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.