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March 2005 | Draft (Uncompleted)

Features of Mandelbrot and Julia Sets

 

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Preliminary (uncompleted) version of Annex 4 to
Sustainability through the Dynamics of Strategic Dilemmas
in the light of the coherence and visual form of the Mandelbrot set


Definitions

Axes: In Figure 1, positive "x" values lie above the horizontal axis and imaginary "y" values are left and right. These appear to be widely used axes, common in early mathematics. In fact the "imaginary" nature of the "y" axis (right and left) is a mathematical device to enable the representation of complex numbers. The axes are therefore not of the same kind.

Complex points (numbers): The M-set is a mathematical set, a collection of numbers. These numbers are different than the real numbers with which people are familiar and are termed "complex numbers". They have a "real" part and an "imaginary" part. The real part is an ordinary number, for example, -2. The imaginary part is a real number times a special number called i, for example, 3i. An example of a complex number would be -2 + 3i. The number i was invented because no real number can be squared (multiplied by itself) and result in a negative number. This means that you can not take the square root of a negative number and get a real number. When you take the square root of a number, you find a number that can be squared to get that number. The number i is defined to be the square root of -1. This means that i squared is equal to -1. So when you square an imaginary number you can get a negative number. For example, 3i squared is -9.

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Points: The axes permit various complex numbers to be positioned in relation to one another in a systematic manner:

Fixed point: The M-set has only one fixed point, the origin. A point whose value under a mapping function is itself. Many J-sets have fixed points. "indifferent" fixed points, which are neither attractive or repellent, arise for values of c right on the cardioid including at the touching points of buds.

Thus we have three regions associated with f(z)=z*z. Two regions tend to two of the fixed points of f, namely 0 and infinity. We define a fixed point to be an "attracting fixed point" if all points near the fixed point have orbits that have that fixed point as a limit. Thus 0 and infinity are attracting fixed points, and the regions that tend to them are their "basins of attraction." The fixed point 1 is not attracting since it has points nearby whose orbits do not tend to 1. The third region is the unit circle. No point there has an orbit that has an attracting fixed point as a limit.

Peitgen elaborate that fixed points are repelling when the complex derivative at that point has absolute value greater than 1, attracting when less than 1, and "indifferent" when equal to 1

For the special case where c=0+0i, the Julia set is simply a (nonfractal) circle with radius 1

Fixed pointy in peoples lives -- home birth placeetc

Origin: This is the point at which the axes cross, defined as (x=0; y= 0).

Birth place, start of argument, etc

Complex points: These are the positions of complex numbers taken up in terms of the axes.

Points of an argument. Where we have got to in our relationship

Critical point: This is the starting point for the process of generation of the M-set and is in this case equivalent to the origin.

Critical starting point in discussion? Kairos Dramatic moment Critical popint in a negotiation

Iterative generation of M-set: Chaos occurs in objects like quadratic equations when they are regarded as dynamical systems by treating simple mathematical operations like taking the square root, squaring, or cubing and repeating the same procedure over and over, using the output of the previous operation as the input for the next (iteration). This procedure generates a list of real or complex numbers that are changing as the procedure continues -- a dynamic system. The M-set is a set of points that fail to escape to infinity under an iterated point process.

It is amazing that the orbit of 0 "knows" the shape of the filled Julia set for x2 + c. The reason that 0 is so special stems from the fact that 0 is the critical point of x2 + c. That is, the derivative of x2 + c is 2x, and this derivative only vanishes at x=0.

Indeed, it is not possible to determine whether certain c-values lie in the Mandelbrot set. We can only iterate a finite number of times to determine if a point lies in M . Certain c-values close to the boundary of M have orbits that escape only after a very large number of iterations. A second question is: How do we know that the orbit of 0 under x2 + c really does escape to infinity? Fortunately, there is an easy criterion which helps:

Computation of complex points: The M-set is defined as the set of points c in the complex plane for which the iteratively defined sequence zn+1=zn2 + c : does not tend to infinity (where zo=0 and c=x + iy). Each such number being transformed into its corresponding image point (a pixel for display purposes)

In other words, our map is z ? z2 + c, where c is the first complex number in our iteration. So the sequence goes c, 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.

