Complex
Quadratic Dynamics: A Study of the Mandelbrot and Julia Sets |
|||||||||||||||||||||||||||||||||||||||||||||||||||

Chaos Theory June 2001 |
|||||||||||||||||||||||||||||||||||||||||||||||||||

This exploration is primarily concerned with a specific family of complex quadratic polynomial equations which are used to generate the Mandelbrot and Julia sets. The goal of this project is to analyze these equations from the perspective of dynamical systems theory. Amazingly, even the simplest forms of low dimensional non-linear systems generate dynamic behavior that is extremely complicated. In general, complex dynamical systems can exhibit three distinct modes of long term behavior. For example, given a system of a given state, all trajectories of in the phase space mapping might asymptotically approach a single value. This system can be characterized by what is known as an attracting fixed point. If the system is unbounded then trajectories might escape to infinity, in which case infinity can be considered an attracting fixed point. In another scenario, the trajectories in phase space might approach a cycle of a certain frequency. This phenomenon is known as an attracting periodic orbit, or limit cycle. However, there is a third characteristic mode of dynamic behavior that remains bounded, but is neither attracted to a fixed point or to a periodic orbit. This phenomenon is known generally as chaos. The trajectories of a chaotic system are aperiodic, meaning that the pattern never repeats itself. Chaotic systems also exhibit extreme sensitive dependence on initial conditions; thus, practical long-term prediction making is impossible. Even if only by Heisenberg’s uncertainty principle, a dynamic state of position and velocity can never be measured exactly. Before the advent of computer technology, western thinkers assumed all dynamic behavior was either simple order, characterized by fixed points and periodic orbits, or it was completely random. The rediscovery of natural chaotic patterns by experimental mathematicians has not only demolished many antiquated misconceptions of order and randomness, but has also created an entirely new structure of reality. The structure of this new dimension is infinitely complex, a result that stems from the fact that chaotic trajectories remain bounded and at same time are aperiodic. This infinitely complex structure is actually the limit set of a chaotic trajectory, and is often called a fractal. By mapping the flows of chaotic systems, experimenters have discovered these fractal limit sets in the form of chaotic attractors, which are dynamic maps in phase space. A few examples are the Lorenz, Rossler, and Henon attractors. Dynamic systems can also be analyzed by constructing iterative maps of equations. Of these maps, the most famous are definitely the logistic and the Mandelbrot. The Mandelbrot set is probably the most mysterious object in mathematics. Many aspects concerning the analysis of this structure require advanced mathematical tools and concepts. However, the Mandelbrot set is very accessible to anyone who has taken high school level mathematics. Generally speaking, the Mandelbrot is a map in the complex plane which is determined by a simple quadratic equation. Specifically, we are concerned with the following mapping:
Dynamic systems are characterized by a set of parameters, and a by a set of initial conditions. In the case of this complex quadratic function, the c-value is the parameter and the z-value is the initial condition, both of which are complex numbers, meaning they have two parts: one real and one imaginary. To study this equation as a dynamic system, we use an iterative process whereby we input the initial condition, compute the output, and then feed the output back into the original equation. This list of successive iterations is called the orbit of the given initial condition. To
start with, let’s consider the dynamics of If
we observe the orbits of To
map this set using a computer, all you have to do is follow this simple
algorithm. First set an upper bound N, which is the maximum number
of iterations to be performed. Then, for each point c in the plane,
check the first N iterates of
The
c-values that escape to infinity are usually assigned a color which represents
how many iterations it took for the orbit to each the upper bound N. In
other words, the color of a c-value indicates how fast the orbit escapes
to infinity. However, any computer mapping of the Mandelbrot is
only an approximation because it is impossible to compute an infinite
number of iterations. Some orbits of The Mandelbrot set It
can be shown that the entire Mandelbrot lies inside a disk of radius
2 centered at the origin. The escape criterion for
The
next step is to ask, for which c-values does the orbit of
The
second equation means that zc lies on or
inside a circle of radius ½. On this circle | Next,
consider the region of the M-set that has an attracting cycle of period
2. As mentioned above, the orbit of The
bulb-like regions directly attached to the main cardioid are called primary
bulbs, and there are an infinite number of them. Moreover, each
primary bulb consists of c-values for which The
Mandelbrot set graphically depicts for which c-values the orbit of To
obtain this dynamic map of
Conventionally,
the set of complex numbers is usually visualized as a plane. However,
it is sometimes useful in complex analysis to view the complex numbers
on a sphere. Each point in the z-plane corresponds to a point on
the z-sphere, with the origin located at the bottom of the sphere. The
advantage is that we can interpret the top of this sphere as the point “at
infinity.” The complex sphere is the set
It
is easy to calculate that F c (z) = 2 /(1+cz 2). This
means that F To
compute J There
are an infinite number of Julia sets, one for each value of c. The
