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Iteration, the Julia Set, and the Mandelbrot Set

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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.

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

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Linear Iteration

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The seed of an iteration rule is denoted x 0, the next term in the iteration is x 1, then x 2, x 3, and so on. For the orbit 0 → –2 → –3 → –3.5 → –3.75 → … x 0 = 0, x 1 = –2, x 2 = –3, x 3 = –3.5, and so on.

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Linear Iteration

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

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Linear Iteration

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Iteration rule: x → 2x + 1 Seed: x 0 = 0 The orbit is 0 → 1 → 3 → 7 → 15 → 31→ 63 → … The numbers in this orbit grow larger and larger. This orbit tends to infinity.

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Linear Iteration With the iteration rule x → 2x + 1, the orbit of x 0 = –2 tends to (negative) infinity since the orbit is –2 → –3 → –5 → –9 → –17 → –33 → … The orbit of x 0 = –1 is fixed under this iteration rule since x 1 = –1.

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Linear Iteration For 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 x 0 = 2 under this rule begins 2 → –4 → 8 → –16 → 32 → –64 → 128→ –256 → … This orbit tends to positive and negative infinity.

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Linear Iteration Orbits may also cycle. The orbit of x 0 = 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.

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Types of Fixed Points

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

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Types of Fixed Points The linear iteration rule x → –x + 2 has a fixed point at x 0 = 1, but it is neither attracting nor repelling. The orbit of x 0 = 4 is 4 → –2 → 4 → –2 → 4 → –2 → 4 → … The orbit of x 0 = 7 is 7 → –5 → 7 → –5 → 7 → –5 → 7 → …

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Types of Fixed Points If we choose any other seed x 0 for the iteration rule x → –x + 2 then the orbit is x 0 → –x → –(–x ) + 2 = x 0. 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

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Quadratic Iteration A quadratic iteration rule is an iteration rule of the form x → x 2 + c where c is a constant. The fate of the orbit of x → x 2 + c depends on the seed and the parameter c.

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Quadratic Iteration The orbit of zero, under x → x 2 + 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 → …

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Quadratic Iteration The orbit of 0 is called the critical orbit of the iteration rule x → x 2 + c. The value 0 is special because the minimum value of y = x 2 + c occurs at x = 0.

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Quadratic Iteration Fixed points can also be determined algebraically for quadratic iteration rules. To find the fixed points, solve the equation x = x 2 + c

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Quadratic Iteration Solving the equation x = x 2 + c is equivalent to determining the place where the graph of y = x 2 + c crosses the diagonal y = x. The behavior near a fixed point can be determined graphically. fixed point

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Quadratic Iteration The initial seed is a point on the line y = x (or x-axis). The result of an iteration is the y-value on y = x 2 + c associated with that x-value. That y-value becomes the next x-value to be iterated.

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Quadratic Iteration A repelling fixed point y = x y = x 2 + c

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Quadratic Iteration An attracting fixed point y = x y = x 2 + c

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Quadratic Iteration y = x The graph of y = x has a fixed point at x = 0.5. y = x 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

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Quadratic Iteration Orbits 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.

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Quadratic Iteration Finding cycles for quadratic iterations algebraically is usually extremely difficult or impossible. To find the 2-cycle for the rule x → x 2 + c, iterate twice x → x 2 + c → (x 2 + c) 2 + c And then solve the equation x = (x 2 + c) 2 + c To find the 3-cycle, iterate three times and solve the resulting equation.

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Complex Linear Iteration a + b θ

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Complex Linear Iteration If a + b is a complex number with polar angle θ and magnitude r, then a = r cos θ b = r sin θ a + b = r cos θ + r sin θ = r(cos θ + sin θ) This is the polar representation of the complex number a + b.

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Complex Linear Iteration For two complex numbers a + b and c + d: (a + b) + (c + d) = (a + c) + (b + d) e(a + b) = ea + eb If a + b = r 1 (cos θ 1 + sin θ 1 ) and c + d = r 2 (cos θ 2 + sin θ 2 ) (a + b)(c + d) = r 1 r 2 (cos θ 1 cos θ 2 - sin θ 1 sin θ 2 ) + r 1 r 2 (sin θ 1 cos θ 2 + sin θ 2 cos θ 1 ) (a + b)(c + d) = r 1 r 2 (cos (θ 1 +θ 2 ) + (sin (θ 1 +θ 2 ))

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Complex Linear Iteration (a + b)(c + d) = r 1 r 2 (cos (θ 1 +θ 2 ) + (sin (θ 1 +θ 2 )) To multiply two complex numbers geometrically, add their polar angles and multiply their magnitudes.