Iterations: The maximum number of iterations (N) used in testing points in the computation, can be selected as desired, for instance 100. Larger N will give sharper detail but take longer.

Human life is characterized by iterative processes. Physiologically these include breathing and the pumping action of the heart. Vision is based on rapid eye movement (REM). The circadian rhythm of the waking/sleeping cycle can also be understood in this way, as can the cycle of consumption/excretion. Many habits are characteristically iterative, as is engaging in sex. The succession of human generations, through which society (and the planetary surface) is populated, may also be considered iterative. A number of religions hold strong convictions regarding reincarnation, itself an iterative process. Within society there are many regular processes that can be usefully seen as iterative: rituals, regular meetings, festivals, etc that provide benchmark points indicative of its status. The most fundamental debates have an iterative aspect as the same points are explored are explored again and again.

** dilemma: going round and round without resolution / periodic cycle

Surfaces and volumes: The M-set can be represented on a surface or on a volume.

Complex plane: A complex plane can be thought of as being the set of all complex numbers, and then used as a way of visualizing relationships between those numbers using images. The cardioid characteristic of the M-set (as discussed here) emerges through the representation of the M-set on a complex plane.

Complex sphere: The complex numbers may alternatively be represented as points on a complex sphere. The origin (below) would then be one pole and infinity the other pole (above) with the unit sphere being the equator. Lit from "infinity", points on the sphere would leave shadows in unique positions on the complex plane (as described above). The complex plane is thus a projection of the complex sphere.

Scope

Size: It can be shown that the entire M-set lies inside a disk of radius 2, centered at the origin.

Boundedness: A complex number c is in the M-set if the iteration of z2+c (beginning with zero) remains bounded. If |z| exceeds 2, the z sequence diverges. The boundary of the set of complex values of c -- such that z does not escape to infinity -- is very complicated. The M-set therefore lies within |c| less than or equal to 2.

Boundary zone: We may take the view that the process of self-organization takes place at the "edge of chaos", where the system is able to poise itself at a position of optimum fitness, between (ultimately stultifying) stability and the chaos and unpredictability (and therefore unmanageability) of some form of strange attractor. Such a situation is often depicted by the infinite (fractal) variety of the boundaries of the Mandelbrot or J-sets. Systems in the edge of chaos position may show a pattern of what is called punctuated equilibrium in which phases of seeming equilibrium are interspersed with what appears to be chaotic behaviour; so a strange attractors is the focal point of a phase of equilibrium.

Dimension: Dimension is a measurement of how complex an object is. We are used to the idea of a point being zero dimensional, a line being one dimensional, and a solid square being two dimensional. Present conceptions of dimensionality and fractals are practical working definitions and are by no means rigorously defined. In general terms, the M-set has a dimension of 2 because the entire set is contained in a disk which has a dimension of 2. The topological dimension of the M-set is 1 -- the boundary has an empty interior, so the dimension must be less than 2. Despite its complexity, the M-Set has a singular fractal dimension. Whereas other fractals have a non-integer dimension, the M-set fractal dimension is 2.

For some types of functions, the set of numbers that yield chaotic or unpredictable behavior in the plane is called the Julia set These Julia sets are complicated even for quadratic equations. They are examples of fractals - sets which, when magnified over and over, always resemble the original image. The closer you look at a fractal, the more you see exactly the same object. Fractals naturally have a dimension that is not an integer - not 1 or 2, but often somewhere in between.

Human experience may be understood as lying within the world of polarization and duality. The coherence and integrity of human experience -- any sense of unity -- therefore emerge within the framework of that duality.

