Thank you Dr. Lehtihet

I have seen those zoomed pictures in my simulations. As I said this system is sensitive to initial conditions. However it seems it is not bounded. In fact it is not such unbounded too!, since it will come back again. Suppose that you have run it for a specific time (say T_run) and one of the variables (say Mr) has a maximum/minimum value in T1. If you run the system longer, there will be a time (say T2) in which a greater maximum/minimum will happen (this is not a mathematical statement, it is what I have seen). I think by definition we have not a bounded strange attractor and therefore no chaos. If this system is interesting to anyone, I recommend seeing the attached file which is a submitted paper of mine. I have done tons of simulations on these kind of systems and I have seen many strange and interesting trajectories. I appreciate any analysis and possible collaboration.

This is an interesting ODE system. I add a link to my implementation in Mathematica that is arranged such that you can change the initial values and can restrict the visualization to a selectable part of the time interval for closer inspection. Your beautyful almost symmetric (Mr,Mi) diagram don't get reproduced with the initial values given to 5 post-point decimals. Playing with the dMr, dMi, dLr, dLi adjustment devices you can set your exact values which actually created your graphics.

To work with my simulation interactively you need not to have access to Mathematica. You only need to download the free CDF-player from Wolfram Research. It is a rather large program (very powerful too) that will occupy approximately 1GB on your hard-drive.

http://ulrichmutze.de/misc/sajadsProblem.cdf

Thank you Professor Mutze. I have a very slow internet connection at home. I will check the file you have prepared in the university tomorrow. I have uploaded a better figure in the attachment of this answer (note that the initial condition for the right picture is [1 1 1 1]). However I like to know what kind of behavior is that?

I have uploaded my MATLAB codes in the attachment.

Dr. Jafari, may I propose that we communicate on a 'first name basis'? BTW I'm not a Professor.

Unfortunately I have no MATLAB so I can't experiment with your code. Mathematica fails to give for [1,1,1,1] something very similar to your picture. So we probably have to compare integrations over shorter time spans to see whether we at least start right. BTW, I assumed for the right figure the same time span (namely. 100) as the one used for the left graphics.

Just a curiosity, what happens if one rewrite the complex-valued system in modulus-phase formalism?

Did you mean for L^2=(Lr+i Li)^2 or L^2=(Lr+i Li)*(Lr-i Li)?

Dear Fabio,

I meant L^2=(Lr+i Li)^2

Dear Sajad,

you meant "2D over C" "2D over R^2"...

Since you system is non linear you can develop chaos for a certain set of parameters. If you want to recover the 4D corresponding system you can think in terms of non-commutative algebra

and introducing the base {1,e1,e2,e1e2} where 1 is the unit, e1 and e2 are orthogonal versors associated with the directions of M and L and e1e2 is the pseudoscalar (bivector )for the twodimensional space.

The algebra follows the rules (. is the ordinary dot product and ^ is the wedge product)

e1e1=e1.e1+e1^e1=1 (as the outer product "^" of parallel multivectors is zero)

e2e2=e2.e2+e2^e2=1

e1e2=e1.e2+e1^e2=e1^e2=-e2^e1 (the outer product is anticommutative)

for example you have

e1e1e2=e2 , e1e2e1=-e2, e2e1e2=-e1, e1e2e2=e1

(e1e2)^2=-1 (note the correspondence with the square of the imaginary unit)

and

a0+a3 e1e2=e1(a0 e1+a3 e2)

this states that a complex valued field is the geometric product of two vector.

If you develop the algebra on your system you will get the "2D on R2" form...

I think that is possible to explore the "4D on R" way too...

Note: dissipative does not means no chaos!!!

@Fabio. According to normal mathematical usage in the given context 'dimension' is a property of 'linear spaces' and linear spaces refer to a field (in the algebraic sense). So, '2D over C' is OK, whereas '2D over R²' in unidiomatic nonsense. After M has been introduced as a 'complex variable' there is no ambuigity in the meaning of M²! I can't see from your essay that non-commutative algebra can help to create non-trivial results on the system under consideration.

I didn't understand recent comments! In my notation ^ means power. and I didn't use cross (*) or dot (.). Just simple multiply. I know dissipative does not means no chaos. In fact expect that some rare cases (Hamiltonian chaos and conservative chaos) other chaotic systems are dissipative.

@Ulrich. Sorry Ulrich you are right when you say '2D over R²' is a non-sense... I meant a couple of 2D spaces... I asked for M^2 to be sure... Geometric Algebra provides identical results to standard algebra but in a more compact fashion and

with an higher level of abstraction... I believe...

Dear Sajad

In your above figure, you have chosen to use absolute coordinates (Mr, Mi) to show the variations of one of your complex variables. You will get much nicer pictures if you use instead relative coordinates or barycentric coordinates.

Let C = (M + L) /2

The following figure shows (left) the variations of the complex variable M - C and (right) that of L - C.

