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 a rigid pendulum with a sinusoidal torque at its pivot. Drag the bob to set its release, adjust the drive and damping, and record a time trace, a phase portrait and a Poincaré section together; the section takes one sample per drive cycle at a fixed phase. Compare motion that repeats every cycle or every two cycles with sensitive motion, or scan the drive strength with an independent release at each value.

Recorded time $0\,\mathrm{s}$

Principal angle $0\,\mathrm{rad}$

Unwrapped angle $0\,\mathrm{rad}$

Angular velocity $0\,\mathrm{rad\,s^{-1}}$

Relative kinetic energy $0\,J$

Gravitational energy $0\,J$

Mechanical energy $0\,J$

Viscous dissipation $0\,J$

Oscillating torque work $0\,J$

Recorded energy-balance defect $0\,J$

Step-halving state difference $0$

Actual integration step $0\,\mathrm{ms}$

Drive period $0\,s$

Acquired drive cycles $0$

Acquired retained section points $0$

Finite-record section recurrence $\text{Collecting}$

Finite-time tangent growth rate $0\,\mathrm{s^{-1}}$

Acquired tangent-growth window $0\,s$

Step-halving comparison window $0\,s$

Applied pivot torque $0\,\mathrm{N\,m}$

Drive-strength scan progress $0\,\%$

Maximum scan work-energy defect $0\,J$

Set the release, then record the motion

Does a driven pendulum repeat itself if you check it once every drive cycle?

Bob mass

m

1.00\,\mathrm{kg}

Rod length

L

1.00\,\mathrm{m}

Gravity

g

9.81\,\mathrm{m\,s^{-2}}

Dimensionless torque strength

\Gamma

0.90\,

Drive to natural frequency ratio

\omega_d/\omega_0

0.67\,

Viscous damping ratio

\zeta

0.25\,

Drive phase at release

\phi

0.00\,{}^\circ

Release angle

\theta_0

0.00\,{}^\circ

Initial angular velocity

\dot\theta_0

0.00\,\mathrm{rad\,s^{-1}}

Total drive cycles

N

96.00\,

Discard initial drive cycles

N_{\mathrm{discard}}

32.00\,

Drive scan lower bound

\Gamma_{\min}

1.00\,

Drive scan upper bound

\Gamma_{\max}

1.12\,

Maximum numerical step

h_{\max}

10.00\,\mathrm{ms}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

The purple outline sets the release angle, and amber marks the actuator at the pivot. Playback shows only section points recorded after the discarded transient. A scan makes 121 independent releases; at each strength it discards 64 cycles and keeps 24, with all other controls held at their current values. By default the scan covers a narrow interval that resolves the period-doubling region; edit the bounds to look further. A scan starts from a limited set of initial states and can miss attractors that coexist, so compare longer records and finer steps.

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

BackgroundMotion 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 mattersThe 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 misconceptionA section full of points proves the motion is chaotic.

Better mental modelLong 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 misconceptionAny evenly spaced video frames form a Poincaré section.

Better mental modelHere 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 misconceptionOne scan of drive strength shows every attractor.

Better mental modelEach 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

01
Record a repeating motion
Choose Period-one record, 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.
02
Double the period
Choose Period-two record and keep enough cycles.
What to observe: The returns alternate between two nearby clusters.
03
Look for sensitive motion
Choose Sensitive 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.
04
Scan the drive strength
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.