Sets and connectedness -- J-set and M-set

A dynamical system is generally defined mathematically on a configuration space consisting of a topological manifold such as a plane, whether or not it is complex. Non-linear dynamical systems may be defined on such a plane by quadratic functions (of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero). Under suitable conditions, deviations from linearity result in complex chaotic behavior (in which the orbits of the system are attracted to a complex higher-dimensional subset called a strange attractor, or are ergodic). For some types of functions, the set of numbers that yield chaotic or unpredictable behavior in the plane is called the Julia set (J-sets). These sets are complicated even for quadratic equations. They are examples of fractals - sets which, when magnified over and over, always resemble the original image.

Dynamic system
Julia
Mandelbrot

Non-linear system
Any curve, system, or set of equations that cannot be differentiated, namely lines cannot be found to approximate the rate of change of the system at any given point.

     
     
     
     
     
     
     
basic definitions

A Julia set is almost the same thing. It is defined to be : the set of all the complex numbers, z, such that the iteration of f(z) -- > z 2 + c is bounded for a particular value of c. Again, more simply put it is the graph of all the complex numbers z, that do not go to infinity when iterated in f(z) -- > z 2 + c, where c is constant.

 

the definition of the Mandelbrot set is : the set of all the complex numbers, c, such that the iteration of f(z) -- > z 2 + c is bounded (starting with z =0 + 0i). More simply put, the Mandelbrot set is the graph of all the complex numbers c, that do not go to infinity when iterated in f(z) -- > z 2 + c, with a starting value of z =0 + 0i.
Generation / Iteration
The results of a procedure whose result is fed back into the same procedure many times
  Selection of a particular point, then multiplying every other point by it repeatedly, then adding the original point at each iteration. Multiplication of every point on a complex plane by itself, repeatedly, adding the oriinal point at each repetition.
  The set associated with the function z = z2 + c, where c is an arbitrary constant. The J-set therefore iterates z2 + c for fixed c and varying starting z values. Any set containing only those points that remain stable during iteration. The set of points that do not escape to infinity when the function z = z2 + c is iterated, where c is the point itself and z starts at the orgin. The M-set iterates z2 + c with z starting at 0 and varying c
 

Julia sets come from iterating a map; that is, applying a function again and again.

The function that is iterated can be practically anything, as long as it uses complex numbers.

Quadratic Julia sets are generated by the quadratic mapping (2) for fixed c.

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Space J-set is in dynamical or variable space (z-plane). M-set is in parameter space (c-plane)
     
Number There are an infinite number of different J-sets possible, each  
Specification Defined for a given value of c -- a complex number, but for any given Julia set, it is held constant  
  The Julia set J(c) is made of all points z, which do not go to an attractor (or infinity) under iterations.  

Orbits
The trajectory of a point or other object, whether through physical space (eg a planet) or through mathematical space (eg a complex plane). 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. Orbits about an attractor can be super-stable, periodic, or chaotic. Given a specific choice of c and z0, the iterative recursion leads to a sequence of complex numbers z1, z2, z3 -- called the orbit of z0. Depending on the exact choice of c and z0, a large range of orbit patterns are possible. All sequences of z computed through the iterative equation will fall into one of three classes of behaviour:

Behaviour 1

Convergence: the sequence of points {xk} converges to a limit

Values increase without bound (towards infinity), the J-set is then known as of the dust type. For a given fixed c, most choices of z0 yield orbits that tend towards infinity.

Some Julia sets may consist of many disconnected points (called "dust sets"). The further from the origin, the quicker the J-sets break up and fall into Cantor dust. either f(z) can continue to grow without bounds or it will stay bounded.

Points z0 in the complex plane that do not stay bounded with successive iterations of f(z) are said to be in the escape set Ec.

