M086 · Nonlinear dynamics / stroboscopic evidence

Driven Pendulum & Poincaré Section

Apply a sinusoidal pivot torque to a rigid pendulum. Drag its release bob, tune drive and damping, then acquire a synchronized time trace, phase portrait and fixed-phase Poincaré section. Compare period-one and period-two records with sensitive motion, or scan drive strength using independent releases.

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 $0\,J$

RELEASE / ACQUIRE / VERIFY

Does the record repeat at the drive phase?

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\,\%

Purple outline sets release angle; amber marks the pivot actuator. Playback reveals only acquired section points after the discarded transient. A scan uses 121 independent releases, discards 64 cycles and retains 24 per strength, holding all other current controls fixed. The default narrow interval resolves the period-doubling region; edit the bounds to explore further. It samples a finite basin and may miss coexisting attractors; compare longer records and finer steps.

Rigid massless rod and point bob; prescribed sinusoidal torque and viscous pivot damping. Angles remain unwrapped for integration and fold only for the phase and section views. Section points discard the selected initial cycles. Recurrence and renormalized tangent growth are finite-record diagnostics, not proofs of an asymptotic attractor. Step-halving comparison covers the first four drive cycles because long chaotic trajectories can decorrelate.

Start with the essentials

Focus question: What evidence distinguishes a repeating response from sensitive dynamics?
One-sentence intuition: Use both fixed-phase recurrence and renormalized tangent growth, and check the energy ledger and short-window numerical refinement.

Core mathematical model

Driven nonlinear equation

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

Torque strength is normalized by inertia and the squared natural rate.

Fixed-phase section

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

Only acquired points after the selected discarded cycles appear. Folding angle does not reset the dynamics.

Independent work and heat

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

Power quadratures are integrated alongside the motion rather than reconstructed from the energy difference.

Fixed-phase variational flow

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

The drive phase is held fixed when perturbing the state. Tangents are repeatedly renormalized to avoid saturated pair distances.

Finite-time tangent growth

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

The duration readout excludes discarded cycles; one chosen tangent and a finite record do not prove an asymptotic exponent.

Common difficulties

A scattered section is evidence

Typical misconceptionMany section points alone prove chaos.

Better mental modelLong transients and quasiperiodic motion can also make many points; compare tangent growth and longer retained windows.

Sampling phase must stay fixed

Typical misconceptionAny evenly spaced video frames form a Poincaré section.

Better mental modelThese points are synchronized to one complete drive cycle and one forcing phase.

Finite scans depend on protocol

Typical misconceptionOne strength scan lists all attractors.

Better mental modelEach column starts independently from the current initial state and has a finite settling window. Other basins can coexist.

Run the experiment

01
Acquire repetition
Finish the period-one record and inspect the section count.
What to observe: Many acquired returns nearly overlap; the finite recurrence test reports one cycle.
02
Double the return pattern
Use the period-two preset and retain enough cycles.
What to observe: The return alternates between two nearby clusters.
03
Seek sensitive motion
Use the sensitive preset, then compare finer steps and longer records.
What to observe: The section spreads and finite-time tangent growth can be positive, while short-window refinement remains small.
04
Scan with a stated protocol
Scan drive strength, inspect progress, then return to the section.
What to observe: Each strength retains 24 returns after discarding 64 cycles. Changing a physical control invalidates the previous scan.