Liu System Research Articles

Computations of synchronisation conditions in some fractional-order chaotic and hyperchaotic systems

A novel criterion for achieving synchronisation in fractional-order chaotic and hyperchaotic systems is presented. Here, it is proved that the existence of a Lyapunov function in the integer-order differential system implies local stability of the steady state in its fractional-order counterpart. So, our criterion is based on computations of suitable linear feedback controllers of the fractional-order systems according to an appropriate choice of Lyapunov function. Furthermore, a new fractional-order hyperchaotic system is introduced here. The case of hyperchaos in the proposed system is verified by computing its greatest two Lyapunov exponents which are shown to be positive. The new synchronisation criterion is successfully applied to the fractional Liu system, the fractional Samardzija–Greller system, the fractional financial system and a novel fractional-order hyperchaotic system. Numerical results are used to verify the analytical results.

Chaotic characteristics analysis of fractional-order Liu system

Chaotic characteristics analysis of fractional-order Liu system

Topological Horseshoe Analysis and FPGA Implementation of a Classical Fractional Order Chaotic System

Scholars have done extensive research on dynamic analysis and analog circuits implementation of the classical fractional order chaotic system-Liu System (FOLS). However, they did not rigorously prove the existence of FOLS from the perspective of mathematics. And they also did not effectively design digital circuits to generate signals of the fractional order chaotic systems, especially the 2.7-order system. This paper selects an appropriate Poincare section where a first return Poincare map of FOLS was defined. Based on computer-assisted verification method, the conclusion is that the Poincare map is semi-conjugate to a 2-shift map and the topological entropy of the map is no less than ln 2, which rigorously verifies the existence of chaotic behavior in the 2.7-order Liu system. This proof is necessary before the chaotic system is used for information encryption. The next and most significant task is to build a system model through DSP-Builder software and generate chaotic signals using Field Programmable Gate Array chip. The results of oscilloscope consistent with numerical simulations, which lays the foundation for image and video streaming encryption.

Dynamic analysis and synchronization of conformable fractional-order chaotic systems

In this paper, chaotic systems based on the conformable fractional-order derivative are investigated. First, some comments on the discrete Conformable Adomian decomposition method (discrete CADM) proposed recently are made. We demonstrate that the discrete CADM is not globally accurate for conformable fractional-order differential equations and some conclusions about conformable fractional-order chaotic systems in present papers based on the discrete CADM are inappropriate. Second, we restudy the conformable fractional-order simplified Lorenz system. And then, the conformable fractional-order Liu system is introduced and studied. The basic dynamical behaviors of the systems are analyzed in detail, such as dissipation, the equilibrium points and their stability, Lyapunov exponents, bifurcation diagrams, phase portraits and time series of states. Some interesting dynamic behaviors different from the Caputo or Riemann-Liouville fractional-order chaotic system are discovered. It is found that the system order of the commensurate case for chaos is $q \in (0,1]$ . However, for the incommensurate case, dissipation and stability of equilibrium points depend on system order qi and time t which may cause chaotic transients or period-doubling oscillation. In addition, the dynamic behaviors of conformable fractional-order chaotic systems will be static as $t \rightarrow + \infty$ which is named “dynamic death”. Finally, synchronization between conformable fractional-order simplified Lorenz system and conformable fractional-order Liu system is realized.

Difference Synchronization of Identical and Nonidentical Chaotic and Hyperchaotic Systems of Different Orders Using Active Backstepping Design

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

Multi-switching compound synchronization of four different chaotic systems via active backstepping method

This manuscript presents a theoretical and numerical analysis to achieve compound synchronization of four non-identical chaotic systems for different multi-switching states. Multi-switching compound synchronization is achieved for three drive systems and one response system via active backstepping technique. By using Lyapunov stability theory, asymptotically stable synchronization states are established. To elaborate the considered scheme an example of Pehlivan system, Liu system, Qi system and Lu system is discussed. The conclusions drawn from computational and analytical approaches are in excellent agreement.

Modified function projective feedback control for time-delay chaotic Liu system synchronization and its application to secure image transmission

This paper is devoted to synchronize time-delay Liu chaotic system in order to develop a secure image transmission system. An adaptive modified function projective synchronization (MFPS) method is proposed to control the behavior of the uncertain Liu chaotic system in the presence of time-delay state variables and uncertainty in system parameters. The stability and convergence of the present method are analyzed by means of Lyapunov stability theorem. Numerical simulations were used to validate the effectiveness of the proposed MFPS method. Furthermore, a novel image cryptosystem is generated for secure image transmission. Image data is embedded within the response signals and is transmitted to the receiver. The image data can be extracted at the receiver by using of the drive signals. The proposed image cryptosystem was applied to a plain image, and also, performance of the image cryptosystem was evaluated by some mathematical tools and the results were satisfactory, in both cases.

