Regular and chaotic dynamic analysis and control of chaos for a vertically vibrating and rotating circular tube containing a particle

Abstract

The paper presents the detailed nonlinear dynamic analysis of a vertically vibrating and rotating circular tube containing a particle. By adding a harmonic periodic vibration and damping force on this nonlinear system, enriched dynamic behaviors of the nonlinear system are presented. By applying the Lyapunov direct method, the conditions of stability and instability of relative equilibrium position can be determined. A codimension one bifurcation analysis for the autonomous system is carried out near the degenerate point. It is found that Hopf bifurcation occurs in the system by center manifold theory. And by applying various numerical results, such as phase portrait, Poincare map, time history and power spectrum analysis, the behaviors of the periodic and chaotic motion can be presented. The effects of the change of parameters in the system can be found in the bifurcation diagrams and parameter diagram. Further, by using Lyapunov exponents and Lyapunov dimensions we can verify the chaotic behavior. Finally, seven methods, namely, the adding of constant torque, the adding of periodic torque, the adding of impulse, delayed feedback control, adaptive control, bang-bang control, optimal control, are use to control chaos effectively to periodic orbit.