Hopf-pitchfork Bifurcation of High Dimensional maps with Applications to Vibro-impact Systems
Xu Huidong,Xie Jianhua
(Department of Applied Mechanics and Engineering, Southwest Jiaotong University, 610031, Chengdu, China)

Abstract: Hopf-pitchfork bifurcation of high dimensional maps is dealt with here. When a real eigenvalue of the Jacobian matrix of the map at fixed point gets beyond the value +1 and a pair of complex conjugate eigenvalues cross the unit circle simultaneously, the high-dimensional map is reduced to a three-dimensional map by the center manifold theorem. The reduced map is further transformed into its normal form following the theory of normal forms. The two-parameter unfolding of the map near the point of Hopf-pitchfork bifurcation is investigated analytically. The numerical simulation results indicate that the vibro-impact system demonstrates a complicated dynamic behavior.
Keywords: maps, vibro-impact, Hopf-pitchfork bifurcation, chaos.