I am dealing with some 3-D flows which I suspect are chaotic.
Can someone investigate the following three systems and tell me if they are really chaotic (and bounded, not chaotic in transient and unbounded) for sure?
Thank you
Case A:
x' = z
y’ = y-xz
z’ = -z +xy +0.1yz
Case B:
x' = y
y’ = z
z’ = -4y +x^2-1
Case C:
x' = y
y’ = z
z’ = -x -2y -y^2 +2xy -yz
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Some points:
1. All the cases are conservative and dealing with them is more difficult than usual dissipative cases.
2. Parameters are in the most elegant form. I don’t mean finding chaos with varying the parameters. The only variables here are the initial conditions.
3. With OUR CALCULATIONS, they all have positive largest Lyapunov exponent.
Proving something (even on a purely numerical ground) requires some efforts. What I would suggest, if you are not familiar with invariant manifold, is to select different initial conditions (in the Poincare section), check which of them do not (seem to) explode and then looking carefully at the structure of the "last" orbit which does not explode, it should tell a lot about what is going on. This is tedious work that has to be done by changing slightly the initial condition and also running each simulation long enough to be confident that one does not see a transient behaviour.
clearly for initial conditions x_0=y_0=z_0 = 0 Case B is unbounded. Are the
numerical coefficients fixed for every case? The evolution of these systems
will depend on the values of the parameters as well.
Dear Adam and David,
1. All the cases are conservative and dealing with them is more difficult than usual dissipative cases.
2. Parameters are in the most elegant form. I don’t mean finding chaos with varying the parameters. The only variables here are the initial conditions.
Dear J.I. Arana,
With OUR CALCULATIONS, they all have positive largest Lyapunov exponent. However since they are conservative, I am not sure about our results. On the other hand I worry about that they may be unbounded.
Dear Sajad
this article could be interesting for you
.http://inls.ucsd.edu/~rulkov/papers/gen-synch/prE_aux.pdf
Dear Sajad, did you investigate the stability of steady state(s) and behaviour of each system starting from the vicinity from them as well as from a vicinity of nullcline for each variable? I would start analysis from the easiest cases in order to get hints about properties of the system.
I have a preliminary question: how can you be sure that the three systems are all conservative? In model A, the trace of the velocity field (which is equal to the volume contaction rate) is 0.1 y. This system is conservative only if you know that the average of y is zero.
No problem with model B, while in model C the trace is -y and one again must be sure that the average of y vanishes.
Dear Sajad,
I think Antonio Politi has pointed out a very important comment. Specially in case A.
Dear Antonio and Babak,
Based on my simulations, cases A and C are conservative. They have a pair of equal Liapunov exponents with different signs and a zero one: (k,0,-k).
A couple of comments. First in case A, one can reason as in case C, since y
is the time derivative of (x^2/2+y) and so, if the evolution is bounded, y must be, on average, equal to zero.
Second comment: the chaotic component I see from the Poincare section is bounded and I would guess it is surrounded by ordered motion. What happens if you select an initial condition that lies away from the chaotic sea? For instance: x=10,y=-60, z=0? Do such a type of initial conditions wind around, following a quasi periodic motion?
The attached picture is for (-3, 3, 1) as the initial conditions (with MY CALCULATIONS). for (10,-60,0) the answer is unbounded.
At this point, I believe that there must be some critical curve (a homoclinic/heteroclinic curve?) separating out bounded from unbounded motion. I don't know if it is worth making the effort, but the strategy could be that of reconstructing the invariant manifold of some simple periodic orbit (or even a fixed point).
Dear Professor Politi,
This system (case A) is a kind of super-rare flow (conservative, chaotic? (I am not still sure), and more than them with a line of equilibrium). I believe it is worthy to be investigated carefully.
I am not familiar with invariant manifolds.
-----------------------
Can someone simulate it and verify its boundedness and chaos?
Proving something (even on a purely numerical ground) requires some efforts. What I would suggest, if you are not familiar with invariant manifold, is to select different initial conditions (in the Poincare section), check which of them do not (seem to) explode and then looking carefully at the structure of the "last" orbit which does not explode, it should tell a lot about what is going on. This is tedious work that has to be done by changing slightly the initial condition and also running each simulation long enough to be confident that one does not see a transient behaviour.
Dear Sajad Jafari
I am very much agreed with your nice opinion for chaotic system. Really, It is very nice problem, which never reached to its end. A simple example of its is branching process.
If one can draw a triangle on page and divided it by and any line so called median and repeated it many time, so that it can not be reached to its end.
Be aware that there is NO transient regime in conservative system. So, if your systems are actually conservative, you do not have to use long runs... From the picture you showed, the behavior is most likely chaotic. But proving this is not always easy. see Malasoma's works for some clue about that.
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