Consider the famous fractal sets, Mandelbrot and Julia sets. They are based on the idea of choosing two complex numbers Z(0) and C with proper run time and escape-region. They are achieved by repeatedly evaluating the following equation:
Z(n+1) = Z(n)^2+C
For example, in Mandelbrot set, consider a 400×400 mesh when x is in [-2.5 1.5], y is in [-1.5 1.5], run-time is 32 and the escape region is 2.
The final plot is as follows
The yellow part in that figure corresponds to the points in which the value of the function never reaches the escape region. However, the different spectrum of the blue points corresponds to the iterations in which the function crosses the escape region
I have two questions:
a) Is there any study on the transient part of such process (and not steady state)
b) What happens when we don’t consider escape-region?
I think I can answer your questions, if I interpreted them correctly.
Answer to b). The "true" Mandelbrot set M is the set of parameters C for which the infinite sequence Z(n) you describe remains bounded. It can be shown that if such sequence ever gets outside the ball of radius 2 (for some n), then it will escape to infinity. This is why the escape region is used to make the plot, and it is not very relevant.
Answer to a). The parameters exhibiting interesting "transient" dynamics are those that take longer to escape, the interesting part being the behavior of the sequence before it escapes. Roughly, it is like this: taking long for a sequence to escape means that the corresponding parameter is close to M. Thus, the parameters in the boundary of M have sequences exhibiting the "limits" of all the possible transient dynamics. The study of such limits is a central theme in the area of Complex Dynamics.
since transients in a fractal algorithm such as the definition of the Mandelbrot set are only defined in terms of the particular algorithm, they have no other significance (it may be true that many other iterations provably reach the same asymptote, and they will, of course, have different transients);
as far as i know, there is no applicable theory as yet for the sets that are almost asymptotes (it would be nice to have one, since most computational processes cannot wait long enough for an asymptote))
Hello Sajad,
There are a number of ways in which fractal structures can appear in dynamical systems. It is important to differentiate here between two situations: the fractal which appears in the border of the Mandelbrot set is in parameter space (since c is a parameter) and the one which appears after you choose a value for c inside the Mandelbrot set and analyse the system's phase space is called the Julia set.
For the Mandelbrot set, the fractal structure appears as the limit between the values of c in which z = 0 diverges or not. Since I have never formally studied dynamics on parameter space, I cannot discuss on it.
For the Julia set, you have what is called a fractal basin boundary. What occurs here is that there are stable invariant manifolds related to one or more unstable periodic orbits (or fixed points), that have zero measure and affect the dynamics of the system by acting as boundaries between different final states. When you set the parameters you described, you are separating the different dynamical behaviors in order to see the boundary between them.
This situation is the mechanism behind final state sensitivity in chaotic systems, that is, an error in initial conditions near the boundary leads to an error in our knowledge of the system's final state which is not linear, but goes with the fractal dimension of the boundary. (See "Grebogi et al, Final state sensitivity: an obstruction to predictability").
Another famous example in which fractal basin boundaries appear and which can be more understandable is the "Hénon map". (See, for example, "Chaos", by Alligood et al, chapter 4).
Formally answering your questions:
a) In this case, chaos manifests itself as a transient behavior and its relevance depends on your application. I haven't seen transient analysis in the Julia set (although I am certain it exists), but I know it was studied in other systems. (See, for example, "Transient Chaos" by Lai and Tel).
b) If you don't consider the escape region, you cannot distinguish between the orbits that diverge and the ones that do not. Therefore, you will not be able to highlight the boundary between these sets.
Kind regards
- Badges
- Science topic
- Similar topics
- Mathematics