Poincaré section
A driven damped pendulum can swing periodically or chaotically depending on how hard you push it. The left view shows the trajectory in 3D phase space (θ, ω, t mod 2π). The right view shows the Poincaré section — a snapshot of the state each time the driving force completes a full cycle. Periodic motion gives discrete points; chaos gives a fractal strange attractor.
θ'' + γθ' + sin(θ) = A·cos(Ωt) | section at t mod 2π/Ω = 0
The driven damped pendulum
The equation θ'' + γθ' + sin(θ) = A cos(Ωt) describes a pendulum with damping coefficient γ and a periodic driving force of amplitude A and frequency Ω. For small driving, the pendulum settles into periodic oscillation matching the drive. But as A increases past critical values, the motion undergoes period-doubling bifurcations — the pendulum takes two drive cycles to repeat, then four, then eight, cascading to infinity at the onset of chaos. This is one of the cleanest demonstrations of the universal route to chaos discovered by Mitchell Feigenbaum.
What is a Poincaré section?
Henri Poincaré introduced this technique to reduce the complexity of continuous dynamics. Instead of watching the full trajectory, we strobe the system — recording its state only at regular intervals (here, once per drive period). A periodic orbit with period equal to the drive appears as a single point. A period-2 orbit appears as two points. Chaotic motion produces an infinite scatter of points, but not randomly — they fall on a strange attractor with intricate fractal structure. The section reduces a 3D flow to a 2D map while preserving the essential topology.
Strange attractors and fractal structure
In the chaotic regime, the Poincaré section reveals a strange attractor — a set with fractional dimension that the trajectory approaches but never exactly repeats. Nearby trajectories diverge exponentially (positive Lyapunov exponent), yet they remain confined to this fractal set. If you zoom into the attractor, you find layers within layers — a Cantor-set-like structure arising from the repeated stretching and folding of phase space. This sensitive dependence on initial conditions is the hallmark of deterministic chaos.
From order to chaos
Use the driving amplitude slider to watch the transition. At low A, the section shows one point (period-1). Increase A and it splits into two (period-2), then four, then a cascade to chaos. Within the chaotic regime, you may find windows of periodicity — narrow parameter ranges where order briefly re-emerges before chaos resumes. This interleaving of order and chaos is a universal feature of nonlinear dynamics, appearing in systems from dripping faucets to population dynamics to laser physics.