The M-set graphically depicts for which c-values the orbit of fc (0) will have an attracting fixed point (main cardioid), an attracting periodic orbit (primary bulbs), or will diverge to infinity (colored region).
Behaviour 2

Periodic cycle: for some p>0 x0=xp so that the sequence repeats itself

Values collapse (to zero), the J-set is then of the solid type. others from larger "solid" areas that seem all connected.

For some values of c certain choices of z0 yield orbits that eventually go into a periodic loop.

All other points in the complex plane stay bounded as n is taken to infinity -- they are termed prisoners and are said to be in the prisoner set Pc defined for a given c.

Since for c inside M, the iterations remain bounded, pixels corresponding to the Mandelbrot set consume the greatest amount of the computational time. However, iterations inside M evolve differently depending on the value of c. (Behavior of the iterations is related to the appearance of the Julia sets Jc.) For example, for c inside the big cardioid, the iterations converge. For c inside the big circle to the left of the cardioid, the iterations converge to a cycle of period 2. For c inside each wart attached to the cardioid, the iterations converge to a periodic cycle whose period is determined by the corresponding wart.
Behaviour 3

Chaos: none of the above. The points {xk} go from one place to another in apparently chaotic manner. The set of points with chaotic orbits is called the Julia set for a given function f.

Values change, but do not seem be (1) or (2). J-sets are strictly defined as class (3) points, when they drift around to other class (3) points and do not tend towards zero or infinity. Finally, some starting values yield orbits that appear to dance around the complex plane, apparently at random. Points in the Julia set, however, are said to be chaotic, meaning that very small differences in points tend to show wildly different results.Julia sets are all points where the sequence of z values change, often drastically, but do not approach infinity nor zero.

The J-set is of the dendritic type. some form thin, wiggly lines that are all connected but do not outline any shapes ("dendritic" types).

 
 

 

 

 

 
Attractor
A value, or set of values, to which an iterated mathematical function converges, no matter the initial value. While infinity is a point attractor, depending on the choice of c, there may also exist one other attractor in the system. For two dimesional functions, a region of points with attractors is termed a basin of attraction.
  Infinity as a point attractor. If there is no second attractor (i.e., infinity is the only attractor) then the Julia set is a disconnected Cantor dust set.  
  If this second attractor does exist for a particular c, then the Julia set is topologically connected, and is in fact the boundary between the basin of attraction to infinity and the basin of attraction to the finite attractor.  
  , the second attractor may be either a point attractor or a periodic cycle. (A point attractor is essentially a periodic cycle of period 1.) The exact shape of the basin of attraction to this second attractor depends on c.  
Connectedness / (Boundedness?)
  There are an infinite number of different J-sets possible. But unlike the M-set grounded in zero where the black portions are all connected with each other in the complex plane, the different J-sets are disconnected with each other. An M-set is the set of all parameter values whose J-sets are wholly connected.The M-set is a wholly connected archipelago of self-similar islets linked by an array of extremely twisty, ever branching fibers.
Connected 1 Topologically equivalent to a severely deformed circle. Origin trapped inside the set, the set is topologically equivalent to a circle and thus is wholly connected. Mandelbrot has discovered the set M of parameter values c for which Julia sets are connected. This set that now bears his name may also be defined as the set of c's for which iterations {zk} starting with z0 = 0 remain bounded. ?????
Connected 2 Topologically equivalent to a curve (or line) with an infinite series of branches and sub-branches called a dendrite (e.g., the Julia set for c=0+i)" (Elert 22.shtml). Origin is a part of the set, the set is dendritic.  
Disconnected Orbit of the origin eventually escapes to infinity. Disconnected sets are completely disconnected into a countably infinite assembly of isolated points. In addition, these points are arranged in dense groups such that any finite disk surrounding a point contains at least one other point in the set. Such sets are said to be dustlike. As they can be shown to be similar to the Cantor middle thirds set, they are often called Cantor dusts.  
     
