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Determining Lyapunov spectrum and Lyapunov dimension based on the Poincaré map in a vibro-impact system

机译：在振动碰撞系统中基于庞加莱图确定李雅普诺夫谱和李雅普诺夫维数

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摘要

We develop a method to compute the Lyapunov spectrum and Lyapunov dimension, which is effective for both symmetric and unsymmetric vibro-impact systems. The Poincaré section is chosen at the moment after impacting, and the sixdimensional Poincaré map is established. The time between two consecutive impacts is determined by the initial conditions and the impact condition, hence the Poincaré map is an implicit map. The Poincaré map is used to calculate all the Lyapunov exponents and the Lyapunov dimension. By numerical simulations, the attractors are represented in the projected Poincaré section, and the Lyapunov spectrum is obtained. The multi-degree-of-freedom vibro-impact system may exhibit complex quasi-periodic attractors, which can be characterized by the Lyapunov dimension.
机译：我们开发了一种计算Lyapunov谱和Lyapunov维数的方法，该方法对于对称和不对称的振动系统都是有效的。撞击后立即选择“庞加莱”部分，并建立六维庞加莱地图。两次连续碰撞之间的时间取决于初始条件和碰撞条件，因此庞加莱图是一个隐式图。庞加莱图用于计算所有Lyapunov指数和Lyapunov维。通过数值模拟，在投影的庞加莱截面中表示吸引子，并获得Lyapunov谱。多自由度振动冲击系统可能会表现出复杂的准周期吸引子，这可以通过Lyapunov维数来表征。

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