On the boundness of some solutions of the lü system

International Journal of Bifurcation and Chaos

Volumn 22, Issue 1, 2012, Pages

  Zhang, Fuchen ^a   Mu, Chunlai ^a   Li, Xiaowu ^a  

^a CHONGQING UNIVERSITY   (China)

Author keywords

boundedness; L system; Lyapunov function

Indexed keywords

CHAOTIC SYSTEMS;

BOUNDEDNESS; BOUNDNESS;

LYAPUNOV FUNCTIONS;

EID: 84862793585     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127412500150     Document Type: Article

References (40)

1. Agiza, H. N. [2004] "Chaos synchronization of Lü dynamical system," Nonlin. Anal.: Th. Meth. Appl. 58, 11-20.
2. Boichenko, V. A. & Leonov, G. A. [2005] Dimension Theory for Ordinary Differential Equations (Teubner- Verlag, Wiesbaden).
3. Chen, D., Sun, J. &Wu, Q. [2004] "Impulsive control and its application to Lü's chaotic system," Chaos Solit. Fract. 21, 1135-1142.
4. Chen, G. & Lü, J. [2003] Dynamics of the Lorenz System Family: Analysis, Control and Synchronization [M] (Science Press, Beijing) [in Chinese].
5. Coria, L. N. & Starkov, K. E. [2009] "Bounding a domain containing all compact invariant sets of the permanent-magnet motor system," Commun. Nonlin. Sci. Numer. Simulat. 14, 3879-3888.
6. Han, X., Lu, J. & Wu, X. [2004] "Adaptive feedback synchronization of Lü system," Chaos Solit. Fract. 24, 221-227.
7. Krishchenko, A. P. & Starkov, K. E. [2006] "Localization of compact invariant sets of the Lorenz system," Phys. Lett. A 353, 383-388. (Pubitemid 43697231)
8. Krishchenko, A. P. & Starkov, K. [2007] "Estimation of the domain containing all compact invariant sets of a system modelling the amplitude of a plasma instability," Phys. Lett. A 367, 65-72.
9. Krishchenko, A. P. & Starkov, K. E. [2010] "Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations," Commun. Nonlin. Sci. Numer. Simulat. 15, 1159-1165.
10. Leonov, G. A., Abramovich, S. M. & Bunin, A. I. [1985] "Problems of nonlinear and turbulent process in physics," Proc. Second Int. Working Group, Kiev Part II (in Russian), pp. 75-77.
11. Leonov, G. & Reitmann, V. [1987] Attraktoreingrenzuny ur Nichtlineare System (Teubner-Verlag, Leipzing).
12. Leonov, G., Bunin, A. & Koksch, N. [1987] "Attractor localization of the Lorenz system," ZAMM 67, 649-656.
13. Li, C. & Chen, G. [2003] "A note on Hopf bifurcation in Chen's system," Int. J. Bifurcation and Chaos 13, 1609-1615. (Pubitemid 37067813)
14. Li, D. [2006] "TS fuzzy realization of chaotic Lü system," Phys. Lett. A 356, 51-58.
15. Li, D., Lu, J., Wu, X. & Chen, G. [2005] "Estimating the bounds for the Lorenz family of chaotic systems," Chaos Solit. Fract. 23, 529-534.
16. Li, D., Lu, J., Wu, X. & Chen, G. [2006] "Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system," J. Math. Anal. Appl. 323, 844-853. (Pubitemid 44688526)
17. Li, D., Lu, J. & Wu, X. [2009] "Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz-Haken system," Chaos Solit. Fract. 39, 1290-1296.
18. Li, T., Chen, G. & Tang, Y. [2004] "On stability and bifurcation of Chen's system," Chaos Solit. Fract. 19, 1269-1282.
19. Liao, X., Fu, Y. & Xie, S. [2005] "On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization," Sci. China Ser. F Inform. Sci. 48, 304-321. (Pubitemid 44113761)
20. Liao, X. & Yu, P. [2006] "Study on the global property of the smooth Chua's system," Int. J. Bifurcation and Chaos 16, 2815-2841. (Pubitemid 44924963)
21. Lorenz, E. N. [1963] "Deterministic non-periods flows," J. Atmos. Sci. 20, 130-141.
22. Lü, J. & Chen, G. [2002] "A new chaotic attractor coined," Int. J. Bifurcation and Chaos 12, 659-661. (Pubitemid 34862041)
23. Lu, J., Chen, G. & Zhang, S. [2002] "The compound structure of a new chaotic attractor," Chaos Solit. Fract. 14, 669-672. (Pubitemid 34232062)
24. Qin,W. & Chen, G. [2007] "On the boundedness of solutions of the Chen system," J. Math. Anal. Appl. 329, 445-451.
25. Rösser, O. E. [1976] "An equation for continuous chaos," Phys. Lett. A 57, 397-398.
26. Shu, Y., Xu, H. & Zhao, Y. [2009] "Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization," Chaos Solit. Fract. 42, 2852-2857.
27. Starkov, K. E. [2009a] "Bounds for a domain containing all compact invariant sets of the system describing the laser-plasma interaction," Chaos Solit. Fract. 39, 1671-1676.
28. Starkov, K. E. [2009b] "Bounds for compact invariant sets of the system describing dynamics of the nuclear spin generator," Commun. Nonlin. Sci. Numer. Simulat. 14, 2565-2570.
29. Sun, Y. [2009] "Solution bounds of generalized Lorenz chaotic systems," Chaos Solit. Fract. 40, 691-696.
30. Swinnerton-Dyer, P. [2001] "Bounds for trajectories of the Lorenz equations: An illustration of how to choose Lyapunov functions," Phys. Lett. A 281, 161-167. (Pubitemid 32233633)
31. Tigan, G. & Constantinescu, D. [2009] "Heteroclinic orbits in the T and the Lü systems," Chaos Solit. Fract. 42, 20-23.
32. Ueta, T. & Chen, G. [2000] "Bifurcation analysis of Chen's attractor," Int. J. Bifurcation and Chaos 10, 1917-1931.
33. Wang, P., Li, D. & Hu, Q. [2010] "Bounds of the hyperchaotic Lorenz-Stenflo system," Commun. Nonlin. Sci. Numer. Simulat. 15, 2514-2520.
34. Yu, C. & Chen, G. [2006] "Complex dynamics in Chen's system," Chaos Solit. Fract. 27, 75-86. (Pubitemid 40999964)
35. Yu, P. & Liao, X. [2006] "New estimations for globally attractive and positive invariant set of the family of the Lorenz system," Int. J. Bifurcation and Chaos 16, 3383-3390. (Pubitemid 46352363)
36. Yu, P. & Liao, X. [2008] "On the study of globally exponentially attractive set of a general chaotic system," Commun. Nonlin. Sci. Numer. Simulat. 13, 1498-1507.
37. Yu, P. & Liao, X. [2009] "New estimations for globally attractive and positive invariant set of the family of the Lorenz systems," Int. J. Bifurcation and Chaos 11, 3383-3390. (Pubitemid 46352363)
38. Yu, Y. & Zhang, S. [2003] "Hopf bifurcation in the Lü system," Chaos Solit. Fract. 17, 901-906.
39. Zhang, Q., Lü, J. & Chen, S. [2010] "Coexistence of antiphase and complete synchronization in the generalized Lorenz system," Commun. Nonlin. Sci. Numer. Simulat. 15, 3067-3072.
40. Zhang, F., Shu, Y. & Yang, H. [2011] "Bounds for a new chaotic system and its application in chaos synchronization," Commun. Nonlin. Sci. Numer. Simulat. 16, 1501-1508.