碰撞振动系统的一类余维二分岔及T^2环面分岔

A CODIMENSION-2 BIFURCATION AND T^2 TORUS BIFURCATION OF THREE-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEM

建立了三自由度碰撞振动系统的动力学模型及其周期运动的Poincar映射,当Jacobi矩阵存在两对共轭复特征值同时在单位圆上时,通过中心流形-范式方法将六维映射转变为四维范式映射。理论分析了这种余维二分岔问题,给出了局部动力学行为的两参数开折。证明系统在一定的参数组合下,存在稳定的Hopf分岔和T^2环面分岔。数值计算验证了理论结果。

A three-degree-of-freedom vibro-impact system is considered in this paper. The dynamical model and Poincare maps are established. When two pairs of complex conjugate eigenvalues of the Jacobi matrix cross the unit circle simultaneously, by considerable derivation, the six-demensional map can be reduced to a four-dimensional normal form by the center manifold theorem and theory of normal forms. Further by putting in polar coordinates and ignoring the azimuthal components, we investigate theoretically this codimension-two bifurcation which is characterized by so-called Hopf-Hopf degeneracy. The two-parameter unfoldings of local dynamical behavior are put forward. It is proved that there exist the Hopf bifurcation and T2 torus bifurcation under certain parameter combinations. Numerical simulation results indicate that the vibro-impact system presents complicate and interesting T1 invariant circle and invariant T2 torus as two controlling parameters varying near the critical point. Investigation of torus bifurcation of vibro-impact system has important significance for studying global dynamical behavior and route to chaos via quasi-period bifurcation.