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Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system

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  • Zhang, Jian-Gang
  • Li, Xian-Feng
  • Chu, Yan-Dong
  • Yu, Jian-Ning
  • Chang, Ying-Xiang

Abstract

In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.

Suggested Citation

  • Zhang, Jian-Gang & Li, Xian-Feng & Chu, Yan-Dong & Yu, Jian-Ning & Chang, Ying-Xiang, 2009. "Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2150-2168.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:5:p:2150-2168
    DOI: 10.1016/j.chaos.2007.06.131
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    References listed on IDEAS

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    1. Gakkhar, Sunita & Singh, Brahampal, 2005. "Complex dynamic behavior in a food web consisting of two preys and a predator," Chaos, Solitons & Fractals, Elsevier, vol. 24(3), pages 789-801.
    2. Gakkhar, Sunita & Singh, Brahampal, 2006. "Dynamics of modified Leslie–Gower-type prey–predator model with seasonally varying parameters," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1239-1255.
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    5. Merta, Henryk, 2006. "Characteristic time series and operation region of the system of two tank reactors (CSTR) with variable division of recirculation stream," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 279-285.
    6. Merta, Henryk & Pelka, Rafał, 2005. "Chaotic dynamics of a cascade of plug flow tubular reactors (PFTRs) with division of recirculating stream," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1211-1219.
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    Cited by:

    1. Chu, Yan-Dong & Chang, Ying-Xiang & Zhang, Jian-Gang & Li, Xian-Feng & An, Xin-Lei, 2009. "Full state hybrid projective synchronization in hyperchaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1502-1510.
    2. Chu, Yan-Dong & Li, Xian-Feng & Zhang, Jian-Gang & Chang, Ying-Xiang, 2009. "Nonlinear dynamics analysis of a modified optically injected semiconductor lasers model," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 14-27.
    3. Taheri, Alireza Ghomi & Setoudeh, Farbod & Tavakoli, Mohammad Bagher & Feizi, Esmaeil, 2022. "Nonlinear analysis of memcapacitor-based hyperchaotic oscillator by using adaptive multi-step differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).

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