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Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci. (English) Zbl 1200.65102
Summary: In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equilibrium structure of the chaotic system and can be applied to degenerate Hopf bifurcation at the desired location with preferred stability.

65P20 Numerical chaos
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37M20 Computational methods for bifurcation problems in dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
Full Text: DOI
[1] Sprott, J.C., Some simple chaotic flows, Phys. rev. E, 50, 647-650, (1994)
[2] Sprott, J.C., A new class of chaotic circuit, Phys. lett. A, 266, 19-23, (2000)
[3] Sprott, J.C., Simplest dissipative chaotic flow, Phys. lett. A, 228, 271-274, (1997) · Zbl 1043.37504
[4] Lorenz, E.N., Deterministic non-periodic flow, J. atmos. sci., 20, 130-141, (1963) · Zbl 1417.37129
[5] Chen, G.R.; Ueta, T., Yet another chaotic attractor, Int. J. bifurcat. chaos, 9, 1465-1466, (1999) · Zbl 0962.37013
[6] Lü, J.H.; Chen, G.R., A new chaotic attractor coined, Int. J. bifurcat. chaos, 12, 659-661, (2002) · Zbl 1063.34510
[7] Yang, Q.G.; Chen, G.R.; Huang, K.F., Chaotic attractors of the conjugate Lorenz-type system, Int. J. bifurcat. chaos, 17, 3929-3949, (2007) · Zbl 1149.37308
[8] Rössler, O.E., An equation for continuous chaos, Phys. lett. A, 57, 397-398, (1976) · Zbl 1371.37062
[9] van der Schrier, G.; Maas, L.R.M., The diffusionless Lorenz equations; S˘ilnikov bifurcations and reduction to an explicit map, Physica D, 141, 19-36, (2000) · Zbl 0956.37038
[10] Shaw, R., Strange attractor, chaotic behaviour and information flow, Z. naturforsch. A, 36, 80-112, (1981) · Zbl 0599.58033
[11] Yang, Q.G.; Chen, G.R., A chaotic system with one saddle and two stable node-foci, Int. J. bifurcat. chaos, 18, 1393-1414, (2008) · Zbl 1147.34306
[12] Q.G. Yang, Z.C. Wei, G.R. Chen, A unusual 3D autonomous quadratic chaotic system with two stable node-foci, Int. J. Bifur. Chaos 2010, doi:10.1142/S0218127410026320. · Zbl 1193.34091
[13] Munmuangsaen, B.; Srisuchinwong, B., A new five-term simple chaotic attractor, Phys. lett. A, 373, 4038-4043, (2009) · Zbl 1234.37030
[14] Wei, Z.C.; Yang, Q.G., Controlling the diffusionless Lorenz equations with periodic parametric perturbation, Comput. math. appl., 58, 1979-1987, (2009) · Zbl 1189.34118
[15] Chen, D.S.; Wang, H.O.; Chen, G., Anti-control of Hopf bifurcations, IEEE trans. circuits syst. I: FTA, 48, 661-672, (2001) · Zbl 1055.93037
[16] Chen, Z.; Yu, P., Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions, Chaos solitons fractals, 26, 1231-1248, (2005) · Zbl 1093.37508
[17] Hamzi, B.; Kang, W.; Barbot, J.P., Analysis and control of Hopf bifurcations, SIAM J. control optim., 42, 2200-2220, (2004) · Zbl 1069.93014
[18] Kuznetsov, Y.A., Elements of applied bifurcation theory, (1998), Springer-Verlag New York · Zbl 0914.58025
[19] Sotomayor, J.; Mello, L.F.; Braga, D.C., Bifurcation analysis of the watt governor system, Comp. appl. math., 26, 19-44, (2007) · Zbl 1182.70038
[20] J. Sotomayor, L.F. Mello, D.C. Braga, Lyapunov coefficients for degenerate Hopf bifurcations, (2007), arXiv:0709.3949v1 [math.DS]. Available from: <http://arxiv.org/abs/0709.3949>.
[21] Messias, M.; Braga, D.C.; Mello, L.F., Degenerate Hopf bifurcation in chua’s system, Int. J. bifurcat. chaos, 19, 497-515, (2009) · Zbl 1170.34333
[22] Mello, L.F.; Coelho, S.F., Degenerate Hopf bifurcations in the Lü system, Phys. lett. A, 373, 1116-1120, (2009) · Zbl 1228.70014
[23] Takens, F., Unfoldings of certain singularities of vectorfields: generalized Hopf bifurcations, J. diff. eqs., 14, 476-493, (1973) · Zbl 0273.35009
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