可逆性误差法（REM）：行星动力学的一种新的动态快速指示器
The Reversibility Error Method (REM): a new, dynamical fast indicator for planetary dynamics
摘要：
我们描述了可逆误差方法（REM）及其在行星动力学中的应用。REM是基于保守哈密顿系统的相空间轨迹的时间可逆性分析。舍入误差破坏了时间的可逆性，在同一时间间隔内前后积分时，与初始条件的位移与轨迹的动力学性质有关。如果运动是混沌的，在非零最大特征李雅普诺夫指数（mLCE）的意义下，那么REM随时间呈指数增长，如exp λt，而当运动是规则的（准周期性）时，REM随时间呈幂律增长，如t^α，其中α和λ是实系数。我们比较REM与mLCE的一个变体，邻近轨道的平均指数增长因子（MEGNO）。测试集包括限制性三体问题和五个共振行星系：HD 37124，Kepler-60，Kepler-36，Kepler-29和Kepler-26。我们发现这些算法的结果之间非常一致。此外，REM的数值实现是惊人的简单，是基于坚实的理论背景。REM只需要运动方程的辛和时间可逆（对称）积分器。这种方法也是CPU高效的。它可能是特别有用的动力学分析的多个行星系的开普勒样本，其特点是低偏心率轨道和相对较弱的相互作用。作为一个有趣的副作用，我们发现开普勒-29行星系中可能存在稳定的混沌现象。
英文摘要：
We describe the Reversibility Error Method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Characteristic Lyapunov Exponent (mLCE), then REM increases exponentially with time, as exp λt, while when the motion is regular (quasi-periodic) then REM increases as a power law in time, as t^α, where α and λ are real coefficients. We compare the REM with a variant of mLCE, the Mean Exponential Growth factor of Nearby Orbits (MEGNO). The test set includes the restricted three body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26. We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the KEPLER sample, characterized by low-eccentricity orbits and relatively weak mutual interactions. As an interesting side-result, we found a possible stable chaos occurrence in the Kepler-29 planetary system.
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