(3). To find the equilibrium points, set:

$\displaystyle \begin{aligned} -y+x(\mu-x^2-y^2)&=0 \\

x+y(\mu-x^2-y^2)&=0\end{aligned}$

Solving those simultaneously, I get the only equilibrium point is the origin. Now, I assume you can linearize that system and calculate the eigenvalues of the linear system. The eigenvalues will tell you if the origin is a sink and from that conclude if the origin is stable.

Don't know about B.

For C, solve the equations:

$\displaystyle x=x\sqrt{\frac{\mu}{\mu e^{-4\pi \mu}+x^2(1-e^{-4\pi \mu})}}$

and

$\displaystyle x=x\sqrt{\frac{1}{4\pi x^2+1}}$

If x is limited to the positive x-axis, then the only solution is $\displaystyle \sqrt{\mu}$

It can only be periodic if the flow through the Poincare section (the x-axis) returns to the exact same point and that can only happen if some point on the x-axis is a fixed point.

Also, try and get to Mathematica. In version 7, it has a very nice function called "StreamPlot" which plots effortlessly, the flow of these kinds of systems. For example, the code:

Code:

StreamPlot[{-y + x*(\[Mu] - x^2 - y^2),
x + y*(\[Mu] - x^2 - y^2)} /. \[Mu] -> 1,
{x, -2, 2}, {y, -2, 2}]

produced the plot below and it suggests that when $\displaystyle \mu=1$ we have a limit cycle that is the unit circle.