# Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))

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Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z)) 3. Rational maps (z n + /z n )

We’ll investigate chaotic behavior in the dynamical plane (the Julia sets) z 2 + c exp(z) z 2 + /z 2

As well as the structure of the parameter planes. z 2 + c exp(z)z 3 + /z 3 (the Mandelbrot set)

A couple of subthemes: 1.Some “crazy” mathematics 2.Great undergrad research topics

The Fractal Geometry of the Mandelbrot Set

How to count The Fractal Geometry of the Mandelbrot Set

The Fractal Geometry of the Mandelbrot Set How to add How to count

Many people know the pretty pictures...

but few know the even prettier mathematics.

Oh, that's nothing but the 3/4 bulb....

...hanging off the period 16 M-set.....

...lying in the 1/7 antenna...

...attached to the 1/3 bulb...

...hanging off the 3/7 bulb...

...on the northwest side of the main cardioid.

Oh, that's nothing but the 3/4 bulb, hanging off the period 16 M-set, lying in the 1/7 antenna of the 1/3 bulb attached to the 3/7 bulb on the northwest side of the main cardioid.

Start with a function: x + constant 2 and a seed: x 0

Then iterate: x = x + constant 10 2

Then iterate: x = x + constant 10 2 21 2

Then iterate: x = x + constant 10 2 21 2 32 2

Then iterate: x = x + constant 10 2 21 2 32 2 43 2

Then iterate: x = x + constant 10 2 21 2 32 2 43 2 Orbit of x 0 etc. Goal: understand the fate of orbits.

Example: x + 1 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 3 4 5 6

Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 3 4 5 6

Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 4 5 6

Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = 5 6

Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = 6

Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6

Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6 “Orbit tends to infinity”

Example: x + 0 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

Example: x + 0 Seed 0 2 x = 0 0 1 x = 2 3 4 5 6

Example: x + 0 Seed 0 2 x = 0 0 1 2 x = 3 4 5 6

Example: x + 0 Seed 0 2 x = 0 0 1 2 3 x = 4 5 6

Example: x + 0 Seed 0 2 x = 0 0 1 2 3 4 5 6 “A fixed point”

Example: x - 1 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 2 3 4 5 6

Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = 3 4 5 6

Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 4 5 6

Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = 5 6

Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = -1 5 x = 0 6 “A two- cycle”

Example: x - 1.1 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

Example: x - 1.1 Seed 0 2 x = 0 0 x = -1.1 1 x = 2 3 4 5 6

Example: x - 1.1 Seed 0 2 x = 0 0 x = -1.1 1 x = 0.11 2 x = 3 4 5 6

Example: x - 1.1 Seed 0 2 x = 0 0 x = -1.1 1 x = 0.11 2 x = 3 4 5 6 time for the computer!

Observation: For some real values of c, the orbit of 0 goes to infinity, but for other values, the orbit of 0 does not escape.

Complex Iteration Iterate z + c 2 complex numbers

Example: z + i Seed 0 2 z = 0 0 z = 1 2 3 4 5 6

Example: z + i Seed 0 2 z = 0 0 z = i 1 z = 2 3 4 5 6

Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = 3 4 5 6

Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = 4 5 6

Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = 5 6

Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = 6

Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = -1 + i 6 2-cycle

Example: z + i Seed 0 2 1 i -i

Example: z + i Seed 0 2 1 i -i

Example: z + i Seed 0 2 1 i -i

Example: z + i Seed 0 2 -i 1 i

Example: z + i Seed 0 2 1 i -i

Example: z + i Seed 0 2 -i 1 i

Example: z + i Seed 0 2 1 i -i

Example: z + i Seed 0 2 -i 1 i

Example: z + 2i Seed 0 2 z = 0 0 z = 1 2 3 4 5 6

Example: z + 2i Seed 0 2 z = 0 0 z = 2i 1 z = -4 + 2i 2 z = 12 - 14i 3 z = -52 + 336i 4 z = big 5 z = BIGGER 6 Off to infinity

Same observation Sometimes orbit of 0 goes to infinity, other times it does not.

The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. Why do we care about the orbit of 0?

The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. As we shall see, the orbit of the critical point determines just about everything for z 2 + c. 0 is the critical point of z 2 + c.

Algorithm for computing M Each grid point is a complex c-value.

Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. red = fastest escape

Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. orange = slower

Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. yellow green blue violet

Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

The eventual orbit of 0

3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0 3-cycle

The eventual orbit of 0

4-cycle

The eventual orbit of 0 4-cycle

The eventual orbit of 0 4-cycle

The eventual orbit of 0 4-cycle

The eventual orbit of 0 4-cycle

The eventual orbit of 0 4-cycle

The eventual orbit of 0 4-cycle

The eventual orbit of 0 4-cycle

The eventual orbit of 0

5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 5-cycle

The eventual orbit of 0 2-cycle

The eventual orbit of 0 2-cycle

The eventual orbit of 0 2-cycle

The eventual orbit of 0 2-cycle

The eventual orbit of 0 2-cycle

The eventual orbit of 0 fixed point

The eventual orbit of 0 fixed point

The eventual orbit of 0 fixed point

The eventual orbit of 0 fixed point

The eventual orbit of 0 fixed point

The eventual orbit of 0 fixed point

The eventual orbit of 0 fixed point

The eventual orbit of 0 fixed point

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 goes to infinity

The eventual orbit of 0 gone to infinity

One reason for the importance of the critical orbit: If there is an attracting cycle for z 2 + c, then the orbit of 0 must tend to it.

How understand the of the bulbs? periods

How understand the of the bulbs? periods

junction point three spokes attached

Period 3 bulb junction point three spokes attached

Period 4 bulb

Period 5 bulb

Period 7 bulb

Period 13 bulb

Filled Julia Set:

Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.

Example: z 2 Seed: 0 In filled Julia set?

Example: z 2 Seed: 0Yes In filled Julia set?

Example: z 2 Seed: 0Yes 1 In filled Julia set?

Example: z 2 Seed: 0Yes 1 In filled Julia set?

Example: z 2 Seed: 0Yes 1 In filled Julia set?

Example: z 2 Seed: 0Yes 1 Yes In filled Julia set?

Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

Example: z 2 Seed: 0Yes 1 Yes i 2i In filled Julia set?

Example: z 2 Seed: 0Yes 1 Yes i 2i No In filled Julia set?

Example: z 2 Seed: 0Yes 1 Yes i 2i No 5 In filled Julia set?

Example: z 2 Seed: 0Yes 1 Yes i 2i No 5No way In filled Julia set?

Filled Julia Set for z 2 All seeds on and inside the unit circle. i 1

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic”

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

Other filled Julia sets

c = 0

Other filled Julia sets c = -1

Other filled Julia sets c = -1

Other filled Julia sets c = -1

Other filled Julia sets c = -1

Other filled Julia sets c = -1

Other filled Julia sets c = -1

Other filled Julia sets c = -1

Other filled Julia sets c = -1

Other filled Julia sets c = -.12+.75i

Other filled Julia sets c = -.12+.75i

Other filled Julia sets c = -.12+.75i

Other filled Julia sets c = -.12+.75i

Other filled Julia sets c = -.12+.75i

Other filled Julia sets c = -.12+.75i

If c is in the Mandelbrot set, then the filled Julia set is always a connected set.

Other filled Julia sets But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.

Other filled Julia sets c =.3

Other filled Julia sets c =.3

Other filled Julia sets c =.3

Other filled Julia sets c =.3

Other filled Julia sets c =.3

Other filled Julia sets c = -.8+.4i

Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c:

Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: But if the orbit of 0 does not go to infinity, the Julia set is connected (just one piece). If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

Animations: In and out of M arrangement of the bulbs Saddle node Period doubling Period 4 bifurcation

How do we understand the arrangement of the bulbs?

How do we understand the arrangement of the bulbs? Assign a fraction p/q to each bulb hanging off the main cardioid. q = period of the bulb

Where is the smallest spoke in relation to the “principal spoke”? p/3 bulb principal spoke

1/3 bulb principal spoke The smallest spoke is located 1/3 of a turn in the counterclockwise direction from the principal spoke.

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

1/3 bulb 1/3

??? bulb 1/3

1/4 bulb 1/3

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

1/4 bulb 1/3 1/4

??? bulb 1/3 1/4

2/5 bulb 1/3 1/4

2/5 bulb 1/3 1/4 2/5

2/5 bulb 1/3 1/4 2/5

2/5 bulb 1/3 1/4 2/5

2/5 bulb 1/3 1/4 2/5

2/5 bulb 1/3 1/4 2/5

??? bulb 1/3 1/4 2/5

3/7 bulb 1/3 1/4 2/5

3/7 bulb 1/3 1/4 2/5 3/7

3/7 bulb 1/3 1/4 2/5 3/7

3/7 bulb 1/3 1/4 2/5 3/7

3/7 bulb 1/3 1/4 2/5 3/7

3/7 bulb 1/3 1/4 2/5 3/7

3/7 bulb 1/3 1/4 2/5 3/7

3/7 bulb 1/3 1/4 3/7 2/5

??? bulb 1/3 1/4 3/7 2/5

1/2 bulb 1/3 1/4 3/7 1/2 2/5

1/2 bulb 1/3 1/4 3/7 1/2 2/5

1/2 bulb 1/3 1/4 3/7 1/2 2/5

1/2 bulb 1/3 1/4 3/7 1/2 2/5

??? bulb 1/3 1/4 3/7 1/2 2/5

2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

How to count

1/4 How to count

1/3 1/4 How to count

1/3 1/4 2/5 How to count

1/3 1/4 2/5 3/7 How to count

1/3 1/4 2/5 3/7 1/2 How to count

1/3 1/4 2/5 3/7 1/2 2/3 How to count

1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. How to count

1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. 1/101 32,123/96,787 How to count

