Iteration,

the Julia Set,

and the Mandelbrot Set

Iteration Terminology•Iteration – to repeat a process over and over.

•Iteration rule – the process that will be repeated over and over. (Can be numerical or geometric)

•Seed – the place to begin the iteration .

•Orbit of the iteration rule – the list of numbers or geometric figures obtained by successively

applying the iteration rule to the output of the previous iteration.

Linear IterationA linear iteration rule is an iteration rule of the form x → Ax + B where A and B are constants.

Linear IterationIteration rule: Seed: 0Applying the iteration rule produces the following orbit.0 → –2 → –3 → –3.5 → –3.75 → –3.875 → –3.9375 → …

The numbers in the orbit are getting closer and closer to –4.The fate of this orbit is: It approaches –4.

Linear IterationThe seed of an iteration rule is denoted x0, the next term in the iteration is x1, then x2, x3, and so on.

For the orbit0 → –2 → –3 → –3.5 → –3.75 → …

x0 = 0, x1 = –2, x2 = –3, x3 = –3.5, and so on.

Linear IterationIteration rule: Seed: x0 = –4

The orbit is–4 → –4 → –4 → –4 → –4 → …

The orbit stays constant at –4. The number –4 is called a fixed point for this iteration rule.

Linear IterationThe fixed point can be determined by solving the equation x = Ax + B .

Linear IterationIteration rule: Seed: x0 = 5

The orbit is5 → 0.5 → –1.75 → –2.875 → –3.4375 → –3.71875 → –3.859375 → …

The orbit of this seed also approaches –4. The orbit of any seed will approach –4 for this iteration rule.

Linear IterationIteration rule: x → 2x + 1Seed: x0 = 0

The orbit is0 → 1 → 3 → 7 → 15 → 31→ 63 → …

The numbers in this orbit grow larger and larger. This orbit tends to infinity.

Linear IterationWith the iteration rule x → 2x + 1, the orbit of x0 = –2 tends to (negative) infinity since the orbit is

–2 → –3 → –5 → –9 → –17 → –33 → …

The orbit of x0 = –1 is fixed under this iteration rule since x1 = –1.

Linear IterationFor the iteration rule x → –2x, 0 is a fixed point. All other orbits eventually alternate between large positive and large negative values.

The orbit of x0 = 2 under this rule begins2 → –4 → 8 → –16 → 32 → –64 → 128→ –256 → …

This orbit tends to positive and negative infinity.

Linear IterationOrbits may also cycle. The orbit of x0 = 4 for the linear iteration rule x → –x + 2 is

4 → –2 → 4 → –2 → 4 → –2 → 4 → …

This orbit is a cycle of period 2 since the orbit repeats every second iteration.

Types of Fixed PointsAn earlier iteration rule we looked at was . We said the fixed point for this rule was –4.

When the seed x0 = 5 was used, the orbit was

5 → 0.5 → –1.75 → –2.875 → –3.4375 → –3.71875 → –3.859375 → …

This orbit tends to –4.

The orbit of 0 also tended to –4.

Types of Fixed PointsFor the iteration rule , –4 is called an attracting fixed point.

The fixed point for the linear iteration rule x → Ax + B is attracting if all other orbits tend to the fixed point under iteration.

Types of Fixed PointsThe fixed point of the iteration rule x → 2x + 1 is –1.The orbit of x0 = 0 tended to infinity.

The orbit of x0 = –2 tended to (negative) infinity.

The fixed point –1 is called a repelling fixed point.

A fixed point for which all other orbits tend to move away from under iteration is a repelling fixed point.

Types of Fixed PointsThe linear iteration rule x → –x + 2 has a fixed point at x0 = 1, but it is neither attracting nor repelling.

The orbit of x0 = 4 is

4 → –2 → 4 → –2 → 4 → –2 → 4 → …The orbit of x0 = 7 is

7 → –5 → 7 → –5 → 7 → –5 → 7 → …

Types of Fixed PointsIf we choose any other seed x0 for the iteration rule x → –x + 2 then the orbit isx0 → –x0 + 2 → –(–x0 + 2 ) + 2 = x0.

This orbit begins to cycle after two iterations. This orbit is a cycle of period 2. The fixed point is neither attracting nor repelling, it is a neutral fixed point

Quadratic IterationA quadratic iteration rule is an iteration rule of the form x → x2 + c where c is a constant.

The fate of the orbit of x → x2 + c depends on the seed and the parameter c.

