New estimate the bounds for the generalized Lorenz system
Corresponding Author
Fuchen Zhang
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
Correspondence to: Fuchen Zhang, College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China.
E-mail: [email protected]
Search for more papers by this authorGuangyun Zhang
School of Foreign Languages, Southwest Petroleum University, Chengdu 610500, China
Search for more papers by this authorDa Lin
School of Automatic and Electronic Information, Sichuan University of Science and Engineering, Zigong 643000, China
Search for more papers by this authorXiangkai Sun
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
Search for more papers by this authorCorresponding Author
Fuchen Zhang
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
Correspondence to: Fuchen Zhang, College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China.
E-mail: [email protected]
Search for more papers by this authorGuangyun Zhang
School of Foreign Languages, Southwest Petroleum University, Chengdu 610500, China
Search for more papers by this authorDa Lin
School of Automatic and Electronic Information, Sichuan University of Science and Engineering, Zigong 643000, China
Search for more papers by this authorXiangkai Sun
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
Search for more papers by this authorAbstract
The bound of a chaotic system is important for chaos control, chaos synchronization, and other applications. In the present paper, the bounds of the generalized Lorenz system are studied, based on the Lyapunov function theory and the Lagrange multiplier method. We obtain a precise bound for the generalized Lorenz system. The rate of the trajectories is also obtained. Furthermore, we perform the numerical simulations. Numerical simulations are presented to show the effectiveness of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.
References
- 1 Leonov GA. Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. Journal of Applied Mathematics and Mechanics 2001; 65(1): 19–32.
- 2 Leonov GA, Seledzhi S. Stability and bifurcations of phase-locked loops for digital signal processors. International Journal of Bifurcation and Chaos 2005; 15(4): 1347–1360.
- 3 Yang T, Yang LB, Yang C M. Impulsive control of Lorenz system. Physica D: Nonlinear Phenomena 1997; 110(1-2): 18–24.
- 4 Hu J, Chen SH, Chen L. Adaptive control for anti-synchronization of Chua's chaotic system. Physics Letters A 2005; 339(6): 455–460.
- 5 Park JH. Adaptive synchronization of Rossler system with uncertain parameters. Chaos, Solitons and Fractals 2005; 25(2): 333–338.
- 6 Li CD, Liao XF. Lag synchronization of Rossler system and Chua circuit via a scalar signal. Physics Letters A 2004; 329(4-5): 301–308.
- 7 Wang XY, Wang MJ. Projective synchronization of nonlinear-coupled spatiotemporal chaotic systems. Nonlinear Dynamics 2010; 62: 567–571.
- 8 Wang XY, Wang MJ. A hyperchaos generated from Lorenz system. Physica A: Statistical Mechanics and its Applications 2008; 387(14): 3751–3758.
- 9 Luo C, Wang XY. Algebraic representation of asynchronous multiple-valued networks and its dynamics. IEEE/ACM Transactions on Computational Biology and Bioinformatics 2013; 10(4): 927–938.
- 10 Luo C, Wang XY. Dynamics of random boolean networks under fully asynchronous stochastic update based on linear representation. PloS One 2013; 8(6): e66491.
- 11 Wang XY, Wang MJ. Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos: An Interdisciplinary Journal of Nonlinear Science 2007; 17(3): 033106.
- 12 Liu HJ, Wang XY, Zhu QL. Asynchronous anti-noise hyper chaotic secure communication system based on dynamic delay and state variables switching. Physics Letters A 2011; 375(30–31): 2828–2835.
- 13 Pecora LM, Carroll TL. Synchronization in chaotic systems. Physical Review Letters 1990; 64: 821–824.
- 14 Zhang FC, Shu YL, Yang HL, Li XW. Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chao synchronization. Chaos, Solitons and Fractals 2011; 44(1–3): 137–144.
- 15 Zhang FC, Shu YL, Yang HL. Bounds for a new chaotic system and its application in chaos synchronization. Communications in Nonlinear Science and Numerical Simulation 2011; 16(3): 1501–1508.
