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 This system has come from an unpublished paper which we have written (see sprott.physics.wisc.edu/pubs/paper389.pdf).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