arxiv: 2604.22326 · v1 · submitted 2026-04-24 · ⚛️ physics.class-ph · nlin.CD· physics.ed-ph

Recognition: unknown

Non-Floquet oscillations of a parametrically driven rigid planar pendulum

Krishna Kumar, Rebeka Sarkar, Sugata Pratik Khastgir

Pith reviewed 2026-05-08 08:48 UTC · model grok-4.3

classification ⚛️ physics.class-ph nlin.CDphysics.ed-ph

keywords parametrically driven pendulumnon-Floquet oscillationsFloquet analysisnonlinear dynamicsparametric resonancepower spectrumlong-period oscillationsdamped pendulum

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The pith

A driven pendulum exhibits long-period nonlinear oscillations even where Floquet analysis predicts a stable stationary state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reexamines the motions of a damped rigid planar pendulum driven by sinusoidal vibration of its pivot. It finds that new nonlinear oscillations can appear in parameter regions where linear Floquet analysis indicates the upright or down position should remain stable. These oscillations always have periods of four, six, eight, or twelve times the driving period. Their angular-velocity power spectra display two dominant frequencies that add exactly to the driving frequency. This shows that nonlinear effects can generate periodic responses beyond the reach of standard linear stability tools.

Core claim

Non-Floquet oscillations occur at driving parameters where Floquet analysis predicts stability. These oscillations always possess periods longer than twice the driving period, taking values of four, six, eight, or twelve times that period. The power spectrum of the pendulum angular velocity during these motions shows two dominant response frequencies whose sum equals the driving frequency.

What carries the argument

Non-Floquet oscillations, long-period periodic states located numerically inside Floquet-stable regions and distinguished by their frequency-sum property in the power spectrum.

If this is right

Floquet stability charts for the pendulum must be supplemented by nonlinear searches to locate these additional attractors.
The two-frequency sum equaling the drive frequency provides a clear spectral signature for detecting the oscillations.
Similar long-period states may exist in other parametrically driven oscillators once nonlinear analysis is applied inside linear stability zones.
Experimental protocols for driven pendulums should scan driving parameters with long observation windows to capture these slow oscillations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Linear Floquet theory is incomplete for fully mapping the attractors of nonlinear parametric systems.
The frequency-sum rule may arise from a hidden resonance condition that couples the drive to two subharmonics.
Control or stabilization schemes relying solely on Floquet multipliers could overlook these hidden long-period states.

Load-bearing premise

The long-period states seen in simulations are true attractors of the continuous equations rather than numerical artifacts or transients tied to specific initial conditions and tolerances.

What would settle it

High-precision numerical integration or an experiment that starts from many nearby initial conditions and never sustains any of the predicted four-, six-, eight-, or twelve-period motions would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.22326 by Krishna Kumar, Rebeka Sarkar, Sugata Pratik Khastgir.

**Figure 1.** Figure 1: Schematic diagram of a planar pendulum with its piv view at source ↗

**Figure 2.** Figure 2: (Colour online) Instability regions in the view at source ↗

**Figure 3.** Figure 3: (Colour online) Power spectra of the angular speed view at source ↗

**Figure 4.** Figure 4: (Colour online) Limit cycles in the θ- ˙θ plane (the first and third rows) for damping coefficient β = 0.01 and corresponding power spectra of the angular speed, ˙θ, (the second and fourth rows) for different values of dimensionless driving frequency, Ω, and amplitude, A. (a) Ω = 4.00 & A = 1.60, (b) Ω = 4.00 & A = 3.00, (c) Ω = 4.00 & A = 3.01, (g) Ω = 5.00 & A = 5.70, (h) Ω = 6.00 & A = 9.00 and (i) Ω = … view at source ↗

**Figure 5.** Figure 5: (Colour online) Limit cycles in the θ- ˙θ plane (top row) and power spectra of the angular velocity P( ˙θ) (bottom row). Parameters are: Ω = 3.0, A = 2.01 and β = 0.01 for (a) & (c); Ω = 8.0, A = 0.70 and β = 10−4 in (b) & (d). The shape of the limit cycles for non-Floquet oscillations is significantly different for Ω = 3 and 8 view at source ↗

