An even more indepth analysis can be obtained with the use of a Poincaré Section. For this system, it can be calculated by examining the phase space of the system at the a given moment of the drive cycle. 

Observations:Below are some observed Poincaré Sections for various values of the driving amplitude and various angles of observation.
Fig. 1: Poincaré Section of damped, driven pendulum when ω_{D} = 2/3, q = 2, and Φ = 0.0. (a) g = 0.9, the system is periodic; (b) g = 1.07, the system exhibits period doubling; (c) g = 1.15, the system is chaotic.
Fig. 2: Poincaré Section of damped, driven pendulum when ω_{D} = 2/3, q = 2, and Φ = 0.0. (a) g = 1.35, the system is periodic; (b) g = 1.45, the system exhibits period doubling; (c) g = 1.47, the system exhibits period quadrupeling; (d) g = 1.50, the system is chaotic.
Fig. 3: Poincaré Sections for different values of Φ when ω_{D} = 2/3, q = 2, and g = 1.50. (a) Φ = 0.0; (b) Φ = 0.628; (c) Φ = 1.25; (d) Φ = 1.88; (e) Φ = 2.15; (f) Φ = 3.14. It can be observed from (a) and (f) that for angles greater than pi, the resultant Poincaré Sections are antisymmetrical to each other. 