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Bursting phenomena as well as the bifurcation mechanism in controlled Lorenz oscillator with two time scales. (English) Zbl 1242.34058

Summary: A controlled Lorenz model with fast-slow effect has been established, in which there exist order gap between the variables associated with the controller and the original Lorenz oscillator, respectively. The conditions of fold bifurcation as well as Hopf bifurcation for the fast subsystem are derived to investigate the mechanism of the behaviors of the whole system. Two cases in which the equilibrium points of the fast subsystem behave in different characteristics have been considered, leading to different dynamical evolutions with the change of coupling strength. Several types of bursting phenomena, such as fold/fold burster, fold/Hopf burster, near-fold/Hopf burster, fold/near-Hopf buster have been observed. Theoretical analysis shows that the bifurcations points which connect the quiescent state and the repetitive spiking state agree well with the turning points of the trajectories of the bursters. Furthermore, the mechanism of the period-adding bifurcations, resulting in the rapid change of the period of the movements, is presented.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] Barrio, R.; Serrano, S., Physica D: Nonlinear Phenomena, 229, 1, 43 (2007) · Zbl 1120.34322
[2] Hinke, M. O.; Bernd, K., Computers and Graphics, 26, 5, 815 (2002)
[3] Pishkenari, H. N.; Shahrokhi, M.; Mahboobi, S. H., Chaos, Solitons and Fractals, 32, 2, 832 (2007) · Zbl 1138.37311
[4] Galias, Z.; Zgliczyński, P., Physica D: Nonlinear Phenomena, 115, 3-4, 165 (1998) · Zbl 0941.37018
[5] Guellal, S.; Grimalt, P.; Cherruault, Y., Computers and Mathematics with Applications, 33, 3, 25 (1997) · Zbl 0869.65044
[6] Schmutz, M.; Rueff, M., Physica D: Nonlinear Phenomena, 11, 1-2, 167 (1984) · Zbl 0593.58025
[7] Pivka, L.; Wu, C. W.; Huang, Anshan, Journal of the Franklin Institute, 331, 6, 705 (1994) · Zbl 0833.34043
[8] Tlelo-Cuautle, E.; Muñoz-Pacheco, Jesús M., Applied Mathematics and Computation, 190, 2, 1526 (2007) · Zbl 1227.65128
[9] Yassen, M. T., Applied Mathematics and Computation, 131, 1, 171 (2002) · Zbl 1042.49002
[10] Jose, A.; Julio, S.; Puebla, Hector, Physics Letters A, 338, 2, 128 (2005) · Zbl 1136.37365
[11] Awad, E.; Bukhari, F., Applied Mathematics and Computation, 144, 2-3, 337 (2003) · Zbl 1036.49028
[12] Baek, J.; Lee, H.; Kim, S.; Park, M., Physics Letters A, 373, 47, 4368 (2009) · Zbl 1234.34012
[13] Yang, Z. Q.; Lu, Q. S.; Li, L., Chaos, Solitons and Fractals, 27, 3, 689 (2006) · Zbl 1083.37535
[14] Wang, H. X.; Lu, Q. S.; Wang, Q. Y., Communications in Nonlinear Science and Numerical Simulation, 13, 8, 1668 (2008) · Zbl 1221.37199
[15] Bravo de la Parra, R.; Arino, O.; Sánchez, E.; Auger, P., Mathematical and Computer Modelling, 31, 4-5, 17 (2000) · Zbl 1043.92519
[16] Innocenti, M.; Greco, L.; Pollini, L., Automatica, 39, 2, 273 (2003) · Zbl 1011.93072
[17] Han, X.; Jiang, B.; Bi, Q., Acta Physica Sinica, 58, 7, 4408 (2009)
[18] Han, X.; Jiang, B.; Bi, Q., Physics Letters A, 373, 3643 (2009)
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