Consider a chaotic system (flow, not map) which is described by:
x1' = f1(x1, x2, xn)
x2' = f2(x1, x2, xn)
…..
xn' = fn(x1, x2, xn)
Suppose that we project the strange attractor of this system on a 2-D plane of two of its variables (e.g. x1 and x2). Is that possible this projection be a limit cycle?
Let me ask it in another form. I have 2 time series of 2 states of a system. I am plotting them in the state space and I see only a limit cycle (a simple limit cycle, e.g. a circle). However when I watch those 2 signals in the time domain they are not periodic at all and they seem chaotic. Is that possible?
I would say in 3 dimensions it is not possible for a continuous flow because the flow would be confined to the surface of a cylinder, and since it cannot intersect itself, it cannot be a chaotic attractor. In dimensions higher than 3, it should be possible since only one of the n variables is constrained by your requirement, leaving 3 or more to produce the chaos.
It is possible because in chaotic system iteration is important. your system is discrete or continuous? You should plot both of them for same time.
The answer to your first question is yes: take f1 and f2 to depend only on x1 and x2. Then that subsystem can generate a limit cycle regardless of what x3..xn are doing. But this example isn't relevant to your second question. Could the two variables be functionally related, e.g. x1^2+x2^2=const.?
To me, it is possible only if there is a decoupled subsystem within the system. Otherwise, all state variables, in any projection, of a dynamical system, carry the same information about dynamical steady states.
I would say in 3 dimensions it is not possible for a continuous flow because the flow would be confined to the surface of a cylinder, and since it cannot intersect itself, it cannot be a chaotic attractor. In dimensions higher than 3, it should be possible since only one of the n variables is constrained by your requirement, leaving 3 or more to produce the chaos.
Take x'=f(x,y), y'=g(x,y) with a limit cycle. Then multiply them by any positive function of three (or more variables), h(x,y,z), thus, the vector field x'=f(x,y)h(x,yz), y'=g(x,y)h(x,y,z), has the same limit cycle as the original, but now the time deppends on the variable z. Now we simply chose the equations for z to have the desired behaviour in the time scale.
No more answer to your question , you have already all the good possible explanations.
You can use as an example the Lorenz system, which consists of 3 nonlinear differential equations. Depending on the values of the parameters, the solutions of the equations give as a result initially a point attractor, then limit cycle (1 periodicity), torus (2 periodicities) and finally the chaotic (strange) attractor (practically infinite periodicities). Now, in the case of the strange attractor, the 2-dimensional phase portrait does not give a limit cycle.
Thank you all very much. In agreement with Professor Lozi, those answers from Professor Sprott and José Luis Bravo Trinidad are enough for me. Everything is clear now.
I want to use this opportunity! Can somebody do me a favor and answer my 2 other questions in ResearchGate?
"Strange Period Adding instead of usual period doubling"
"Does anyone know some CONSERVATIVE CHAOTIC systems"
you can find them easily by searching those titles.
Thanks again.
Regarding conservative chaos the Henon-Heiles map comes to mind:
http://mathworld.wolfram.com/Henon-HeilesEquation.html
Take a circle map fi = 2*fi mod(2*pi) and x1=cos(fi), x2=sin(fi), and here you are.
A 2-D projection of a chaotic system is always possible but the motion would not be unidorectional around the limit cycle. There would be reversals and accellerations as the behaviour followed its path on the multi-dimensional chaotic attractor.
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