finite-time synchronization can be achieved between three-dimension Lorenz chaotic system and four-dimension Lorenz hyperchaotic system which have certain parameters or uncertain parameters. The electronic circuits of finite-time synchronization using Multisim 12 are designed to verify our conclusion. And the application to the secure communications is also analyzed and discussed.

Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System

The fractional-order hyperchaotic Lorenz system is solved as a discrete map by applying the Adomian decomposition method (ADM). Lyapunov Characteristic Exponents (LCEs) of this system are calculated according to this deduced discrete map. Complexity of this system versus parameters are analyzed by LCEs, bifurcation diagrams, phase portraits, complexity algorithms. Results show that this system has rich dynamical behaviors. Chaos and hyperchaos can be generated by decreasing fractional order q in this system. It also shows that the system is more complex when q takes smaller values. SE and C 0 complexity algorithms provide a parameter choice criteria for practice applications of fractional-order chaotic systems. The fractional-order system is implemented by digital signal processor (DSP), and a pseudo-random bit generator is designed based on the implemented system, which passes the NIST test successfully.

Solution and dynamics analysis of a fractional‐order hyperchaotic system

Numerical solution and chaotic behaviors of the fractional‐order simplified Lorenz hyperchaotic system are investigated in this paper. The solution of the fractional‐order hyperchaotic system is obtained by employing Adomian decomposition method. Lyapunov characteristic exponents algorithm for the fractional‐order chaotic system is designed. Dynamics of the fractional‐order hyperchaotic system are analyzed by means of bifurcation diagrams, Lyapunov characteristic exponents, C0 complexity, and chaos diagram. It shows that this system has rich dynamical behaviors, and it is more complex when the fractional order q is small. It lays a foundation for the practical application of the fractional‐order hyperchaotic systems. Copyright © 2015 John Wiley & Sons, Ltd.

A novel image encryption algorithm based on genetic recombination and hyper-chaotic systems

In this paper, a novel image encryption algorithm based on genetic recombination and hyper-chaotic system is proposed. The basic rules of genetic recombination are employed to scramble images because of its effectiveness. Specifically, the plain image is expanded into two compound images composed of selected four bit-planes and diffuse them at bit-plane level, the compound bit-planes and key streams are reconstructed based on the principles of genetic recombination, then perform traditional diffusion and obtain cipher images. The hyper-chaotic Lorenz system in this algorithm generates pseudorandom sequences in each phase. The experiment results and analysis have proved that the novel image encryption algorithm is effective for image encryption.

Adaptive Hybrid Function Projective Synchronization of General Chaotic Complex Systems With Different Orders

A lot of progress has been made in the research of hybrid function projective synchronization (HFPS) for chaotic real nonlinear systems, while the HFPS of two different chaotic complex nonlinear systems with nonidentical dimensions is seldom reported in the literatures. So this paper discusses the HFPS of general chaotic complex system described by a unified mathematical expression with different dimensions and fully unknown parameters. Based on the Lyapunov stability theory, the adaptive controller is designed to synchronize two general uncertain chaotic complex systems with different orders in the sense of HFPS and the parameter update laws for estimating unknown parameters of chaotic complex systems are also given. Moreover, the control coefficients can be automatically adapted to updated laws. Finally, the HFPS between hyperchaotic complex Lorenz system and complex Chen system and that between chaotic complex Lorenz system and hyperchaotic complex Lü are taken as two examples to demonstrate the effectiveness and feasibility of the proposed HFPS scheme.

Construction of Dynamical Non-linear Components Based on Lorenz System and Symmetric Group of Permutations

This paper is devoted to presenting a new technique for generating S-boxes based on a Lorenz system and S8 group of permutations. Within the algorithm, the initial condition is employed to drive the hyperchaotic Lorenz system to generate a chaotic sequence which is used to construct S-boxes. With different system conditions, many of distinct S-boxes can be obtained dynamically. Some cryptographic properties for a good S-box such as bijection, nonlinearity, SAC, BIC and differential approximation probability are found to hold in the obtained S-boxes from the proposed method. The analytic results indicate that all the criteria for designing robust S-boxes can be achieved. The comparison of the proposed S-box generating method with other chaos-based schemes indicates that our S-boxes have a better performance with respect to some properties. Classification of information security techniques and its applications

BIFURCATIONS AND SYNCHRONIZATION OF THE FRACTIONAL-ORDER SIMPLIFIED LORENZ HYPERCHAOTIC SYSTEM

In this paper, dynamics of the fractional-order simplied Lorenz hyperchaotic system is investigated. Modied Adams-Bashforth-Moulton method is applied for numerical simulation. Chaotic regions and periodic windows are identied. Dierent types of motions are shown along the routes to chaos by means of phase portraits, bifurcation diagrams, and the largest Lyapunov exponent. The lowest fractional order to generate chaos is 3.8584. Synchronization between two fractional-order simplied Lorenz hyperchaotic systems is achieved by using active control method. The synchronization performances are studied by changing the fractional order, eigenvalues and eigenvalue standard deviation of the error system.

Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation

Abstract In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system (Jia, 2007) with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007) has the Lyapunov exponents L1 = 0.3684,L2 = 0.2174,L3 = 0 and L4 =−12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as DKY =3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L1 = 0.4195,L2 = 0.2430,L3 = 0.0145,L4 = 0 and L5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as DKY =4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE), which is greater than the maximal Lyapunov exponent (MLE) of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.

