Research On Symmetry, Bifurcation And Chaos Of Multi-Degree-Of-Freedom Vibro-Impact Systems

This dissertation considers two typical three-degree-of-freedom vibro-impact systems with symmetric two-sided rigid constraints(i.e.,symmetric vibro-impact systems),and studies bifurcations and chaos in these non-smooth dynamic systems.It is shown there are some important differences in the dynamic behaviour between the symmetric vibro-impact systems and the general ones.Based on the theoretic researchs and engineering applications on the stability, bifurcation of periodic motion and chaos in vibro-impact systems,charpter one surveys the recent achievements,developments and unsolved problems.Chapter two considers the symmetric period n-2 motion of the symmetric vibro-impact system,and sduties the symmetric property of the Poincarémap of the system.The existence conditions of the symmetric period n-2 motion are obtained,and the solutions of the symmetric period n-2 motion is deduced.The Poincarésection and its symmetric section are determined,and a symmetric transformation is established.Subsequently,the Poincarémap of the system is constructed.It is shown that the Poincarémap exhibits some symmetric property, and as a result,the Poincarémap P can be expressed as the second iteration of another implicit map Q_γ.The symmetric period n-2 fixed point and the antisymmetric period n-2 fixed point are defined,and they correspond to the symmetric period n-2 motion and the antisymmetric period n-2 motion, respectively.Based on bifurcation theories of the fixed point of the map,it is proved that period-doubling bifurcation,Hopf-flip bifurcation and pitchfork-flip bifurcation are suppresed by the symmetry of the Poincarémap.For the two antisymmetric fixed points,it is shown that they have the same stability since their Jacobian matrice have the same eigenvalues.The Jacobian matrix of the Poincarémap is also computed in detail.In numerical simulations,the eigenvalues of the Jacobian matrix is the basis of analysing various bifurcation phenomena. Numerical simulation shows that the symmetric period n-2 fixed point may have both pitchfork bifurcation and Hopf bifurcation.Chapter three mainly focuses on the routes to chaos after pitchfork bifurcation in symmetric vibro-impact systems.Using the limit set theory in dynamical systems,the symmetry of the attractors in symmetric vibro-impact systems is investigated.The conception of the limit set of the implicit map is induced,and the attrator is defined as a asympotic stableω-limit set.Special interesting is given in the condition of the transformation from the asymmetric limit set to symmetric limit set,and the following conclusion is obtained:If the intersection of theω-limit set and its conjugate limit set is non-empty,then the co-limit set is a symmetric limit set.By numerical simulations,not only asymmetric conjugate chaotic attractors and symmetric chaotic attractor,but also asymmetric conjugate quasi-periodic attractors and symmetric quasi-periodic attractor,are obtained.It is also shown that in some parameter region,attractors may undergo symmetry-breaking and symmetry-restoring processes repeatedly.For the the Poincarémap P,attractors may grow in size abruptly and has explosion property. Due to the birth of a larger symmetric attractor,this phenomenon is called symmetry-restoring.At the same time,for the implicit map Q_γ,the two conjugate attractors collide to and merge into each other suddenly,and these two conjugate attractors evolve into the new symmetric attractor.The above two explanations are true for both quasi-periodic attractors and chaotic attractors.Chapter four discusses codimension two bifurcations of the period n-2 motion in symmetric vibro-impact systems.Since the symmetry characteristic of the Poincarécan be captured by the implicit map Q_γ,then the study on the normal form of the Poincarémap P near the bifurcation point is translated into the study on the normal form of the map Q_γ.Hopf-Hopf bifurcation,Hopf-pitchfork bifurcation and bifurcation satisfying 1:2 resonance conditions of the Poincarémap P are correspond to Hopf-Hopf bifurcation,Hopf-flip bifurcation and bifurcation satisfying 1:4 resonance conditions of the map Q_γ,respectively.It is shown that for the corresponding codimension two bifurcation,the normal form of the Poincarémap P are same to that of the map Q_γ,but the coefficients of the associated normal form are different,which makes the different area bound in the folding portraits near the codimension two bifurcation.By numerical simulations, Hopf-Hopf bifurcation,Hopf-pitchfork bifurcation and bifurcation satisfying 1:2 resonance conditions of the Poincarémap P are obtained.In addition,three isolated stable Hopf circles of the Poincarémap P are also obtained,and the evolving sequence is:A unstable symmetriC period n-2 fixed point→a symmetric semi-stable Hopf circle→three stable Hopf circles.No reasonable explanations can be given for this phenomenon so far.Chapter five represents studies on the computation method of Lyapunov exponents in symmetric vibro-impact systems.Due to the symmetry of the Poincarémap P,a implicit map Q_γis established to compute all the Lyapunov exponents.Based on the map Q_γ,the method of computing Lyapunov exponents from time series in smooth dynamic systems is applied to the symmetric vibro-impact systems.Once all the Lyapunov exponents are obtained,Lyapunov dimension can be calculated,and offers a measurement of the degree of the singularity of the chaotic attractor.It is also effective to distinguish long periodic motion and chaotic motion making use of Lyapunov exponents.In symmetric vibro-impact systems,if different Poincarésections are chosen for constructing Poincarémap,they are equivalent for computing Lyapunov exponents.