On disappearance of chaos in fractional systems

In a seminal paper, Grigorenko and Grigorenko [15], numerically studied fractional order dynamical systems (FODS) of the form Dαixi=fi(x1,x2,x3),0≤αi≤1,(i=1,2,3); and showed the existence of chaos in case of fractional Lorenz system when Σ=α1+α2+α3≤3. Since then voluminous numerical work has been done to explore various FODS, in this regard. It is now an established fact that Σ acts as a chaos controlling parameter.In the present article we take a survey of present literature on chaotic behavior of fractional order dynamical systems. Further we numerically explore fractional Chen, Rössler, Bhalekar–Gejji, Lorenz and Liu systems and observe that chaos always disappear if Σ ≤ 2. The existing examples in the literature along with the systems that we have analyzed lead us to conjecture non-existence of chaos if Σ ≤ 2; which in some sense is a generalization of classical Poincaré–Bendixon theorem for FODS.

FPGA implementation of two fractional order chaotic systems

This paper discusses the FPGA implementation of the fractional-order derivative as well as two fractional-order chaotic systems where one of them has controllable multi-scroll attractors. The complete hardware architecture of the Grünwald-Letnikov (GL) differ-integral is realized with different memory window sizes. As an application of the proposed circuit, a complete fractional-order FPGA implementation of Liu chaotic system is introduced with different fractional-orders. Moreover, a fractional-order controllable heart and V-shape multi-scrolls chaotic systems are verified in the case of symmetric and asymmetric cases. Different interesting attractors are realized under various parametric changes with distinct step sizes for different fractional-orders. To verify the chaotic behavior of many generating attractors, the Maximum Lyapunov Exponent (MLE) is calculated for such systems. The designs have been simulated using Xilinx ISE 14.5 and realized on Xilinx FPGA Virtex 5 XC5VLX50T. The achieved throughputs are: 4.4Gbit/s for GL, 1.986Gbit/s for Liu system, and 2.921Gbit/s for V-Shape multi-scroll attractor.

Robust Synchronisation of Uncertain Fractional-Order Chaotic Unified Systems

Abstract Fractional-order chaotic unified systems include a variety of fractional-order chaotic systems such as Chen, Lorenz, Lu, Liu, and financial systems. This paper describes a sliding mode controller for synchronisation of fractional-order chaotic unified systems in the presence of uncertainties and external disturbances, and affirms the stability of the controller (which is composed of error dynamics). Moreover, the synchronisation of two separate fractional-order chaotic systems is studied. For this aim, fractional integral sliding surface is defined. Then the sliding mode control rule for stability of error dynamic is presented based on the Lyapunov stability theorem. Simulation results, obtained by using MATLAB, show that the proposed sliding mode has employed an appropriate approach against uncertainties and to reduce the chattering phenomenon that often occurs with sliding mode controllers.

Multi-switching synchronization between T system and Liu system

In this paper, multi-switching synchronization between two chaotic systems with fully unknown parameters has been studied. The proposed scheme presents a generalized way to achieve different synchronizations for different switching states of two chaotic systems. Investigations are accomplished by using adaptive control method and Lyapunov stability theory. To analyze the proposed methodology, example of chaotic T system and Liu system have been considered. Theoretical results are validated by numerical simulations.

Hybrid chaos synchronization between two different hyperchaotic systems via two approaches

This paper deals with the hybrid synchronization between two non-identical 4D hyperchaotic systems; Modified hyperchaotic Pan system and hyperchaotic Liu system via nonlinear control technique, based on two analytical approaches; Lyapunov’s second method and Gardano’s method. We compare between the two approaches and notice the Lyapunov stability theory is need to modify a quadratic Lyapunov function at some time, and furthermore, it’s failing in other cases. For this reason, this Letter is using a new analytical approach which doesn’t need any modified, what distinguishes this method and makes it better than the Lyapunov approach. Theoretical and numerical simulations are given to illustrate and verify the results.

Combined synchronization of time-delayed chaotic systems with uncertain parameters

The article presents combined synchronization among time delayed chaotic systems in the presence of uncertain parameters using nonlinear control method. Control functions are designed to achieve combined synchronization using Lyapunov-Krasovskii function for stability analysis. The synchronization among three and four time delayed chaotic systems have been shown as examples of combined synchronization. Double delay Rossler system, the advanced Lorenz system, time delay Liu and Chen systems have been taken to show the combined synchronization. Numerical simulation and graphical results are carried out using Runge–Kutta method for delay-differential equations, which show that the designing of control functions are very effective and reliable and can be applied for combined synchronization among time-delayed chaotic systems.

