
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
λ_{1,2} = x ± (x^{2} +
b)^{1/2}
principal axis of this ellipse coinside with eigenvectors of the matrix
and deformation of the initial circle is determined by the
λ_{1,2} values. Thus for a = 1.4 and b = 0.3
the fixed point x_{2} = y_{2} = 0.884 is unstable
with λ_{1} = 0.156 and λ_{2} = 1.92 .
For a = 8, b = 1 correspondingly x = y = ± 2.83 and
λ_{1} = ± 5.83 and λ_{2} =
± 0.17 .
To the left you see direct and inverse mappings of small vicinities
(with radiuses R) of the fixed points.
