Yue Yuan, Miao Pengcheng, Xie Jianhua, Celso Grebogi
Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China.
Institute for Complex Systems and Mathematical Biology King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom.
Chaos. 2016 Nov;26(11):113121. doi: 10.1063/1.4968552.
Quasiperiodic chaos (QC), which is a combination of quasiperiodic sets and a chaotic set, is uncovered in the six dimensional Poincaré map of a symmetric three-degree of freedom vibro-impact system. Accompanied by symmetry restoring bifurcation, this QC is the consequence of a novel intermittency that occurs between two conjugate quasiperiodic sets and a chaotic set. The six dimensional Poincaré map P is the 2-fold composition of another virtual implicit map Q, yielding the symmetry of the system. Map Q can capture two conjugate attractors, which is at the core of the dynamics of the vibro-impact system. Three types of symmetry restoring bifurcations are analyzed in detail. First, if two conjugate chaotic attractors join together, the chaos-chaos intermittency induced by attractor-merging crisis takes place. Second, if two conjugate quasiperiodic sets are suddenly embedded in a chaotic one, QC is induced by a new intermittency between the three attractors. Third, if two conjugate quasiperiodic attractors connect with each other directly, they merge to form a single symmetric quasiperiodic one. For the second case, the new intermittency is caused by the collision of two conjugate quasiperiodic attractors with an unstable symmetric limit set. As the iteration number is increased, the largest finite-time Lyapunov exponent of the QC does not converge to a constant, but fluctuates in the positive region.
准周期混沌（QC）是准周期集和混沌集的组合，在一个对称的三自由度振动冲击系统的六维庞加莱映射中被发现。伴随着对称性恢复分岔，这种QC是在两个共轭准周期集和一个混沌集之间出现的一种新型间歇性的结果。六维庞加莱映射P是另一个虚拟隐映射Q的2重复合，从而产生了系统的对称性。映射Q可以捕获两个共轭吸引子，这是振动冲击系统动力学的核心。详细分析了三种类型的对称性恢复分岔。第一，如果两个共轭混沌吸引子合并在一起，由吸引子合并危机引起的混沌 - 混沌间歇性就会发生。第二，如果两个共轭准周期集突然嵌入到一个混沌集中，QC是由三个吸引子之间的一种新的间歇性引起的。第三，如果两个共轭准周期吸引子直接相互连接，它们会合并形成一个单一的对称准周期吸引子。对于第二种情况，新的间歇性是由两个共轭准周期吸引子与一个不稳定对称极限集的碰撞引起的。随着迭代次数的增加，QC的最大有限时间李雅普诺夫指数不会收敛到一个常数，而是在正区域波动。