  We do not have to calculate the whole Julia set, we only have to examine the orbits of specific points: These specific points are the critical points, i.e. all points where the first derivation vanishes (f'(z)=0), as defined above. The orbits of the critical points define the type of the Julia set: If all orbits stay limited, then the Julia set is connected. If at least one orbit tends to converge to infinity, then the Julia set is dust like.  
Graphic representation
J-sets and M-sets have a close relation to each other.
    The image around a selected point in an M-set resembles the image of the associated J-set.
     
     
     
     
     
     
     

 

Julia set (J-set)

Quadratic J-sets are the family of sets generated by the special quadratic case form f(z) = z2 + c. Here z represents a variable of the form x+iy (x and y real numbers) which can take on all values in the complex plane. This is sometimes referred to as a quadratic map, and is a type of dynamical system. Each point in the complex plane corresponds to a different J-set derived from a function z representing a variable (of the form a+ib) which can take on all values in the complex plane.. *** Points in the Fatou set tend to stick together; that is, points close to each other will follow similar paths, drawing closer to either infinity or zero. http://www.mcgoodwin.net/julia/juliajewels.html

With respect to human behaviour and understanding, a J-set might be usefully described as a "pattern". A distinction can then be made between three kinds of pattern:

    1. Essentially unstable patterns that persist only briefly, if at all, and may only be briefly assumed to have any existence. These are the behaviours which seem to be part of an enduring pattern but more or less quickly prove not to be. Equally they are the modes of thought which may breiefly appear to be consistent, but quickly prove not to be.
    2. These are patterns which are essentially habitual and unvarying, consistent with a single general pattern of behaviour of which they are an exemplification.

Connectedness

Mandelbrot set (M-set)

The M-set is an answer to any query regarding the existence of an organizing principle for the infinite number of possible J-sets -- namely an organizing principle that classifies these J-sets.

applet:

Form and features

The Natural definition of a fractal: A geometric figure or natural object that combines the following characteristics: (a) its parts have the same form or structure as the whole, except that they are at a different scale and may be slightly deformed; (b) its form is extremely irregular or fragmented, and remains so, whatever the scale of examination; (c) it contains "distinct elements" whose scales are very varied and cover a large range. This looks a little more within our grasp to prove. The first part is what we like to call self-similarity, which the Julia set demonstrates easily (no matter how far in we zoom, we see the same structures), but the Mandelbrot set is harder to see. However, on closer examination, we see the master Mandelbrot set repeated over and over leading us to believe that the first part is true. Part (b) is highly obvious for our sets, as is part (c).

Self-similarity:

Symmetry: The bilateral symmetry results from the imaginary components of the complex numbers while the vertical, head and tail, directions result from the real number components.

Other M-sets: Any iterated function can be used to build a M-set. The original M-set (as discussed here) uses iterated squaring. M-sets with iterated cubing, third power, fourth power, and fifth power can also be produced. Such sets exhibit an n-fold rotational symmetry where n is one smaller than the iterated power used to generate the set.

Cardioid: This is the main body of the set as represented in Figure 1. Attached to it are "bulbs".

"Bulbs" (or "circles", or "disks") around the cardioid: "Head": This region corresponds to the large bulb directly attached above the main cardioid. This has an attracting cycle of period 2. As mentioned above, the orbit of fc (0) has an attracting 2-period at c = -1. In fact, it can be easily proved that the region of 2-period orbits is bounded by a circle given by | c + 1 | = 1/4, which is a circle of radius 1/4 centered at z = -1. In other words, all c-values within this bulb will generate an orbit of fc (0) that approaches a cycle of period 2.

Filaments / Tendrils:

Orbits: Super-stable orbits quickly converge to a fixed point. In other words, each primary bulb corresponds to a different period, and all c-values in the same primary bulb generate orbits which approach a cycle of the same period p.

Period of attracting cyle: A complex number c is in the M-set if the iteration of z2+c (beginning with zero) remains bounded. It is in a (hyperbolic) component of period n if the iteration attracts to a peroidic orbit of period n. These coponents are the disks budding off of the M-set or cardioids for mini-M-sets.

