I get $\displaystyle 1/2\pm \sqrt{3}/2$ for the eigenvalues: The origin is a source but that's true only for solutions near the origin. Further qualitative analysis would show that solutions close to the origin spiral outward to the equilibrium solution which is periodic and solutions outside the equilibrium solution spiral towards it. Here's some Mathematica code to illustrate these two scenarios. Me personally, I'd always do it numerically just to check things.

Code:

tmax = 50;
xinit = 0.2;
yinit = 0.2;
sol = NDSolve[{Derivative[1][x][t] == y[t],
Derivative[1][y][t] ==
-x[t] + (1 - x[t]^2)*y[t],
x[0] == xinit, y[0] == yinit},
{x[t], y[t]}, {t, 0, tmax}]
p1 = ParametricPlot[Evaluate[{x[t], y[t]} /.
sol], {t, 0, tmax}, PlotStyle -> Red]
tmax = 50;
xinit = 2.5;
yinit = 2.5;
sol = NDSolve[{Derivative[1][x][t] == y[t],
Derivative[1][y][t] ==
-x[t] + (1 - x[t]^2)*y[t],
x[0] == xinit, y[0] == yinit},
{x[t], y[t]}, {t, 0, tmax}]
p2 = ParametricPlot[Evaluate[{x[t], y[t]} /.
sol], {t, 0, tmax}, PlotStyle -> Blue]
Show[{p1, p2}, PlotRange ->
{{-3, 3}, {-3, 3}}]