help with this coupled differential equations

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jazz_1969
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help with this coupled differential equations

Postby jazz_1969 » Sun May 05, 2013 7:56 pm UTC

Hi!

I've been reading the comics for a long time now!
I have this problem with a set of coupled differential equations I came along while working in a research project and thought about asking for help.

This are the equations:

S1 = (p1 + q11 · Y1 + q12 · Y2) · (1 − Y1 − Y2)
S2 = (p2 + q22 · Y2 + q21 · Y1) · (1 − Y1 − Y2)

where S1 and S2 are the derivative functions with respect to time of Y1 and Y2 respectively.
p1, p2, q11, q22, q12 and q21 are all constants of the model.

Any help would be really appreciated.

Thanks a lot!!

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Voekoevaka
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Re: help with this coupled differential equations

Postby Voekoevaka » Sun May 05, 2013 8:50 pm UTC

I see two fixed points in your equation. The first one depends of the parameters :
(Y1, Y2)=( (p1q22-p2q12)/(q11q22-q12q21), (p2q11-p1q21)/(q11q22-q12q21) )
The second fixed point is not a point, but a line : the line of equation Y2=1-Y1.

The jacobian matrix of the problem is :
| -p11 - q11 Y1 + q11 (1 - Y1 - Y2) - q12 Y2 _ -p11 - q11 Y1 + q12 (1 - Y1 - Y2) - q12 Y2 |
| -p2 - q21 Y1 + q21 (1 - Y1 - Y2) - q22 Y2_m -p2 - q21 Y1 + q22 (1 - Y1 - Y2) - q22 Y2 _|

Depending on the parameters, this point will be attractive or repulsive.

About the line : I think we can replace the variables Y1 and Y2 by X1=Y1+Y2 and X2=Y1-Y2. Studying the evolution around this line will be calculating the jacobian with X1=0.
I'm a dozenalist and a believer in Tau !

jazz_1969
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Joined: Sun May 05, 2013 7:51 pm UTC

Re: help with this coupled differential equations

Postby jazz_1969 » Sun May 05, 2013 9:38 pm UTC

Thank you!

Yes we see the same things, in fact we’ve studied the fixed point and line. We call that line “saturation”. The model doesn’t make sense for Y1 or Y2 <0 or Y1+Y2>1 and all parameters p and q should me positive.

When we tried to calculate the stability, attractiveness and repulsion of the fixed point and line we came across a problem though, we use a Taylor series development and we neglect the quadratic terms so we can work with a linearized system. We’ve been told that this is not… ok to do when dealing with non isolated fixed points (that is, with the line, which we consider a set of non isolated fixed points).

The analysis indicates it is a stable attractive line (with the limitations already described). We also arrived to this conclusions using a perturbation analysis (with the same limitations).

But what we’re trying to figure out, from a mathematical point of view is if this set of equations has a solution in the time domain, that is, if it is possible to find Y1(t) and Y2(2)

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eta oin shrdlu
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Re: help with this coupled differential equations

Postby eta oin shrdlu » Mon May 06, 2013 2:09 am UTC

This won't give you a complete closed-form solution, except in special cases where the eigenvalues of Q are nice, but:

Suppose you had some function s(t) such that ds/dt = 1 - Y1(t) - Y2(t). What could you say about d(Y1)/ds and d(Y2)/ds?

jazz_1969
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Re: help with this coupled differential equations

Postby jazz_1969 » Mon Jun 17, 2013 10:52 pm UTC

Thanks a lot Eta Oin Shrdlu.
You say this won't give a closed form solution and for what we've been researching we think so too. But can this be proven? This would give us a lot more credibility when writting the report for people who doesn't know maths. They wouldn't understand the proof anyways, but just stating we have it (and that being true) would be much more professional.

Thanks!

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eta oin shrdlu
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Re: help with this coupled differential equations

Postby eta oin shrdlu » Tue Jun 18, 2013 6:05 am UTC

jazz_1969 wrote:Thanks a lot Eta Oin Shrdlu.
You say this won't give a closed form solution and for what we've been researching we think so too. But can this be proven? This would give us a lot more credibility when writting the report for people who doesn't know maths. They wouldn't understand the proof anyways, but just stating we have it (and that being true) would be much more professional.

Thanks!
I don't have a proof of nonexistence, but I haven't looked very hard.

However: Have you followed the suggestion of my previous post? You should be able to simplify the problem enough that problems such as the question of stability of your "saturation" line and the solution of the flow field are easily answered. What have you done so far? You should be able to reduce the problem to a single first-order differential equation.

MostlyHarmless
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Re: help with this coupled differential equations

Postby MostlyHarmless » Tue Jun 18, 2013 7:15 am UTC

You have a center manifold of this system (Y1=1-Y2). From this, it should be easy to approximate the dynamics of the system off of (but close to) this manifold. (Hint: look at the eigenvectors of the Jacobian when Y1=1-Y2.) Since that is only a one dimensional system, it shouldn't be too hard to find out what the system does near the center manifold.

(If you feel like being technical, the graph of Y1=1-Y2 is a center manifold of any fixed point on that line -- not of the system. I haven't checked, but the other fixed point is probably hyperbolic and therefore governed by the linear terms of your taylor series. The rest of the system might be more complicated, but you'll probably have a pretty good idea of what happens by describing behavior around the fixed point and that center manifold.)

Edit: Take a look at the Poincare-Bendixson Theorem. There aren't many things that your system can do, so you don't have to solve anything explicitly to understand how it works. This system can't be chaotic, so there are only three possibilities: trajectories go off to infinity, they approach an equilibrium, or they approach a periodic orbit.


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