#### Exercise 5: Damped, Driven Rigid Pendulum

Next add a driving torque to your pendulum model. Once you’re sure that your model is accurately computing the damped, driven pendulum’s dynamics, systemically explore the behavior of the model by varying the driving torque amplitude τ d . Use the same physical parameters and initial conditions as before, including values of b = 0.2 and ω = 3 rad/s for the damping strength and angular frequency of the driving torque, respectively. Then, vary the magnitude of the driving torque amplitude from 2 Nm to 10 Nm, in 1 Nm increments, producing time-dependent and phase space plots for each value of τ d. For the time-dependent graphs of angular displacement and angular velocity, only a few periods (perhaps 5-6) are necessary to plot, but for the phase space trajectories it will be most interesting if you take the computations out to a few hundred thousand time steps or more. Of particular interest will be the phase space trajectories. The rigid pendulum is a type of nonlinear system, and therefore the dynamics actually become chaotic for certain physical parameters and initial conditions. To best observe the chaotic behavior, restrict the values of the angular displacement to the range - π to +π , by including two if-then constructions at the very end of the loop in which you implement the Euler-Cromer algorithm. If the angular displacement θ becomes less than -π then its value is increased by 2π . If θ becomes greater than π then its value is decreased 2π. Since θ is an angular variable, values that differ by 2π correspond to the same physical position of the pendulum. This restriction is not necessary, but will be convenient for your analysis. Describe in detail the dynamical behavior of your model as you vary the driving torque. Can you identify the behavior that corresponds to chaotic?