majority of these sets are extremely complicated. Before moving
on, we will first consider two simple Julia sets. First, consider
J Except
for the two exceptions at c = 0 and –2, all Julia sets have fractal
boundaries. As mentioned above, there are an infinite variety of
Julia sets. However, any given Julia set falls into one of two
categories. The discovery of this fundamental dichotomy dates back
to 1919 when G. Julia and P. Fatou proved that for each c-value, the
Julia set is either a Examples of Cantor Julia sets The
simplest example of a Cantor set is generated by the middle-thirds process. First
take the line segment S
If
we choose c = 0, then the corresponding Julia set is the unit circle. If
we choose any other c-value which lies inside the main cardioid region,
the corresponding Julia set will be a deformed circle. What happens
to the filled Julia set, K A similar
bifurcation occurs as the c-value crosses into any of the primary bulbs. The
only difference is that different primary bulbs are characterized by different
periods. Consider the shaped of the filled Julia sets as this transition
occurs. For c-values inside the large cardioid of the Mandelbrot set,
K In this
way, it is easy to see that the location of the c-value in the Mandelbrot
set immediately gives one an idea of what the corresponding Julia set will
look like. How is it that the parameter map of the orbits of Each
primary bulb in the Mandelbrot set corresponds to a certain period, call
it p. The period of any given primary bulb can be determined in several
ways. The analytic approach is to compute the orbit of As mentioned above, the filled Julia set for a c-value in a primary bulb contains a number of regions which are connected at juncture points. The number of regions connected by a single juncture point equals the period of the bulb. However, the simplest method to determine the period of a bulb requires only a visual interpretation of the Mandelbrot set. At the tip of each primary bulb, there is an antenna like structure. These antennas consist of a number of spokes which emanate from a central junction point. Amazingly, the period of a bulb attached to the main cardioid is equal to the number of spokes on the antenna attached to that bulb.
Let us
now consider the periodic dynamics of K The most
obvious method to determine the value of q/p is to compute K Now that we understand how to determine the value of the rotation numbers and what they represent, let’s observe how they are arranged. Remember, each primary bulb can be labeled with a specific rotation number. Consider the sequence of rotation numbers generated by traveling along the main cardioid in the counterclockwise direction. Amazingly, this sequence is exactly the rational numbers between zero and one. Start at the cusp, which is where the cardioid intersects with the positive real axis, and call this point zero. As we travel along the cardioid in the counterclockwise direction, we visit each bulb corresponding to rotation number q/p in exactly the same order as the rational numbers, ending at one when we arrive back at the cusp. This is truly one of the most amazing natural properties of the Mandelbrot set. There is also an amazing pattern that been discovered which is based solely on the period numbers p. In this example, call the main cardioid the “period-1 bulb.” Also, note that the period-2 bulb is the largest primary bulb and extends to the left of the cardioid. Now, observe that the largest bulb between the period-1 and period-2 bulb is the period-3 bulb, either at the top of the bottom of the Mandelbrot set. Next, observe that the largest bulb between period-2 and period-3 is period-5. If the pattern isn’t obvious yet, consider the largest bulb between period-3 and period-5, this bulb is period-8. The sequence generated by this process is the Fibonacci sequence. If you have a nice Mandelbrot explorer you can verify that the pattern continues on. ( 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . ) The intricate structure of the Mandelbrot arises because of its fractal nature. In one sense, the boundary of this set is finite; however, the length of the perimeter of this boundary is infinite. The boundary exhibits infinite complexity in that increasing the amount of magnification only brings out more and more intricate details. The only limit to how far we can explore magnifications of this boundary is the computing speed of our present technology. Fractals are different from ordinary objects in two important ways: dimensionality and self-similarity. Firstly, consider the concept of dimension. A general definition of the dimension of a set is the minimum number of coordinates required to describe every point in the set. This is often referred to as the topological dimension. For example, a line segment is one-dimensional, a planar surface is two-dimensional, and a spherical object is three-dimensional. The topological dimension of a set is always a positive integer. However, fractals exhibit self-similarity across all scales of magnification. This feature leads to paradoxes when we try to define the topological dimension of the set. In result, the discovery of fractals has led to new definition of dimension. The similarity dimension, or fractal dimension, can be defined as follows. Suppose the set S may be subdivided into k congruent pieces, each of which may be magnified by a factor of M to yield the whole set S. Then the fractal dimension D of S is
There
are several possible notions of similarity dimension such as the Hausdorff
dimension, the capacity dimension, the box dimension, and the correlation
dimension. In many cases, the fractal dimension of a set is not an
integer, but is often some value between whole numbers. In general
a fractal can be defined as a subset of However