Quadratic IterationThe orbit of zero, under x → x2 + c, has different fates for different values of c.When c = 1, the orbit of 0 tends to infinity:

0 → 1 → 2 → 5 → 26 → 677 → …When c = 0, the orbit remains fixed at 0:

0 → 0 → 0 → 0 → …When c = −1, the orbit cycles with period 2:

0 → −1 → 0 → −1 → …When c = −2, the orbit of 0 is eventually fixed:

0 → −2 → 2 → 2 → 2 → …

Quadratic IterationThe orbit of 0 is called the critical orbit of the iteration rule x → x2 + c.

The value 0 is special because the minimum value of y = x2 + c occurs at x = 0.

Quadratic IterationFixed points can also be determined algebraically for quadratic iteration rules.

To find the fixed points, solve the equationx = x2 + c

Quadratic IterationSolving the equation x = x2 + c is equivalent to determining the place where the graph of y = x2 + c crosses the diagonal y = x.

The behavior near a fixed point can be determined graphically.

fixed point

Quadratic IterationThe initial seed is a point on the line y = x (or x-axis). The result of an iteration is the y-value on y = x2 + c associated with that x-value. That y-value becomes the next x-value to be iterated.

Quadratic IterationA repelling fixed point y = xy = x2 + c

Quadratic IterationAn attracting fixed point

y = xy = x2 + c

Quadratic Iterationy = x

The graph of y = x2 + 0.25 has a fixed point at x = 0.5.

y = x2 + 0.25

Appears to be attracting from the left

Appears to be repelling to the right

The fixed point 0.5 is neither attracting nor repelling, it is a neutral fixed point

Quadratic IterationOrbits of a quadratic iteration may be attracted to a fixed point or they may be repelled from it. Orbits may also cycle or tend to cycles with different periods.

Quadratic IterationFinding cycles for quadratic iterations algebraically is usually extremely difficult or impossible.

To find the 2-cycle for the rule x → x2 + c, iterate twicex → x2 + c → (x2 + c)2 + c

And then solve the equationx = (x2 + c)2 + c

To find the 3-cycle, iterate three times and solve the resulting equation.

Complex Linear Iteration•A complex number is a number of the form a + b .𝑖•The magnitude of a complex number is the distance of the complex number from the origin.

The magnitude of a + b is 𝑖•The polar angle of a complex number is the

angle formed by the positive x-axis and line connecting the complex number to the origin.

a + b𝑖θ

Complex Linear IterationIf a + b is a complex number with polar angle 𝑖 θ and magnitude r, then

a = r cos θb = r sin θ

a + b = r cos 𝑖 θ + r sin 𝑖 θ = r(cos θ + sin 𝑖 θ)This is the polar representation of the complex number a + b .𝑖

Complex Linear IterationFor two complex numbers a + b and c + d :𝑖 𝑖

(a + b ) + (c + d ) = (a + c) + (b + d) 𝑖 𝑖 𝑖e(a + b ) = ea + eb 𝑖 𝑖

If a + b = 𝑖 r1(cos θ1 + sin 𝑖 θ1)

and c + d = r𝑖 2(cos θ2 + sin 𝑖 θ2)

(a + b )(c + d ) = r𝑖 𝑖 1r2(cos θ1cos θ2 - sin θ1 sin θ2)

+ r𝑖 1r2(sin θ1cos θ2 + sin θ2 cos θ1)

(a + b )(c + d ) = r𝑖 𝑖 1r2(cos (θ1+θ2) + (sin (𝑖 θ1+θ2))

Complex Linear Iteration(a + b )(c + d ) = 𝑖 𝑖 r1r2(cos (θ1+θ2) + (sin (𝑖 θ1+θ2))

To multiply two complex numbers geometrically, add their polar angles and multiply their magnitudes.

Complex Linear IterationIteration rule: x → 2xSeed: x0 = 1 + 𝑖The orbit is

1 + → 2 + 2 → 4 + 4 → 8 + 8 → 16 + 16 → …𝑖 𝑖 𝑖 𝑖 𝑖The orbit moves farther and farther away from the origin. This orbit tends to infinity.

1 + 𝑖 2 + 2𝑖

4 + 4𝑖

Complex Linear IterationIteration rule: x → x𝑖Seed: x0 = a + b 𝑖The orbit isa + b → –b + a → –a – b → b – a → a + b → …𝑖 𝑖 𝑖 𝑖 𝑖Which is a 4-cycle in the complex plane.(Magnitude is the same, but each point is rotated 90°)

a + b𝑖–b + a𝑖

–a – b𝑖b – a𝑖

Complex Linear IterationIteration rule: x → ( + ) x𝑖Seed: 11 → + → → − + → − → …𝑖 𝑖 𝑖This orbit tends to 0.

