In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilitie...In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.展开更多
This paper introduces the finding of a unified Lorenz-like system.By gradually tuning the only parameter d,the reported system belongs to Lorenz-type system in the sense defined by Clikovsky.Meanwhile,this system belo...This paper introduces the finding of a unified Lorenz-like system.By gradually tuning the only parameter d,the reported system belongs to Lorenz-type system in the sense defined by Clikovsky.Meanwhile,this system belongs to Lorenz-type system,Lu-type system,Chen-type system with d less than,equivalent to and greater than 1.5,respectively,according to the classification defined by Yang.However,this system can only generate a succession of Lorenz-like attractors.Some basic dynamical properties of the system are investigated theoretically and numerically.Moreover,the tracking control of the system with exponential convergence rate is studied.Theoretical analysis and computer simulation show that the proposed scheme can allow us to drive the output variable x\ to arbitrary reference signals exponentially,and the guaranteed exponential convergence rate can be estimated accurately.展开更多
基金Projected supported by the National Natural Science Foundation of China (Grant No. 11202155)the Fundamental Research Funds for the Central Universities, China (Grant No. K50511700001)
文摘In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.
基金Supported by the Research Foundation of Education Bureau of Hunan Province of China under Grant No.13C372Jiangsu Provincial Natural Science Foundation of China under Grant No.14KJB120007the Outstanding Doctoral Dissertation Project of Special Funds under Grant No.27122
文摘This paper introduces the finding of a unified Lorenz-like system.By gradually tuning the only parameter d,the reported system belongs to Lorenz-type system in the sense defined by Clikovsky.Meanwhile,this system belongs to Lorenz-type system,Lu-type system,Chen-type system with d less than,equivalent to and greater than 1.5,respectively,according to the classification defined by Yang.However,this system can only generate a succession of Lorenz-like attractors.Some basic dynamical properties of the system are investigated theoretically and numerically.Moreover,the tracking control of the system with exponential convergence rate is studied.Theoretical analysis and computer simulation show that the proposed scheme can allow us to drive the output variable x\ to arbitrary reference signals exponentially,and the guaranteed exponential convergence rate can be estimated accurately.