Thank you Dr. Lehtihet

Can you explain the reason of the behavior of those trajectories?

Dear Sajad,

I agree with Ulrich, this is a very interesting ODE system. Unfortunately, at this time, I do not have any good explanation for the strange trajectories observed. The behavior seems to be strongly dependent on the initial conditions. However, this behavior needs first to be checked carefully for any artifact introduced by the numerical integration scheme. In fact, for some initial conditions, we get an overflow.

Dear all,

actually the system behaves not too differently from what I would have expected. The main observation is that large |M| makes a large |dL/dt| and also makes the direction of dL/dt change very fast (note that the phase of M² changes twice as fast as that of M). So the large loops in the (Mr,Mi) diagram get passed at very high speed (as my demo shows if the number of visualization points is set to a lower value). If one would visualize also the time spent by the system state in the various regions of the diagram the large loops would be virtually non-existent. Due to the fast direction change in dL/dt one expects a Brownian motion like Zig-Zag tendency. What makes things difficult to understand is that the same arguments hold for M and L exchanged. This obviously avoids the explosive behaviour which holds if the initial values (and thus all values) of M and L are real.

Sajad says, he sees no chaos in the system. This might be right with a particular (narrow?) definition of chaos in mind. With a wide understanding I would look to the points where the M trajectory intersects some sphere |M| = some constant and very probably would find a 'chaotic point set'.

Dear Ulrich,

I agree with you. In fact, if I have expressed my concern regarding possible artifact introduced by the numerical scheme, that is precisely because of these high speeds. For instance, a constant-time-step integrator might not work properly as the predefined step might become inadequate if the integration is done over a long time interval. If you make two runs with this type of integrator using two distinct time steps, you might get in some cases quite different results for the same initial conditions. Moreover, using a variable-step integrator will not solve completely the problem. The high-speed loops you mentioned will force your integrator to reduce drastically the time step until the integration becomes unworkable.

BTW there is no need for complex numbers to observe the mentioned problems. All you have to do is to choose your initial conditions right on the real axis instead of the complex plan. You will quickly get an overflow.

I found the linked image and would conclude that Sajad found a clear instance of hyperchaos.

http://www.scholarpedia.org/article/File:Pobur.jpg

Dear Ulrich,

If you like chaotic-looking pictures, here is one obtained with Sajad's ODE.

The figure on the left gives the motion of M in barycentric coordinates and that on the right gives M in absolute coordinates.

Dear Sajad,

do you see chaos now?

Dear Halim,

could you send me the production data for your very nice pictures so that I can compare them with the output of the Mathematica integrator?

BTW did you come across hyperchaos so far?

BTW² I did not understand your remark concerning the case Mi=Li=0. My remark to this case was that it is exploding (i.e. growing soon exponentially) and I wondered why nonvanishing imaginary parts are so effective in avoiding such exponential growth. I see these 'fast large loops' as a kind of frustrated exponential growth.

Thank you Dr. Lehtihet

I have seen those zoomed pictures in my simulations. As I said this system is sensitive to initial conditions. However it seems it is not bounded. In fact it is not such unbounded too!, since it will come back again. Suppose that you have run it for a specific time (say T_run) and one of the variables (say Mr) has a maximum/minimum value in T1. If you run the system longer, there will be a time (say T2) in which a greater maximum/minimum will happen (this is not a mathematical statement, it is what I have seen). I think by definition we have not a bounded strange attractor and therefore no chaos. If this system is interesting to anyone, I recommend seeing the attached file which is a submitted paper of mine. I have done tons of simulations on these kind of systems and I have seen many strange and interesting trajectories. I appreciate any analysis and possible collaboration.

Dear Ulrich,

Sure. The initial conditions have been set in the following way. First, I write M and L under the form Aexp(ia) and Bexp(ib), respectively.

For the above example, I took A = B = 0.1 a = 76° and b = 90°

It seems that most of the time, there will be no erratic looking output if you choose b = a + 180° .

Concerning your question regarding non-vanishing imaginary parts, they actually do not avoid the exponential exploding. They simply delay it. You might have already noticed that the circles become larger and larger. For some special initial conditions, the circles may become tangent to the real axis. When this happens, you will get a quick overflow.

I will give some other remarks later concerning the ODE itself.

.

.

Dear Sajad,

Thanks for the paper. I do like very much the story you describe and I now understand the signification of the names you gave to your variables. I also think that the barycentric coordinates might be more appropriate to represent your outputs.

Before I read your paper, I felt uncomfortable with your ODE because I could not relate it to a physical dynamic system. Now that I have read your paper, I can see what you are trying to model. However, I feel even more uncomfortable with your ODE. This has to do with the consistency of the units.

For example, when you write dM/dt = 1 + L² - alpha*M, I can understand that alpha = 0.01 is expressed in some units representing the inverse of a time. But how about 1 and L² in what units are they expressed ?

