Symmetry And Bifurcations Of Vibro-Impact Systems

This dissertaton mainly presents a study on the symmetric period n-2 motion and the symmetry of the Poincare map of the vibro-impact system. It is proved that -1 couldn't be the eigenvalue of Jacobian matrix of Poincare map at .fixed point by the analysis of the symmetry of. Poincare map for one-degree-of-freedom impact oscillator. A large number of numerical simulation shows that one-degree-of-freedom impact oscillator between two rigid sides could only have pitchfork bifurcations, while the two-degree-of-freedom vibro-impact system could have pitchfork bifurcations and Hopf bifurcations.Chapter 1 surveys some recent achievements and developments of the study on the stability, bifurcation and chaos of vibro-impact system. The main contents and results of the dissertation are introduced at the same time.In chapter 2, an one-degree-of-freedom impact oscillator between two rigid sides is considered. The symmetric period n-2 motion and its Poincare map are derived analytically. It is proved that -1 couldn't be the eigenvalue of Jacobian matrix of Poincare map at fixed point by the analysis of the symmetry of the system. The numerical simulation shows that the one-degree-of-freedom impact oscillator between two rigid sides could only have pitchfork bifurcations. Symmetric fixed point could be transformed into two antisymmetrical fixed points via pitchfork bifurcations, and they have the same stability. They give birth to two synchronous period-doubling sequencies respectively,and bring about two antisymmetrical chaotic attractors by themselves.Chapter 3 investigates the periodic motion and Poincare maps of a two-degree-of-freedom vibro-impact system. The stability of the periodic motion is determined by the eigenvalues of the Jacobian matrix. It is shown that there exist Hopf bifurcations and period-doubling bifurcations in the vibro-impact system under suitable system parameters. The quasi-periodic responses of the...