高维映射的Hopf-pitchfork分岔及其在碰撞振动系统中的应用

Hopf-pitchfork Bifurcation of High Dimensionalmaps with Applications to Vibro-impact Systems

研究了高维映射的Hopf-pitchfork分岔。通过中心流形理论,将高维映射降阶为一个三维映射,再通过范式方法将降阶后的三维映射转化为范式映射。理论分析了三维范式映射在Hopf-pitchfork分岔点附近的参数开折。通过对分岔点附近含间隙振动系统的分岔行为进行数值仿真,验证了理论结果。

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.