A Chaotic System Based Image Encryption Algorithm using Plaintext-related Confusion

A new plaintext-related image encryption system based on the hyper-chaotic Lorenz system is proposed in this paper. In the proposed image encryption system, the hyper-chaotic Lorenz system is employed to generate six pseudo-random matrixes, where, two of them use the “add-modulus” operations to carry out the plaintext-unrelated image diffusion, and the other four matrixes are used to confuse the plaintext-related image. In the image confusion, each pixel will swap its location with another pixel, and the target location is determined by some elements in the plain image and the four matrixes. The proposed image encryption system can resist the chosen/known plaintext attacks due to the application of the plaintext-related image confusion in it. The simulation results show that the proposed encryption system also have the characters of fast encryption/decryption speed, large key space, strong key sensitivity, strong plaintext sensitivity, good statistical properties of cipher images, and large information entropy, etc. Thus, the proposed system can be used in practical communications.

Increased-order generalized synchronization of chaotic and hyperchaotic systems

This paper presents increased-order generalized synchronization (GS) of chaotic and hyperchaotic systems with different order based on active control technique. By this technique, we design suitable control functions to achieve GS between (i) a new three-dimensional (3D) chaotic system and four-dimensional (4D) hyperchaotic Lorenz system and (ii) four-dimensional hyperchaotic Lorenz system and five-dimensional (5D) hyperchaotic Lorenz system. The corresponding numerical simulation results are presented to verify the effectiveness of this technique.

Comments on “Color image encryption using Choquet fuzzy integral and hyper chaotic system”

An image encryption system whose cipher code stream only controlled by the secret key, but has nothing to do with the plaintext, is vulnerable to chosen plaintext attacks. Recently, an image encryption scheme using Choquet fuzzy integral and hyper chaotic Lorenz system has been proposed Liu et al. (2013), which employed plaintext-independent cipher code stream, then should be subjected to chosen plaintext attack. This paper cryptanalyzed the aforesaid encryption scheme using chosen plaintext attack, and pointed out that even if possessing good pseudorandom cipher code generation method, the encryption system is still insecure with unreasonable designed encryption scheme.

Generalized Projective Synchronization of Diverse Structures Hyperchaotic Systems with Unknown Parameters

In this paper, the generalized projective synchronization for a general class of hyperchaotic systems is investigated. A systematic, powerful and concrete scheme is developed to investigate the generalized projective synchronization between the drive system and response system based on the feedback control approach. The hyperchaotic Chen system and hyperchaotic Lorenz system are chosen to illustrate the proposed scheme. Numerical simulations are provided to show the effectiveness of the proposed schemes.

Adaptive modified synchronization of hyperchaotic systems with fully unknown parameters

In this paper, we demonstrate that the adaptive hybrid synchronization behavior can coexist in two identical and different hyperchaotic systems with terms of parametric uncertainty. By using rigorous mathematical theory, the controller is designed based on Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. The adaptive hybrid synchronization between two identical systems (hyperchaotic Lu system) and different systems (hyperchaotic Lorenz and hyperchaotic Chen systems) are taken as two illustrative examples to show the effectiveness of the proposed method. Theoretical analysis and numerical simulations are shown to verify the results.

Control and Synchronization of Chaotic and Hyperchaotic Lorenz Systems via Extended Backstepping Techniques

We propose novel controllers for stabilization and tracking of chaotic and hyperchaotic Lorenz systems using extended backstepping techniques. Based on the proposed approach, generalized weighted controllers were designed to control chaotic behaviour as well as to achieve synchronization in chaotic and hyperchaotic Lorenz systems. The effectiveness and feasibility of the proposed weighted controllers were verified numerically and showed their robustness against noise.

Algorithms for Lorenz System Manifold Computation

A new algorithm is presented to compute both one dimensional stable and unstable manifolds of planar maps. It is proved that the gradient of the global manifold can be predicted by the known points on the manifold with a gradient prediction scheme and it can be used to locate the image or preimage of the new point quickly. The performance of the algorithm is demonstrated with hyper chaotic Lorenz system. DOI : http://dx.doi.org/10.11591/telkomnika.v12i4.4740

Finite-Time Combination-Combination Synchronization of Hyperchaotic Systems and its Application in Secure Communication

Global finite time synchronization of a class of combination-combination chaotic systems via master-slave coupling is investigated. A nonlinear feedback controller and a continuous generalized linear state-error feedback controller with simple structure are introduced in the synchronization scheme. They are applied to a practical master-slave synchronization scheme for combination-combination systems, which consist of the Chen chaotic system, hyperchaotic Chen system and hyperchaotic Lorenz system. Numerical simulations are provided to illustrate the effectiveness of the new synchronization criteria.

Synchronizing chaotic systems based on an arbitrary function matrix

In this paper, we introduce a type of chaotic synchronization, where two chaotic systems are synchronized based on a function matrix. In particular, each drive system state synchronizes with a certain combination of response system states. Based on linear system theory and the pole placement technique, the scheme is given and illustrated with hyperchaotic Lorenz system and Lorenz chaotic system. Numerical simulations are carried out to verify the effectiveness of the proposed scheme.