---
title: "Driven Pendulum & Poincaré Section Interactive Lab | Sandbox Physics"
description: "Does a driven pendulum repeat itself if you check it once every drive cycle? For periodic motion, yes. Period-one motion is back in the same state at every check, and period-two motion alternates between two states. If the samples scatter and a tiny difference in the starting state keeps growing, the motion is sensitive to where it started, at least over the stretch you recorded."
canonical_url: "https://sandboxphysics.com/labs/driven-pendulum/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/driven-pendulum/"
last_modified: "2026-10-25"
---

M086 · Nonlinear dynamics / stroboscopic sampling

# Driven Pendulum & Poincaré Section

**Does a driven pendulum repeat itself if you check it once every drive cycle?** For periodic motion, yes. Period-one motion is back in the same state at every check, and period-two motion alternates between two states. If the samples scatter and a tiny difference in the starting state keeps growing, the motion is sensitive to where it started, at least over the stretch you recorded.

Drive the pendulum and sample it once per cycle. Compare repeating motion, period doubling and irregular returns, then scan the drive strength.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The pendulum is a point bob on a rigid, massless rod, driven by a given sinusoidal torque and slowed by viscous damping at the pivot. The integration tracks the full, unwrapped angle; the angle is folded into one turn only in the phase and section views. The section leaves out the initial cycles you choose to discard. The recurrence test and the renormalized tangent growth describe this finite record only; they do not prove what the long-term attractor is. The step-halving comparison covers only the first four drive cycles, because over longer times two chaotic runs can lose all resemblance to each other.

Physics tutorial

## A section turns motion into a return record

**Background** Motion can look intricate even when it repeats. Sample it once per drive cycle, always at the same phase of the forcing, and the rhythm of the drive drops out, so successive returns can be compared directly.

**Why it matters** The University of Maryland Phys410 lectures on nonlinear mechanics cover the driven, damped pendulum and its period doubling. This Lab uses the same dimensionless measure of torque and lets you choose how long to observe and how many early cycles to discard.

Start with the essentials

- **Focus question**: What evidence distinguishes a repeating response from sensitive dynamics?
- **One-sentence intuition**: Check whether the fixed-phase samples repeat and whether a tiny perturbation grows. Then confirm that work and heat account for the energy and that a halved time step agrees over a short window.

### Core mathematical model

#### Equation of the driven pendulum

$$
\ddot\theta+2\zeta\omega_0\dot\theta+\omega_0^2\sin\theta=\Gamma\omega_0^2\cos(\omega_dt+\phi)
$$

The drive strength is the torque divided by the moment of inertia and by the square of the natural angular frequency.

#### Sampling at a fixed drive phase

$$
t_n=nT_d,\quad T_d=2\pi/\omega_d,\quad (\theta_n\bmod 2\pi,\dot\theta_n)
$$

Only points recorded after the discarded cycles appear. Folding the angle into one turn changes the plot, not the motion.

#### Work and heat computed separately

$$
W=\int\tau_d\dot\theta\,\mathrm{d}t,\quad Q=\int2\zeta I\omega_0\dot\theta^2\,\mathrm{d}t,\quad E-E_0=W-Q
$$

The drive power and the damping power are integrated along with the motion. They are not inferred from the change in energy.

#### How a small perturbation evolves

$$
\delta\ddot\theta+2\zeta\omega_0\delta\dot\theta+\omega_0^2\cos\theta(t)\,\delta\theta=0
$$

The perturbation changes the angle and angular velocity but not the drive phase. It is rescaled again and again, so its growth does not level off the way the distance between two separate trajectories does.

#### Growth rate over a finite time

$$
\lambda_T=\frac1T\sum_j\log\frac{\|\delta z_j^-\|}{\|\delta z_j^+\|},\quad z=(\theta,\dot\theta/\omega_0)
$$

The time shown leaves out the discarded cycles. One chosen perturbation followed over a finite record cannot prove the long-term exponent.

### Common difficulties

#### A scattered section is evidence, not proof

**Typical misconception** A section full of points proves the motion is chaotic.

**Better mental model** Long transients and quasiperiodic motion can also produce many points. Compare the tangent growth, and keep a longer window of cycles.

#### Sampling phase must stay fixed

**Typical misconception** Any evenly spaced video frames form a Poincaré section.

**Better mental model** Here the samples are taken exactly once per drive cycle, always at the same phase of the forcing.

#### A scan depends on how it is run

**Typical misconception** One scan of drive strength shows every attractor.

**Better mental model** Each column starts fresh from the current initial state and has only a limited time to settle. Other attractors, with their own basins, can exist at the same strength.

### Run the experiment

1. 01

   #### Record a repeating motion

   Choose Repeats every cycle, acquire the whole section and check how many section points there are.

   **What to observe:** Many returns land almost on top of each other, and the recurrence test over this record reports a period of one cycle.

2. 02

   #### Double the period

   Choose Repeats every two cycles and keep enough cycles.

   **What to observe:** The returns alternate between two nearby clusters.

3. 03

   #### Look for sensitive motion

   Choose Irregular motion, then compare finer time steps and longer records.

   **What to observe:** The section points spread out and the tangent growth rate can be positive, while the step-halving difference over the short window stays small.

4. 04

   #### Scan the drive strength

   Open Recording and diagnostics, click Scan drive strength, watch the progress, then return to the section view.

   **What to observe:** At each strength the first 64 cycles are discarded and the next 24 returns are kept. Changing any physical setting voids the previous scan.

Physical conclusion

A scattered section with positive tangent growth is evidence of sensitive motion in this record. It does not establish the global attractor or cover every initial condition.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)