Local dynamics at a fixed point

We can write a complex multiplier l (in the polar coordinate system) as
l = r exp(if ).
Then iterations (or images) of a point (z_o + e ) in the vicinity of a fixed point z_o = f(z_o) are
z_k = f^ok(z_o + e ) = z_o + l ^ke + O(e²) ~ z_o + r^k e^ikfe .
That is, if we put coordinate origin to z_o , after every iteration point z_k+1 is rotated by angle f with respect to the previous position z_k and its radius is scaled by r = |l|.

For f = 2p m/n points z_k jump exactly m rays in the counterclockwise direction at each iteration and make n-rays "star" or "petals" structures discussed on the previous page. These structures are more "visible" for r = 1 + d , |d | << 1 (e.g. near the main cardioid border).

Attracting fixed point

For r < 1 all points in the vicinity of attractor z_o move smoothly to z_o . You can see "star" structures made by orbit of the critical point.

Repelling fixed point

	For c outside the main cardioid, r > 1 and the fixed point z_o becomes repelling (and it lies in J). Connected J set separates basin of attracting cycle and basin of infinit point. Therefore in the vicinity of z_o rotations by 2p m/n generate n-petals structures made of these two basins. Points in petals are attracted by periodic cycle and points in narrow whiskers go to infinity.

You see below, that rotational symmetry near repeller z_o keeps for "dendrite" and Cantor dust J-sets too.

Spiral structures in the Julia sets

It is evident, that if f = 2p m/n + d , then mapping
z_k = f^ok(z_* + e ) ~ z_* + r^k e^ikfe
generates spiral structures in the neighbourhood of the fixed point z_* . Some of these spirals are shown below.

Next we can investigate stability of fixed points and period 2 orbit of quadratic mappings analytically.

Contents Previous: The fixed points and periodic orbits Next: Attracting fixed point and period 2 orbit
updated 17 August 2003