- 16 Pogromsky A, Santoboni G, Nijmeijer H. An ultimate bound on the trajectories of the Lorenz system and its applications. Nonlinearity 2003; 16: 1597–1605.
- 17 Leonov GA. Lyapunov dimension formulas for Henon and Lorenz attractors. St Petersburg Mathematical Journal 2001; 13(3): 453–464.
- 18 Leonov GA. Lyapunov functions in the attractors dimension theory. Journal of Applied Mathematics and Mechanics 2012; 76(2): 129–141.
- 19 Wang P, Zhang YH, Tan SL, Wan L. Explicit ultimate bound sets of a new hyperchaotic system and its application in estimating the Hausdorff dimension. Nonlinear Dynamics 2013; 74: 133–142.
- 20 Leonov GA, Boichenko VA. Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors. Acta Applicandae Mathematica 1992; 26(1): 1–60.
- 21 Mu CL, Zhang FC, Shu YL, Zhou SM. On the boundedness of solutions to the Lorenz-like family of chaotic systems. Nonlinear Dynamics 2012; 67(2): 987–996.
- 22 Zhang FC, Mu CL, Li XW. On the boundness of some solutions of the Lu system. International Journal of Bifurcation and Chaos 2012; 22(01): 1–5.
- 23 Qin WX, Chen GR. On the boundedness of solutions of the Chen system. Journal of Mathematical Analysis and Applications 2007; 329: 445–451.
- 24 Leonov GA, Bunin AI, Koksch N. Attractor localization of the Lorenz system. Zeitschrift fr Angewandte Mathematik und Mechanik 1987; 67: 649–656.
- 25 Algaba A, Fernandez-Sanchez F, Merino M, Rodríguez-Luis AJ. Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system. Chaos: An Interdisciplinary Journal of Nonlinear Science 2013; 23(3): 033108.
- 26 Chen GR. The Chen system revisited. Dynamics of Continuous, Discrete and Impulsive Systems 2013; 20: 691–696.
- 27 Chen YM, Yang QG. The nonequivalence and dimension formula for attractors of Lorenz-type systems. International Journal of Bifurcation and Chaos 2013; 23(12):1350200.
- 28 Algaba A, Fernandez-Sanchez F, Merino M, Rodríguez-Luis AJ. The Lu system is a particular case of the Lorenz system. Physics Letters A 2013; 377(39): 2771–2776.
- 29 Chen GR, Lu JH. Dynamical Analysis, Control and Synchronization of the Lorenz Systems Family, Science Press: Beijing, 2003.
- 30
Zheng Y,
Zhang XD. Estimating the bound for the generalized Lorenz system. Chinese Physics B 2010; 19(1): 010505.
10.1088/1674-1056/19/1/010505 Google Scholar
- 31
Leonov GA,
Kuznetsov NV. Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. International Journal of Bifurcation and Chaos 2013; 23(1):1330002.
10.1142/S0218127413300024 Google Scholar
- 32 Bragin VO, Vagaitsev VI, Kuznetsov NV, Leonov GA. Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits. Journal of Computer and Systems Sciences International 2011; 50(4): 511–543.
- 33 Leonov GA, Kuznetsov NV, Vagaitsev VI. Localization of hidden Chua's attractors. Physics Letters A 2011; 375(23): 2230–2233.
- 34 Leonov GA, Kuznetsov NV, Vagaitsev VI. Hidden attractor in smooth Chua systems. Physica D: Nonlinear Phenomena 2012; 241(18): 1482–1486.
- 35 Leonov GA, Kuznetsov NV. Time-varying linearization and the Perron effects. International Journal of Bifurcation and Chaos 2007; 17(4): 1079–1107.
- 36 Kuznetsov NV, Mokaev TN, Vasilyev PA. Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor. Communications in Nonlinear Science and Numerical Simulation 2014; 19(4): 1027–1034.
- 37 Kuznetsov NV, Leonov GA. On stability by the first approximation for discrete systems. 2005 International Conference on Physics and Control, St. Petersburg, Russia, 2005, Phys Con 2005, Proceedings 2005; (1514053): 596–599.
