用于双摆周期轨道的变分延拓
Variational Continuation for Double Pendulum Periodic Orbits
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中文总结 AI 辅助
本研究提出基于自动微分的海森矩阵变分延拓方法，用于数值延拓动力系统周期轨道，成功实现双摆周期轨道从不动点的延拓，发现了两摆质量不同时静止的周期轨道。
中文摘要 AI 辅助
我们提出一种基于海森矩阵的方法，用于数值延拓动力系统中的周期轨道。将环（周期轨道候选）参数化为傅里叶级数；基于环与物理微分方程的偏差定义损失函数。与以往依赖手动推导雅可比矩阵的工作不同，我们的方法利用机器学习领域常用的自动微分技术，自动化了该过程。延拓方向可由损失景观的平坦方向（具有零特征值的方向）确定，使周期轨道的搜索高效且具有导向性。我们的方法无需积分器，能精确初始化不稳定不动点附近的振荡，可高效检测轨道族交点与次谐波分岔。作为演示，我们展示了双摆周期振荡从不动点的完整延拓，呈现了轨道族上的分岔并对周期轨道分支进行分类。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，这在现有文献中尚未被报道。
英文摘要
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.
发表机构
- Massachusetts Institute of Technology（麻省理工学院）
- The NSF AI Institute for Artificial Intelligence and Fundamental Interactions（美国国家科学基金会人工智能与基础交互研究所）
机构由 AI 辅助整理，请以论文原文为准。
