Consider a general 3D system like:
x' = f1(x,y,z)
y' = f2(x,y,z)
z' = f3(x,y,z)
Can we predefine f1, f2 and f3 in such a way that if there is any strange attractor, it will have some certain topological properties, like its place and shape?
However, here I don't mean the type of strange attractor (Lorenz like, Chen like, … or number of scrolls). I want to design a chaotic system which is for example around a specific curve (like a circle) in the 3D space.
Thank you Professor Letellier. I think some of the works you and Professor Gilmore have done is related to what I want (e.g. Connecting curves for dynamical systems). However here I don't mean the type of strange attractor (Lorenz like, Chen like, … or number of scrolls). I want to design a chaotic system which is for example around a specific curve (like a circle) in the 3D space.
I do not know any means in order to obtain what you want, since some simple formulas can give very specific (strange shapes) see for exampel the Mira-Gumowsky attractors.
http://demonstrations.wolfram.com/StrangeAttractorOfGumowskiMira/
Dear Dr. Dubeibe
Let me disagree with you.
1. Differing in one term is not some little difference. Consider:
x’ = y
y’ = x
and
x’ = y
y’ = -x
They are different only in a sign and one has exponential unbounded answer while the other has a limit cycle.
2. That randomness you mentioned is in the time domain. In the state space we encounter an ordered topology.
http://www.actaphys.uj.edu.pl/vol42/pdf/v42p0235.pdf
The authors aimed to construct a chaotic system which is spherical in state space.
Unfortunately, this is still an open question. There is some known families (for instance, the Lorenz-like systems) of systems, that is, they have similar algebraic forms (mostly, they differ by their nonlinearities) and all produce the same type (topology) of chaotic attractors. Thus, the Chen system produces the same kind of attractors than the Lorenz system. So basically, we know that if you modify a given system by replacing its nonlinearity, you will not change too much the topology of the attractor produced.
Now the inverse problem (as you posed it) is not yet investigated.
I don't think so. Just look at the Duffing and Ueda systems, they just differ in one term and the phase space plots are quite different. On the other hand, thats one of the main characteristics of a chaotic system: a long term "random" evolution.
You can have a look to the PhD thesis (http://infoscience.epfl.ch/record/32820) and related papers by Oscar De Feo. He defined a method to construct chaotic attractors starting from a given limit cycle.
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