Chaos‐based fast colour image encryption scheme with true random number keys from environmental noise

The paper proposes a chaos-based colour image encryption scheme, the highlight is that the randomly sampled noise signal is applied to serve as the initial values of a chaotic system. The 256-bit hash value of noise is transformed into the one-time initial values of the Liu system. The sequences generated by Liu system are subjected to three batteries of TestU01. Exclusive OR, the only operation, is applied to diffuse the pixels, and some measures are taken to speed up the encryption process. Finally, some statistical tests are performed to assess reliability and efficiency of the proposed cryptosystem in terms of time complexity and security.

Three Important Phenomena of Chaos Synchronization Between Two Different Hyperchaotic Systems via Adaptive Control Method

This paper presents three important phenomena of chaos synchronization between two different hyperchaotic systems using nonlinear adaptive control strategy. In detailed, complete synchronization, anti- synchronization and hybrid synchronization with nine unknown parameters. Modified hyperchaotic Pan system is consider as drive and hyperchaotic Liu system as response system. Stabilization of error dynamics for each phenomenon is realized by satisfying Lyapunov's second method as a main tool. Theoretical analysis and numerical simulations are shown to verify the results.

Multimedia Cryptography based on Liu and Chen Systems

Information security is an important matter in communication systems such as internet communication, multimedia systems, medical imaging, and especially military communication. This paper presents two types of multimedia cryptography; text and image cryptography depending on using Chen and Liu systems. The two systems together are used to generate a secret key to encipher multimedia. First by changing the position of characters in original text and changing position of pixels in original image, then changing the character value itself and pixel value depending on random numbers generated by Liu and Chen systems .The experimental and simulation results show that the proposed system is effective and has a high security level. Due to its key space, high key sensitivity, high entropy , low correlation, strong against differential analysis and finally low time complexity since the  same set of numbers must be generated depending in the same key schedule to return original multimedia and this is impossible without knowing exact initial conditions, parameters at the same schedule. &nbsp

Stability analysis on a class of nonlinear fractional-order systems

The stability of a class of commensurate and incommensurate nonlinear fractional-order systems is studied. First, two comparison inequalities for incommensurate fractional-order systems are proposed. Based on that, a stability criterion regarding a class of incommensurate fractional-order system is given. And then, three stability criteria are presented concerning a typical class of commensurate nonlinear fractional-order systems. After that, two global stability criteria concerning commensurate and incommensurate nonlinear fractional-order systems are provided, respectively. Finally, the fractional-order Liu system and the fractional-order Chen system are taken as examples to show how to apply the proposed results to stabilization problems.

Controlling Liu Chaotic System with Feedback Method and Its Circuit Realization

In the paper, the Liu system with a feedback controller is discussed. The influence of the feedback coefficient of the controlled system is studied through Lyapunov exponents spectrum and bifurcation diagram. Various attractors are demonstrated not only by numerical simulations but also by circuit experiments. Only one feedback channel is used in our study, which is useful in communication. The circuit experiments show that our study has significance in practical applications.

Application of similar Liu system in underwater weak signal detection

Weak signal detection is a vital technology in underwater acoustic communication with strong noise background. In this area, non-autonomous Duffing system is still widely used, and a lot of researches focus on enhancing the ability to detect weak signal and to find out the detection limitation of the Duffing system. Moreover, great achievements have already made. But problems still exist such as non-convergence of the periodic state of the Duffing system and its narrow detection domain. Unfortunately, researches on weak signal detection by using other systems are still rare. In order to solve the above problems, a new three-dimensional similar Liu chaotic system for weak signal detection is proposed. A thorough theoretical analysis for the similar Liu chaotic system is given, and its equilibrium point and the Lyapunov index are deduced and analyzed in detail. The major conclusion is that the variable x of the new system becomes a deformation signal when the input signal amplitude is greater than a certain critical value, the variables y and z converge to zero, and the Lyapunov exponents are less than zero at the same time. This means that no matter how strong the input signal is, the detection can be achieved by using a similar Liu chaotic system as long as its amplitude exceeds the threshold value. The periodic convergence and wide area detection of the similar Liu chaotic system are proved by the Matlab simulation, the Multisim circuit simulation, and the actual circuit test. This new system solves the two problems of the period convergence and narrow detection domain for the traditional Duffing system. The periodic state and chaotic state are easy to distinguish when detected. The periodic state can be maintained when the signal amplitude changes from short distance to long distance in a new system. The spectral signal-to-noise ratio range increases up to -46:57 dB in the similar Liu chaotic system. The characteristics of the new system are only effected by its structure and parameters. The system does not rely on the external factors, and it can be extended. By using some switching devices, the conversion between the chaotic state and periodic state can be realized in the practical engineering applications with a higher detection accuracy. The new design concept of the similar Liu chaotic system shows a very high practical value. It will lay a certain foundation for the underwater acoustic communication of the ocean internet of things in the future.

Image Encryption Based on Hyperchaotic Liu system Algorithm

Image Encryption Based on Hyperchaotic Liu system Algorithm