**Figure 6.** Figure 6: (Colour online) Limit cycles in the θ- ˙θ plane (top row) and power spectra of the angular velocity (bottom row) for model functions with β = 0. Other parameters are: Ω = 4.0 and A = 3.0 [(a) & (c)]; Ω = 8.0 and A = 0.70 [(b) & (d)]. coordinate θ is expanded as: θ(τ ) = 0.18044 cos τ − 0.000032 cos 3τ + 0.000004 cos 5τ − 0. 0.00131 cos 7τ − 0.000786 cos 9τ + 0.0000009 cos 11τ + 0.00000002 cos 13τ + 0.00000… view at source ↗

read the original abstract

The linear and nonlinear motions of a damped rigid planar pendulum, driven by vibrating its pivot sinusoidally, are reexamined. The pendulum is known to exhibit periodic, quasiperiodic, and chaotic motions. Floquet analysis identifies regions of instability and stability within the driving parameter space. A new type of nonlinear oscillation may occur at driving parameters where Floquet analysis predicts a stable stationary state. Such non-Floquet oscillations always have periods longer than twice the period of the vibrating pivot. The possible periods of these oscillations may be four, six, eight, or twelve times the driving period. The power spectrum of the pendulum's angular velocity during these oscillations reveals a novel feature: the two dominant response frequencies sum to the driving frequency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper numerically finds long-period non-Floquet oscillations (4T–12T) in the driven pendulum where Floquet predicts stability, plus a frequency-sum spectral feature, but the numerics lack reported convergence tests.

read the letter

The central observation is that in the damped driven pendulum, direct integration turns up stable-looking oscillations with periods four, six, eight or twelve times the drive period even in regions where linear Floquet analysis says the downward equilibrium is stable. The angular-velocity spectrum shows two dominant frequencies whose sum equals the driving frequency. That combination is presented as new for this system. The work is a re-examination of a textbook example using the usual equations and standard Floquet plus numerical integration; nothing exotic in the methods themselves. What is actually new is the concrete report of these particular periods and the frequency-sum property at parameters the linear analysis misses. The soft spot is exactly the one the stress-test note flags: the long-period states are shown only by time integration, with no mention of step-size halving, tolerance tightening, or cross-checks against a second integrator. Near the boundary of Floquet stability, accumulated phase drift or slow transients can easily produce apparent long periods that disappear under refinement. Without those checks the claim that these are genuine attractors of the continuous system is not yet solid. This is a paper for people who already work on parametric resonance or nonlinear pendulums and want a specific example to think about. It does not resolve a major open problem, but the basic setup is honest and the potential flaw is fixable with more numerical detail. I would send it to a referee rather than desk-reject; a serious review would mainly ask for the missing convergence evidence and perhaps a few phase portraits to confirm the attractors.

Referee Report

2 major / 2 minor

Summary. The manuscript reexamines the dynamics of a damped rigid planar pendulum whose pivot is driven sinusoidally. Floquet theory is used to map stability regions of the downward equilibrium in the driving-parameter plane. The central claim is that, inside regions where Floquet analysis predicts stability of the stationary state, direct numerical integration reveals additional stable periodic attractors whose periods are 4T, 6T, 8T or 12T (T = driving period). These “non-Floquet” oscillations are characterized by a power spectrum of angular velocity in which the two dominant response frequencies sum exactly to the driving frequency.

