Abstract
Hopf bifurcation of a unified chaotic system – the generalized Lorenz canonical form (GLCF) – is investigated. Based on rigorous mathematical analysis and symbolic computations, some conditions for stability and direction of the periodic obits from the Hopf bifurcation are derived.
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References
Sparrow, C.: The Lorenz Equations: Bifurcation, Chaos, and Strange Attractor. New York, Springer-Verlag (1982)
Guckenheimer, J., Holmes, P.J.: Nonlinear oscillations dynamical systems and bifurcations of vector fields. New York, Springer-Verlag (1983)
Kuznetsov, Y.A.: Elements of applied bifurcation theory. New York, Springer-Verlag (1995)
Luo, D., Wang, X., Zhu, D., Han, M.: Bifurcation theory and methods of dynamical systems. Singapore, World Scientific (1997)
Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurcation Chaos 9, 1465–1466 (1999)
Ueta, T., Chen, G.: Bifurcation analysis of Chen’s equation. Int. J. Bifurcation Chaos 10, 1917–1931 (2000)
Zhou, T.S., Chen, G., Tang, Y.: Chen’s attractor exists. Int. J. Bifurcation Chaos 14, 3167–3178 (2004)
Li, T.C., Chen, G., Tang, Y.: On stability and bifurcation of Chen’s system. Chaos Solitons Fractals 19, 1269–1282 (2004)
Chang, Y., Chen, G.: Complex dynamics in Chen’s system. Chaos Solitons Fractals, in press
Zhou, T.S., Tang, Y., Chen, G.: Complex dynamical behaviors of the chaotic Chen’s System. Int. J. Bifurcation Chaos 13(9), 2561–2574 (2003)
Yu, X., Xia, Y.: Detecting unstable periodic orbits in Chen’s chaotic attractor. Int. J. Bifurcation Chaos 10, 1987–1991 (2001)
Čelikovský, S., Chen, G.: Hyperbolic-type generalized Lorenz system and its canonical form In: Proceedings of the 15th Triennial World Congress of IFAC. Barcelona, Spain, July 2002
Čelikovský, S., Chen, G.: On the generalized Lorenz canonical form. Chaos Solitons Fractals 26, 1271–1276 (2005)
Zhou, T.S., Chen, G., Čelikovský, S.: Si’lnikov chaos in the generalized Lorenz canonical form of dynamics systems. Nonlinear Dyn. 39, 319–334 (2005)
Bi, Q., Yu, P.: Symbolic computation of normal forms for semi-simple cases. J. Comput. Appl. Math. 102(2), 195–220 (1999)
Chow, S.N., Li, C.Z., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge, Cambridge University Press (1994)
Lynch, S.: Dynamical systems with application using maple. Boston, Birkhauser (2001)
Yu, P.: Simplest normal forms of Hopf and generalized Hopf bifurcations. Int. J. Bifurcation Chaos 9(10), 1917–1939 (1999)
Yu, P.: Symbolic computation of normal forms for resonant double Hopf bifurcations using a perturbation technique. J. Sound Vib. 247(4), 615–632 (2001)
Čelikovský, S., Chen, G.: On a generalized Lorenz canonical form of chaotic systems. Int. J. Bifurcation Chaos 12, 1789–1812 (2002)
Chen, G.: Beyond the Lorenz System, In: Proceedings of Chinese Conference on Dynamics, Nonlinear Vibrations, and Motion Stability. Nanjing, China, Oct. 28–31, (2004)
Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurcation Chaos 12, 659–661 (2002)
Shimizu, T., Morioka, N.: On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Physics Letters A 76, 201–204 (1976)
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Li, T., Chen, G., Tang, Y. et al. Hopf Bifurcation of the Generalized Lorenz Canonical Form. Nonlinear Dyn 47, 367–375 (2007). https://doi.org/10.1007/s11071-006-9036-x
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DOI: https://doi.org/10.1007/s11071-006-9036-x