Existence of Approximate Solutions to Nonlinear Lorenz System under Caputo-Fabrizio Derivative

In this article,we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative(CFFD).The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii.Also,we enriched our work by establishing a stable result based on the Ulam-Hyers(U-H)concept.Also,the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method.We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders.The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier.Also,the suggested method results for the proposed system under the mentioned derivative are new.Furthermore,the adopted technique has some useful features,such as the lack of prior discrimination required by wavelet methods.our proposed method does not depend on auxiliary parameters like the homotopy method,which controls the method.Our proposed method is rapidly convergent and,in most cases,it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.

Local Dynamics Analysis of a Four-Dimensional Hyperchaotic Lorenz System

The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stability under different parameter conditions are analyzed by applying Routh-Hurwitz criterion. The results indicate that the system may exist one, three and five equilibrium points for different system parameters. Based on the central manifold theorem and normal form theorem, the pitchfork bifurcation and Hopf bifurcation are studied respectively. By using the Hopf bifurcation theorem and calculating the first Lyapunov coefficient, the Hopf bifurcation of this system is obtained as supercritical for certain parameters. Finally, the results of theoretical parts are verified by some numerical simulations.

Existence of Supercritical Hopf Bifurcation on a Type-Lorenz System

In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.

A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system

We propose a new image encryption algorithm on the basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded in the proposed algorithm to enhance the security. Such an algorithm is detailed in terms of security analyses, including correlation analysis, information entropy analysis, run statistic analysis, mean-variance gray value analysis, and key sensitivity analysis. The experimental results demonstrate that the proposed image encryption scheme has the advantages of large key space and high security for practical image encryption.

Theoretical Basis and Application of an Analogue-Dynamical Model in the Lorenz System

The theoretical basis and application of an analogue-dynamical model （ADM） in the Lorenz system is studied. The ADM can effectively combine statistical and dynamical methods in which the small disturbance of the current initial value superimposed on the historical analogue reference state can be regarded as a prediction objective. Primary analyses show that under the condition of appending disturbances in model parameters, the model errors of ADM are much smaller than those of the pure dynamical model （PDM）. The characteristics of predictability on the ADM in the Lorenz system are analyzed in phase space by conducting case studies and global experiments. The results show that the ADM can quite effectively reduce prediction errors and prolong the valid time of the prediction in most situations in contrast to the PDM, but when model errors are considerably small, the latter will be superior to the former. To overcome such a problem, the multi-reference-state updating can be applied to introduce the information of multi-analogue and update analogue and can exhibit exciting performance in the ADM.

Hybrid TS fuzzy modelling and simulation for chaotic Lorenz system

The projection of the chaotic attractor observed from the Lorenz system in the X-Z plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined. In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi-Sugeno （TS） fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex （two-wing） butterfly attractors observed from this system in the three dimensional （3D） X-Y-Z space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.

A new four-dimensional hyperchaotic Lorenz system and its adaptive control

Based on the Lorenz chaotic system, this paper constructs a new four-dimensional hyperchaotic Lorenz system, and studies the basic dynamic behaviours of the system. The Routh-Hurwitz theorem is applied to derive the stability conditions of the proposed system. Furthermore, based on Lyapunov stability theory, an adaptive controller is designed and the new four-dimensional hyperchaotic Lorenz system is controlled at equilibrium point. Numerical simulation results are presented to illustrate the effectiveness of this method.

Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems

This paper presents the problem of generating four-wing （eight-wing） chaotic attractors. The adopted method consists in suitably coupling two （three） identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings （eight wings） of these novel attractors axe located around the four （eight） equilibria with two （three） pairs of unstable complex-conjugate eigenvalues.

Stochastic impulsive control for the stabilization of Lorenz system

This paper derives some sufficient conditions for the stabilization of Lorenz system with stochastic impulsive control. The estimate of the upper bound of impulse interval for asymptotically stable control is obtained. Some differences between the system with stochastic impulsive control and with deterministic impulsive control are presented. Computer simulation is given to show the effectiveness of the proposed method.

Adaptive control and synchronization of an uncertain new hyperchaotic Lorenz system

This paper is involved with the adaptive control and synchronization problems for an uncertain new hyperchaotic Lorenz system. Based on the Lyapunov stability theory, the adaptive control law is derived such that the trajectory of hyperchaotic Lorenz system with unknown parameters can be globally stabilized to an unstable equilibrium point of the uncontrolled system. Furthermore, an adaptive control approach is presented to the synchronizations between two identical hyperchaotic systems, particularly between two different uncertain hyperchaotic systems. Numerical simulations show the effectiveness of the presented method.

Estimating the bound for the generalized Lorenz system

A chaotic system is bounded, and its trajectory is confined to a certain region which is called the chaotic attractor. No matter how unstable the interior of the system is, the trajectory never exceeds the chaotic attractor. In the present paper, the sphere bound of the generalized Lorenz system is given, based on the Lyapunov function and the Lagrange multiplier method. Furthermore, we show the actual parameters and perform numerical simulations.