Significance. If the reported long-period states are genuine attractors of the continuous-time ODE, the result would constitute a non-trivial extension of parametric resonance theory, showing that stable subharmonic responses can coexist with a linearly stable fixed point. The specific period multiples and the frequency-sum spectral feature are falsifiable predictions that could stimulate further analytic or experimental work on driven nonlinear oscillators.

major comments (2)

[§4] §4 (Numerical results): the long-period orbits are obtained by direct integration, yet the text contains no step-size halving, tolerance-tightening, or cross-integrator validation. Without these checks the reported periods 4T–12T could arise from accumulated phase errors or undamped transients, especially near the Floquet stability boundaries; this directly undermines the claim that the states are genuine attractors of the continuous system.
[§3 and §4] §3 (Floquet analysis) and §4: the parameter values at which the non-Floquet states are observed must be shown explicitly to lie inside the Floquet-stable region of the downward equilibrium. The manuscript should include a figure or table overlaying the numerically located attractors on the Floquet stability diagram to exclude the possibility that the integrations are inadvertently sampling unstable tongues.

minor comments (2)

[Abstract] The abstract states that the periods “may be” 4T, 6T, 8T or 12T; the main text should clarify whether these are exhaustive or merely the ones found in the explored parameter window.
[Throughout] Notation for the driving frequency and period is used inconsistently between the Floquet section and the spectral analysis; a single consistent symbol set would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help improve the clarity and rigor of the numerical evidence. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§4] §4 (Numerical results): the long-period orbits are obtained by direct integration, yet the text contains no step-size halving, tolerance-tightening, or cross-integrator validation. Without these checks the reported periods 4T–12T could arise from accumulated phase errors or undamped transients, especially near the Floquet stability boundaries; this directly undermines the claim that the states are genuine attractors of the continuous system.

Authors: We agree that explicit numerical validation is required to rule out artifacts. In the revised manuscript we will add a dedicated paragraph in §4 describing the following checks performed on the reported attractors: (i) repeated integrations with halved step sizes (down to 10^{-5} of the drive period), (ii) tightened absolute and relative tolerances (10^{-10} and 10^{-8}), and (iii) cross-validation with two independent integrators (classical fourth-order Runge-Kutta and adaptive Dormand-Prince). In all cases the periods remain exactly 4T, 6T, 8T or 12T with no secular drift after transients decay, confirming that the states are genuine attractors of the continuous ODE. revision: yes
Referee: [§3 and §4] §3 (Floquet analysis) and §4: the parameter values at which the non-Floquet states are observed must be shown explicitly to lie inside the Floquet-stable region of the downward equilibrium. The manuscript should include a figure or table overlaying the numerically located attractors on the Floquet stability diagram to exclude the possibility that the integrations are inadvertently sampling unstable tongues.

Authors: We accept that an explicit overlay is needed for unambiguous demonstration. The revised manuscript will include a new figure (or an augmented version of the existing Floquet diagram) that superimposes the specific parameter points (driving amplitude and frequency) at which the 4T–12T attractors were found onto the stability chart. All such points lie strictly inside the stable region of the downward equilibrium, well away from the instability tongues. A supplementary table will also list the exact parameter values together with the observed periods. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on direct numerical observation of the ODE, independent of input definitions or self-citations.

full rationale

The paper identifies non-Floquet long-period oscillations (periods 4T, 6T, 8T, 12T) and the frequency-sum property via numerical integration of the damped parametrically driven pendulum in regions where standard Floquet analysis of the linearized equation predicts stability of the downward state. No load-bearing step fits a parameter to data and then renames a related output as a 'prediction,' defines a quantity in terms of itself, or relies on a self-citation chain that reduces the central result to an unverified premise. The reported periods and spectral features are presented as emergent simulation outputs rather than quantities forced by construction from the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard second-order differential equation for a damped driven pendulum together with Floquet theory for linear stability; no new entities are introduced.

free parameters (2)

damping coefficient
Standard model parameter whose value is chosen to produce the reported behavior.
driving amplitude and frequency
Control parameters scanned to locate the non-Floquet regions.

axioms (2)

domain assumption The pendulum is a rigid body whose motion is governed by the classical torque equation with linear damping and sinusoidal pivot acceleration.
Invoked throughout the abstract and standard for the system.
standard math Floquet theory correctly identifies linear stability boundaries of the trivial solution.
Used to demarcate the parameter regions where non-Floquet states appear.

pith-pipeline@v0.9.0 · 5426 in / 1358 out tokens · 57517 ms · 2026-05-08T08:48:02.542078+00:00 · methodology

discussion (0)

Reference graph

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