I have asked a similar question before. I thank those participated and helped; however I didn’t find my answer. Consider the system described in the following ODEs:
dx/dt = 1 +z^2 -w^2 -0.01x
dy/dt = 2zw -0.01y
dz/dt = -1 +x^2 -y^2 -0.01z
dw/dt = 2xy -0.01w
Can anybody please simulate the above system and tell me what kind of solution it has? With my simulations, that system is sensitive to initial conditions and has some strange trajectories I cannot understand. It seems that trajectories go to infinity, but they always come back.
Is this kind of behavior familiar for some one?
Please see the attached figure. Note that in that, I have simulated the system for 100 sec and my initial conditions have been [1, 1, -1, 1].
Thanks,
Sajad
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There are some points I should add:
1. I used ode113 in MATLAB. It uses and an integrator with adaptive step-size.
2. This system has come from an unpublished paper which we have written (see sprott.physics.wisc.edu/pubs/paper389.pdf). The original system is:
dM/dt = 1 +L^2 -0.01M
dL/dt = -1 +M^2 -0.01L
and I have considered M & N as complex variables (M = x + iy & L = z + iw).
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There are some more additional points I should add:
1. This system is an excellent example to challenge the power and accuracy of different integrating methods. Just simulate it and run it for100, 1000, and 10000 sec. Observe time series and some state plots (I recommend seeing x time series and the trajectory in x-y plane). You can see that different methods and step-sizes can affect the results noticeably.
2. This system is obviously dissipative. However I don’t think being dissipative ensures being bounded for a system (am I right?).
3. I have attached extra materials to this question (see Dance of the ballerinas.rar). I think they are interesting, but I don’t know why we can see such results and how we can interpret. I appreciate any help and possible collaboration on that.
sprott.physics.wisc.edu/pubs/paper389.pdf
Dear Sajad,
I still haven't tried to simulate the system, but I have a preliminary question about it: are the variables x, y, z and w constrained in some way (i.e. by physical considerations)? Or they can actually diverge?
If some or all of them can diverge, then maybe the system has some critical points at infinity. Trajectories that go to infinity and then come back could signal the presence of asymptotic saddle points. Anyway, I don't think this behaviour could be related to chaos.
I hope these general considarations can help in some way, in any case I will try and simulate the system.
Cheers
Hi Sajad,
Your system is interesting. My result is the same as yours. When I change the viewing angle in 3D, the trajectory travels mainly on two planes for any combination of 3 variables among x y z w.
SK
Add your answer Similar behavior is illustrated in my thesis (available on the www) and in the (1957=the year of the Lorenz attractor -6), a model of coupled dynamo conceived to explain the random reversal of the hearth poles. In the Langyage of Rufus Bowen, the Rikitake system is like the Lorenz system, but with the critical point invariant under the same (axial) symmetry than the equations pushed to infinity, This type of behavior also appears in some conservative systems tha are relevant to celestial mechanics (Sitnikov). I Used some theorems of L.P. Shil'nikov that I generalized in my Poincare Insrtiture paper on such matters. Is your example any important for any application or otherwise. Idid not write, nor know where to find most of what I know about that. Contact me at [email protected] if you need more info (we would need to skyp or the like, etc...). Best regards, Charles
I agree with Charles Tresser who gave you important informations. You can also ask Prof Christophe Letellier who is specialist of the topology of strange attractors [email protected]
This may not be the case for your system, but there are dynamical systems in 4-D, even dissipative ones, whose measure is accurately Gaussian and thus stretches to infinity in all dimensions. There is currently a $500 prize being offered to anyone who can supply a proof or provide convincing evidence to the contrary for some such systems: http://williamhoover.info/snook2014.pdf
Dear Sajad your system is dissipative there for it is stable and its trajectories do not go to infinity. Dr. Tresser said it could be similar to Silinikov chaos but base on my knowledge Silinikov chaos is a type of homoclinic chaos, system with one saddle fix point with specific condition between real and imaginary part.
Thanks to new answers, I have modified the question again. Please see the new attachment of the question. I think it is beautiful.
Hi Sajad,
The ballerina is indeed beautiful. By plotting (Mr, Mi, [blue]) and (Lr, Li, [red]) together, we can observe six roots - three for M and three for L.
You have to think about these questions:
1. Does your system have an energy which can be written down and which is dissipated in some sense?
2. Is there a bounded absorbing set for your system?. Note that by definition a bounded absorbing set (or a ball of radius independent of B) is a set/ball to which all trajectories emanating from a bounded set B of (initial) vectors tend to as time go to infinity.
In either case if this is something you want to achieve in a first step, both can help you prove the existence a maximal bounded attractor which consists of all those closed (and bounded) beautiful trajectories displayed in your picture. Both of the points in 1), 2) can help describe your system dynamically in a more precise way.
It is possible express L(t) and M(t) by means of an expansion in Taylor series, in function of t and of exponent n.
So it is possible calculate for every n the exact values of l[n] and m[n] in function of l[0] and m[0] that are the initial conditions. For n --> infinity one can obtain the exact solution.
Here m[0] = x[0] + I y[0], l[0] = z[0] + I w[0].
x(t), y(t) are the real and imaginary parts of M(t) and z(t) and w(t) are the real and imaginary parts of L(t).
In the attached file there is an example of Mathematica calculation with a generalization of the constant 0.01 (al and be).
It is possible generalize the calculation by means of iterations for every n.
Ciprian G. Gal gave you a good answer. You can also try the following method:
The Poincaré sections and the first return.
The idea is to give us a plane S in phase space E and trace the intersection of the flow with S. This produces a sequence of points in S that constitute the Poincare section of the flow. The sequence itself is a dynamic system. The discreet flow is called first return (I do not know if it's the right terminology in English). The advantage of this method of Poincaré sections is that it allows to switch from one flow of continuous time in an n-dimensional phase space to a discrete time flow in a phase space in two dimensions. In addition, the topological properties of the flow are maintained (Hamiltonian or dissipative nature, stability, chaos) modulo the information loss from the section.
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