振动筛系统的Hopf-Hopf-Flip分岔与混沌演化

建立了振动筛系统的动力学模型和周期运动的六维Poincaré映射,基于Poincaré映射方法和数值仿真分析了此系统在余维三分岔点附近的动力学行为。研究了其Jacobian矩阵两对复共轭特征值和一负实特征值同时穿越单位圆情况下的Hopf-Hopf-Flip分岔,该系统在此类余维三分岔点附近存在周期运动的Hopf分岔、Flip分岔、环面分岔以及"五角星形"概周期吸引子,揭示了环面倍化以及分形出"五角星形"概周期吸引子并向混沌演化的两种非常规过程,它对于振动筛系统的动力学优化设计提供了理论参考。

HOPF-HOPF-FLIP BIFURCATION AND ROUTES TO CHAOS OF A SHAKER SYSTEM

The dynamical model and six-dimensional Poincaré maps of a shaker system are established in this paper firstly.Then,using Poincaré maps and numerical integral method,this paper investigates its local codimension-3 bifurcation,concerning the case of two complex conjugate pairs of eigenvalues and a negative eigenvalue of linearized map escaping the unit circle simultaneously.Local behaviors of the system,near the point of Hopf-Hopf-Flip bifurcation,are studied,where Hopf bifurcation occurs,as well as Flip bifurcation,torus bifurcation and "pentagram" attractor in projected Poincaré sections.The routes to chaos via torus-doubling bifurcation and gradual fractalization of torus represented by "pentagram" attractor are analyzed by numerical simulation.The system parameters of a shaker may be optimized by studying the stability and bifurcation of periodic motion.