Bifurcation: Time-dependent systems are capable of abrupt changes in their topological form called bifurcations as the underlying parameters cross critical values. Bifurcations result in abrupt catastrophic change in the topology of the flow under continuous variation of the time-dependent parameters.

Attractor / Repellor: Strange attractors have orbits which chaotically trip from one basin to another. The above period 3 set of attractors can be modeled physically with an iron pendulum suspended between three magnets. It is a strange attractor and the final resting position is indeterminate even if the bob starts close to one particular magnet.

Rotation:

Features: Many of the detailed features of the M-set have been given colloquial names, usually descriptive in relation to natural phenomena [more].

Colloquial names for the entire M-set R2 and for various parts of it are derived from the resemblance of the continent to a sitting Buddha (when rotated 90 degrees so that west is up). The term buddhabrot in particular is used to refer to plots in which all iterates, as well as the parameter value, are plotted in the same image; with suitable color mappings such plots are said to resemble a sitting Buddha even more than the normal-style plot. See also topknot. From the Mandelbrot Set Glossary and Encyclopedia, 2004 Robert P. Munafo.

Colour:

Mapping in higher dimensions: The classic M-set is a map in the complex plane. The M-set can also be mapped in higher dimensions.


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Attraction to Shape http://home.earthlink.net/~sroof/Abraxas/sar/shape/shape.htm

The Mandelbrot set is then the set of complex c values for which the z orbit remains bounded. From Julia and Fatou, it is known that the basin of attraction of any finite attractor will contain the critical point (see, for example, Devaney devaney), so the Mandelbrot set catalogues the parameter values for which a finite attractor exists. Other initial conditions may not fall in the basin of attraction of a finite attractor even if one exists; thus the Mandelbrot set is the maximum region in parameter space for which orbits can remain bounded. [more]

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Ben Moore. Fractls. http://krone.physik.unizh.ch/~moore/complex/fractals2.html

Has a: A Mandelbrot Set with 8 Accompanying Julia Sets in a Constellation Diagram

The Mandelbrot set for the quadratic mapping f: z --> z2 = c is shown below for all parameters c = x = iy in the range x = [-2, 1/2] y = [-2, 2]. Some wholly connected Julia sets were also added and their approximate location in parameter space indicated. This type of arrangement is known as a constellation diagram.

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Glenn Elert. The Chaos Hypertextbook. 1995-2003 http://hypertextbook.com/chaos/

Strange and Complex by Glenn Elert

http://hypertextbook.com/chaos/23.shtml

The factor that determines whether a Julia set is wholly connected or wholly disconnected is the parameter value. Thus it would be instructive to plot the behavior of the Julia sets for all parameter values. The resulting construction would be the complex analog of a bifurcation drawing. At first glance, this seems a daunting task. Plotting every possible Julia set and then examining it to determine whether it was connected or not would take an eternity. Luckily for us, however, we need only study the behavior of one point in the complex plane. This trick was discovered by the Polish-American mathematician Benoit Mandelbrot and in his honor the set of all parameter values whose Julia sets are wholly connected is called a Mandelbrot set. The Mandelbrot set for the quadratic mapping f: z --> z2 = c is shown below for all parameters c = x = iy in the range x = [-2, 1/2] y = [-2, 2]. Some wholly connected Julia sets were also added and their approximate location in parameter space indicated. This type of arrangement is known as a constellation diagram.

In higher dimensions, however, attraction and repulsion are not limited to points. An iterative map can collapse on to any structure possible in that dimension. Attractors and repellers can form paths, surfaces, volumes, and their higher dimensional analogs. For example, the two-dimensional map f: (x, y) --> (x, y/2) attracts all points asymptotically to the x-axis. Likewise, a two-dimensional object can act as a repeller. Such is the case for the map f: (x, y) --> (x2 - y2, 2xy). Points inside the unit circle head for the origin while those outside fly off to infinity. Points on the circle remain there and thus for this map the unit circle can be considered a fixed repeller.

