×

zbMATH — the first resource for mathematics

Controlling the diffusionless Lorenz equations with periodic parametric perturbation. (English) Zbl 1189.34118
Summary: Diffusionless Lorenz equations (DLE) are a simple one-parameter version of the well-known Lorenz model, which was obtained in the limit of high Rayleigh and Prandtl numbers, physically corresponding to diffusionless convection. A simple control method is presented to control chaos by using periodic parameter perturbation in DLE. By using the generalized Melnikov method, the parameter conditions could be obtained to guide the controlled DLE to a low-periodic motion. Moreover, the existence conditions of periodic orbits and homoclinic orbits in the system are given. Some results of the numerical simulation are also explained clearly by a rigorous analysis.

MSC:
34H10 Chaos control for problems involving ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34E15 Singular perturbations, general theory for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Sparrow, C., The Lorenz equations: bifurcation, chaos, and strange attractor, (1982), Springer New York
[2] 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
[3] Maas, L.R.M., A simple model for the three-dimensional thermally and wind-driven Ocean circulation, Tellus A, 46, 671-680, (1984)
[4] Wiggins, S.; Holmes, P., Periodic orbits in slowly varying oscillators, SIAM J. math. anal., 18, 592-611, (1987) · Zbl 0619.34041
[5] Wiggins, S.; Holmes, P., Homoclinic orbits in slowly varying oscillators, SIAM J. math. anal., 18, 612-629, (1987) · Zbl 0622.34041
[6] Li, J.; Zhang, J., New treatment on bifurcation of periodic solutions and homoclinic orbits at high \(r\) in the Lorenz equations, SIAM J. appl. math., 53, 1059-1071, (1993) · Zbl 0781.34031
[7] Huang, D., Periodic orbits and homoclinic orbits of the diffusionless Lorenz equations, Phys. lett. A, 309, 248-253, (2003) · Zbl 1009.37010
[8] Wu, Z.M.; Xie, J.Y.; Fang, Y.Y.; Xu, Z.Y., Controlling chaos with periodic parametric perturbations in Lorenz system, Chaos solitons fractals, 32, 104-112, (2007) · Zbl 1138.37314
[9] Yu, X., Controlling chaos using input – output linearization approach, Internat. J. bifur. chaos, 7, 1659-1664, (1997) · Zbl 0965.93030
[10] Yang, H.T.; Yan, J.J., Design of sliding mode controller for Lorenz chaotic system with ninlinear input, Chaos solitons fractals, 19, 891-898, (2004) · Zbl 1064.93010
[11] Zeng, Y.; Singh, S.N., Adaptive control of chaos in Lorenz system, Dyn. control, 7, 143-154, (1997) · Zbl 0875.93191
[12] Chen, S.H.; Zhang, Q.J.; Xie, J.; Wang, C.P., A stable-manifold-based method for chaos control and synchronization, Chaos solitions fractals, 20, 947-954, (2004) · Zbl 1050.93032
[13] Chen, M.Y; Zhou, D.H.; Yun, S., Nonlinear feedback control of Lorenz system, Chaos solitions fractals, 21, 295-304, (2004) · Zbl 1049.93036
[14] Guémez, J.; Gutiérrez, J.M.; Iglesias, A., Stabilization of periodic and quasiperiodic motion in chaotic systems through changes in the system variables, Phys. lett. A, 190, 429-433, (1994)
[15] Guémez, J.; Gutiérrez, J.M.; Iglesias, A., Suppression of chaos through changes in the system variables: transient chaos and crises, Physica D, 79, 164-173, (1994) · Zbl 0888.58029
[16] Guémez, J.; Gutiérrez, J.M.; Iglesias, A., Suppression of chaos through changes in the system variables though Poincaré and Lorenz return maps, Internat. J. bifur. chaos, 7, 1305-1313, (1996) · Zbl 0874.58041
[17] Mirus, K.A.; Sprott, J.C., Controlling chaos in low and high-dimensional systems with periodic parametric perturbations, Phys. rev. E, 59, 5313-5324, (1999)
[18] Olver, P.J., Applications of Lie groups to differential equations, (1986), Springer New York · Zbl 0656.58039
[19] Arnold, V.I., Mathematical methods of classical mechanics, (1978), Springer New York · Zbl 0386.70001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.