Synchronization of noise-perturbed generalized Lorenz system by sliding mode control

Synchronization of a noise-perturbed generalized Lorenz system by using sliding mode control method is investigated in this paper. Two sliding mode control methods are proposed to synchronize the noise-perturbed generalized Lorenz system. Numerical simulations are also provided for the illustration and verification of the methods.

Identification and Control of Hyperchaotic Lorenz System Based on Quantum Inspired PSO and Wavelet Neural Network

For the identification and control problem of hyperchaotic Lorenz system,a method of feedback compensation control based on quantum inspired PSO and wavelet neural network(PSOQI-Wavelet Neural Network)is proposed.Firstly,the quantum principle obtained from Quantum PSO(QPSO)has been combined with standard PSO to form a new hybrid algorithm called PSO with Quantum Infusion(PSO-QI).Then,the parameters of wavelet neural network were optimized with PSO-QI and feedback compensation control for hyperchaotic Lorenz system is implemented using optimized PSOQI-Wavelet Neural Network.The numerical simulation results showed that this method has better precision and can quickly track given hyperchaotic Lorenz system.

Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation

The chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to the circuit unit, an electronic circuit is designed to realize a 3.8-order generalized Lorenz chaotic system. Furthermore, synchronization between two fractional-order systems is achieved by utilizing a single-variable feedback method. Circuit experiment simulation results verify the effectiveness of the proposed scheme.

Hyperchaos behaviors and chaos synchronization of two unidirectional coupled simplified Lorenz systems

To design a hyperchaotic generator and apply chaos into secure communication, a linear unidirectional coupling control is applied to two identical simplified Lorenz systems. The dynamical evolution process of the coupled system is investigated with variations of the system parameter and coupling coefficients. Particularly, the influence of coupling strength on dynamics of the coupled system is analyzed in detail. The range of the coupling strength in which the coupled system can generate hyperchaos or realize synchronization is determined, including phase portraits, Lyapunov exponents, and Poincaré section. And the critical value of the system parameter between hyperchaos and synchronization is also found with fixed coupled strength. In addition, abundant dynamical behaviors such as four-wing hyperchaotic, two-wing chaotic, single-wing coexisting attractors and periodic orbits are observed and chaos synchronization error curves are also drawn by varying system parameter c. Numerical simulations are implemented to verify the results of these investigations.

Estimating the ultimate bound and positively invariant set for a generalized Lorenz system

A generalized Lyapunov function was employed to investigate the ultimate bound and positively invariant set of a generalized Lorenz system.We derived an ellipsoidal estimate of the ultimate bound and positively invariant set for the generalized Lorenz system,for all the positive values of system parameters a,b,and c.Our results extend the related result of Li,et al.[Li DM,Lu JA,Wu XQ,et al.,Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,Journal of Mathematical Analysis and Application,2006,323(2):844-653].

Stability Analysis and Design of Impulsive Control Lorenz Systems Family

Lorenz 系统家庭统一 Lorenz 系统，陈系统和 L ü系统是一个典型混乱家庭。在这篇论文，我们与变化时间的推动间隔认为冲动的控制是 Lorenz 混乱系统家庭。由建立一套不平等的一个有效工具，我们分析冲动的控制 Lorenz 系统家庭的 asymptotic 稳定性并且获得一些新不太保守的条件。基于稳定性分析，我们与变化时间的推动间隔设计一个新奇冲动的控制器。解说性的例子被提供显示出我们的方法的可行性和有效性。获得的结果能不仅被用来为 Lorenz 系统家庭设计冲动的控制，而且能被扩大到另外的混乱系统。

Quantifying local predictability of the Lorenz system using the nonlinear local Lyapunov exponent

非线性局部Lyapunov指数(NLLE)可以用来度量混沌系统的局地可预报性。本文基于NLLE方法研究了Lorenz吸引子在相空间上的局地可预报性的空间分布特征。结果表明,在吸引子两翼的内、外边缘的局地可预报性期限较高,而吸引子中部地区的局地可预报性期限则较低。然而,局地可预报性期限的分布却没有呈现有组织的均一结构,相邻两点的局地可预报性期限可能差别很大。局地可预报性的来源被认为与吸引子上的局地动力学有关,由所在位置和在当前状态的持续时间决定。

Chaos Synchronization of Uncertain Lorenz System via Single State Variable Feedback

This paper treats the problem of chaos synchronization for uncertain Lorenz system via single state variable information of the master system. By the Lyapunov stability theory and adaptive technique, the derived controller is featured as follows: 1) only single state variable information of the master system is needed;2) chaos synchronization can also be achieved even if the perturbation is occurred in some parameters of the master chaotic system. Finally, the effectiveness of the proposed controllers is also illustrated by the simulations as well as rigorous mathematical proofs.

Limit Cycles near Stationary Points in the Lorenz System

在静止的点附近的 Lorenz 系统的限制周期数字地被分析。在线性 Lorenz 系统的阶段空间的一架飞机被用来在限制周期附近定位轨道的合适的起始的点。数字结果在静止的点附近显示出一个马厩和一个不稳定的限制周期。稳定的限制周期比不稳定的小并且以前没在文学被报导。另外，在 Lorenzsystem 的所有限制周期理论上被证明不平面。