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More on Julia & Mandelbrot Sets, Chaos

More on Julia & Mandelbrot Sets, Chaos. Glenn G. Chappell CHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 481/681 Lecture Notes Wednesday, April 7, 2004. Review: Fractals. We have looked at three types of fractals: Turtle Fractals

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More on Julia & Mandelbrot Sets, Chaos

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  1. More on Julia & Mandelbrot Sets, Chaos Glenn G. ChappellCHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 481/681 Lecture Notes Wednesday, April 7, 2004

  2. Review:Fractals • We have looked at three types of fractals: • Turtle Fractals • Generated by recursive routines that give commands to a “turtle”. • Can be modeled by string productions/L-systems. • Such fractals are always curves at heart. • Strange Attractors • Attractors of an IFS. • More on the next few slides … • Julia Sets, the Mandelbrot Set, etc. • More to come … CS 481/681

  3. Review:Strange Attractors [1/3] • An iterated function system (IFS) consists of one or more functions (“maps”) that are repeatedly applied to their own outputs. • IFS’s of various sorts are heavily used in mathematical modeling. • The orbit of a point under an IFS is the set of all points that the IFS sends it to. • The attractor of an IFS is the set of points that the IFS moves points toward. • For example, the IFS on the real number line with the single map x→x/2, has the point x = 0 as its attractor. • If an attractor is a fractal, then it is said to be a strange attractor. CS 481/681

  4. Review:Strange Attractors [2/3] • If an IFS has a single map, and this map is an affine function, then it never has a strange attractor. • We can create IFS’s with strange attractors by: • Using multiple affine maps • See ifs1.cpp. • Using non-affine maps • We have not done this yet. CS 481/681

  5. Review:Strange Attractors [3/3] • To draw an approximation of an attractor, begin with a random point, and draw its orbit. That is: • Choose a point. • Repeat: • Draw the point. • Send the point through the IFS. • If there are multiple maps, then this means choosing a random map and applying it. • In some IFS’s points approach the attractor rather slowly. In these you may want to send a point through the IFS many times before you begin drawing. • See ifs1.cpp for example code. CS 481/681

  6. Review:Julia Sets [1/2] • We have seen how to describe fractals as IFS attractors, as well as how to draw these. • There is another way to create fractals using IFS’s: • The fractal is the set of all points that are not “sent to infinity” by the IFS. • A point is “sent to infinity” if its orbit is not contained in any bounded region. CS 481/681

  7. Review:Julia Sets [2/2] • Using this method, we can draw interesting fractals in the complex plane. • The complex plane is just the collection of all complex numbers, represented using Cartesian coordinates. • So the point (x, y) corresponds to the complex number x + yi. • The Idea • Pick a fixed complex number c = a + bi. • Our IFS has one map: z→z2 + c. • This map is not affine! • The not-sent-to-infinity set for this IFS is called a filled Julia set (or “filled-in Julia set”). • For every complex number, there is a corresponding filled Julia set. CS 481/681

  8. Review:The Mandelbrot Set • Suppose we do the same process as with a filled Julia set, except that the number we add to the square is the number we started the iteration with. • In other words, our map is z → z2 + c, where c is the first complex number in our iteration. • So the sequence goesc, c2 + c, (c2 + c)2 + c,((c2 + c)2 + c)2 + c, etc. • When we use this process, the not-sent-to-infinity set is called the Mandelbrot set. • There are many Julia sets, but only one Mandelbrot set. • The Mandelbrot set is the collection of all complex numbers that lie in their own filled-in Julia sets (right?). CS 481/681

  9. More on Julia & Mandelbrot Sets:Computation • How do we compute maps like z → z2 + c when z and c are complex? • Write complex numbers as real & imaginary parts, and work it out. • For example, let z = x + yi, and let c = a + bi. • Then, • So, representing each complex number as two real numbers, we have two maps, both of which are applied each time: • x→x2 – y2 + a • y→ 2xy + b (be sure to use the oldx-value here!) CS 481/681

  10. More on Julia & Mandelbrot Sets:How to Draw [1/3] • How do we draw a not-sent-to-infinity set? • Given a point, we need some way to figure out whether it is sent to infinity. • In general, this is very difficult. • So we find ways of making good guesses. • Often the region around a set is actually more interesting than the set itself. • So we look at a way of coloring the region around a set. CS 481/681

  11. More on Julia & Mandelbrot Sets:How to Draw [2/3] • An Idea • Pick a threshold distance. We will assume that, if a point eventually ends up farther than this from zero, then the point is heading to infinity. • Given a point, apply the appropriate process to the point until: • The distance from zero exceeds the threshold. In this case, we guess that the point is outside the set. • OR: Some iteration limit is met. In this case, we guess that the point lies in the set. • Now apply this to every point in our drawing region. Point in the set are one color (traditionally, black); points outside the set are a different color. • Note: There are other methods; you may want to look into them. CS 481/681

  12. More on Julia & Mandelbrot Sets:How to Draw [3/3] • It is often of interest how long a point outside the set “dwells” inside the threshold. • This “dwells count” can result in a very nice way to color the region around Mandelbrot & Julia Sets. • Simply choose a color based on the count. • Threshold distance 2 works well with the Mandelbrot set. • It has been proven that if a point ever exceeds distance 2, then it is definitely heading to infinity. • So this threshold gives you a way to be sure a point is not in the M-set. However, it does not give you a way to be sure the point is in the M-set. • As you view smaller & smaller regions, getting decent images will require increasing the iteration limit. CS 481/681

