Fractional-order Lorenz System Research Articles

On a function projective synchronization scheme for non-identical Fractional-order chaotic (hyperchaotic) systems with different dimensions and orders

Referring to fractional-order systems, this paper investigates the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions and different orders. By taking a master system of dimension n and a slave system of dimension m, the method enables each master system state to be synchronized with a linear combination of slave system states, where the scaling factor of the linear combination can be any arbitrary differentiable function. The approach presents some useful features: i) it enables commensurate and incommensurate non-identical fractional-order systems with different dimension n<m or n>m to be synchronized; ii) it can be applied to a wide class of chaotic (hyperchaotic) fractional-order systems for any differentiable scaling function; iii) it is rigorous, being based on two theorems, one for the case n<m and the other for the case n>m. Two different numerical examples are reported, involving chaotic/hyperchaotic fractional-order Lorenz systems (three-dimensional and four-dimensional master/slave, respectively) and hyperchaotic/chaotic fractional-order Chen systems (four-dimensional and three-dimensional master/slave, respectively). The examples clearly highlight the capability of the conceived approach in effectively achieving synchronized dynamics for any differentiable scaling function.

A Novel Image Encryption Algorithm Based on a Fractional-Order Hyperchaotic System and DNA Computing

In the era of the Internet, image encryption plays an important role in information security. Chaotic systems and DNA operations have been proven to be powerful for image encryption. To further enhance the security of image, in this paper, we propose a novel algorithm that combines the fractional-order hyperchaotic Lorenz system and DNA computing (FOHCLDNA) for image encryption. Specifically, the algorithm consists of four parts: firstly, we use a fractional-order hyperchaotic Lorenz system to generate a pseudorandom sequence that will be utilized during the whole encryption process; secondly, a simple but effective diffusion scheme is performed to spread the little change in one pixel to all the other pixels; thirdly, the plain image is encoded by DNA rules and corresponding DNA operations are performed; finally, global permutation and 2D and 3D permutation are performed on pixels, bits, and acid bases. The extensive experimental results on eight publicly available testing images demonstrate that the encryption algorithm can achieve state-of-the-art performance in terms of security and robustness when compared with some existing methods, showing that the FOHCLDNA is promising for image encryption.

Stability and stabilization of a class of fractional-order nonlinear systems for $$\varvec{0&lt;}\,{\varvec{\alpha }} \,\varvec{&lt; 2}$$ 0 &lt; α &lt; 2

This paper investigates the stability and stabilization problem of fractional-order nonlinear systems for \(0<\alpha <2\). Based on the fractional-order Lyapunov stability theorem, S-procedure and Mittag–Leffler function, the stability conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with \(0<\alpha <2\) are proposed. Finally, typical instances, including the fractional-order nonlinear Chen system and the fractional-order nonlinear Lorenz system, are implemented to demonstrate the feasibility and validity of the proposed method.

Chaos control and function projective synchronization of fractional-order systems through the backstepping method

We study the chaos control and the function projective synchronization of a fractional-order T-system and Lorenz chaotic system using the backstepping method. Based on stability theory, we consider the condition for the local stability of nonlinear three-dimensional commensurate fractional-order system. Using the feedback control method, we control the chaos in the considered fractional-order T-system. We simulate the function projective synchronization between the fractional-order T-system and Lorenz system numerically using MATLAB and depict the results with plots.

Different Generalized Synchronization Schemes Between Integer-Order and Fractional-Order Chaotic Systems with Different Dimensions

This paper addresses the problem of generalized synchronization (GS) between different dimensional integer-order and fractional-order chaotic systems. Based on the stability theory of linear continuous time dynamical systems, stability results of linear fractional order systems and nonlinear controllers, different criterions are derived to achieve generalized synchronization. The effectiveness of the proposed control schemes are verified by considering two examples: fractional-order chaotic Lorenz and hyperchaotic Lorenz systems and hyperchaotic Chen and fractional-order chaotic Chen systems.

Dynamical properties and complexity in fractional-order diffusionless Lorenz system

In this paper, dynamics and complexity of the fractional-order diffusionless Lorenz system which is solved by the developed discrete Adomian decomposition method are investigated numerically. Dynamical properties of the fractional-order diffusionless Lorenz system with the control parameter and derivative order varying is analyzed by using bifurcation diagrams, and period-doubling route to chaos in different cases is observed. The complexity of the system is investigated by means of Lyapunov characteristic exponents, multi-scale spectral entropy algorithm and multiscale Renyi permutation entropy algorithm. It can be observed that the three methods illustrate consistent results and the system has rich complex dynamics. Interestingly, complexity decreases with the increase of derivative order. It shows that the fractional-order diffusionless Lorenz system is a good model for real applications such as information encryption and secure communication.

