The damped driven pendulum is modeled using :

d²(x)/d(t²) + b*d(x)/d(t) + sin(x) = F*cos(x).  (4)

Numerically simulate (4) with b=0.22 and F=2.7

a) Starting from any reasonable initial condition, perform a phase portrait analysis. Show that the time series has an erratic appearance, and interpret it in terms of the pendulum's motion.

b) Plot the Poincare section by sampling the system whenever  t=2*pi*k, where k is an integer.

 d^2(x)/d(t^2) + sin(x)=0  (1)

d^2(x)/d(t^2) + x = 0 (2)

d^2(x)/d(t^2) + ( x - (x)^3/6) = 0 (3)

1) Compare the results of numerical simulations of (1), (2), (3) to see how closely the period of the periodic orbits relate.

a) Perform a phase portrait ( (x)'(t) vs. x ) analysis for (1), (2), and (3).

b) Consider the initial conditions x(0)= x₀ and x'(0)=0. For what intervals of x₀ do the periodic orbits of (2...