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Complex Quadradic Dynamics: A Study of the Mandelbrot and Julia Sets [text]

The Mandelbrot set is a map in the parameter plane that graphically demonstrates the behavior of fc (0) for all c-values.

The Mandelbrot set, M, includes all c-values for which the orbit of z o = 0, under the mapping of fc, remains bounded.

Using the equations above, one can easily see that c = z - z2 . Equivalently, c = x(q) = 1/2eqi - 1/4e2qi . This parameterized curve traces out the large central region of the Mandelbrot set. The interior of this region contains all c-values for which fc has an attracting fixed point. This region is often referred to as the main cardioid of the Mandelbrot set.

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,

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.

7, 5, 8, 3, 7, 4, 5, 6, 7 --- 3, 4, 5

2 -------------------------------- 7

7, 5, 8, 3, 7, 4, 5, 6, 7 -- 3, 4, 5

What if our life is an animation of 3D slices taken from a 4D continuum?

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 f c (z) = z2 + c, for a fixed c, is the Julia set. The filled black region represents the z-values that behave orderly, either as periodic cycles or fixed points. The colored region represents the z-values that will escape to infinity. The Julia set, J c, is the boundary between these two competing basins of attraction.

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 f c (z) by modulating both the parameter values and the initial condition values at the same time. In any case, it is certain that the frontier of Chaos Theory is a field of infinite fractal possibilities.

Chaotic processes are characterised by a critical dependence on initial conditions and other external influences. The state of a system can be represented by a point in a multidimensional 'phase space', and within this there are 'basins of attraction' towards different types of behaviour. A characteristic of chaotic systems is that the basins of attraction have a complicated interface like that between the dark (member) and light (non-member) areas of the Mandelbrot Set. As with the Mandelbrot Set, the interface has similar form when viewed at any level of resolution, so the basins of attraction are intimately mixed no matter how precisely the co-ordinates are specified. This means there is no limit to the smallness of the disturbance, or deviation of an initial setting, that can drastically change the system behaviour. The behaviour is unpredictable in principle, irrespective of the precision of the methods of measurement and manipulation.

 

Bohm introduces a new concept in which he describes the Implicate Order as a kind of *generative order.* He notes that "This order is primarily concerned not with the outward side of development, and evolution in a sequence of successions, but with a deeper and more inward order out of which the manifest form of things can emerge *creatively.*" Bohm believes that the generative order "proceeds from an origin in free play which then unfolds into ever more crystallized forms." Generative order can be seen in the work of an artist. Bohm uses the example of Mandelbrot's mathematically-derived fractals to illustrate more scientifically this cosmic generativity. "Fractals involve an order of similar differences which include changes of scale as well as other possible changes." Bohm notes that "By choosing different base figures and generators, but each time applying the generator on a smaller and smaller scale, Mandelbrot is able to produce a great variety of shapes and figures--All are filled with infinitesimal detail and are evocative of the types of complexity found in natural forms."

The Mandelbrot set of degree d  2, denoted by Md, is defined as the set of parameters c for which any of the following equivalent conditions holds:

Origin's orbit under iteration of the polynomial Pc (i.e., the sequence {0, c, cd + c, (cd + c)d + c, . . . }) is bounded.

Julia set of Pc is connected.

Julia set of Pc contains the origin (the only critical point of Pc).

Figure 4: Progressive emergence of M-set through succession of iterations
Iteration 1 Iteration 2
Iteration 4 Iteration 5
Iteration 7 Iteration 8
Iteration 10 Iteration 11
Iteration 13 Iteration 14
Iteration 16 Iteration 17
Iteration 19 Iteration 20
Iteration 22 Iteration 23

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Figure 5: Illustration of different colouring conventions inside the M-set
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