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. 

     

Poincare Section   =  

g   =         q   =        

Reinitialize

Observations:

          Below are some observed Poincaré Sections for various values of the driving amplitude and various angles of observation.

(a)

(b)

(c)

Fig. 1:  Poincaré Section of damped, driven pendulum when ωD = 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.

 

(a)

(b)

(c)

(d)

Fig. 2:  Poincaré Section of damped, driven pendulum when ωD = 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.

 

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 3:  Poincaré Sections for different values of Φ when ωD = 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 anti-symmetrical to each other.