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In this homework, you use the application at this web address to explore a force-dampened pendulum, which is described by these equations
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This system was first introduced to you in the lecture for 3.1
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Notice that this is very similar to the equations you used in quiz 3.2, except for this term
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This is the forcing term, which allows for much different dynamics
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This is the application found at that web address
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The dynamics of this system are governed by the equation I just showed you
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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
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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
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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.
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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
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That is, therell be no forcing term in this initial problem
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Once we set these, we click Setup and Start Animation
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This should be very familiar to us
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The pendulum is slowing down and closing at smaller and smaller angles for every iteration
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In addition, in the state space trajectory, we see it slowly spiraling in to the fixed point (0,0)
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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
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To speed this up, lets go ahead and stop the animation, set the max trajectory length to 5,000, and generate a full trajectory
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As you can see, the dynamics have in fact converged to a fixed point for this parameter set
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Lets try this same experiment, but for a slightly different initial condition
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When we generate a full trajectory, we see that this is also a fixed point
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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)
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This allows us to answer question 1: this is a fixed point attractor