Periodic points of complex quadratic mappings

Let

:$f\_c(z)\; =\; z^2+c\backslash ,$

be the complex quadratic mapping, where $z$ and $c$ are complex numbers.

Notationally, $f^\{(k)\}\; \_c\; (z)$ is the $k$-fold composition of $f\_c$ with itself (not to be confused with the $k$th derivative of $f\_c$)—that is, the value after the k-th iteration of the function $f\; \_c.$ Thus

:$f^\{(k)\}\; \_c\; (z)\; =\; f\_c(f^\{(k-1)\}\; \_c\; (z)).$

Periodic points of a complex quadratic mapping of period $p$ are points $z$ of the dynamical plane such that

:$f^\{(p)\}\; \_c\; (z)\; =\; z,$

where $p$ is the smallest positive integer for which the equation holds at that z.

We can introduce a new function:

:$F\_p(z,f)\; =\; f^\{(p)\}\; \_c\; (z)\; -\; z,$

so periodic points are zeros of function $F\_p(z,f)$: points z satisfying

:$F\_p(z,f)\; =\; 0,$

which is a polynomial of degree $2^p.$

The degree of the polynomial $F\_p(z,f)$ describing periodic points is $d\; =\; 2^p$ so it has exactly $d\; =\; 2^p$ complex roots (= periodic points), counted with multiplicity.

File:Multiplier4 f.png

File:Mandelbrot set Components.jpg

File:Critical orbit 3d.png. It tends to weakly attracting fixed point with abs(multiplier) = 0.99993612384259]]

The multiplier (or eigenvalue, derivative) $m(f^p,z\_0)=\backslash lambda$ of a rational map $f$ iterated $p$ times at cyclic point $z\_0$ is defined as:

:$m(f^p,z\_0)\; =\; \backslash lambda\; =\; \backslash begin\{cases\}$

f^{p \prime}(z_0), &\mbox{if }z_0 \ne \infty \\

\frac{1}{f^{p \prime} (z_0)}, & \mbox{if }z_0 = \infty \end{cases}

where $f^\{p\backslash prime\}\; (z\_0)$ is the first derivative of $f^p$ with respect to $z$ at $z\_0$.

Because the multiplier is the same at all periodic points on a given orbit, it is called a multiplier of the periodic orbit.

The multiplier is:

• a complex number;
• invariant under conjugation of any rational map at its fixed point;Alan F. Beardon, Iteration of Rational Functions, Springer 1991, {{ISBN|0-387-95151-2}}, p. 41
• used to check stability of periodic (also fixed) points with stability index $abs(\backslash lambda).\; \backslash ,$

A periodic point isAlan F. Beardon, Iteration of Rational Functions, Springer 1991, {{ISBN|0-387-95151-2}}, page 99

• attracting when
• super-attracting when $abs(\backslash lambda)\; =\; 0;$
• attracting but not super-attracting when
• indifferent when $abs(\backslash lambda)\; =\; 1;$
• rationally indifferent or parabolic if $\backslash lambda$ is a root of unity;
• irrationally indifferent if $abs(\backslash lambda)=1$ but multiplier is not a root of unity;
• repelling when $abs(\backslash lambda)\; >\; 1.$

Periodic points

• that are attracting are always in the Fatou set;
• that are repelling are in the Julia set;
• that are indifferent fixed points may be in one or the other.[http://www.ijon.de/mathe/julia/some_julia_sets_1_en.html Some Julia sets by Michael Becker] A parabolic periodic point is in the Julia set.

Let us begin by finding all finite points left unchanged by one application of $f$. These are the points that satisfy $f\_c(z)=z$. That is, we wish to solve

: $z^2+c=z,\backslash ,$

which can be rewritten as

: $\backslash \; z^2-z+c=0.$

Since this is an ordinary quadratic equation in one unknown, we can apply the standard quadratic solution formula:

: $\backslash alpha\_1\; =\; \backslash frac\{1-\backslash sqrt\{1-4c\}\}\{2\}$ and $\backslash alpha\_2\; =\; \backslash frac\{1+\backslash sqrt\{1-4c\}\}\{2\}.$

So for $c\; \backslash in\; \backslash mathbb\{C\}\; \backslash setminus\; \backslash \{1/4\backslash \}$ we have two finite fixed points $\backslash alpha\_1$ and $\backslash alpha\_2$.

Since

: $\backslash alpha\_1\; =\; \backslash frac\{1\}\{2\}-m$ and $\backslash alpha\_2\; =\; \backslash frac\{1\}\{2\}+m$ where $m\; =\; \backslash frac\{\backslash sqrt\{1-4c\}\}\{2\},$

we have $\backslash alpha\_1\; +\; \backslash alpha\_2\; =\; 1$.

Thus fixed points are symmetrical about $z\; =\; 1/2$.

File:JuliaRay3.png

File:Fixed.png

File:Julia0bb.jpg for F(z) = z*z with marked fixed point]]

Here different notation is commonly used:[http://www.math.nagoya-u.ac.jp/~kawahira/works/cauliflower.pdf On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178. ] {{webarchive|url=https://web.archive.org/web/20110717033213/http://www.math.nagoya-u.ac.jp/~kawahira/works/cauliflower.pdf |date=2011-07-17 }}

:$\backslash alpha\_c\; =\; \backslash frac\{1-\backslash sqrt\{1-4c\}\}\{2\}$ with multiplier $\backslash lambda\_\{\backslash alpha\_c\}\; =\; 1-\backslash sqrt\{1-4c\}$

and

:$\backslash beta\_c\; =\; \backslash frac\{1+\backslash sqrt\{1-4c\}\}\{2\}$ with multiplier $\backslash lambda\_\{\backslash beta\_c\}\; =\; 1+\backslash sqrt\{1-4c\}.$

Again we have

:$\backslash alpha\_c\; +\; \backslash beta\_c\; =\; 1\; .$

Since the derivative with respect to z is

:$P\_c\text{'}(z)\; =\; \backslash frac\{d\}\{dz\}P\_c(z)\; =\; 2z\; ,$

we have

:$P\_c\text{'}(\backslash alpha\_c)\; +\; P\_c\text{'}(\backslash beta\_c)=\; 2\; \backslash alpha\_c\; +\; 2\; \backslash beta\_c\; =\; 2\; (\backslash alpha\_c\; +\; \backslash beta\_c)\; =\; 2\; .$

This implies that $P\_c$ can have at most one attractive fixed point.

These points are distinguished by the facts that:

• $\backslash beta\_c$ is:
• the landing point of the external ray for angle=0 for $c\; \backslash in\; M\; \backslash setminus\; \backslash left\backslash \{\; 1/4\; \backslash right\backslash \}$
• the most repelling fixed point of the Julia set
• the one on the right (whenever fixed point are not symmetrical around the real axis), it is the extreme right point for connected Julia sets (except for cauliflower).[http://www.ibiblio.org/e-notes/MSet/Attractor.htm Periodic attractor by Evgeny Demidov] {{webarchive|url=https://web.archive.org/web/20080511195806/http://www.ibiblio.org/e-notes/MSet/Attractor.htm |date=2008-05-11 }}
• $\backslash alpha\_c$ is:
• the landing point of several rays
• attracting when $c$ is in the main cardioid of the Mandelbrot set, in which case it is in the interior of a filled-in Julia set, and therefore belongs to the Fatou set (strictly to the basin of attraction of finite fixed point)
• parabolic at the root point of the limb of the Mandelbrot set
• repelling for other values of $c$

An important case of the quadratic mapping is $c=0$. In this case, we get $\backslash alpha\_1\; =\; 0$ and $\backslash alpha\_2=1$. In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

We have $\backslash alpha\_1=\backslash alpha\_2$ exactly when $1-4c=0.$ This equation has one solution, $c=1/4,$ in which case $\backslash alpha\_1=\backslash alpha\_2=1/2$. In fact $c=1/4$ is the largest positive, purely real value for which a finite attractor exists.

We can extend the complex plane $\backslash mathbb\{C\}$ to the Riemann sphere (extended complex plane) $\backslash mathbb\{\backslash hat\{C\}\}$ by adding infinity:

:$\backslash mathbb\{\backslash hat\{C\}\}\; =\; \backslash mathbb\{C\}\; \backslash cup\; \backslash \{\; \backslash infty\; \backslash \}$

and extend $f\_c$ such that $f\_c(\backslash infty)=\backslash infty.$

Then infinity is:

• superattracting
• a fixed point of $f\_c$:R L Devaney, L Keen (Editor): Chaos and Fractals: The Mathematics Behind the Computer Graphics. Publisher: Amer Mathematical Society July 1989, {{ISBN|0-8218-0137-6}} , {{ISBN|978-0-8218-0137-6}}$$f\_c(\backslash infty)=\backslash infty=f^\{-1\}\_c(\backslash infty).$$

File:Bifurcation1-2.png]]

File:Bifurcation of periodic points from period 1 to 2 for fc(z)=z*z +c.gif

Period-2 cycles are two distinct points $\backslash beta\_1$ and $\backslash beta\_2$ such that $f\_c(\backslash beta\_1)\; =\; \backslash beta\_2$ and $f\_c(\backslash beta\_2)\; =\; \backslash beta\_1$, and hence

:$f\_c(f\_c(\backslash beta\_n))\; =\; \backslash beta\_n$

for $n\; \backslash in\; \backslash \{1,\; 2\backslash \}$:

:$f\_c(f\_c(z))\; =\; (z^2+c)^2+c\; =\; z^4\; +\; 2cz^2\; +\; c^2\; +\; c.$

Equating this to z, we obtain

:$z^4\; +\; 2cz^2\; -\; z\; +\; c^2\; +\; c\; =\; 0.$

This equation is a polynomial of degree 4, and so has four (possibly non-distinct) solutions. However, we already know two of the solutions. They are $\backslash alpha\_1$ and $\backslash alpha\_2$, computed above, since if these points are left unchanged by one application of $f$, then clearly they will be unchanged by more than one application of $f$.

Our 4th-order polynomial can therefore be factored in 2 ways:

: $(z-\backslash alpha\_1)(z-\backslash alpha\_2)(z-\backslash beta\_1)(z-\backslash beta\_2)\; =\; 0.\backslash ,$

This expands directly as $x^4\; -\; Ax^3\; +\; Bx^2\; -\; Cx\; +\; D\; =\; 0$ (note the alternating signs), where

: $D\; =\; \backslash alpha\_1\; \backslash alpha\_2\; \backslash beta\_1\; \backslash beta\_2,\; \backslash ,$

: $C\; =\; \backslash alpha\_1\; \backslash alpha\_2\; \backslash beta\_1\; +\; \backslash alpha\_1\; \backslash alpha\_2\; \backslash beta\_2\; +\; \backslash alpha\_1\; \backslash beta\_1\; \backslash beta\_2\; +\; \backslash alpha\_2\; \backslash beta\_1\; \backslash beta\_2,\; \backslash ,$

: $B\; =\; \backslash alpha\_1\; \backslash alpha\_2\; +\; \backslash alpha\_1\; \backslash beta\_1\; +\; \backslash alpha\_1\; \backslash beta\_2\; +\; \backslash alpha\_2\; \backslash beta\_1\; +\; \backslash alpha\_2\; \backslash beta\_2\; +\; \backslash beta\_1\; \backslash beta\_2,\; \backslash ,$

: $A\; =\; \backslash alpha\_1\; +\; \backslash alpha\_2\; +\; \backslash beta\_1\; +\; \backslash beta\_2.\backslash ,$

We already have two solutions, and only need the other two. Hence the problem is equivalent to solving a quadratic polynomial. In particular, note that

: $\backslash alpha\_1\; +\; \backslash alpha\_2\; =\; \backslash frac\{1-\backslash sqrt\{1-4c\}\}\{2\}\; +\; \backslash frac\{1+\backslash sqrt\{1-4c\}\}\{2\}\; =\; \backslash frac\{1+1\}\{2\}\; =\; 1$

and

: $\backslash alpha\_1\; \backslash alpha\_2\; =\; \backslash frac\{(1-\backslash sqrt\{1-4c\})(1+\backslash sqrt\{1-4c\})\}\{4\}\; =\; \backslash frac\{1^2\; -\; (\backslash sqrt\{1-4c\})^2\}\{4\}=\; \backslash frac\{1\; -\; 1\; +\; 4c\}\{4\}\; =\; \backslash frac\{4c\}\{4\}\; =\; c.$

Adding these to the above, we get $D\; =\; c\; \backslash beta\_1\; \backslash beta\_2$ and $A\; =\; 1\; +\; \backslash beta\_1\; +\; \backslash beta\_2$. Matching these against the coefficients from expanding $f$, we get

: $D\; =\; c\; \backslash beta\_1\; \backslash beta\_2\; =\; c^2\; +\; c$ and $A\; =\; 1\; +\; \backslash beta\_1\; +\; \backslash beta\_2\; =\; 0.$

From this, we easily get

:$\backslash beta\_1\; \backslash beta\_2\; =\; c\; +\; 1$ and $\backslash beta\_1\; +\; \backslash beta\_2\; =\; -1$.

From here, we construct a quadratic equation with $A\text{'}\; =\; 1,\; B\; =\; 1,\; C\; =\; c+1$ and apply the standard solution formula to get

: $\backslash beta\_1\; =\; \backslash frac\{-1\; -\; \backslash sqrt\{-3\; -4c\}\}\{2\}$ and $\backslash beta\_2\; =\; \backslash frac\{-1\; +\; \backslash sqrt\{-3\; -4c\}\}\{2\}.$

Closer examination shows that:

:$f\_c(\backslash beta\_1)\; =\; \backslash beta\_2$ and $f\_c(\backslash beta\_2)\; =\; \backslash beta\_1,$

meaning these two points are the two points on a single period-2 cycle.

We can factor the quartic by using polynomial long division to divide out the factors $(z-\backslash alpha\_1)$ and $(z-\backslash alpha\_2),$ which account for the two fixed points $\backslash alpha\_1$ and $\backslash alpha\_2$ (whose values were given earlier and which still remain at the fixed point after two iterations):

:$(z^2+c)^2\; +\; c\; -z\; =\; (z^2\; +\; c\; -\; z)(z^2\; +\; z\; +\; c\; +1\; ).\; \backslash ,$

The roots of the first factor are the two fixed points. They are repelling outside the main cardioid.

The second factor has the two roots

:$\backslash frac\{-1\; \backslash pm\; \backslash sqrt\{-3\; -4c\}\}\{2\}.\; \backslash ,$

These two roots, which are the same as those found by the first method, form the period-2 orbit.[http://www.ibiblio.org/e-notes/MSet/Attractor.htm Period 2 orbit by Evgeny Demidov] {{webarchive|url=https://web.archive.org/web/20080511195806/http://www.ibiblio.org/e-notes/MSet/Attractor.htm |date=2008-05-11 }}

Again, let us look at $c=0$. Then

: $\backslash beta\_1\; =\; \backslash frac\{-1\; -\; i\backslash sqrt\{3\}\}\{2\}$ and $\backslash beta\_2\; =\; \backslash frac\{-1\; +\; i\backslash sqrt\{3\}\}\{2\},$

both of which are complex numbers. We have $|\; \backslash beta\_1\; |\; =\; |\; \backslash beta\_2\; |\; =\; 1$. Thus, both these points are "hiding" in the Julia set.

Another special case is $c=-1$, which gives $\backslash beta\_1\; =\; 0$ and $\backslash beta\_2\; =\; -1$. This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

File:Periodic points of f(z) = z*z-0.75 for period =6 as intersections of 2 implicit curves.svg

The degree of the equation $f^\{(n)\}(z)=z$ is 2ⁿ; thus for example, to find the points on a 3-cycle we would need to solve an equation of degree 8. After factoring out the factors giving the two fixed points, we would have a sixth degree equation.

There is no general solution in radicals to polynomial equations of degree five or higher, so the points on a cycle of period greater than 2 must in general be computed using numerical methods. However, in the specific case of period 4 the cyclical points have lengthy expressions in radicals.[https://arxiv.org/abs/0802.2565 Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram]

In the case c = –2, trigonometric solutions exist for the periodic points of all periods. The case $z\_\{n+1\}=z\_n^2-2$ is equivalent to the logistic map case r = 4: $x\_\{n+1\}=4x\_n(1-x\_n).$ Here the equivalence is given by $z=2-4x.$ One of the k-cycles of the logistic variable x (all of which cycles are repelling) is

:$\backslash sin^2\backslash left(\backslash frac\{2\backslash pi\}\{2^k-1\}\backslash right),\; \backslash ,\; \backslash sin^2\backslash left(2\backslash cdot\backslash frac\{2\backslash pi\}\{2^k-1\}\backslash right),\; \backslash ,\; \backslash sin^2\backslash left(2^2\backslash cdot\backslash frac\{2\backslash pi\}\{2^k-1\}\backslash right),\; \backslash ,\; \backslash sin^2\backslash left(2^3\backslash cdot\backslash frac\{2\backslash pi\}\{2^k-1\}\backslash right),\; \backslash dots\; ,\; \backslash sin^2\backslash left(2^\{k-1\}\backslash frac\{2\backslash pi\}\{2^k-1\}\backslash right).$

{{Reflist}}

• Geometrical properties of polynomial roots
• Alan F. Beardon, Iteration of Rational Functions, Springer 1991, {{ISBN|0-387-95151-2}}
• Michael F. Barnsley (Author), Stephen G. Demko (Editor), Chaotic Dynamics and Fractals (Notes and Reports in Mathematics in Science and Engineering Series) Academic Pr (April 1986), {{ISBN|0-12-079060-2}}
• [https://web.archive.org/web/20040828174339/http://www.math.sunysb.edu/cgi-bin/thesis.pl?thesis02-3 Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002]
• [http://hdl.handle.net/10090/3895 The permutations of periodic points in quadratic polynominials by J Leahy]

{{Wikibooks|Fractals }}

• [http://cosinekitty.com/mandel_orbits_analysis.html Algebraic solution of Mandelbrot orbital boundaries by Donald D. Cross ]
• [http://www.mrob.com/pub/muency/brownmethod.html Brown Method by Robert P. Munafo]
• [https://arxiv.org/abs/hep-th/0501235 arXiv:hep-th/0501235v2] V.Dolotin, A.Morozov: Algebraic Geometry of Discrete Dynamics. The case of one variable.

{{DEFAULTSORT:Periodic Points Of Complex Quadratic Mappings}}

Category:Complex dynamics

Category:Fractals

Category:Limit sets