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Variational Continuation for Double Pendulum Periodic Orbits
Leo Yao, Ziming Liu, Max Tegmark
TL;DR
Periodic-orbit searches require increasingly precise initialization and can struggle with longer or higher-dimensional orbits. This paper uses a Hessian-based loss-landscape method in loop space to continue orbit families, demonstrating classified double-pendulum bifurcations and previously missing orbit types.
Problem
Existing periodic-orbit initializations do not incorporate orbit-family structure, while longer or higher-dimensional orbits become increasingly difficult to converge because of precision errors.
Method
The method represents loops with a Fourier series, identifies periodic orbits as zero-loss loops, and uses Hessian flat directions to initialize continuation along connected orbit families.
Results
The method surveys and classifies continuous families of double-pendulum periodic orbits, including off-normal-mode branches and orbits where both masses are never simultaneously at rest.
Takeaways & Limitations
Following bifurcation branches makes more complicated periodic motions interpretable as combinations of simple oscillations and informs discovery of new orbit candidates.
Takeaways & Limitations
The Rprop optimizer scales unfavorably for higher-dimensional systems compared with recent Jacobian-free methods, although Hessian probing can be paired with other convergence methods.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper addresses the lack of efficient methods for mapping complete periodic-orbit spaces by introducing Hessian-based variational continuation and demonstrating it on the double pendulum. It surveys orbit families, bifurcations, and previously unreported periodic motions.
- The proposed method combines automatic differentiation with Hessian eigendecompositions to navigate periodic-orbit solution space and map orbit families efficiently.
- The double-pendulum demonstration reveals bifurcation structures and periodic orbits in which both masses are never simultaneously at rest.
- Shooting methods become exponentially harder for longer or higher-dimensional orbits because close trajectories diverge and numerical errors accumulate.
- Variational methods avoid this convergence problem by varying a closed loop locally toward the equations of motion rather than integrating initial conditions.
- Periodic-orbit families form connected one-dimensional branches that terminate at fixed points or divergent periods or phase-space coordinates.
- Existing double-pendulum studies had not conducted a systematic or full numerical survey of periodic orbits and bifurcations.
II. METHOD
The method represents candidate periodic orbits as closed loops and minimizes their local mismatch with the dynamical equations. Zero loss identifies periodic orbits, while the loop-space Hessian exposes directions for continuing connected solutions.
- A periodic-orbit candidate is a closed loop with period T and trajectory z(t) satisfying z(T)=z(0).
- The loss averages the squared mismatch between the loop tangent and the physical evolution over the entire trajectory.
- The loop is parameterized finitely, using a Fourier series, so the loss becomes a function of parameter vector θ.
- Zero loss corresponds to a loop that locally matches physical evolution everywhere, with both loss and parameter gradients equal to zero.
B. Hessian Analysis
The method uses Hessian curvature and flat directions of a variational loss to identify and continue nearby periodic orbits in loop space.
- The Hessian describes local loss curvature around a parametrized loop, with eigenvalues measuring curvature magnitudes and eigenvectors giving associated directions.
- A zero Hessian eigenvalue identifies a perturbation direction that preserves zero loss and therefore remains on the periodic-orbit space.
- For nontrivial periodic orbits, one flat direction represents time-phase shifting, while additional zero eigenvalues represent further connected perturbation directions.
- Restricting known or trivial perturbations until one zero eigenvalue remains extracts the direction of an adjacent connected orbit family.
- A finite loop-space step along the flat direction initializes a nearby orbit that can be converged, benefiting from the variational method’s larger convergence radius.
C. Hessian Applications
Hessian analysis replaces brute-force periodic-orbit searches with geometric continuation, enabling systematic initialization, branch exploration, and bifurcation detection.
- Hessian analysis replaces grid-search approaches by exploiting the geometric structure of periodic solutions.
- Fixed-point initialization: For unstable fixed points, sweeping the period and locating a zero minimum eigenvalue systematically discovers small oscillations and their eigenvector initializations.
- Branch continuation: The larger loop-space convergence radius permits larger continuation steps and longer orbit-family exploration than integrator methods when precision issues arise.
- Bifurcation detection: A family crossing produces a two-dimensional Hessian nullspace, detectable when a second eigenvalue approaches zero.
- Bifurcation detection: Period-doubling bifurcations are exposed by doubly winding the loop, which permits odd-frequency perturbations and a second flat direction.
D. Implementation
The double pendulum loop is represented by a finite Fourier series, with automatic differentiation and gradient-based optimization used to solve the variational problem.
- A loop consists of a period T and closed trajectory z(t), parameterized by a Fourier decomposition with finite maximum frequency cutoff K.
- Computations use PyTorch float64 precision and autograd for Hessian computation.
- Figure 4 depicts period sweeps for branch initialization, loop-space continuation with an orthogonal constraint, and crossing detection through eigenvalue drops.
- The implementation uses Rprop to adapt parameter step sizes from gradient signs while minimizing integrated error.
- Optimization targets ℓ < 10^-10, increasing K when higher-energy or longer-period orbits require additional Fourier resolution.
III. RESULTS
Period sweeps identify double-pendulum normal modes through Hessian eigenvalue drops, and the resulting eigenvectors provide precise periodic-orbit initializations, including period-doubled cases.
- ΔT = 0.01 controls detected-period precision, and a K = 16 sweep at fixed points completes in a few minutes.
- T ≈3.39 and T ≈8.21 occur at zero-eigenvalue drops for the stable both-masses-down fixed point, while T ≈5.28 occurs around each one-up saddle point.