  13. More on Julia & Mandelbrot Sets:Interface Issues [1/2] • Drawing Mandelbrot & Julia sets (and other fractals) can be very time-consuming. • Thus, the standard GLUT approach (drawing the whole image at each display call) may not be appropriate. • One Solution • Keep an off-screen image, which starts blank and is slowly filled in. • The display function draws this image. • If the image is incomplete, then the idle function determines the color of a few more points in the image, and posts a redisplay event. CS 481/681

  14. More on Julia & Mandelbrot Sets:Interface Issues [2/2] • Zooming in and out is a nice feature. • Zoom in on a point the user clicks on? • Or have the user outline a rectangle, and use this region as the new drawing region. • Since images take time to render, it can be nice to render a lower-resolution version first. • If a bit of thought is put into this, then the pixels the low-resolution image can become some of the pixels in the eventual high-resolution image. CS 481/681

  15. More on Julia & Mandelbrot Sets:Other Maps • These ideas can be applied using just about any other map on the complex numbers. • Suppose f is a function that takes two complex numbers and returns a single complex number. • We can use z → f(z, c) in place of z → z2 + c. • An obvious map to try next is z → z3 + c. CS 481/681

  16. Chaos:Introduction • A concept that has had a profound effect on much of the scientific thinking of the late 20th century is “chaos”. • Chaos refers to the complex, difficult-to-predict behavior found in nonlinear systems. • A function is nonlinear if it is not linear. • We generally toss out affine functions, too. • The first recorded instance of someone noticing chaotic behavior in a simple system was Edward Lorenz in 1960, while studying mathematical models of weather. • Before this, the “standard wisdom” was that systems with simple descriptions always had simple behavior; now we know that this is not true. CS 481/681

  17. Chaos:The Butterfly Effect • One of the important properties of chaos is “sensitive dependence on initial conditions”, informally known as “the Butterfly Effect”. • This term possibly originated from the title of a talk Lorenz gave to the AAAS in 1972: “Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?”. • Sensitive dependence on initial conditions means that a very small change in the initial state of a system can have a large effect on its later state. • In particular, you can eventually get the large effect, no matter how small the initial change was. CS 481/681

  18. Chaos:Discovery • Again, chaos was first noticed in 1960. • Lorenz was studying a mathematical model of weather. He was confused by the apparent fact that, in two runs of his program, the same input data was giving completely different output. • The different results turned out to be caused by a very slight difference in the input data: a 6-decimal-place value in one input (0.506127) was truncated to 3 decimal places (0.506) in the other. • This was still confusing, since the change seemed insignificant. However, we now know that weather is a chaotic system; the Butterfly Effect is real. • Lorenz found a simpler system that exhibited the same behavior. This system (the Lorenz Butterfly) is now one of the standard examples of a chaotic system. • Next we build up some background and define “chaos”. CS 481/681

  19. Chaos:IFS’s & Dynamical Systems • Recall: An iterated function system (IFS) consists of one or more functions, repeatedly applied to their own outputs. • The domains & codomains of all the functions must be the same. • The IFS’s we consider here will each consist of a single function. • Simple example: • f(x) = x+5. Initial value: x0 = 2. • Then x1 = f(x0) = 2+5 = 7. • x2 = f(f(x0)) = f(x1) = 7+5 = 12. And so on … (no chaos here). • A related concept is that of a “continuous dynamical system”. • In an IFS (also called a discrete dynamical system), we describe the state of the system at discrete time intervals (0, 1, 2, …). • In a continuous dynamical system, we describe the state of the system all the time, usually with a system of differential equations. • In practice, continuous systems are often converted to discrete systems in order to approximate their solutions. So we can reasonably confine our attention to IFS’s. CS 481/681

  20. Chaos:Orbits & Periodicity • Suppose we have an IFS with exactly one function: f:S → S. • Recall: Given a point x in S, the orbit of x is the collection of all points that the IFS takes x to. • That is, the orbit of x contains x, f(x), f(f(x)), f(f(f(x))), etc. • x is a periodic point if repeatedly applying f eventually takes x back to itself. • Put another way: A periodic point is a point whose orbit is finite. CS 481/681

  21. Chaos:Definition • An IFS with one function f:S→S is exhibits chaos (equivalently, it is chaotic) if it has the following properties: • Sensitive dependence on initial conditions. • Periodic points are dense in S. • This means that, no matter where you are in S, there is a periodic point nearby (as near as you want). • f is topologically transitive. • This means that, if you pick two points x, y in S, then there is a point near x whose orbit takes it near y (again, both occurrences of “near” mean “as near as you want”). • Some researchers have used other definitions of chaos; this definition appears to be the one most commonly used. • Some informal essays on chaos suggest that sensitive dependence is the defining characteristic of chaos. But this is not true; consider f(x) = 2x. CS 481/681

  22. Chaos:DEMO • Demo time! • In class, chaos.cpp was demonstrated. CS 481/681

  23. Chaos:Notes • Chaotic systems generally have strange attractors. • Think “pretty pictures”.  • When we say a chaotic system is “unpredictable”, we do not mean that it is nondeterministic. • A deterministic system is one that always gives the same results for the same input/initial values. • Real-world chaotic systems are “unpredictable” in practice because we can never determine our input values exactly. Thus, sensitive dependence on initial conditions will always mess up our results, eventually. • A chaotic system does not need to be very complex [for example, f(x) = ax(1–x) is pretty simple], but it does need to be nonlinear. • So methods that approximate functions by linear/affine functions (e.g., Newton’s Method) can sometimes give qualitatively different behavior than that which they purport to approximate. CS 481/681

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