Multivariate permutation entropy and its application for complexity analysis of chaotic systems

Abstract To measure the complexity of multivariate systems, the multivariate permutation entropy (MvPE) algorithm is proposed. It is employed to measure complexity of multivariate system in the phase space. As an application, MvPE is applied to analyze the complexity of chaotic systems, including hyperchaotic Henon map, fractional-order simplified Lorenz system and financial chaotic system. Results show that MvPE algorithm is effective for analyzing the complexity of the multivariate systems. It also shows that fractional-order system does not become more complex with derivative order varying. Compared with PE, MvPE has better robustness for noise and sampling interval, and the results are not affected by different normalization methods.

Robust finite-time control of fractional-order nonlinear systems via frequency distributed model

This paper studies the application of frequency distributed model for finite-time control of a class of fractional-order nonlinear systems. Firstly, a class of fractional-order nonlinear systems with model uncertainties and external disturbances are introduced, and a new frequency distributed model with theoretical inference is presented. Secondly, a novel fast terminal sliding surface is proposed and its stability to origin is proved based on the frequency distributed model and Lyapunov stability theory. Furthermore, based on finite-time stability and sliding mode control theory, a robust control law to ensure the occurrence of the sliding motion in a finite time is designed for stabilization of the fractional-order nonlinear systems. Finally, two typical examples of three-dimensional nonlinear fractional-order Lorenz system and four-dimensional nonlinear fractional-order Chen system are employed to demonstrate the validity of the proposed method.

A modified sliding mode approach for synchronization of fractional-order chaotic/hyperchaotic systems by using new self-structuring hierarchical type-2 fuzzy neural network

This paper presents a new adaptive sliding mode control approach for the synchronization of the uncertain fractional-order chaotic systems. A self-structuring hierarchical type-2 fuzzy neural network (SHT2FNN) is proposed for estimation of uncertainties. Also the switching control action in the conventional sliding mode scheme is replaced by combination type-2 fuzzy neural network (T2FNN) with hyperbolic tangent function. In SHT2FNN, the number of rules is determined by a proposed algorithm. Adaptation laws of all trainable parameters of T2FNN and the consequent parameters of SHT2FNN, are derived based on Lyapunov stability analysis. The simulation results on two kind systems: Genio-Tesi and Coullet System and fractional-order hyper-chaotic Lorenz system, confirm the efficacy of the proposed scheme in synchronization of the uncertain fractional-order hyperchaotic and fractional-order chaotic systems.The proposed controller is robust against the approximation error and external disturbance. The proposed self-structuring algorithm in this paper is simple and it can be applied in the high dimensional problems. Furthermore, the proposed algorithm can delete unimportant rules. Adjusting the structure of the T2FNN in the hierarchical form ensures that the estimation error is very small so it can be negligible. Furthermore, the proposed strategy guarantees the robustness of controller.

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.

Adaptive function projective combination synchronization of three different fractional-order chaotic systems

In this paper, we investigate the adaptive function projective combination synchronization of three fractional-order chaotic systems: the fractional-order Lorenz system, the fractional-order Chen's system and the fractional-order Lü system. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of three fractional-order chaotic systems is designed, and a stability criterion for the above-mentioned systems is attained. The corresponding theoretical analysis and numerical simulations are shown to verify the feasibility and effectiveness of the proposed synchronization technique.

Mittag–Leffler stability of fractional-order Lorenz and Lorenz-family systems

Based on the stability theory of fractional-order systems, using the fractional-order Lyapunov direct method, in this paper, the Mittag–Leffler stability for fractional-order Lorenz and fractional-order Lorenz-family system is investigated, respectively. We propose the definition of the Mittag–Leffler stability; some sufficient conditions for the Mittag–Leffler stability are obtained by selecting properly the parameters and the initial values of system. Finally, numerical simulations are performed to verify the theoretical analysis.

