For a dynamical system the attractor is where it will end up eventually.
This can be a fixed point like a ball bearing in a bowl.
Or an orbit or cyclic attractors like a planet around the sun.
Or we may have a system that never returns in the same place like the
toy "cosmic balls" - this we call a strange attractor
To be more precise,
if we consider the system's state space i.e. the list of all
possible states for the system, then a strange or chaotic attractor is a set of states in a system's state space with
very special properties. First of all, the set is an attracting set. So
that the system, starting with its initial condition in the appropriate
basin, eventually ends up in the set. Even if the system is perturbed off
the attractor, it eventually returns. Second, and most important, once
the system is on the attractor nearby states diverge from each other exponentially
It is easy to generate strange attractors in the plane:just try a
formula of the type:
x = f(x,y)
y = g(x,y)
with different values of constants and starting points until you get an
image (orbit) that remains always in a chosen "window" for example
-4<x,y>4 For example
this image shows 30
attractors(not all strange) found in this way.
Chaotic attractors have
geometric structures that are fixed and unchanging---despite the fact that the
trajectories moving within them appear unpredictable. In this sense, the
chaotic attractor's geometric shape is the order underlying the apparent
The geometric shape of a chaotic attractor implements a kind of dough
kneading. The local separation of trajectories corresponds to stretching
the dough and the global attraction property corresponds to folding the stretched
dough back onto itself. One result of the stretch-and-fold aspect of chaotic
attractors is that they are fractals---some
cross-section of them reveals similar structure on all scales.
To understand this let's consider the following formula:
x = x*y+a*x-y with a=0.357057
y = x+y
The attractor shape shown at the top of this document, reminds a seashell and is
typical of attractors. You can better appreciate the elegance of the shape
and its apparent 3D pattern in this color rendering
of the same attractor.
To understand how this shape is created, we consider a red checkerboard
200x200 and deform the squares using the formula.
of three images shows how the checkerboard is deformed in the three
first applications of the formula. The attractor is superimposed and is
the result of continuing this procedure many many times or, in scientific terms ,the limit set. But you can see
that also after 3 passes, the final shape is starting to appear.
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