In this homework, you use the application at this web address to explore a force-dampened pendulum, which is described by these equations
This system was first introduced to you in the lecture for 3.1
Notice that this is very similar to the equations you used in quiz 3.2, except for this term
This is the forcing term, which allows for much different dynamics
This is the application found at that web address
The dynamics of this system are governed by the equation I just showed you
The top left pane shows a physical pendulum; the top right pane shows a state space portrait; the bottom left pane shows a time-series plot, and the bottom right pane allows you to control the different parameters of this algorithm
The purpose of this homework is to explore this new dynamical system, and relate the physical, state space, and time-series representations of the dynamics
While Ill only go over one of the problems the mandatory one the rest of the homework will provide a guided tour through these dynamics.
The first problem asks us to start at an initial condition of p/4 with no initial velocity, with the drag coefficient, or coefficient of friction, of 0.1, and both drives to zero
That is, therell be no forcing term in this initial problem
Once we set these, we click Setup and Start Animation
This should be very familiar to us
The pendulum is slowing down and closing at smaller and smaller angles for every iteration
In addition, in the state space trajectory, we see it slowly spiraling in to the fixed point (0,0)
If we look at the time-series plot, we see that the dynamics are very slowly converging down to what appears to be a fixed point
To speed this up, lets go ahead and stop the animation, set the max trajectory length to 5,000, and generate a full trajectory
As you can see, the dynamics have in fact converged to a fixed point for this parameter set
Lets try this same experiment, but for a slightly different initial condition
When we generate a full trajectory, we see that this is also a fixed point
Notice that 3.1 started right next to the unstable fixed point at p, but since it was not exactly p, it converged back to the stable fixed point at (0,0)
This allows us to answer question 1: this is a fixed point attractor