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 requ...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.展开更多
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 stab...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.展开更多
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 demonstr...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.展开更多
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...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.展开更多
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 cur...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.展开更多
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...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.展开更多
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 s...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.展开更多
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 s...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.展开更多
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...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.展开更多
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 ...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.展开更多
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 att...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.展开更多
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 gener...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.展开更多
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.Firs...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.展开更多
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 th...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.展开更多
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 coup...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.展开更多
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 invari...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].展开更多
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 contr...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.展开更多
基金support of Taif University Researchers Supporting Project No. (TURSP-2020/162),Taif University,Taif,Saudi Arabiafunding this work through research groups program under Grant No.R.G.P.1/195/42.
文摘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.
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 61004078 and 60971022)the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2009GQ009 and ZR2009GM005)+1 种基金the China Postdoctoral Science Foundation (Grant No. 20100481293)the Special Funds for Postdoctoral Innovative Projects of Shandong Province, China (Grant No. 201003037)
文摘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.
基金jointly supported by the National Natural Science Foundation of China (Grant Nos. 40805028, 40675039 and 40575036)the Meteorological Special Project (GYHY200806005)the National Science and Technology Support Program of China (2006BAC02B04 and 2007BAC29B03)
文摘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.
基金Project partially supported by the Natural Science Foundation of Educational Committee of Anhui Province, China (Grant No 2006kj250B).
文摘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.
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China (Grant No. 10872165)
文摘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.
基金supported by the National Natural Science Foundation of China (Grant Nos 70571030 and 90610031)the Advanced Talent Foundation of Jiangsu University of China (Grant No 07JDG054)
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant No. 60674059)Research Fund of University of Science and Technology Beijing, China (Grant No. 00009010)
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant No 70571059)
文摘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.
基金National Natural Science Foundation of China(No.11261001)。
文摘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.
基金supported by the Natural Science Foundation of Hebei Province,China (Grant Nos A2008000136 and A2006000128)
文摘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.
基金Projects(61073187,61161006) supported by the National Nature Science Foundation of ChinaProject supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry,China
文摘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.
文摘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].
基金supported by the Key Project of the Ministry of Education under Grant No.107099the Key Youth Science and Technology Foundation of University of Electronic Science and Technology of China under Grant No.L08010201JX0720
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant No 10472054, and the Specialized Research Fund for the Doctoral Program of Higher Education (No 20040003070).