Points of a small circle around (x, y) are mapped into an ellipse
around (x', y'). E.g. for real eigenvalues of the matrix J
λ1,2 = x ± (x2 +
principal axis of this ellipse coinside with eigenvectors of the matrix
and deformation of the initial circle is determined by the
λ1,2 values. Thus for a = 1.4 and b = 0.3
the fixed point x2 = y2 = -0.884 is unstable
with λ1 = 0.156 and λ2 = -1.92 .
For a = -8, b = 1 correspondingly x = y = ± 2.83 and
λ1 = ± 5.83 and -λ2 =
± 0.17 .
To the left you see direct and inverse mappings of small vicinities
(with radiuses R) of the fixed points.