OBSERVATION CHAMBER / ACTIVE

Double Pendulum Poincaré Observatory

Observe a continuous-time double pendulum through a deterministic Poincaré section and compare regular-island and chaotic-sea candidates without claiming a formal chaos proof.

dynamics / Poincaré section Observation Model

01 / OBSERVE THE SECTION

One continuous double pendulum, sampled at one crossing

CANVAS 2D / READY

ROUTE 86 / DYNAMICS / POINCARÉ SECTION

Drag a bob · arrows edit angles · Space play/pause · S step · R reset

Section contract θ₂ = 0 mod 2π with an upward crossing ω₂ > 0. The crossing state is found by linear interpolation between fixed RK4 samples.

The left plot shows (wrap(θ₁), ω₁) section points. The right panel shows the same continuous state. A finite pattern is a candidate only; it is not a formal Lyapunov result or a proof of chaos.

02 / SET THE MODEL

Angles, speed, and scale

REGULAR CANDIDATE

CHAMBER STATUS

SYSTEM: ONLINE
MODE: CONTINUOUS / POINCARÉ SECTION
SIGNAL: READY TO OBSERVE

Initial θ₁0.20 radangle from the downward vertical Initial θ₂−0.15 radangle from the downward vertical Initial ω₁0.00 rad/sangular velocity Initial ω₂0.00 rad/sangular velocity Gravity g9.81 m/s²model scale Length L1.00 mboth arms; m₁ = m₂ = 1 kg Horizon32.0 sfixed step dt = 1/240 s

Paused / ready. Section points: 0.

Equal point masses and equal arms are used. There is no damping, drive, collision, or external measurement. The model uses continuous equations and fixed-step RK4.

03 / READ THE INVARIANTS

Section points and energy consistency

FINITE

Section points

θ₂ = 0 mod 2π, ω₂ > 0

Current θ₁

0.200 rad

unwrapped state

Current θ₂

−0.150 rad

unwrapped state

Last section x

—

wrap(θ₁)

Last section y

—

ω₁ rad/s

Elapsed

0.00 s

fixed RK4 time

Initial energy E_init

−28.929 J

energy shell: E(t) ≈ E_init; equilibrium reference E_eq

Current energy E(t)

−28.929 J

recomputed each step

ΔE = E(t) − E_init

0.000e+0 J

signed difference from E_init

Max normalized drift

0.000e+0

ΔE / max(|E_init|, 3gL)

Clamp guard

|ω| cap events

Self-test

—

running deterministic checks

04 / CHECK THE NUMERICS

Deterministic diagnostics

SELF-TEST

Integrator

RK4 / dt = 1/240 s

Section

θ₂ = 0 mod 2π / upward

Display axes

wrap(θ₁) / ω₁

Renderer

Canvas2D

Self-test cases

Equilibrium, acceleration, RK4 determinism, crossing interpolation, wrap, energy, symmetry, time reversal, finite and drift bounds.

Show generated report

No report yet.

A regular island, mixed layer, or chaotic-sea pattern is a finite observation candidate. This page does not calculate a formal Lyapunov exponent and does not claim a proof of chaos.

MODEL NOTE

Continuous time becomes a readable section.

The state is (θ₁, θ₂, ω₁, ω₂). The fixed RK4 samples are continuous-time states; only the upward crossings of θ₂ = 0 mod 2π are placed on the section plot.

SECTION NOTE

Direction separates the crossing.

Only an upward crossing with ω₂ > 0 is accepted. The plotted point uses linear interpolation between the previous and next fixed samples, then wraps θ₁ to [−π, π).

LIMIT NOTE

Finite patterns are candidates, not verdicts.

Regular islands, mixed layers, and chaotic-sea candidates can appear in a finite window. Energy drift and finite guards are shown so that numerical consistency is visible beside the pattern.

FORMULA / CONTRACT

Double-pendulum equations and energy shell

dθ₁/dt = ω₁, dθ₂/dt = ω₂

dω₁/dt = [−g(2m₁+m₂)sinθ₁ − m₂g sin(θ₁−2θ₂) − 2m₂sinδ(l₂ω₂² + l₁ω₁²cosδ)] / [l₁(2m₁+m₂−m₂cos2δ)]

dω₂/dt = 2sinδ[l₁ω₁²(m₁+m₂) + g(m₁+m₂)cosθ₁ + l₂m₂ω₂²cosδ] / [l₂(2m₁+m₂−m₂cos2δ)]

E(t) = ½(m₁+m₂)l₁²ω₁² + ½m₂l₂²ω₂² + m₂l₁l₂ω₁ω₂cosδ − (m₁+m₂)gl₁cosθ₁ − m₂gl₂cosθ₂

Here δ = θ₁ − θ₂, angles are measured from the downward vertical. The default REGULAR preset starts at E_init = −28.929 J; the downward equilibrium reference is E_eq = −29.430 J for m₁ = m₂ = 1 kg, L = 1 m, g = 9.81 m/s². The drift readout uses E_init, not E_eq.

HOW TO OBSERVE

Three finite comparisons

Start with REGULAR CANDIDATE and inspect the section points for a narrow, repeatable island.
Switch to MIXED, then CHAOTIC SEA CANDIDATE. Compare point spread and the energy-drift readout over the same horizon.
Use STEP to inspect one 1/60 s UI advance. Use CLEAR SECTION to keep the current state while starting a fresh point record.

OBSERVATION POLICY

This observatory runs a deterministic, browser-local continuous-time double-pendulum model. Angles, calculations, section points, reports, and images stay local. It does not measure visitors or a real pendulum.

Route

This chamber belongs to Signals & Networks.

SIGNAL Back to Signals & Networks Route.