## Differential equations in the phase plane

The case of D = 0 requires more mathematics, as do cases in which periodic orbits (limit cycles) exist surrounding an unstable focus. The way that one demonstrates the existence of such limit cycles is to show that the steady state is an unstable focus but that points in the phase plane far away from the origin move towards it. A variety of good texts at the next level exist; I suggest that you poke around at a book store and spend time looking through different ones. A particularly simple example, which is called the Hopf bifurcation, corresponds to a pair of differential equations in polar coordinates for angle (0) and radius from the origin (r):

— = a — r2 dt where c = 0 and a are constants. In this case, the angular velocity is a constant c. The dynamics of radius are more interesting. If a < 0, then dr/dt < 0 and we see that the origin is a stable focus. However, if a > 0, the origin is unstable and the circle r = ^fa is stable. The parameter a passing through 0, a stable focus becoming unstable, and the appearance of a periodic orbit is called a Hopf bifurcation.

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