Fuzzy prediction-based feedback control of fractional order chaotic systems

In this paper, approaches to stabilize a fractional order chaotic system on its original equilibrium point are proposed using the Takagi–Sugeno (T–S) fuzzy models and prediction-based feedback controls. A sufficient condition of asymptotical stability for fractional order T–S fuzzy system is derived based on direct Lyapunov stability theory. The proposed approach offers a systematic design procedure for stabilizing a large class of fractional order chaotic systems in the literature about chaos research. The effectiveness of the approach is tested on fractional order Rössler system and fractional order Lorenz system.

Characteristic Analysis and DSP Realization of Fractional-Order Simplified Lorenz System Based on Adomian Decomposition Method

By adopting Adomian decomposition method, the fractional-order simplified Lorenz system is solved and implemented on a digital signal processor (DSP). The Lyapunov exponent (LE) spectra of the system is calculated based on QR-factorization, and it accords well with the corresponding bifurcation diagrams. We analyze the influence of the parameter and the fractional derivative order on the system characteristics by color maximum LE (LEmax) and chaos diagrams. It is found that the smaller the order is, the larger the LEmaxis. The iteration step size also affects the lowest order at which the chaos exists. Further, we implement the fractional-order simplified Lorenz system on a DSP platform. The phase portraits generated on DSP are consistent with the results that were obtained by computer simulations. It lays a good foundation for applications of the fractional-order chaotic systems.

Research on particle swarm optimization algorithm with characteristic of quantum parallel and its application in parameter estimation for fractional-order chaotic systems

Parameter estimation for fractional-order chaotic systems is a multi-dimensional optimization problem, which is one of the important issues in fractional-order chaotic control and synchronization. With the characteristic of quantum parallel, a new quantum parallel particle swarm optimization algorithm is proposed for solving the problem of parameter estimation in fractional-order chaotic systems. A new method of quantum coding is presented with quantum parallel characteristic which can make the calculation number of each generation increase exponentially. On the basis of this method, a particle evolution equation composed of quantum current rotation angle, individual optimal rotation angle, and global optimum rotation angle is proposed, which can restraint the behavior of particles in quantum space, and also can improve the search capability of the algorithm. Numerical simulations of the fractional-order Lorenz system and the fractional-order Chen system are conducted and the results demonstrate the effectiveness, robustness and versatility of the proposed algorithm.

Analysis of a Nonlinear System with a Random Parameter

In this paper, a nonlinear system with random parameter, which is called stochastic fractional-order complex Lorenz system, is investigated. The Laguerre polynomial approximation method is used to study the system. Then, the stochastic fractional-order system is reduced into the equivalent deterministic one with Laguerre approximation. The ensemble mean and sample responses of the stochastic system can be obtained.

Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems

In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation methods, including phase portraits, bifurcation diagrams, the largest Lyapunov exponent and power spectrum diagrams. Some interesting phenomena, such as inverse period-doubling bifurcation and intermittent chaos, are found to exist in the proposed systems.

Analysis and circuit design of a fractional-order Lorenz system with different fractional orders

The paper first discusses the recently reported fractional-order Lorenz system, analyzes it by using the frequency-domain approximation method and the time-domain approximation method, and finds its chaotic dynamics when the order of the fractional-order system varies from 2.8 to 2.9 in steps of 0.1. Especially for the fractional-order Lorenz system of the order as low as 2.9, the results obtained by the frequency-domain method are consistent with those obtained by the time-domain method. Some Lyapunov exponent diagrams, bifurcation diagrams, and phase orbits diagrams have also been shown to verify the chaotic dynamics of the fractional-order Lorenz system. Then, an analog circuit for the fractional-order Lorenz system of the order as low as 2.9 is designed to confirm its chaotic dynamic, the results from circuit experiment show that it is chaotic.

Synchronization of Fractional-Order Systems of the Lorenz type: The Non- adaptive Case

This paper presents a synchronization method for fractional order systems of the Lorenz type with known parameters. First we include a study of the isolated behavior of the fractional order Lorenz system (FOLS). Then a master and a slave FOLS are considered and appropriate control signals are designed and applied to the slave system, in order to achieve the synchronization of both systems. These control signals were designed applying the extended Routh-Hurwitz criterion to fractional order systems (FOSs). Simulation results show that synchronization is achieved when all control signals are used, as well as when a reduced number of control signals are employed.

A New Fractional-Order Chaotic Complex System and Its Antisynchronization

We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.