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Complex Linear Iteration Iteration rule: x → 2x Seed: x 0 = 1 + The orbit is 1 + → → → → → … The orbit moves farther and farther away from the origin. This orbit tends to infinity

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Complex Linear Iteration Iteration rule: x → x Seed: x 0 = a + b The orbit is a + b → –b + a → –a – b → b – a → a + b → … Which is a 4-cycle in the complex plane. (Magnitude is the same, but each point is rotated 90°) a + b –b + a –a – b b – a

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Complex Linear Iteration 1

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Complex iteration rule: x → Ax where A = a + b If the magnitude of a + b is greater than 1, then when we multiply a number by a + b, the resulting complex number 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.

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Complex Linear Iteration Complex iteration rule: x → Ax where A = a + b If the magnitude of a + b is less than 1, then each successive 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.

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

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The Squaring Rule Iteration rule: x → x 2 y = x y = x 2

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The Squaring Rule Iteration rule: x → x 2 If x 0 = 0 or 1, the orbit is fixed If 0 < | x 0 | < 1, the orbit tends to 0 If | x 0 | >1, the orbit goes to infinity If x 0 = –1, the orbit is eventually fixed at 1.

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The Squaring Rule Iteration rule: x → x 2 Seed: x 0 = r(cos θ + sin θ) x 0 = r(cos θ + sin θ) x 1 = r 2 (cos 2θ + sin 2θ) x 2 = r 4 (cos 4θ + sin 4θ) x 3 = r 8 (cos 8θ + sin 8θ). x n = r 2n (cos 2 n θ + sin 2 n θ)

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The Squaring Rule Complex squaring iteration does not differ very much from the real case for most seeds. If r > 1, the orbit tends to infinity since r 2n will get larger and larger. If r < 1, the orbit tends to 0 since r 2n will get very small. If r = 1, the entire orbit lies on the circle of radius 1.

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The Julia Set Orbits 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.

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The Julia Sets The filled Julia set for the squaring iteration.

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The Julia Set Seeds 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.

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Julia Sets of Quadratic Iterations Instead of using the squaring iteration rule, x → x 2, the more general quadratic iteration rule, x → x 2 + c, can be used.

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Julia Sets of Quadratic Iterations It 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 → x 2 + 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.

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Julia Sets of Quadratic Iterations Suppose that x n is a complex number and denotes the nth point along an orbit. According to the Triangle Inequality | x n +1 | = | x n 2 + c | ≥ | x n | 2 - |c| If | x n | > 2, then | x n | 2 - |c| > 2| x n |- |c|. And if | x n | > |c|, then 2| x n |- |c| > | x n |.

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Julia Sets of Quadratic Iterations So if | x n | > 2 and | x n | > |c|, then | x n+1 | > | x n |. 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.

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Julia Sets of Quadratic Iterations The 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.

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Julia Sets of Quadratic Iterations Compute the filled Julia set by hand for x → x 2 − 1 Escape value = 2

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Julia Sets of Quadratic Iterations On a TI-83: Enter the seed, then press enter. (For complex seeds, use on the bottom row.) Press (This gives x 1 ) Pressing enter again gives x 2. 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 ANSx2x2 − ENTER 1

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Julia Sets of Quadratic Iterations c = –1

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Julia Sets of Quadratic Iterations c = –0.75

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Julia Sets of Quadratic Iterations c = 0.25

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Julia Sets of Quadratic Iterations c = – This is sometimes called the fractal rabbit.

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Julia Sets of Quadratic Iterations This is a fractal because it is self similar.

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Julia Sets of Quadratic Iterations c =

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Julia Sets of Quadratic Iterations c = – This is fractal dust. Under more iterations, the set of points will becomes smaller and smaller, but there are some points that will never escape.

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

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Julia Sets of Quadratic Iterations x → x 2 – x 0 = 0 x 1 = – x 2 = – x 3 = – x 4 = – x 5 = – x 6 = x 7 = – The orbit tends to a 3-cycle.

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Julia Sets of Quadratic Iterations – – The cycle gives the number of pieces in the set.

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Julia Sets of Quadratic Iterations

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If 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.

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The Mandelbrot Set The 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 → x 2 + c does not go to infinity.

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The Mandelbrot Set

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This 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.

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The Mandelbrot Set

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Rather 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.

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The Mandelbrot Set for (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; else color = colors[n]; SetPixel(hdc, hcenter+cr*scale, vcenter+ci*scale, color); }

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

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References Devaney, 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.

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