The Van der Pol Equation Is a Model of An Electronic Circuit

Assignment Task

1. The van der Pol equation,first arose in studying the first transistor, namely. The vacuum tube, when subjected to large currents. 

• Using µ = 0 . 1, use Maple to plot solution curves (on separate graphs!) when x (0) = 1 , x ^′ (0) = 0; x (0) = 1, x ^′ (0) = 0 . 5; x (0) = 1, x ^′ (0) = 1 . 5; and x (0) = 1, x ^′ (0) = 2 . Make sure your t range is large enough so that the long-term behaviour is clear ( t ∈ [0 , 1] is wildly inadequate).
• Repeat part (a) using µ = If your t -range is large enough, Maple should complain.
• Maple complained in the previous part because the ODE is “stiff,” and the Runge-Kutta 4-5 Method does not like stiff ODEs (DEsolve’s default ). You can get around this by specifying the method Maple should use (in the form method=). The “rosenbrock” method is a common choice for stiff

2. The diagram below shows a double pendulum, which consists of two weightless rods of length l joined together and attached to a fulcrum at the top, with a mass, m, at the The angles the rods make with the vertical are θ₁ and θ₂, as shown in the diagram.

The motion of a double pendulum is described by the following four symultaneous equations

where p ₁ and p ₂ represent the two momentums. We will use Maple’s dsolve and odeplot commands to try to understand how a double pendulum behaves.

• Enter the above system into Maple as a comma-separated list (a sequence, in Maple-speak).
• For the rest of this question we will be using m = 1 kg, l = 1 m, and g = 9.8 Substitute these values into the system using the eval command. 
• Use Maple to solve the system numerically with initial conditions θ ₁ = 2 , θ ₂ = 0.1, p ₁ = p ₂ =
• Use odeplot to plot θ ₁ and θ ₂ against Make sure that the curve is smooth, rather than jagged. In fact, for all subsequent parts, make sure your curves are smooth. What do you think of the graph?
• Use odeplot to plot θ ₂ against θ ₁ . Identify any unexpected Remember that θ ₁ and θ ₂ are angles in radians.
  • odeplot allows you to plot expressions in the DE variables. For example, if θ ₁ and θ2 are known to Maple as theta1 and theta2,
  • odeplot(DEsys, [t^2, theta1(t)*theta2(t)], t=0..10)
  • will plot θ ₁ (t)θ ₂ (t) against t ² . Use this fact to plot the locations of the mass.
• The odeplot command also allows you to animate what you are plotting over Do so with the plot of the last part
• Create an animation of the mass’ location where the initial conditions are θ ₁ = 2 , θ ₂ = 0.1001, p ₁ = p ₂ = 0. Do this plot in red (the next part explains the change in colour). 
• Use the display command to plot the two animations How long does it take for the two paths to differ noticeably? Note that the time scale is in seconds.

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