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
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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.
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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
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Next: Attracting fixed point and period 2 orbit
updated 17 August 2003