Local dynamics at a fixed point

We can write a complex multiplier λ (in the polar coordinate system) as
    λ = ρ exp(iφ ).
Then iterations (or images) of a point (zo + ε ) in the vicinity of a fixed point zo = f(zo) are
    zk = f ok(zo + ε ) = zo + λ kε + O(ε 2) ~ zo + ρ k eikφε .
That is, if we put coordinate origin to zo , after every iteration point zk+1 is rotated by angle φ with respect to the previous position zk and its radius is scaled by ρ = |λ|.

For φ = 2π m/n points zk 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 ρ = 1 + δ , |δ | << 1 (e.g. near the main cardioid border).

Attracting fixed point

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

Repelling fixed point

For c outside the main cardioid, ρ > 1 and the fixed point zo 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 zo rotations by 2π 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 zo keeps for "dendrite" and Cantor dust J-sets too.

Spiral structures in the Julia sets

It is evident, that if φ = 2π m/n + δ , then mapping
    zk = f ok(z* + ε ~ z* + ρ k eikφε
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