Driven damped pendulum: θ̈ + γθ̇ + sin θ = F·cos(ωt). The Poincaré section (right) strobes the phase portrait at the drive period, revealing period-doubling routes to chaos.
Period-1 orbit: single dot in Poincaré. Period-2: two dots. Chaos: dense cloud. Try F≈1.5 for chaos, F≈1.15 for period-2. Classic example of the Feigenbaum cascade.