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Exercise 5.2: Van der Pol equation.

Consider the van der Pol equation in relaxed oscillation:

The initial conditions are close to a limit cycle, and, in particular, to a slowly varying part of the cycle. This strains step size selection algorithms because the solution is so easy to approximate. With the parameter of 1000 the oscillations exhibits regions of very sharp change where the step size must be small to resolve the solution and the problem is not stiff. Elsewhere, the solution varies slowly and the problem is stiff; the problem is very stable here and all nontrivial solutions tend to vary rapidly to the limit cycle. The limit cycle has a period of about 1615.5 and the interval has been chosen to exhibit several of the internal boundary layers (regions of sharp change).

a) Use VODE with MF =21 (BDF method) to approximate the solution on and generate the plots y .vs. t, .vs. t, and .vs. y (phase plane). Identify the intervals over which the problem appears to be stiff.

b) Use VODE with MF =10 (Adams-Moulton method) to approximate the solution on . What happens?

c) Repeat with RKSUITE.

This problem is very hard for a code intended for non-stiff problems as it changes type from stiff to non-stiff and back as the integration proceeds.