Julia Set Quadratic Maps Relation To Mandelbrot Set

This is sometimes referred to as a quadratic map, and is a type of dynamical system. Given a specific choice of and , the above recursion leads to a sequence of complex numbers , , ... called the orbit of .

Depending on the exact choice of and , a large range of orbit patterns are possible. For a given fixed , most choices of yield orbits that tend towards infinity. (That is, the modulus grows without limit as increases.) For some values of certain choices of yield orbits that eventually go into a periodic loop.

Finally, some starting values yield orbits that appear to dance around the complex plane, apparently at random. (This is an example of chaos.) These starting values make up the Julia set of the map, denoted . Some authors also define the filled-in Julia set, denoted , which is the set of all with yield orbits which do not tend towards infinity. The "normal" Julia set is the edge of the filled-in Julia set.

If the modulus of z_n becomes larger than 2 for some n then it is guaranteed that the orbit will tend to infinity. This test makes it straightforward to plot Julia sets for quadratic maps using a computer

2 Relation to Mandelbrot set

Julia sets are closely related to the Mandelbrot set which is the set of all values of c for which z_n = z_n-1² + c does not tend to infinity through application of the recursion with z₀ = 0. Like the Mandelbrot set, the Julia set is often plotted with different colors signifying the number of iterations carried out before the modulus of z becomes larger than 2.

The Mandelbrot set is, in a way, an index of all Julia sets, For any point on the complex plane (which represents a value of c) a corresponding Julia set can be drawn. We can imagine a movie of a point moving about the complex plane with its corresponding Julia set. When the point lies in the Mandelbrot set, the Julia set is connected. Otherwise, the Julia set is a Cantor dust of unconnected points.

If c is on the boundary of the Mandelbrot set, and is not a waist, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. For instance:

3 Attractors and repellers

As stated above, for any given , the majority of orbits tend toward infinity. For this reason, infinity is described as an attractor of the system. The set of all points with orbits that are "attracted to" infinity makes up the basin of attraction to infinity.

Depending on the choice of , there may also exist one other attractor in the system. While infinity is a point 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 .

If this second attractor does exist for a particular , then the Julia set is topologically connected, and is in fact the boundery between the basin of attraction to infinity and the basin of attraction to the finite attractor. If there is no second attractor (i.e., infinity is the only attractor) then the Julia set is a disconnected Cantor dust set.

One of Gaston Julia's important results is that every basin of attraction always contains at least one critical pointIn chemistry, a critical point is the conditions ( temperature, pressure) at which the liquid state of the matter ceases to exist. As a liquid is heated, its density decreases while the density of the vapor being formed increases. The liquid and vapor den of the map (provided the map is rationalIn mathematics, a rational function is a ratio of polynomials. For a single variable x a typical rational function is therefore P ''x Q ''x where P and Q are polynomials in x as indeterminate, and Q isn't the zero polynomial. Any non-zero polynomial Q is - which the quadratic map is). Since the quadratic map has two critical points (0 and infinity), there can only ever be at most 2 basins of attraction. Since it turns out that infinity is always an attractor, one can determine whether a second attractor exists simply by examining the orbit of 0. This is why the Mandelbrot set can be drawn by examining the behaviour of quadratic maps at 0.

Since (in general) the Julia set is the boundary between basins of attraction, the Julia set is sometimes described as being a repeller because all orbits tend away from it.

4 Generalisations

The Julia set can be defined for any iterated map of the complex plane to itself, or collection of maps. The formal mathematical definition is that the Julia set is the smallest closed, completely invariant point set for such a map or collection of maps which contains at least 3 points. It can also be defined (informally) as the set of points for which nearby points do not exhibit similar behaviour under repeated iterations of the map(s).

Julia sets can also be defined for any n-dimensional space, not just the complex plane.

Julia sets typically (though not always) have a fractal structure, and Julia sets can be associated with fractals such as the Sierpinski triangleThe Sierpinski triangle also called the Sierpinski gasket is a fractal, named after Waclaw Sierpinski. An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows: #Start with any triangle in a plane. The canonical and the Cantor setThe Cantor set introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. The Cantor set is defined by repeatedly removing the middle thirds of line segments. One starts by removing.

The complementIn set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement . Relative complement If A and B are sets, then the relative complement of a A in B also known as the set theoretic of the Julia set (i.e. the set of points for which nearby points do exhibit similar behaviour) is sometimes called the Fatou set, although some authors use the term Fatou set or Fatou dust for a disconnected Julia set.

Some common generalisations include:

The cubic map, , or (with greater generality), , where and are complex-valued parameters (as is in the quadratic mapping). The resulting Mandelbrot set is 4-dimensional. The map has critical points (infinity, zero, , ). As with the quadratic map, infinity is always an attractor. However, zero is always attracted to infinity. It thus remains to check the orbits of and .
Newton's method of approximation can be used to solve various equations in the complex plane. Specifically, to solve some equation , we iterate (where is the first derivative of with respect to .) The method is intended to work starting from a fairly good approximation of the solution. Graphing with arbitrary start points, we typically find parts of the Julia set at points equidistant between roots (so the algorithm "can't decide" which one to go far). The most common example used is the cube roots of unity (i.e., ). To form a Mandelbrot set, a parameterised equation is needed.
In a similar way, Halley's method can be applied, generally resulting in more complex images.
Certain of the complex transcendental functions (such as Sin, Cos, Exp, Log, etc.) can be iterated to produce Julia sets. For example, where is an adjustable parameter. (Note that transcendental functions are not rational maps - so Julia's result about critical points does not necessarily hold.)

5 Examples of Julia sets for quadratic maps

6 Reverse Julia sets

It is also possible to generate Julia sets using a method derived from the Iterated function system (IFS) "random game" method. Instead of using an "escape time" method to find the points that do not belong to the Julia set, the "random game" method uses the reverse formula to track the convergence of a single point towards the edge of the Julia set.

For example, for the Mandelbrot set, the reverse formula is

At each iteration one of the two roots is selected at random. After a sufficiently large number of iterations the current position of z is plotted. The process is then repeated with a different random sequence of roots.

The process of "reversing" the mapping essentially exchanges the attractors and repellers; the Julia set is now an attractor, with infinity and possibly the usual periodic cycle now being repellers.

Note that certain parts of the Julia set - notably the "waists" - are quite hard to reach with the reverse Julia algorithm, since the "points" or "tips" of the Julia set are more strongly attractive.

7 External links

Fractals

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1 Quadratic maps