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
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
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
You see below, that rotational symmetry near repeller zo
keeps for "dendrite" and Cantor dust J-sets too.
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.
Spiral structures in the Julia sets
It is evident, that if φ = 2π m/n + δ ,
zk = f ok(z* +
ε ~ z* + ρ k
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.
Previous: The fixed points and periodic orbits
Next: Attracting fixed point and period 2 orbit
updated 17 August 2003