多自由度碰撞振动系统的对称性、分岔与混沌研究
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摘要
本文以两个典型的三自由度对称碰撞振动模型为研究对象,系统地研究了该类非光滑动力系统的分岔和混沌等动力学行为。研究表明:有对称双侧刚性约束的碰撞振动系统(对称碰撞振动系统)的动力学行为与一般的碰撞振动系统相比有很大的不同。
第一章从碰撞振动系统的周期运动的稳定性、分岔以及混沌等方面的理论研究和工程应用背景出发,综述了部分研究成果、最新发展动态和存在的主要问题。
第二章研究了三自由度对称碰撞振动系统的对称周期n-2运动以及Poincaré映射的对称性。讨论了其对称周期n-2运动的存在性条件,并得到了对称周期n-2运动的解析解。确定了系统的Poincaré截面及其对称截面,建立了一个对称变换,并构造了Poincaré映射。研究发现系统的Poincaré映射具有一定形式的对称性,在这种对称性质作用下,Poincaré映射P可以表示为另外一个映射Q_γ的二次复合。定义了对称不动点和反对称不动点的概念,它们分别对应于碰撞振动系统的对称周期n-2运动和反对称周期n-2运动。利用Poincaré映射的线性化矩阵的特征值,结合映射不动点的分岔理论,证明了碰撞振动系统的对称不动点(对称周期n-2运动)不可能发生余维一的周期倍化分岔,以及Hopf-flip分岔和pitchfork-flip分岔。证明了两个反对称不动点(反对称周期n-2运动)的Poincaré映射的线性化矩阵具有相同的特征值,从而说明它们具有相同的稳定性。计算了对称不动点(对称周期n-2运动)的Poincaré映射的线性化矩的特征值。通过数值模拟发现对称不动点可能发生音叉分岔和Hopf分岔。
第三章研究了一类三自由度对称碰撞振动系统在音叉分岔后通向混沌之路。利用动力系统中的极限集理论,研究了对称碰撞振动系统吸引子的对称性。讨论了从非对称极限集转化为对称极限集的条件,并得到以下结论:若ω-极限集与其共轭极限集的交集非空,则该ω-极限集是对称极限集。数值模拟不仅得到了非对称的共轭混沌吸引子和对称混沌吸引子,还得到了非对称的共轭拟周期吸引子和对称拟周期吸引子。在一定的参数区间上,吸引子则可能反复经历失去对称性和恢复对称性的过程。对于映射P而言,拟周期吸引子明显地具有“爆发”的特征,即吸引子的形状突然变大,并延伸到与自身对称的区域,从而完成了从非对称极限集到对称极限集的演化,因此这个过程可称为吸引子的“恢复对称性”分岔。对于映射Q_γ而言,则可以认为是两个共轭的拟周期吸引子互相碰撞并同时突然融合到对方体内,完成了从非对称极限集到对称极限集的演化,从而生成一个外形尺寸更大的具有对称性的拟周期吸引子。以上这两种解释不仅对于拟周期吸引子是有效的,对于其它类型的吸引子(例如混沌吸引子)同样是有效的。
第四章研究了三自由度对称碰撞振动系统的余维二分岔。因为Poincaré映射P的对称性能够通过映射Q_γ表示,对于映射P在分岔点处的范式研究,可转化为对映射Q_γ的范式研究。讨论了Poincaré映射P在几种余维二分岔点处的范式映射,包括:Hopf-Hopf分岔,Hopf-pitchfork分岔,以及1:2共振的情况。以上这三种情况分别对应于映射Q_γ的Hopf-Hopf分岔,Hopf-flip分岔,以及1:4共振的情况。在这三种余维二分岔的情况下,虽然映射P及其所对应的映射Q_γ的范式的形式是完全相同的,但是范式映射的系数却是不同的。这当然导致了它们范式映射在余维二分岔点处的开折图中的区间边界的不同。对模型一的Hopf-Hopf分岔和1:2共振的情况,以及模型二的Hopf-pitchfork分岔进行了数值分析。还通过数值模拟发现了三个孤立的稳定Hopf圈共存的现象,其演化顺序为:一个不稳定对称不动点→一个半稳定的对称Hopf圈→三个稳定Hopf圈。目前还不能对这种现象给出较好的理论解释。
第五章研究了对称碰撞振动系统的Lyapunov指数的计算方法。对于具有对称双侧刚性约束的碰撞振动系统,利用其Poincaré映射P的对称性质,可以通过构造一个虚拟隐式映射Q_γ来计算所有Lyapunov指数。当得到虚拟隐式映射Q_γ之后,便可以引用光滑系统中采用时间序列的方法来计算Lyapunov指数。当得到全部的Lyapunov指数之后,就可以计算Lyapunov维数,并可以之来衡量混沌吸引子的奇异性。利用Lyapunov指数来区别长周期运动和混沌运动也是十分有效的。本章还说明了在碰撞振动系统中取不同的Poincaré截面构造Poincaré映射P对于研究系统的局部动力学行为是等效的。
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.