Animations: Mandelbulbs Spiralling fingers

1/2

How to add 1/2 1/3 2/5

How to add 1/2 1/3 2/5 3/7

+= 1/2 + 1/3 = 2/5

+= 1/2 + 2/5 = 3/7

22 1/2 0/1 Here’s an interesting sequence:

22 1/2 0/1 Watch the denominators 1/3

22 1/2 0/1 Watch the denominators 1/3 2/5

22 1/2 0/1 Watch the denominators 1/3 2/5 3/8

22 1/2 0/1 Watch the denominators 1/3 2/5 3/8 5/13

22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13

22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13 8/21

22 1/2 0/1 The Fibonacci sequence 1/3 2/5 3/8 5/13 8/21 13/34

The Farey Tree

How get the fraction in between with the smallest denominator?

The Farey Tree How get the fraction in between with the smallest denominator? Farey addition

The Farey Tree

.... essentially the golden number

Another sequence (denominators only) 1 2

Another sequence (denominators only) 1 2 3

Another sequence (denominators only) 1 2 3 4

Another sequence (denominators only) 1 2 3 4 5

Another sequence (denominators only) 1 2 3 4 5 6

Another sequence (denominators only) 1 2 3 4 5 6 7

sequence 1 2 3 4 5 6 7 Devaney

The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS: Have fun! Mandelbrot set explorer; Applets for investigating M-set; Applets for other complex functions; Chaos games, orbit diagrams, etc.

Farey.qt Farey tree D-sequence Continued fraction expansion Far from rationals Other topics Website

Continued fraction expansion Let’s rewrite the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,..... as a continued fraction:

Continued fraction expansion 1212 = 1212 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

Continued fraction expansion 1313 = 1212 + 1111 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

Continued fraction expansion 2525 = 1212 + 1111 + 1111 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

Continued fraction expansion 3838 = 1212 + 1111 + 1111 1111 + the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 5 13 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 8 21 1111 + the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 13 34 1111 + 1111 + the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 13 34 1111 + 1111 + essentially the 1/golden number the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

We understand what happens for = 1a1a + 1b1b + 1c1c 1d1d + 1e1e + 1f1f + 1g1g + where all entries in the sequence a, b, c, d,.... are bounded above. But if that sequence grows too quickly, we’re in trouble!!! etc.

The real way to prove all this: Need to measure: the size of bulbs the length of spokes the size of the “ears.”

There is an external Riemann map : C - D C - M taking the exterior of the unit disk to the exterior of the Mandelbrot set.

takes straight rays in C - D to the “external rays” in C - M 01/2 1/3 2/3 external ray of angle 1/3

Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4

Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4 Then the external ray of angle p/q lands at the “root point” of a period k bulb in the Mandelbrot set.

0 1/3 2/3 0 is fixed under angle doubling, so lands at the cusp of the main cardioid.

0 1/3 2/3 1/3 and 2/3 have period 2 under doubling, so and land at the root of the period 2 bulb. 2

0 1/3 2/3 And if lies between 1/3 and 2/3, then lies between and. 2

0 1/3 2/3 So the size of the period 2 bulb is, by definition, the length of the set of rays between the root point rays, i.e., 2/3-1/3=1/3. 2

0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/7.... 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15

0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/7.... 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15

0 1/3 2/3 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

0 1/3 2/3 3/15 and 4/15 have period 4, and are between 1/7 and 2/7.... 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15

1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/7.... 3/154/15

So what do we know about M? All rational external rays land at a single point in M.

So what do we know about M? All rational external rays land at a single point in M. Rays that are periodic under doubling land at root points of a bulb. Non-periodic rational rays land at Misiurewicz points (how we measure length of antennas).

So what do we know about M? “Highly irrational” rays also land at unique points, and we understand what goes on here. “Highly irrational" = “far” from rationals, i.e.,

So what do we NOT know about M? But we don't know if irrationals that are “close” to rationals land. So we won't understand quadratic functions until we figure this out.

MLC Conjecture: The boundary of the M-set is “locally connected” --- if so, all rays land and we are in heaven!. But if not......

The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS Have fun!

A number is far from the rationals if:

This happens if the “continued fraction expansion” of has only bounded terms.

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