Chaotic Pendulums

Compare A and B with initial perturbation ε: angle in radians for double/driven pendulum, position in metres for spring/Atwood. The log graph shows tip separation in metres, not a mixed-unit state norm or a Lyapunov exponent. Zero is explicit. Chaos depends on energy and forcing; linear Atwood is a non-chaotic control.

Double pendulum: point masses/massless rods. Spring: Cartesian Hooke force with r₀/k/m, gravity downward9.81m/s²; zero spring length is singular and stops integration. Atwood: opposite relative displacements, acceleration g(m₁−m₂)/(m₁+m₂) and viscous damping γ (s⁻¹), not full pulley/cable geometry. Driven: ω₀=√(g/L), θ̈=ω₀²[−sinθ+Acos(Ωω₀t)]−γω₀θ̇; A/Ω/γ dimensionless. This replaces the unsupported Tilt-a-Whirl model.

Fixed RK4 steps; Step advances one step. Work backlog is retained. Physical changes reset, view changes preserve time. Hidden tools pause; paused views draw on demand. 3D shows the same planar paths z=0, without invented altitude. Marker sizes illustrative; PNG in both views. Long chaotic paths are not precision predictions.