Lu Chaotic System Research Articles

Performance Analysis of Chaotic Neural Network and Chaotic Cat Map Based Image Encryption

Nowadays, chaotic systems are used quite often, especially in image encryption applications. Hypersensitivity to the initial conditions, limited field-changing signs and irregular movements make these systems one of the critical elements in scientific matters such as cryptography. These systems are divided in two parts as discrete time and continuous time in terms of their dimensions and properties. Gray level image encryption applications generally use one-dimensional and color image encryption applications generally use multi-dimensional chaotic systems. In this study, Tent Map, Cat Map, Lorenz, Chua, Lu chaotic systems were used for chaotic neural network based image coding application and Logistic Map and 3D Cat Map chaotic systems were used for 3D chaotic Cat Map based image encryption application. The encrypted image and the original image were examined by various analysis methods. As a result of these analyzes, it was seen that both applications gave very successful results. Analyzes have shown that the chaotic neural network based image coding algorithm is more secure and successful.

Research on weak fault signal detection technology based on lv system

The detection of weak fault signals is realized by applying non-resonant parametric excitation to Lu chaotic system. Periodic signals far larger than the characteristic frequency of the system are taken as input, the sharp change of the system attractor is observed, and the existence of weak signals is detected. The conditions that the parameters of the detection system should meet are deduced by using the averaging method and Lyapunov method. Numerical research shows that the detection system can quickly reach a stable state. When the amplitude of the detected weak signals changes slightly, the attractor state changes abruptly, and the system is sensitive to the input. Through the "butterfly effect" of the system, feature signals that are difficult to extract by traditional methods can be detected. And the system has strong anti-interference performance.

Stability Analysis, Control of Simple Chaotic System and its Hybrid Projective Synchronization with Fractional Lu System

In this article, the stability analysis and chaos control of Simple chaotic system have been discussed. A new lemma based on Caputo derivative is used during the stability analysis of the fractional order Simple chaotic system through Lyapunov stability theory. During hybrid projective synchronization the Simple chaotic system is considered as drive system and Lu chaotic system is taken as response system. Nonlinear control method has been used to analyse the hybrid projective synchronization of fractional systems. For numerical simulations Adams-Bashforth-Moulton method has been used and results obtained are presented graphically.

Image encryption algorithm using S-box and dynamic Hénon bit level permutation

For the secure transmission of data through the medium of internet, images have significant importance. Image encryption provides secure transmission of images by converting recognizable form of image into an unrecognizable form. Chaos is considered as a natural required ingredient for cryptography applications, by providing unpredictability, sensitivity of initial state and erogodicity. Therefore from the last decade, a number of chaos-based cryptosystems have been developed for the protection of transmitted images’ content. In this paper, a chaos based algorithm is developed and experimented on six different standard empirical images. The proposed cryptosystem is based on substitution-permutation network (SPN) with cipher block chaining (CBC) mode of operation. A novel algorithm is proposed for the construction of substitution box by using chaotic sine map, which is applied on a block-input of bytes, followed by a permutation based on discretized Henon map, which is applied on a block-input of bits instead of bytes. The hyper chaotic Lu system, which is nonlinear and produces discrete values with long orbits, is used as pseudorandom generator to set new values to control parameters of discretized Henon map for bit-permutation for each block. Moreover, proposed bit-permutation is applied by a matrix formulation which accelerates the bit permutation process for a block-input. Security analysis and results obtained from simulations show that cryptosystem is good resistant to various well-known attacks and have good key space therefore is reliable for secure transmission of images.

Parameter estimation and synchronization of hyper chaotic Lu system with disturbance input and uncertainty using two under-actuated control signals

In this paper, synchronization of two hyper chaotic Lu systems with unknown factors, such as uncertainty and disturbance, and less control signals with respect to system’s dimensions by applying an adaptive sliding mode controller is presented. In this research, firstly, an integral sliding mode controller is designed for synchronization of two hyper chaotic Lu systems with known parameters. Secondly, a controller is used to synchronize two hyper chaotic Lu systems with unknown parameters; in this case, unknown parameters are estimated using an adaptive rule. As a measure of performance, stability of the proposed control method is proved using the Lyapunov stability method. Numerical simulations validate the stabilization and synchronization.

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.

Multiswitching combination synchronisation of non-identical fractional-order chaotic systems

In this paper, multiswitching combination synchronisation (MSCS) scheme has been investigated in a class of three non-identical fractional-order chaotic systems. The fractional-order Lorenz and Chen systems are taken as the drive systems. The combination of multidrive systems is then synchronised with the fractional-order Lu chaotic system. In MSCS, the state variables of the two drive systems synchronise with different state variables of the response system, simultaneously. Based on the stability of fractional-order chaotic systems, the MSCS of three fractional-order non-identical systems has been investigated. For the synchronisation of three non-identical fractional-order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method.

Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems

In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.

The observer-based synchronization and parameter estimation of a class of chaotic system via a single output

Observer-based synchronization and parameter estimation of chaotic systems has been an interesting and important issue in theory and various fields of application. In this paper first we investigate the observer-based synchronization of a class of chaotic systems, and then discuss its parameter estimation via a single output. We assume that only the sum of the first and second state variables is available. By constructing a proper observer, some novel criteria for observer-based synchronization and parameter estimation are proposed via a scalar input. The Lu chaotic system is taken as an example to demonstrate the efficiency of the proposed approach.

Hybrid adaptive function projective synchronization of a hyper Lüchaotic system

In this paper, a hybrid adaptive function projective synchronization of a hyper Lu chaotic system is investigated. Firstly, according to the Lyapunov stability theory and Pecora-Carroll synchronization criterion, the controller and adaptive rate of these two kinds of synchronization methods are analyzed and designed. Then a new hybrid adaptive projective synchronization method is proposed. Finally, in order to prove the effectiveness of the proposed scheme, the comparison between those controllers is studied by simulation example.

Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems

A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS), is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.

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.

Impulsive synchronization of Lü chaotic systems via the hybrid controller

Abstract In this paper, the impulsive synchronization problem of Lu chaotic systems is investigated by using a novel hybrid controller. The proposed hybrid controller is composed of a single controller and impulsive controller. Using the impulsive theory and the novel hybrid controller, some sufficient conditions are derived for the synchronization of Lu chaotic systems. Numerical simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed control scheme has a fast convergence rate compared with the conventional single controller method and impulsive controller method.

Singularity‐conquering tracking control of a class of chaotic systems using Zhang‐gradient dynamics

This study investigates the tracking-control problems of the Lorenz, Chen and Lu chaotic systems. Note that the input–output linearisation method cannot solve these tracking-control problems because of the existence of singularities, at which such chaotic systems fail to have a well-defined relative degree. By combining Zhang dynamics and gradient dynamics, an effective controller-design method, termed Zhang-gradient (ZG) method, is proposed for tracking control of the three chaotic systems. This ZG method, with singularities conquered, is capable of solving the tracking-control problems of the chaotic systems. Both theoretical analyses and simulative verifications substantiate that the tracking controllers based on the ZG method can achieve satisfactory tracking accuracy and successfully conquer singularities encountered during the tracking-control process.

Adaptive synchronization of uncertain unified chaotic systems via novel feedback controls

This paper is concerned with the robust adaptive synchronization problem for unified chaotic systems with parameter variation, system uncertainties, and external disturbances. By combining adaptive method and positive time-varying integral functions, novel adaptive synchronization schemes are proposed. It is shown that the synchronization errors between the drive and response chaotic systems converge to zero asymptotically. Finally, illustrative examples about Lorenz chaotic system, Chen chaotic system, and Lu chaotic system are provided to demonstrate the effectiveness and applicability of the proposed design method.

Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control

This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily.

Stabilizations of Strange Attractors by Feedback Linearization

This study deals with the control of the Lorenz, Chen and Lu chaotic system. Feedback linearization was successfully implemented on these chaotic systems. Feedback linearization was successful to transform the three attractor systems to a general system that simplify the linear control analysis. Hence, a linear controller is designed for the feedback linearized general system. Furthermore, some numerical simulations were carried out for the closed loop systems. These simulations show that the developed controller design method is effective in stabilizing and regulating the response.

Hybrid Synchronization of Lu and Bhalekar-Gejji Chaotic Systems Using Nonlinear Active Control

Abstract Hybrid synchronization based on the active control theory between two different chaotic systems, i.e. Lu and Bhalekar-Gejji is reported in this paper. Lu chaotic system is taken as master and Bhalekar-Gejji chaotic system as slave. Stabilization of error dynamics is achieved by satisfying the Lyapunov stability conditions. Control is designed by using the relevant variables of master and slave systems. Simulation is presented for verification of proposed scheme.

Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points

Based on three-dimensional (3D) Lu chaotic system, we introduce a four-dimensional (4D) nonlinear system with infinitely many equilibrium points. The Lyapunov-exponent spectrum is obtained for the 4D chaotic system. A hyperchaotic attractor and a chaotic attractor are emerged in this 4D nonlinear system. Furthermore, to verify the existence of hyperchaos, the chaotic dynamics of this 4D nonlinear system is also studied by means of topological horseshoe theory and numerical computation.

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lu chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen’s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen’s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.