4.4: Attracting Fixed point and Period 2 orbit
- Page ID
- 101795
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The main cardioid equation
There are always two fixed points z∗ = f(z∗ ) for a quadratic map
f(z∗ ) - z∗ = z∗2 + c - z∗ = 0, (1)
z1,2 = 1/2 ∓ (1/4 - c)½. (2)
A fixed point with multiplier λ = f '(z∗ ) = 2z∗ is attracting if
|λ| < 1, |z∗| < 1/2,
i.e. z∗ lies inside the u = ½ exp(iφ) circle. The multiplier on the circle is
λ = exp(iφ). (3)
It follows from (1) that c = z∗ - z∗2 and corresponding c lies inside the cardioid
c = u - u2 = ½ exp(iφ) - ¼ exp(2iφ),
Re(c) = ½ cos(φ) - ¼ cos(2φ),
Im(c) = ½ sin(φ) - ¼ sin(2φ).
The M-set in the "quadratic" parametrization
We get one more useful "quadratic" parametrization if we use
c = 1/4 - a2. (4)
As since a2 = (-a)2 the M is symmetric with respect to a = 0. After substitution of (4) into (2) we get:
z∗ = 1/2 ± a.
z∗ is attracting if |1/2 ± a| < 1/2, i.e. a lies inside one of the circles
½ eiφ ± 1/2 .
So the (4) transformation converts the main cardioid in two circles.
Internal angles theory
From (3) it follows that if a fixed point lies at the u = ½ exp(i2πm/n) value then under iterations its neighbourhood is rotated by the φ = 2πm/n "internal angle". On the main cardioid the corresponding point lies near the m/n bulb at
cφ = ½ eiφ - ¼ e2iφ .
In the "quadratic" parametrization
aφ = ½ eiφ - 1/2 .
Therefore aφ lays on the r = 1/2 circle at the angle φ with respect to the real axis.
Period 2 orbit
The equation for the period 2 orbit zo = f o2(zo ) = f(f(zo )) is
(zo2 + c)2 + c - zo = (zo2 + c - zo )(zo2 + zo + c + 1) = 0.
The roots of the first factor are the two fixed points z1,2 . They are repelling outside the main cardioid. The second factor has two roots
z3,4 = -1/2 ± (-3/4 - c)½.
These two roots form period-2 orbit. Since z3 z4 = c + 1 the multiplier of the orbit is
λ = f '(z3 ) f '(z4 ) = 4z3 z4 = 4(c + 1).
Therefore the orbit is attracting while |c + 1| < 1/4 or c lies within the ¼ exp(iφ) - 1 circle. This is exactly equation of the biggest 1/2 bulb to the left of the main cardioid.
I.e. the main cardioid and the 1/2 bulb are connected and touch each other in one point z = -3/4.
You see the points z1-4 positions for c = -0.71+0.1i (inside the main cardioid). Two roots z3 , z4 are symmetrical with respect to the point z = -1/2.
Repeller z2 lies in Julia set. Is it "very often" the extreme right point for connected Js ("very often" because it is not true e.g. for "cauliflower").