+ 𝑖 + 𝑖

1

𝑖

Complex Linear IterationComplex iteration rule: x → Ax where A = a + b 𝑖If the magnitude of a + b is greater than 1, then when 𝑖we multiply a number by a + b , the resulting complex 𝑖number has greater magnitude.

The orbit of any nonzero number moves further and further from the origin and these orbits tend to infinity.

The origin is a repelling fixed point for this iteration rule.

Complex Linear IterationComplex iteration rule: x → Ax where A = a + b 𝑖If the magnitude of a + b is less than 1, then each 𝑖successive multiplication results in a complex number with smaller magnitude.

The orbit of any nonzero number moves closer and closer to the origin and these orbits tend to 0.

The origin is an attracting fixed point for this iteration rule.

Complex Linear IterationComplex iteration rule: x → Ax where A = a + b 𝑖If the magnitude of a + b is equal to 1, then the 𝑖magnitude of the seed is not changed. Multiplication rotates the given point by the polar angle of a + b . 𝑖The origin is a neutral fixed point for this iteration rule.

The Squaring RuleIteration rule: x → x2

y = x

y = x2

The Squaring RuleIteration rule: x → x2

•If x0 = 0 or 1, the orbit is fixed

•If 0 < | x0| < 1, the orbit tends to 0

•If | x0| >1, the orbit goes to infinity

•If x0 = –1, the orbit is eventually fixed at 1.

The Squaring RuleIteration rule: x → x2

Seed: x0 = r(cos θ + sin 𝑖 θ)

x0 = r(cos θ + sin 𝑖 θ)

x1 = r2(cos 2θ + sin 2𝑖 θ)

x2 = r4(cos 4θ + sin 4𝑖 θ)

x3 = r8(cos 8θ + sin 8𝑖 θ)...xn = r2n(cos 2nθ + sin 2𝑖 nθ)

The Squaring RuleComplex squaring iteration does not differ very much from the real case for most seeds.If r > 1, the orbit tends to infinity since r2n will get larger and larger.If r < 1, the orbit tends to 0 since r2n will get very small.If r = 1, the entire orbit lies on the circle of radius 1.

The Julia SetOrbits can be categorized into two types: the orbit tends to infinity or it does not.

If the orbit tends to infinity, the orbit “escapes.”

The collection of all seeds that do not escape is called a filled Julia set.

For the squaring iteration, the filled Julia set consists of all those seeds on and inside the circle of radius 1 centered at the origin.

The Julia SetsThe filled Julia set for the squaring iteration.

The Julia SetSeeds inside the circle of radius 1 tend to the attracting fixed point at the origin and those that lie on the circle have orbits that stay on the circle forever.

The circle of radius 1 is called the Julia set for this iteration rule.

The Julia set is the boundary between the seeds whose orbits escape and those whose orbits do not.

Julia Sets of Quadratic IterationsInstead of using the squaring iteration rule, x → x2, the more general quadratic iteration rule, x → x2 + c, can be used.

Julia Sets of Quadratic IterationsIt is important to be able to determine the fate of an orbit. If the orbit does not escape, it is in the filled Julia set. If it does escape, it is not in the set.

An orbit will escape under the iteration rule x → x2 + c if its magnitude ever exceeds an escape value.

The escape value is the larger of 2 and the magnitude of the given value of c.

Julia Sets of Quadratic IterationsSuppose that xn is a complex number and denotes the nth point along an orbit.According to the Triangle Inequality

| xn +1| = | xn2 + c | ≥ | xn|2 - |c|

If | xn| > 2, then

| xn|2 - |c| > 2| xn |- |c|.

And if | xn| > |c|, then

2| xn|- |c| > | xn|.

Julia Sets of Quadratic IterationsSo if | xn | > 2 and | xn | > |c|, then | xn+1| > | xn |.

The sequence is continuously increasing and the orbit of the seed must escape to infinity.

Therefore, the escape value is the larger of 2 and the magnitude of c.

Julia Sets of Quadratic IterationsThe first step in computing a filled Julia set is to determine the escape value.

Next, divide the complex plane into a grid of complex numbers. Each point on the grid represents a seed.

