Chemical reactions in multidimensional systems are often described by a rank-1 saddle, whose stable and unstable manifolds intersect in the normally hyperbolic invariant manifold (NHIM). Trajectories started on the NHIM in principle never leave this manifold when propagated forward or backward in time. However, the numerical investigation of the dynamics on the NHIM is difficult because of the instability of the motion. We apply a neural network to describe time-dependent NHIMs and use this network to stabilize the motion on the NHIM for a periodically driven model system with two degrees of freedom. The method allows us to analyze the dynamics on the NHIM via Poincaré surfaces of section (PSOS) and to determine the transition-state (TS) trajectory as a periodic orbit with the same periodicity as the driving saddle, viz. a fixed point of the PSOS surrounded by near-integrable tori. Based on transition state theory and a Floquet analysis of a periodic TS trajectory we compute the rate constant of the reaction with significantly reduced numerical effort compared to the propagation of a large trajectory ensemble.

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神经网络方法用于周期驱动系统的常双曲不变流形上的动力学

多维系统中的化学反应通常由1级鞍形描述，该鞍形的稳定和不稳定歧管在正常双曲线不变歧管（NHIM）中相交。原则上，从NHIM开始的轨迹在向前或向后传播时永远不会离开该流形。但是，由于运动的不稳定性，很难对NHIM上的动力学进行数值研究。我们应用神经网络来描述与时间相关的NHIM，并使用该网络来稳定具有两个自由度的周期驱动模型系统在NHIM上的运动。该方法使我们能够通过截面的庞加莱表面（PSOS）分析NHIM上的动力学，并确定过渡状态（TS）轨迹为周期性轨道，其周期性与行驶鞍座相同，即。PSOS的固定点，周围几乎是不可分割的花托。基于过渡状态理论和周期性TS轨迹的Floquet分析，我们计算了反应速率常数，与大轨迹系的传播相比，数值努力明显减少。