第一章从碰撞振动系统的周期运动的稳定性、分岔以及混沌等方面的理论研究和工程应用背景出发,综述了部分研究成果、最新发展动态和存在的主要问题。
第二章研究了三自由度对称碰撞振动系统的对称周期n-2运动以及Poincaré映射的对称性。讨论了其对称周期n-2运动的存在性条件,并得到了对称周期n-2运动的解析解。确定了系统的Poincaré截面及其对称截面,建立了一个对称变换,并构造了Poincaré映射。研究发现系统的Poincaré映射具有一定形式的对称性,在这种对称性质作用下,Poincaré映射P可以表示为另外一个映射Q_γ的二次复合。定义了对称不动点和反对称不动点的概念,它们分别对应于碰撞振动系统的对称周期n-2运动和反对称周期n-2运动。利用Poincaré映射的线性化矩阵的特征值,结合映射不动点的分岔理论,证明了碰撞振动系统的对称不动点(对称周期n-2运动)不可能发生余维一的周期倍化分岔,以及Hopf-flip分岔和pitchfork-flip分岔。证明了两个反对称不动点(反对称周期n-2运动)的Poincaré映射的线性化矩阵具有相同的特征值,从而说明它们具有相同的稳定性。计算了对称不动点(对称周期n-2运动)的Poincaré映射的线性化矩的特征值。通过数值模拟发现对称不动点可能发生音叉分岔和Hopf分岔。
第三章研究了一类三自由度对称碰撞振动系统在音叉分岔后通向混沌之路。利用动力系统中的极限集理论,研究了对称碰撞振动系统吸引子的对称性。讨论了从非对称极限集转化为对称极限集的条件,并得到以下结论:若ω-极限集与其共轭极限集的交集非空,则该ω-极限集是对称极限集。数值模拟不仅得到了非对称的共轭混沌吸引子和对称混沌吸引子,还得到了非对称的共轭拟周期吸引子和对称拟周期吸引子。在一定的参数区间上,吸引子则可能反复经历失去对称性和恢复对称性的过程。对于映射P而言,拟周期吸引子明显地具有“爆发”的特征,即吸引子的形状突然变大,并延伸到与自身对称的区域,从而完成了从非对称极限集到对称极限集的演化,因此这个过程可称为吸引子的“恢复对称性”分岔。对于映射Q_γ而言,则可以认为是两个共轭的拟周期吸引子互相碰撞并同时突然融合到对方体内,完成了从非对称极限集到对称极限集的演化,从而生成一个外形尺寸更大的具有对称性的拟周期吸引子。以上这两种解释不仅对于拟周期吸引子是有效的,对于其它类型的吸引子(例如混沌吸引子)同样是有效的。
第四章研究了三自由度对称碰撞振动系统的余维二分岔。因为Poincaré映射P的对称性能够通过映射Q_γ表示,对于映射P在分岔点处的范式研究,可转化为对映射Q_γ的范式研究。讨论了Poincaré映射P在几种余维二分岔点处的范式映射,包括:Hopf-Hopf分岔,Hopf-pitchfork分岔,以及1:2共振的情况。以上这三种情况分别对应于映射Q_γ的Hopf-Hopf分岔,Hopf-flip分岔,以及1:4共振的情况。在这三种余维二分岔的情况下,虽然映射P及其所对应的映射Q_γ的范式的形式是完全相同的,但是范式映射的系数却是不同的。这当然导致了它们范式映射在余维二分岔点处的开折图中的区间边界的不同。对模型一的Hopf-Hopf分岔和1:2共振的情况,以及模型二的Hopf-pitchfork分岔进行了数值分析。还通过数值模拟发现了三个孤立的稳定Hopf圈共存的现象,其演化顺序为:一个不稳定对称不动点→一个半稳定的对称Hopf圈→三个稳定Hopf圈。目前还不能对这种现象给出较好的理论解释。
第五章研究了对称碰撞振动系统的Lyapunov指数的计算方法。对于具有对称双侧刚性约束的碰撞振动系统,利用其Poincaré映射P的对称性质,可以通过构造一个虚拟隐式映射Q_γ来计算所有Lyapunov指数。当得到虚拟隐式映射Q_γ之后,便可以引用光滑系统中采用时间序列的方法来计算Lyapunov指数。当得到全部的Lyapunov指数之后,就可以计算Lyapunov维数,并可以之来衡量混沌吸引子的奇异性。利用Lyapunov指数来区别长周期运动和混沌运动也是十分有效的。本章还说明了在碰撞振动系统中取不同的Poincaré截面构造Poincaré映射P对于研究系统的局部动力学行为是等效的。
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.
引文
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