Compute the orbit of each grid point. If a point on an orbit ever has a magnitude greater than the escape value, then this orbit tends to infinity and is not in the filled Julia set. If that point does not tend to infinity it is in the set.

Julia Sets of Quadratic IterationsCompute the filled Julia set by hand for

x → x2 − 1

Escape value = 2

Julia Sets of Quadratic IterationsOn a TI-83:

Enter the seed, then press enter. (For complex seeds, use on the bottom row.)

Press (This gives x1)

Pressing enter again gives x2.

Continue pressing enter until the magnitude of the answer is > 2 or you have pressed enter 10 times.

If the orbit does not exceed the escape value, plot that point.

Draw Julia Set

2nd ANS x2 − ENTER1𝑖

Julia Sets of Quadratic Iterationsc = –1

Julia Sets of Quadratic Iterationsc = –0.75

Julia Sets of Quadratic Iterationsc = 0.25

Julia Sets of Quadratic Iterationsc = –0.12 + 0.75𝑖

This is sometimes called the fractal rabbit.

Julia Sets of Quadratic Iterations

This is a fractal because it is self similar.

Julia Sets of Quadratic Iterationsc = 0.27 + 0.53𝑖

Julia Sets of Quadratic Iterationsc = –0.75 + 0.2𝑖This is fractal dust. Under more iterations, the set of points will becomes smaller and smaller, but there are some points that will never escape.

Julia Sets of Quadratic IterationsThe orbit of 0 is called the critical orbit and it plays a role in determining the shape of the filled Julia set.

Julia Sets of Quadratic Iterationsx → x2 – 0.12 + 0.75𝑖

x0 = 0

x1 = –0.1200 + 0.7500𝑖x2 = –0.6681 + 0.5700 𝑖

x3 = –0.0015 + 0.0116 𝑖

x4 = – 0.1201 + 0.7500𝑖x5 = –0.6680 + 0.5698 𝑖

x6 = 0.0016 + 0.0113 𝑖

x7 = –0.1201 + 0.7500𝑖The orbit tends to a 3-cycle.

Julia Sets of Quadratic Iterations

–0.6680 + 0.5698 𝑖–0.1201 + 0.7500𝑖

0.0016 + 0.0113 𝑖

The cycle gives the number of pieces in the set.

Julia Sets of Quadratic Iterations

Julia Sets of Quadratic IterationsIf the orbit of 0 does not cycle, but escapes to infinity, then the corresponding filled Julia set for that c-value is fractal dust. When the orbit of 0 does not go to infinity, the filled Julia set is one connected piece, and its boundary, the Julia set, is often a fractal.

The Mandelbrot SetThe set of all c-values for which the orbit of 0 does not escape is called the Mandelbrot set.

It is the set of all c-values for which the corresponding Julia set is connected.

It is the set of c-values for which the corresponding orbit of 0 under x → x2 + c does not go to infinity.

The Mandelbrot Set

The Mandelbrot SetThis set has a very intricate geometry and there is a connection between the position on the Mandelbrot set and the shape of the Julia set, as well as the fate of the orbit of 0.

The Mandelbrot Set

The Mandelbrot SetRather than studying the Mandelbrot set itself, quite often the region very near to the set is studied.

Here, the orbit of 0 escapes to infinity. Sometimes, though, these orbits escape slowly. The number of iterations needed for the orbit to surpass some value is counted and a color is assigned to the point based on that value. The results can be quite aesthetically pleasing.

The Mandelbrot Setfor (cr = left; cr<=right; cr+=step)

{for (ci = top; ci<=bottom; ci+=step)

{zr = zi = zro = n = 0;while (n <= nmax && (zr * zr + zi * zi)< escape_value){zr = zr * zr - zi * zi + cr;zi = 2 * zi * zro + ci;zro = zr;n++;}if (n == nmax + 1) color = 0;elsecolor = colors[n];SetPixel(hdc, hcenter+cr*scale, vcenter+ci*scale, color);}

}

For more information on the Julia set and the Mandelbrot set, check out the following website.

http://math.bu.edu/DYSYS/explorer/index.html

ReferencesDevaney, R. & Choate, J. (2000). Chaos: A tool kit of

dynamics activities. Emeryville, CA: Key Curriculum Press.Devaney, R. (2000). The Mandelbrot and Julia sets: A

tool kit of dynamics activities. Emeryville, CA: Key Curriculum Press.

All graphics produced by Ron Koehn.