Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping.

*(English)*Zbl 1237.70094Summary: The dynamic behavior of a symmetric gyro with linear-plus-cubic damping, which is subjected to a harmonic excitation, is studied in this paper. The Liapunov direct method has been used to obtain the sufficient conditions of the stability of the equilibrium points of the system. By applying numerical results, time history, phase diagrams, PoincarĂ© maps, Liapunov exponents and Liapunov dimensions are presented to observe periodic and chaotic motions. Besides, several control methods, the delayed feedback control, the addition of constant motor torque, the addition of period force, and adaptive control algorithm (ACA), have been used to control chaos effectively. Finally, attention is shifted to the synchronization of chaos in the two identical chaotic motions of symmetric gyros. The results show that one can make two identical chaotic systems to synchronize through applying four different kinds of one-way coupling. Furthermore, the synchronization time is also examined.

##### MSC:

70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |

70Q05 | Control of mechanical systems |

34H10 | Chaos control for problems involving ordinary differential equations |

34C28 | Complex behavior and chaotic systems of ordinary differential equations |

34D99 | Stability theory for ordinary differential equations |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |