Periodic points of complex quadratic mappings

This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.

These periodic points play a role in the theories of Fatou and Julia sets.

Definitions

Let

f_{c}(z)=z^{2}+c\,

be the complex quadric mapping, where $z$ and $c$ are complex-valued.

Notationally, $\ f_{c}^{{(k)}}(z)$ is the $\ k$ -fold composition of $f_{c}\,$ with itself—that is, the value after the k-th iteration of 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 $2^p.$

Stability of periodic points (orbit) - multiplier

Stability index of periodic points along horizontal axis

boundaries of regions of parameter plane with attracting orbit of periods 1-6

Critical orbit of discrete dynamical system based on complex quadratic polynomial. It tends to weakly attracting fixed point with abs(multiplier)=0.99993612384259

The multiplier ( or eigenvalue, derivative ) $m(f^p,z_0)=\lambda \,$ of a rational map $f\,$ iterated p times, at cyclic point $z_{0}\,$ is defined as:

m(f^p,z_0)=\lambda = \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 $f^{p\prime} (z_0)$ 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;^[1]
used to check stability of periodic (also fixed) points with stability index $abs(\lambda). \,$

A periodic point is^[2]

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

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.^[3] A parabolic periodic point is in the Julia set.

Period-1 points (fixed points)

Finite fixed points

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,\,

which can be rewritten as

\ z^{2}-z+c=0.

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

\alpha _{1}={\frac {1-{\sqrt {1-4c}}}{2}}

and

\alpha _{2}={\frac {1+{\sqrt {1-4c}}}{2}}.

So for $c\in C\setminus [1/4,+\inf ]$ we have two finite fixed points $\alpha _{1}\,$ and $\alpha _{2}\,$ .

Since

\alpha _{1}={\frac {1}{2}}-m

and

\alpha _{2}={\frac {1}{2}}+m

where

m={\frac {{\sqrt {1-4c}}}{2}}

then $\alpha_1 + \alpha_2 = 1 .\,$ .

Thus fixed points are symmetrical around $z = 1/2.\,$

This image shows fixed points (both repelling)

Complex dynamics

Fixed points for c along horizontal axis

Fatou set for F(z)=z*z with marked fixed point

Here different notation is commonly used:^[4]

\alpha _{c}={\frac {1-{\sqrt {1-4c}}}{2}}

with multiplier

\lambda _{{\alpha _{c}}}=1-{\sqrt {1-4c}}\,

and

\beta _{c}={\frac {1+{\sqrt {1-4c}}}{2}}

with multiplier

\lambda_{\beta_c} = 1+\sqrt{1-4c}.\,

Using Viète's formulas one can show that:

\alpha_c + \beta_c = 1 .

Since the derivative with respect to z is

P_c'(z) = \frac{d}{dz}P_c(z) = 2z ,

then

P_c'(\alpha_c) + P_c'(\beta_c)= 2 \alpha_c + 2 \beta_c = 2 (\alpha_c + \beta_c) = 2 . \,

This implies that $P_{c}\,$ can have at most one attractive fixed point.

These points are distinguished by the facts that:

$\beta _{c}\,$ is :
- the landing point of external ray for angle=0 for $c\in M\setminus \left\{{\frac {1}{4}}\right\}$
- the most repelling fixed point, belongs to 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).^[5]
$\alpha _{c}\,$ is:
- the landing point of several rays
- attracting when c is in main cardioid of Mandelbrot set, then it is in interior of Filled-in Julia set, it means belongs to Fatou set ( strictly to basin of attraction of finite fixed point )
- parabolic at the root point of the limb of Mandelbrot set
- repelling for other c values

Special cases

An important case of the quadratic mapping is $c=0$ . In this case, we get $\alpha _{1}=0$ and $\alpha _{2}=1$ . In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

Only one fixed point

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

Infinite fixed point

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

{\mathbb {{\hat {C}}}}={\mathbb {C}}\cup \{\infty \}

and extending polynomial $f_{c}\,$ such that $f_c(\infty)=\infty. \,$

Then infinity is :

superattracting
a fixed point of polynomial $f_c: \,$ ^[6]

f_c(\infty)=\infty=f^{-1}_c(\infty).\,

Period-2 cycles

Bifurcation from period 1 to 2 for complex quadratic map

Period-2 cycles are two distinct points $\beta _{1}$ and $\beta _{2}$ such that $f_{c}(\beta _{1})=\beta _{2}$ and $f_{c}(\beta _{2})=\beta _{1}$ .

We write $f_c(f_c(\beta_n)) = \beta_n:$

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 $\alpha _{1}$ and $\alpha _{2}$ , computed above, since if these points are left unchanged by one application of $f$ , then clearly they will be unchanged by 2 applications (or more).

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

First method of factorization

(z-\alpha _{1})(z-\alpha _{2})(z-\beta _{1})(z-\beta _{2})=0.\,

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

D = \alpha_1 \alpha_2 \beta_1 \beta_2, \,

C = \alpha_1 \alpha_2 \beta_1 + \alpha_1 \alpha_2 \beta_2 + \alpha_1 \beta_1 \beta_2 + \alpha_2 \beta_1 \beta_2, \,

B = \alpha_1 \alpha_2 + \alpha_1 \beta_1 + \alpha_1 \beta_2 + \alpha_2 \beta_1 + \alpha_2 \beta_2 + \beta_1 \beta_2, \,

A=\alpha _{1}+\alpha _{2}+\beta _{1}+\beta _{2}.\,

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

\alpha _{1}+\alpha _{2}={\frac {1-{\sqrt {1-4c}}}{2}}+{\frac {1+{\sqrt {1-4c}}}{2}}={\frac {1+1}{2}}=1

and

\alpha _{1}\alpha _{2}={\frac {(1-{\sqrt {1-4c}})(1+{\sqrt {1-4c}})}{4}}={\frac {1^{2}-({\sqrt {1-4c}})^{2}}{4}}={\frac {1-1+4c}{4}}={\frac {4c}{4}}=c.

Adding these to the above, we get $D=c\beta _{1}\beta _{2}$ and $A=1+\beta _{1}+\beta _{2}$ . Matching these against the coefficients from expanding $f$ , we get

D=c\beta _{1}\beta _{2}=c^{2}+c

and

A=1+\beta _{1}+\beta _{2}=0.

From this, we easily get

\beta _{1}\beta _{2}=c+1

and

\beta _{1}+\beta _{2}=-1

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

\beta _{1}={\frac {-1-{\sqrt {-3-4c}}}{2}}

and

\beta _{2}={\frac {-1+{\sqrt {-3-4c}}}{2}}.

Closer examination shows that :

f_{c}(\beta _{1})=\beta _{2}

and

f_c(\beta_2) = \beta_1,

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

Second method of factorization

We can factor the quartic by using polynomial long division to divide out the factors $(z-\alpha_1)$ and $(z-\alpha_2),$ which account for the two fixed points $\alpha _{1}$ and $\alpha _{2}$ (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 ). \,

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

-\frac{1}{2} \pm (-\frac{3}{4} - c)^\frac{1}{2}. \,

These two roots, which are the same as those found by the first method, form the period-2 orbit.^[7]

Special cases

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

\beta _{1}={\frac {-1-i{\sqrt {3}}}{2}}

and

\beta_2 = \frac{-1 + i\sqrt{3}}{2},

both of which are complex numbers. We have $|\beta _{1}|=|\beta _{2}|=1$ . Thus, both these points are "hiding" in the Julia set. Another special case is $c=-1$ , which gives $\beta _{1}=0$ and $\beta _{2}=-1$ . This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

Cycles for period greater than 2

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.^[8]

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

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

References

↑ Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, p. 41
↑ Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, page 99
↑ Some Julia sets by Michael Becker
↑ On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178.
↑ Periodic attractor by Evgeny Demidov
↑ 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
↑ Period 2 orbit by Evgeny Demidov
↑ Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram

External links

Wikibooks has a book on the topic of: Fractals

Algebraic solution of Mandelbrot orbital boundaries by Donald D. Cross
Brown Method by Robert P. Munafo
arXiv:hep-th/0501235v2 V.Dolotin, A.Morozov: Algebraic Geometry of Discrete Dynamics. The case of one variable.

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Periodic points of complex quadratic mappings

Definitions

Stability of periodic points (orbit) - multiplier

Period-1 points (fixed points)

Finite fixed points

Complex dynamics

Special cases

Only one fixed point

Infinite fixed point

Period-2 cycles

First method of factorization

Second method of factorization

Special cases

Cycles for period greater than 2

References

Further reading

External links