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Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz-Haken system. (English) Zbl 1197.37034
Summary: To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for the hyperchaotic Lorenz-Haken system using a technique combining the generalized Lyapunov function theory and optimization. For the Lorenz-Haken system, we derive a four-dimensional ellipsoidal ultimate bound and positively invariant set. Furthermore, the two-dimensional parabolic ultimate bound with respect to $x-z$ is established. Finally, numerical results to estimate the ultimate bound are also presented for verification. The results obtained in this paper are important and useful in control, synchronization of hyperchaos and their applications. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

37D45Strange attractors, chaotic dynamics
34C11Qualitative theory of solutions of ODE: growth, boundedness
34C14Symmetries, invariants (ODE)
34D35Stability of manifolds of solutions of ODE
Full Text: DOI
[1] Lorenz, E. N.: Deterministic non-periods flows, J atoms sci 20, 130-141 (1963)
[2] Celikovsky, S.; Chen, G.: On the generalized Lorenz canonical form, Chaos, solitons & fractals 26, 1271-1276 (2005) · Zbl 1100.37016
[3] Cenys, A.; Tamaservicius, A.; Baziliauskas, A.; Krivickas, R.; Lindberg, E.: Hyperchaos in coupled colpitts oscillators, Chaos, solitons & fractals 17, No. 2 -- 3, 349-353 (2003) · Zbl 1036.94505
[4] Rössler, O. E.: An equation for the hyperchaos, Phys lett A 71, No. 2,3, 155-157 (1979) · Zbl 0996.37502 · doi:10.1016/0375-9601(79)90150-6
[5] Cafagna, D.; Grassi, G.: New 3D-scroll attractors in hyperchaotic Chua’s circuits forming a ring, Int J bifurcat chaos 13, No. 10, 2889-2903 (2003) · Zbl 1057.37026 · doi:10.1142/S0218127403008284
[6] Barbara, C.; Silvano, C.: Hyperchaotic behaviour of two bi-directionally Chua’s circuits, Int J circ theory appl 30, No. 6, 625-637 (2002) · Zbl 1024.94522 · doi:10.1002/cta.213
[7] Park, J. H.: Adaptive synchronization of hyperchaotic Chen system with uncertain parameters, Chaos, solitons & fractals 26, 959-964 (2005) · Zbl 1093.93537
[8] Zhou, T.; Tang, Y.; Chen, G.: Complex dynamical behaviors of the chaotic Chen’s system, Int J bifurcat chaos 9, 2561-2574 (2003) · Zbl 1046.37018 · doi:10.1142/S0218127403008089
[9] Leonov, G.: Bound for attractors and the existence of homoclinic orbit in the Lorenz system, J appl math mech 65, 19-32 (2001) · Zbl 1025.34048
[10] Leonov, G.; Bunin, A.; Koksch, N.: Attractor localization of the Lorenz system, Zamm 67, 649-656 (1987) · Zbl 0653.34040 · doi:10.1002/zamm.19870671215
[11] Li, D.; Lu, J.; Wu, X.; Chen, G.: Estimating the bounds for the Lorenz family of chaotic systems, Chaos, solitons & fractals 23, 529-534 (2005) · Zbl 1061.93506 · doi:10.1016/j.chaos.2004.05.021
[12] Li, D.; Lu, J.; Wu, X.; Chen, G.: Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, J math anal appl 323, 844-853 (2006) · Zbl 1104.37024 · doi:10.1016/j.jmaa.2005.11.008
[13] Liao, X.: On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci China ser E inform sci 34, 1404-1419 (2004)
[14] Lefchetz, S.: Differential equations: geometric theory, (1963) · Zbl 0107.07101
[15] Ning, C. Z.; Haken, H.: Detuned lasers and the complex Lorenz equation: subcritical and supercritical Hopf bifurcations, Phys rev 41, No. 7, 326 (1990)
[16] Fang, J.: Synchronizing hyperchaos and controlling hyperchaos, Chinese sci bull 40, 988-993 (1995)
[17] Colet, P.; Roy, R.; Weisenfeld, K.: Controlling hyperchaos in a multimode laser model, Phys rev E 50, 3453-3457 (1995)