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Dynamics of a new Lorenz-like chaotic system. (English) Zbl 1202.34083
Authors’ abstract: The present work is devoted to giving new insights into a new Lorenz-like chaotic system. The local dynamical entities, such as the number of equilibria, the stability of the hyperbolic equilibria and the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations and the local manifold character, are all analyzed when the parameters are varied in the space of parameters. The existence of homoclinic and heteroclinic orbits of the system is also rigorously studied. More exactly, for \(b\geq 2a>0\) and \(c>0\), we prove that the system has no homoclinic orbit but has two and only two heteroclinic orbits.

MSC:
34C28 Complex behavior and chaotic systems of ordinary differential equations
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[1] Lorenz, E.N., Deterministic nonperiodic flow, J. atmospheric sci., 20, 130-141, (1963) · Zbl 1417.37129
[2] Sparrow, C., The Lorenz equations: bifurcation, chaos, and strange attractor, (1982), Springer-Verlag New York
[3] Chen, G.; Ueta, T., Yet another chaotic attractor, Internat. J. bifur. chaos, 9, 1465-1466, (1999) · Zbl 0962.37013
[4] Lü, J.; Chen, G., A new chaotic attractor coined, Internat. J. bifur. chaos, 12, 659-661, (2002) · Zbl 1063.34510
[5] C˘elikovský, S.; Chen, G., On a generalized Lorenz canonical form of chaotic systems, Internat. J. bifur. chaos, 12, 1789-1812, (2002) · Zbl 1043.37023
[6] Yang, Q.; Chen, G.; Zhou, T., A unified Lorenz-type system and its canonical form, Internat. J. bifur. chaos, 16, (2006), 2855-1871 · Zbl 1185.37088
[7] Yang, Q.; Chen, G., A chaotic system with one saddle and two stable node-foci, Internat. J. bifur. chaos, 18, 1393-1414, (2008) · Zbl 1147.34306
[8] Chua, L.O.; Itoh, M.; Kovurev, L.; Eckert, K., Chaos synchronization in chua’s circuits, J. circuits syst. comput., 3, 93-108, (1993)
[9] Zhou, T.; Chen, G.; Tang, Y., Complex dynamical behaviors of the chaotic chen’s system, Internat. J. bifur. chaos, 13, 2561-2574, (2003) · Zbl 1046.37018
[10] Li, C.; Chen, G., A note on Hopf bifurcation in chen’s system, Internat. J. bifur. chaos, 13, 1609-1615, (2003) · Zbl 1074.34045
[11] Li, T.; Chen, G.; Tang, Y., On stability and bifurcation of chen’s system, Chaos solitons fractals, 19, 1269-1282, (2004) · Zbl 1069.34060
[12] Lü, J.; Zhou, T.; Chen, G.; Zhang, S., Local bifurcations of the Chen system, Int. J. bifur. chaos, 12, 2257-2270, (2002) · Zbl 1047.34044
[13] Ueta, T.; Chen, G., Bifurcation analysis of chen’s equation, Internat. J. bifur. chaos, 10, 1917-1931, (2000) · Zbl 1090.37531
[14] Yu, Y.; Zhang, S., Hopf bifurcation in the Lü system, Chaos solitons fractals, 17, 901-906, (2003) · Zbl 1029.34030
[15] Yu, Y.; Zhang, S., Hopf bifurcation analysis in the Lü system, Chaos solitons fractals, 21, 1215-1220, (2004) · Zbl 1061.37029
[16] Huang, K.; Yang, G., Stability and Hopf bifurcation analysis of a new system, Chaos solitons fractals, 39, 567-578, (2009) · Zbl 1197.34096
[17] Rössler, O.E., An equation for continuous chaos, Phys. lett. A, 57, 397-398, (1976) · Zbl 1371.37062
[18] Vanderschrier, G.; Maas, L., The diffusionless Lorenz equations; S˘ilnikov bifurcations and reduction to an explicit map, Physica D, 141, 19-36, (2000) · Zbl 0956.37038
[19] Alvarez, G.; Li, S.; Montoya, F.; Pastor, G.; Romera, M., Breaking projective chaos synchronization secure communication using filtering and generalized synchronization, Chaos solitons fractals, 24, 775-783, (2005) · Zbl 1068.94002
[20] Sprott, J.C., Some simple chaotic flows, Phys. rev. E, 50, 2, R647-R650, (1994)
[21] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys. rev. lett., 64, 8, 821-824, (1990) · Zbl 0938.37019
[22] Pecora, L.M.; Carroll, T.L., Driving systems with chaotic signals, Phys. rev. A, 44, 4, 2374-2383, (1991)
[23] Vane˘c˘ek, A.; C˘elikovský, S., Control systems: from linear analysis to synthesis of chaos, (1996), Prentice-Hall London · Zbl 0874.93006
[24] Li, T.; Chen, G., On homoclinic and heteroclinic orbits of chen’s system, Internat. J. bifur. chaos, 16, 3035-3041, (2006) · Zbl 1149.34030
[25] Tigan, G.; Constantinescu, D., Heteroclinic orbits in the \(T\) and the Lü systems, Chaos solitons fractals, 42, 20-23, (2009) · Zbl 1198.37029
[26] Kuznetsov, Y.A., Elements of applied bifurcation theory, (2004), Springer-Verlag New York · Zbl 1082.37002
[27] Rubinger, R.M.; Nascimento, A.W.M.; Mello, L.F.; Rubinger, C.P.L.; Manzanares Filho, N.; lbuquerque, H.A., Inductorless chua’s circuit: experimental time series analysis, Math. probl. eng., 2007, (2007), [Article ID 83893] · Zbl 1152.78325
[28] Mello, L.F.; Messias, M.; Braga, D.C., Bifurcation analysis of a new Lorenz-like chaotic system, Chaos solitons fractals, 37, 1244-1255, (2008) · Zbl 1153.37356
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