Weather and economy are two examples where the out come cannot be predicted accurately. A trajectory of a two-body problem, for example, consists of conic sections; ellipses, parabolas, and hyperbolas. Chaotic attractor, on the other hand, often has features repeated on many length or time scales.
The term "fractal" was coined by B.Mandelbrot, a mathematician at IBM, in 1960. It is generally acknowledged that fractals have some or all of the following properties: complicated structure at a wide range of length scales, repetition of structures at different length scales (self-similarity), and a "fractal dimension" that is not an integer. Fractal basin boundary--the boundary between chaotic and periodic motions in initial condition or parameter space, may also have fractal properties.
The construction of Cantor set is simple and it can explain what is meant by fractal dimension. Take a line of unit length and divide into three equal parts, remove the middle 1/3, and divide the remaining parts, in a similar way, each into three parts. The process can be repeated, every time throwing away the middle portion. We can see what happens after 'n' steps or iterations. ! N Î 1 1 2 1/3 4 1/9 8 1/27 --- ---- 2 n 1/3 n. After n steps the total number of divisions become N = 2 n, while the size of each division becomes Î = 1/ 3 n. The Dimension 'd' of the set is defined as: d = limit (log N) / log (1/Î) = limit (log 2 n) / (log 3 n) = log2/lo3 = <1 Î --¥ n-- ¥
SEIRPINSKY GASKET:
Construct a triangle T with vertices A, B and C. Take the midpoints of each side of the triangle and form a triangle. We can remove this triangle from the main triangle T. We will be left with three triangles of the same size that was removed or cut off. We can repeat the process for each of these three triangles and every time removing the central triangle. Finally, as the triangle size decreases, we get what is called a Seirpinsky gasket where the initial points asymptotically approach the attractor at an exponential rate. Any part of the gasket is self-similar to any other part. Seirpinsky gasket is shown in the figure.
MANDELBROT SET:
Consider the following equation, which after few iterations it gives rise to necklace shaped complex picture. Zn+1 = Zn 2 - C where Z and C are complex having a real part and an imaginary part: C = C real + C imaginary and Z = X + i Y Z(n+1) = X(n+1) + i Y(n+1) = f(Zn) = (Xn + i Yn) 2 - (C real + i C im) = (Xn 2 - Yn 2 - Creal ) + I(2 Xn Yn - Cim) X Sq = Sqr(X) Y Sq = Sqr(Y) Y = 2XY - Cim X = X Sq - Y Sq - Creal It is important to keep these statements in the above order. We must first compute Y and then X, because X appears from the equation for Y. Not only the components of C play a role, but also the initial value of Z, Zo = Xo + I Yo. Thus, we have four quantities to change, Xo, Yo, Creal and Cim. For obtaining a two-dimensional fractal we may restrict to two variables, say, Xo and Yo as initial values for the complex value Zo. We can choose a value for C and start the iteration process. As a result, X and Y will change and with them, the new value for Z is obtained. After a given number of iterations, we color the point corresponding to Z according to the result. The method is then repeated for the next value of Zo. The Mandelbrot formula is so constituted it has only two attractors. One of them is " infinite".
By the attractor infinity, we mean that the series of numbers f(Z) exceeds any chosen value. Since for complex numbers there is no meaningful concept larger or smaller. We understand by this the square of the modulus I f (Z) I 2 exceeds any given value after sufficiently many steps, say 100, so that the comparison does not take too long. If the modulus stays below the value of 100.0 after this number of steps, we can say the attractor to be finite or effectively zero. In fact, the situation is somewhat complicated. Only in some cases f (Z) = 0, will be a fixed point that lies near the origin and satisfies f (Z) = Z. Further there are also attractors that are not single points, but which consists of 2, 3, 4, or more points. For an attractor with period 3 f (f (f (Z))) = Z. Despite these complications, we concentrate on only one important thing: that the attractor be finite, that is, that the series should not exceed the value 100. The Mandelbrot set is shown in the figure.
JULIA SET:
Julia set for the map f (Z) = Z 2 + C is shown the figure and as you enlarge or magnify we can observe the self-similarity of the fractal. When we consider two different complex numbers C1 and C2, we can generate two different Julia sets and thus two different graphics. The enclosed figure is generated for two values of the complex parameter C: The constant is set at C1 = -0.17 + 0.78I and variable C2 = 0.32 + 0.043i.