用自动微分找双摆周期轨道，能高效发现新解和分岔。
Variational Continuation for Double Pendulum Periodic Orbits
- 用傅里叶级数参数化轨道，损失函数基于微分方程偏差。
- 无需数值积分器，可精确初始化不稳定平衡点附近的振荡。
- 能发现新周期轨道与子谐波分岔，适合研究混沌系统动力学。
我们提出一种基于海森矩阵的数值延续方法，用于动态系统中的周期轨道。将一个环（周期轨道候选）参数化为傅里叶级数，定义损失函数以衡量该环偏离物理微分方程的程度。与以往依赖手工推导雅可比矩阵的方法不同，本方法利用自动微分技术实现过程自动化。通过损失函数景观中的平坦方向（零特征值方向）确定延续方向，使周期轨道搜索更高效且有引导性。该方法无需数值积分器，能精确初始化不稳定的固定点附近的振荡，并高效检测轨道族交点及子谐波分岔。作为演示，我们完成了从固定点出发的完整双摆周期振荡延续，揭示了轨道族中的分岔现象，并对周期轨道分支进行了分类。特别地，我们发现了两个摆锤质量从未同时静止的周期轨道，据我们所知，此类解此前在文献中尚未被报道。
原文摘要 · Abstract (English)
We present a Hessian-based approach to numerically continue periodic orbits in dynamical systems. A loop (periodic orbit candidate) is parametrized as a Fourier series; a loss function is defined based on the deviation of the loop from the physical differential equations. Unlike previous work relying on hand-derived Jacobians, our method automates the process by leveraging automatic differentiation, a common machine learning technique. The continuation direction can be determined by the flat directions of the loss landscapes (directions with zero eigenvalues), making the search of periodic orbits efficient and guided. Our method is integrator-free, precisely initializes oscillations around unstable fixed points, and efficiently detects orbit family intersections and subharmonic bifurcations. As a demonstration, we present full continuations of periodic double pendulum oscillations from fixed points, showing bifurcations along orbit families and categorizing branches of periodic orbits. In particular, we find periodic orbits where both pendulum masses are never simultaneously at rest, which to our knowledge has been missing in the literature.
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