As long as the units are inconsistent, I will have a hard time understanding your model let alone interpreting any results you will get. On the other hand, now that I have read your paper, I can think of a way to scale your ODE and perhaps to avoid unbounded trajectories.

.

Dear Halim,

as I see it, you see difficulties with inconsistent units that don't exist. The 1 in the equations say that L and M have dimension 0 (or, 'are dimension-less') meaning that M is not 'affection strength of Majnun' but 'affection strength of Majnun divided by the affection strength of some reference lover in some reference situation' and for L correspondingly. Also t is not 'time' but 'time divided by some reference time span, say hour '. Then it is OK (even mandatory) to have dimension-less numbers as coefficients.

Dear Ulrich,

Perhaps I did not express myself correctly. My problem is not with the scaling itself. Obviously, the way the ODE is written, it must be understood as given in a dimensionless format. My problem is to know HOW the scaling has been done so that I can (a) understand the physical meaning of the model and (b) interpret the outputs.

Maybe the answer is already in the paper.

Dear Sajad,

in the appended link there is a remake of your Fig5 b (on page 3 of the Mathematica stuff) that has no recognizable differences to your version. Of course, the Mathematica output is a bit larger and shows more detail. So, at least here, MATLAB and Mathematica agree.

http://ulrichmutze.de/misc/sajadsFig5b.pdf

Dear Sajad,

I had fun reading your manuscript.

There is one topic, however, with which I'm not comfortable in the present form: your description of the role of complex variables.

1. It is not true that M and its complex conjugate, call it Mc, are indistinguishable in your model: if one changes M to Mc, one changes dL/dt to dLc/dt. This changes the direction in which Layla moves in her emotonal state space. In the direction of more love or more hate? This should make a difference at least for the interpretation of the process. Moreover, it makes a difference for the dynamics: In one case |L| may grow whereas in the other case |L| may shrink which changes the influence on M.

2. Closely related to what was just said, one should discuss what connection between the phase of M and the phase of dL/dt (and L and dM/dt) the model implies and whether this connection is plausible. For this to be complete, one had to explain the emotional meaning also for phases between 0 and -180 degrees.

Thank you very much Ulrich

I will post some interesting materials soon

I think there are other inconsistencies in the paper. In particular, the way in which Z² is being used in the ODE model does not make any sense to me considering the interpretation given in the text for the magnitude |Z|. More precisely, the text seems to refer to |Z|² whereas the ODE makes use of Z².

Dears

1. I agree with what you said. However in that paper the main thing is proposing vectors, instead of scalars, as a variable in such models (the rest is an story, like other similar works). It has some benefits, and certainly some limitations. One can improve it. However this is not the topic here. The problem is analyzing those ODEs.

2. Please see the attached pack. I think that is interesting.

Dear Sajad,

using compex variables implicates much more than using 2-dimensional real vectors. This is the point which I don't see appropriately handled in your paper. May I remind you: normal algebraic expressions written in complex arithmetics (e.g. using M² and not |M|²) automaticly are conformal mappings R² -> R² (or put differently, are solutions of the elliptic Cauchy Rieman partial differential equations) wheras functions written in real arithmetics in the two components satisfy no such equations. So by using compex variables for the emotional state you give it not only an additional dimension, you also impose additional mathematical structure that influence the working of your model in an essential manner.

When you say, 'in that paper the main thing is proposing vectors instead of scalars' you admit that the main features of your model, namely that M² is not a mere nonlinearity, but mainly works through the fact that the phase of M² is twice the phase of M is an unintended and unrecognized side effect. This would be a lost opportunity.

PS: as I recognize now, Halim, makes the same point!

The power of complex functions in ODEs can be seen from a variation of Sajad's balerina equation in which square is replaced by another complex function.

I consider

dM/dt = f(L)

dL/dt = f(M)

for M(0) = i, L(0) = 1, f(z) := 2(cos(z) - 1), 0

Here is my most recent simulation tool

in which the values are preset as to reproduce Halims graphics posted with the words

This signals unexpected superior performance of my

asynchronous leapfrog integrator see

https://www.researchgate.net/post/Who_knows_of_an_integrator_for_the_general_first_order_ODE_which_is_second_order_and_needs_only_one_evaluation_of_the_equations_rhs_per_time_step

Correction (2013-06-26).

In my simulation, I have c=0.1 instead of c=-0.1. (See Sajad's manuscript for the definition and the meaning of c). So I had a slightly different system from Sajad's and the comparison with Runge-Kutta 4 is no longer soooo spectacular in favor of leapfrog. Nevertheless leapfrog is a a bit better, which is surprising since a 4th order method is being compared with a 2nd order one. Also the system with the 'wrong' coefficient has a very interesting behavior. Therfore I don't change the appended file; it still documents the behavior of this modified system.