- The corresponding eigenvectors initialize oscillations with a phase-space step of Δθ = 0.5°.
- Relative integration deviations remain within 2% of the phase-space step size, providing precise initial conditions around stable and unstable fixed points.
- An additional eigenvalue dip at T ≈6.78 represents the same normal mode with two oscillations in one loop and is removed by an odd-frequency antisymmetry constraint.
B. Symmetric Orbit Propagation
The method propagates symmetric periodic-orbit families from normal modes by stepping along Hessian zero-eigenvalue directions while constraining convergence. It reaches long-period orbits near vertical configurations, where loop loss remains controlled despite exponentially growing integrator error.
- Propagation procedure: Symmetric orbit families are propagated by repeatedly converging loops, taking the Hessian, and stepping along a zero-eigenvalue direction.Training dynamics are constrained perpendicular to the step direction so propagation follows the orbit family.
- Computational scaling: K = 1024 was required for the longest converged orbits, with runtimes increasing from seconds at K = 16 to a few hours.Memory requirements were primarily due to Hessian size.
- Continuation results: All four orbit families were propagated to masses within 5° of vertical while maintaining loop loss below the convergence condition.These solutions approached period divergence across the energy spectrum.
- Continuation results: Integrator error scaled exponentially with orbit period T and diverged near vertical configurations, whereas variational convergence remained successful.The comparison highlights robustness to long periods and chaotic phase-space dynamics.
C. Bifurcation Detection
Bifurcations are detected from additional zero eigenvalues of the Hessian after removing directions associated with time evolution and branch continuation. Applying the same analysis to twice-wound loops reveals subharmonic period-doubling branches.
- Crossing detection: Off-branch eigenvalues remove the zero modes for time evolution and branch direction, so additional zero crossings identify orbit-family intersections.Sharp drops in the minimum off-branch eigenvalue indicate discrete crossing points.
- Period doubling: Twice-wound loops expose additional eigenvalue minima corresponding to subharmonic perturbations and period-doubling bifurcations.The construction doubles component frequencies and the period while setting odd-frequency components to zero.
- Branch generation: Hessian eigenvectors provide loop perturbations that preserve zero loss and generate directions for branching periodic-orbit families.Higher-dimensional nullspaces require separating the perturbation subspace from the original orbit branch.
D. Interpreting Orbits
Continuous orbit families organize higher-energy motions as extensions of low-energy normal modes and connect more complicated motions through bifurcation branches. Branching perturbations provide a way to decompose and categorize these motions.
- Branching families: Eigenvector directions at bifurcation points lead to branching families and additional connected orbits.This allows orbits to be classified by the families on which they lie.
- Normal-mode extensions: Higher-energy periodic solutions smoothly extend and deform the low-energy normal modes along continuous orbit families.Out-of-phase oscillations retain a recognizable relationship to the low-energy mode as energy increases.
- Orbit interpretation: Branching-family motions can be interpreted as combinations of a base orbit and oscillatory perturbations from it.Bifurcating orbits arise from perturbations at the same and higher multiples of the period.
- Orbit interpretation: Following bifurcation structures from fixed points and normal modes enables systematic categorization and decomposition of complicated motions.The resulting path connects each orbit to simpler base and branching families.
- Branch propagation: Training dynamics are constrained near bifurcations because unconstrained optimization favors the simpler known branch over the new bifurcation branch.Linear extrapolation is used to accelerate the slower final convergence in the constrained direction.
F. Phase Space Exploration
The exploration recovers many connected orbit branches and reveals a phase-space structure consistent with orbit-family theory. It also identifies additional branches and bounds the current survey by convergence failures and incomplete searches of nonsymmetric families.
- Phase-space survey: Many branches were successfully extended, with most approaching period divergences or another termination condition.The spectrum includes a closed-loop orbit family and an additional bifurcation point.
- Phase-space survey: The converged branch and bifurcation structure matches orbit-family theory, including looping, period-diverging, crossing, and time-reversed branches.The reported structures include ID1, ID2, OD1, and OB branches.
- Additional bifurcations: An additional bifurcation branch, OD1B, was found along the out-of-phase period-doubling branch after separate runs diverged from the same family.Increasing Hessian nullspace dimension near bifurcations makes computed zero-eigenvalue directions precision-dependent.
- Limitations: The in-phase and out-of-phase bifurcation branches stopped during propagation because convergence failed.A constraint based on recent orbit-family direction is proposed for future continuation.
- Limitations: Bifurcation points along nonsymmetric orbit families were not explicitly searched, although the method can be extended to additional families up to numerical constraints.The theoretical periodic-orbit structure is infinite.
IV. CONCLUSION
The method uses Hessian-probed loss landscapes and variational loop continuation to systematically explore periodic-orbit families and bifurcations. Applied to the ideal double pendulum, it discovers and classifies complex orbits while highlighting both the promise and dimensional-scaling limitation of the optimization approach.
- Hessian probing identifies loss-landscape structure, connected orbit families, and bifurcations for continuation in loop space.The propagation step can be separated from the optimizer used to converge new loop initializations.
- Rprop combined with linear extrapolation converges double-pendulum orbits of varying complexity across phase space.
- The survey discovers off-normal-mode orbits and systematically classifies them into continuous families.
- Machine-learning optimizers can be applied to physical-system problems, with parametrizations expressing constraints such as propagation direction.
- Rprop scales unfavorably in higher-dimensional systems compared with recent Jacobian-free methods, although alternative convergence methods and Hessian approximations remain compatible.
- Following orbit-family branches makes complicated motions more interpretable as combinations of simpler oscillations.