Use the Runge-Kutta algorithm, with a Runge-Kutta step size t near 0.2, to study the driven damped pendulum --0. 1u-sin(x) + cos(wt) cos x (This is different from the DDP discussed in class, because this force is horizontal, not azimuthal) a) Use quiescent initial conditions x-u-0 at time t-0. For -0.6 and -0.9 plot v(t) up tot 500, show that the transient has died away and that the attractor is period 1. Show also that the attractor is not sinusoidal. b) Now plot u(t) for quiescent initial conditions x(t-: 0)-v(t-0)-0 and -1.22 and 0.9. Show that the behavior is not periodic. Construct the Poincar section (the slice of the 3-d phase space at t-integer multiples of 2T/w) using the time-series data out to at leastt 20,000. It is most convenient to do this by choosing your Runge-Kutta time step to be such that you can easily identify these times, eg. M-2/40. Make sure you plot your x values modulo 2 Use the Runge-Kutta algorithm, with a Runge-Kutta step size t near 0.2, to study the driven damped pendulum --0. 1u-sin(x) + cos(wt) cos x (This is different from the DDP discussed in class, because this force is horizontal, not azimuthal) a) Use quiescent initial conditions x-u-0 at time t-0. For -0.6 and -0.9 plot v(t) up tot 500, show that the transient has died away and that the attractor is period 1. Show also that the attractor is not sinusoidal. b) Now plot u(t) for quiescent initial conditions x(t-: 0)-v(t-0)-0 and -1.22 and 0.9. Show that the behavior is not periodic. Construct the Poincar section (the slice of the 3-d phase space at t-integer multiples of 2T/w) using the time-series data out to at leastt 20,000. It is most convenient to do this by choosing your Runge-Kutta time step to be such that you can easily identify these times, eg. M-2/40. Make sure you plot your x values modulo 2