The system of differential equations:is called the van der Pol system. It arises in the st...

The system of differential equations:

is called the van der Pol system. It arises in the study of non-linear semiconductor circuits, where y represents a voltage and x the current. It is in the Gallery menu.

a) Find the equilibrium points for the system. Use pplane6 only to check your computations.

b) For various values of μ. in the range 0<μ 5, find the equilibrium points, and find the type of each, i.e, is it a nodal sink, a saddle point, …? You should find that there are at least two cases depending on the value of μ. Don’t worry too much about non-generic cases. Use pplane6 only to check your computations.

c) Use pplane6 to illustrate the behavior of solutions to the system in each of the cases found in b). Plot enough solutions to illustrate the phenomena you discover. Be sure to start some orbits very close to (0,0), and some near the edge of the display window. Put arrows on the solution curves (by hand after you have printed them out) to indicate the direction of the motion. (The display window (−5,5, −5,5) will allow you to see the interesting phenomena.)

d) For μ = 1 plot the solutions to the system with initial conditions x(0) = 0, and y(0) = 0.2. Plot both components of the solution versus t. Describe what happens to the solution curves as t → ∞.