Consider the van der Pol system x? = y, y? = ?x + ?(1 ?

Consider the van der Pol system x? = y, y? = ?x + ?(1 ? x2 )y, where we now allow the parameter ? to be any real number. (a) Show that the origin is the only critical point. Determine its type and stability property, and how these depend on ?. (b) Let ? = ?1; draw a phase portrait and conclude that there is a periodic solution that surrounds the origin. Observe that this periodic solution is unstable. Compare your plot with Figure 7.5.4. (c) Draw a phase portrait for a few other negative values of ?. Describe how the shape of the periodic solution changes with ?. (d) Consider small positive or negative values of ?. By drawing phase portraits, determine how the periodic solution changes as ? ? 0. Compare the behavior of the van der Pol system as ? increases through zero with the behavior of the system in Problem 16. Problems 18 and 19 extend the consideration of the Rosenzweig