these definitions of a fractal and dimension can only be rigorously applied
to fractals that are simply self-similar. A few examples are the Koch
Curve, the Sierpinski Triangle, and the Cantor middle-thirds set. These
fractals are called affine self-similar, which means that the set Unlike the affine self-similar fractals, the Mandelbrot set is quasi self-similar. Magnifying any region of the boundary will produce endless self-similar details; however, the similarity is only statistical and the details on different scales are never exactly the same. This type of statistical self-similarity can be found in all forms of nature. Since the Mandelbrot is not affine self-similar our regular definition of similarity dimension makes no sense. If we we’re to speak generally, we could say that the Mandelbrot set has a dimension of 2. This is because the entire set is contained in a disk which has a dimension of 2. The topological dimension of the Mandelbrot is 1. This is because the boundary has an empty interior, so the dimension must be less than 2, and is therefore 1. In any case, all of our present conceptions of dimensionality and fractals are practical working definitions and are by no means rigorously defined. The Mandelbrot
set can also be mapped in higher dimensions. The classic Mandelbrot
set is a map in the complex plane. Each coordinate in the plane is
essentially 2 dimensional because a complex number, z, has two parts and
is written z = a +
The normal Mandelbrot and Julia formulas can be extended to use quaternions instead of complex numbers. Complex numbers can be displayed on a plane because they have two parts. A quaternion Mandelbrot set mapping requires four dimensions to be viewed because each value has four parts. In addition, each c-value of the four-dimensional Mandelbrot corresponds to a specific four-dimensional Julia set. Presently, we are unable to visualize these higher dimensional objects directly. One method of analyzing a 4D quaternion object is to "take slices." If we take slices of a 3D object we will get a bunch of 2D objects. If we take slices of a 4D object we will get a bunch of 3D objects. To get an idea of what the fourth-dimensional object looks like, take all the slices and create an animation of the 3D objects as the slice moves through the quaternion set. This method utilizes time as the fourth dimension, which is exactly what Einstein did to describe the four-dimensional space-time continuum. Some examples of 3D slices of Quaternion fractals Some more examples of 3D slices of Quaternion fractals What if our life is an animation of 3D slices taken from a 4D continuum? My interpretation of this question goes something like this. If there is only one space-time continuum, then everything is determined and there is no free-will. This may be the case, however, I think there is a large amount of evidence to the contrary. There are only two other possibilities. The first possibility is that there are an infinite number of continuums, an idea similar to the Many-Worlds Interpretation of Quantum Theory. The other possibility is that only the moment really exists, and the past and the future only exist as potential probabilities. It may be that these three possibilities do not contradict each other, in which case reality would have to be infinite-dimensional. For example, perhaps our 4D continuum is only one slice of a 5D continuum, then there are actually an infinite number of 4D continuums. Maybe the 5D dimensional continuum is only one slice of an even higher dimensional continuum, and so forth and so on to infinity. As another example, consider that the many moments that we experience in time are actually lower dimensional slices of one higher dimensional moment. In other words, if we could see a higher dimensional continuum without having to take lower dimensional slices, we could see all the past and all the future at one glance. Indeed, we create time by taking these slices. In the case of quaternion fractals, if we could see in four dimensions we wouldn’t need to create animations of lower dimensional slices. The implications of these examples suggest that there is only one infinite dimensional moment which is eternally beyond all finite dimensional continuums, of which there are infinitely many. At the same time, all the past and future exist only as potential probabilities created by slicing the infinite dimensional moment of eternal now in an infinite number of ways. The exploration
of these higher dimensional fractal objects represents the newest frontier
of mathematics as well as computer technology. Amazingly, these hyperspatial
images can be generated simply by using the basic quadratic equation The maps
of this quadratic equation are not necessarily chaotic attractors in the
normal sense of the term. In general, a chaotic attractor is the limit
set of an aperiodic trajectory. This region remains bounded, but the
pattern never repeats so the attractor will always have a fractal structure. The
dynamic mapping of In addition,
it can be proved that for any c-value, the repelling periodic points of In closing,
let us make sure we clearly understand that the Mandelbrot and Julia sets
are different ways of looking at the same thing. For the Mandelbrot,
we hold z constant at 0 and check all c-values. For the Julia set,
we hold c constant and check all z-values. Perhaps with more sophisticated
computer graphics technology we will be able to analyze the mapping of Sources Abraham, Ralph. Devaney, Robert. Devany, Robert and Linda Keen, editors. Peitgen, Heintz-Otto and Dietmar Saupe, editors. Strogatz, Steven H. · images of two-dimensional fractals were created using a program named Xaos. · images of quaternion fractals were taken from the following website, http://www.hypercomplex.org/quats.htm |
|||||||||||||||||||||||||||||||||||||||||||||||||||