Variational Continuation for Double Pendulum Periodic Orbits

Leo Yao; Ziming Liu; Max Tegmark

cs.LGSep 4, 2026

Variational Continuation for Double Pendulum Periodic Orbits

Authors: Leo Yao, Ziming Liu, Max Tegmark

Organizations: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts

Abstract

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.

Figures & tables

Figure 1: Converged double pendulum periodic orbits in the Poincaré section with $\dot{\theta}_{1}=0$ ; bifurcating branches intersect and extend off continuous orbit families between fixed points. Fixed points circled in black. In-Phase , Out-Phase , Down-Up , and Up-Down start from fixed points and run to period divergences. Bifurcation branches abbreviated with I and O for base branch, B for branching at same period, D for period-doubling, and numbered for uniqueness.

Figure 2: An example of a periodic orbit spectrum consisting of multiple orbit families. At period

T_{1}

, a crossing orbit family creates a bifurcation in the orbit spectrum, shown in purple. At period

T_{2}

, a period-doubling bifurcation occurs due to a crossing of an orbit family with twice the period, shown in dashed blue.

Figure 3: For an arbitrary trajectory

\mathbf{z}(t)

in phase space, the loop tangent

\frac{\mathrm{d}\mathbf{z}(t)}{\mathrm{d}t}

may not necessarily match the physical evolution

\mathbf{f}(\mathbf{z}(t))

. For a periodic orbit, these vectors must match across the entire loop.

Figure 4: On left, a branch of orbits

\mathcal{L}

emerging from a fixed point at period

T_{0}

(above) is found by sweeping period

T

at the fixed point, and detecting a drop in the minimal eigenvalue

\lambda

(below). At center, the increased radius of convergence (shaded) of the variational method allows for exploration further along orbit families, with an orthogonal loop space constraint. On right, a crossing of orbit families at

\mathcal{L}^{\prime}

(above) is found by detecting a drop in the off-branch eigenvalue

\lambda

(below).

Normal Mode	Integrated Deviation (degrees)
Down-Down, In-Phase	$1.76\times 10^{-4}$
Down-Down, Out-Phase	$1.02\times 10^{-3}$
Down-Up	$0.0982$
Up-Down	$0.0976$

Table 1: For each of the detected normal modes, a step in loop space corresponding to a phase space deviation of

0.5^{\circ}

is taken, an initial condition is obtained by evaluating the Fourier parametrization at

t=0

, and this phase space condition is integrated for the detected period

T

. The deviation of the integration result from the original condition is small compared to the step size of

0.5^{\circ}

, implying a precise initialization corresponding to a periodic orbit.

Figure 5: Minimum eigenvalue for the period

T

with both masses down (

\theta_{1}=0

\theta_{2}=0

); eigenvalue minimums give out of phase and in-phase normal modes and correct corresponding periods

T

Figure 6: Loop and integrator losses for in-phase oscillations; variational convergence succeeds even for longer periods

T

as masses approach vertical, while integrator error diverges.

Figure 7: Minimum off-branch eigenvalue for out-of-phase oscillations; additional eigenvalue minimums give same period and period-doubling bifurcations leading to new orbit families.

Figure 8: Out-of-phase oscillations at various energies along the continuous family; higher energy oscillations smoothly extend and deform the shapes of the low-energy normal mode.

Figure 9: Bifurcating orbits are created by combinations of perturbations at same and higher multiples of period, which can be interpreted by following propagated branches.

Figure 10: Comparison of loop and integrator losses during training with and without extrapolation; linear search greatly speeds up slow exponential convergence.

Dynamical Systems

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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Locating periodic solutions of chaotic dynamical systems normally requires an initial guess close enough to the target orbit for numerical continuation or gradient-based search to converge. We show that Physics-Informed Neural Networks (PINNs) trained on sparse, noisy observations without initial conditions recover periodic orbits of the gravitational three-body problem, including orbit families absent from the training data. The method rests on a second-order ODE formulation, fixed-frequency Fourier features, percentile-based adaptive refinement, and a trainable scaling parameter, each validated on forward problems. Across two 100-seed ensembles,

23

25\%

of runs converge to families not present in the training data. We then ask what determines which family emerges. Two

χ^2

tests give a consistent answer: changing the training data source significantly shifts the distribution of recovered families (

p < 0.001

, Cramér's

V = 0.339

), whereas switching between the two initialization distributions tested does not (

p = 0.620

V = 0.094

). The random seed selects which family a given run recovers; the distribution the weights are drawn from does not shift the aggregate frequencies, but the training data does. The evidence is empirical: we do not characterize the loss landscape analytically, and PINNs remain slower than conventional integrators on well-posed initial-value problems. What the experiments establish is that the recovered orbits are verifiable rather than merely plausible: the identified ones refine to genuine periodic solutions, a network trained on Lagrange data recovers the figure-eight choreography (Li--Liao class I.A.1, matched to seven significant digits in

T^*

), and one trained on figure-eight data recovers a Broucke--Hadjidemetriou--Hénon orbit closing to

δ_T < 10^{-9}

Abstract

Dynamical Systems

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