Trapping region for a nonlinear ODE system

[Aldo Leal]

I need to find a trapping region for the next nonlinear ODE system

$u'=-u+v*u^2$
$v'=b-v*u^2$

for $b>0$.

What theory i need to use or which code in Mathematica o Matlab could help me to find the optimal trapping region.

 

[tommyxu3]

You should type # twice or $ twice to make latex:
$u'=-u+v*u^2$
$v'=b-v*u^2$
for $b>0,$
for the symbol $'$ is easy to neglect in normal type.

 

[epenguin]

What theory i need to use or which code in Mathematica o Matlab could help me to find the optimal trapping region.

Iscoclines.

Should, well could, give you a trapping region(s).

Maybe you could then find a smaller trapping region looking at (u + v)'

 

[Aldo Leal]

But... the isoclines just for themselves doesn't give a trapping region.

I mean, if you plot the phase plane you'll see that there are a part where all the vectors go to infinity in the y axis, so it is impossible to take the y-axis as a part of the trapping region.

What a can't understand is your proposition of (u+v)' as a trapping region, how this can work?

 

[epenguin]

I had never met the term 'trapping region' before; it is fairly self-evident but I looked it up. The definition does not say that a trapping region has to be finite!
That being so what trapping region/s can you identify, using isoclines?
(No reason there can't be an infinite number of them, perhaps some are more significant than others - minimal and maximal trapping regions in some sense.)

About the second question, in the spirit of showing what you have tried, you surely could say what (u + v)' is.
At a point where this is 0, what are solution curves doing?
Combining these two there is something significant you can say about the solution family, plus you can get a significantly reduced (though still infinite) trapping region.

 

[HOI]

I would use the term "stable region" rather than "trapping region".

 

[Aldo Leal]

It's just that I've finding a optimal $$b*=0.900316$$ and $$b=1$$, i.e, $$b\in(0.009316,1)$$ for the values of $$b$$. Now I've to proove a trapping region for those values of $$b$$. I've been analyzing the eigenvalues using the Hopf Bifurcation theory but I can't catch that trapping (attraction) region.

 

[epenguin]

One of us may have failed to understand the question. Is this homework, or study question?
I understood trapping region of the u,v 2-dimensional space, in almost self-evident meaning or definition in Wikipedia.
That would not have a value of b as an answer (well it might have different answers perhaps for different ranges of b), rather and answer could (but not necessarily must) be in terms of b.

For instance and fairly trivially if I am not mistaken the whole negative half-plane is a trapping region, so is the positive half-plane. If a point starts in one then obeying the given d.e.'s it never leaves it, but there are smaller trapping regions of interest.

I understood the question as something fairly qualitative and not difficult or advanced.

 

[Aldo Leal]

But the fixed point is in the positive half-plane actually now you can see this pictures I'll post it here and you'll see that there is a limit cycle in the positive half-plane, more specific, in the first quadrant so... I need to build a trapping region in order to... at least in averge the divergence points inward taking out the fixed http://file:///C:/Users/PankeP%C3%BCnke/Desktop/limit11.png point...

 

[Aldo Leal]

It is a part of my thesis investigation but I can't figure it out a trapping region

 

[epenguin]

Please give us a thesis title or other indication so that we understand something about the objective.

In any case this problem seems an elementary and intuitive preliminary to the analysis of this d.e.
I am presuming the definition of 'trapping region' is that of Wikipedia: "In applied mathematics, a trapping region of a dynamical system is a region such that every trajectory that starts within the trapping region will move to the region's interior and remain there as the system evolves."
Something you'd so obviously look for I wonder it needs to be exalted to status of a 'concept' with its definition. It is related to other concepts such as 'Lyapunov function' which describes essentially an infinite nested series of trapping regions (mathematicians excuse my uncouthness). A gold standard of global stability which it may or may not be possible to find.

If it's for a thesis you surely need to note what's happening qualitatively over the whole plane, even if only one part is of particular interest.
So, sketch the u (say horizontal) and v axes and the isoclines. This divides the plane into several different regions. Inside each region draw a horizontal arrow pointing in the direction of increase of u according to the d.e. (I.e. sign of u'). You'll have the direction every point is evolving - to within 180°. Then do the same with vertical al arrows for v evolution direction, then together it's defined to within 90°. For good measure along each isocline do a series of little horizontal or vertical arrows - those are exactly the direction of evolutions at points on the isoclines. (You can already do some sketch of solutions, especially if you take into account magnitude indications from the equations. And I suggested do an (x + y)' line - in what directions (to within 180°) do points either side of this line evolve? And on it? However, the things before parentheses are the usual minimum.)

To me, the definition quoted means there can be and are here an infinite number of trapping regions. The u = 0 line creates three such regions - the line and left and right half planes. At u = 0, u' = 0 and the point can never leave the line. To the left and right it can never cross the line. The upper quadrants are also trapping regions aren't they? And if I draw a horizontal line from 0 to - ∞ in the lower left quadrant, everything above it is a trapping region. But not if I do that in the upper quadrant according to the definition. Can you see a subregion of the upper quadrant that is a trapping region? Overall you can find an infinite number of curves between the limits mentioned that define (infinite) trapping regions. You can even define an infinite number of discontinuous curves that will delimit trapping regions.

Maybe you are working to a different definition or we are at cross purposes, as it seems strange someone ready for a math thesis needs me to explain curve sketching. We'd like to see your plots that you mentioned.

 

[Aldo Leal]

Your definition is totally correct, the missedpoint is that the trapping region must contain the fixed point, especifcically for this system we got $$(b,1/b)$$ as a fixed point, which in other words is the intersection between de isoclines.

Now we can se a kind of a "triangle" that take inside both isoclines and the fixed point mentioned before. Now we got to demonstrate that de divergence in that region is negative for all pointo except for the fixed point. You'll see in the image that there is a limit cycle.

I have to tell you that this is a degenerated glycolysis model from Sel'kov the original one got a trapping region that it is easy to find it but we cand assure that we can keep the limit cycle whenn we do a=0 in the original Sel'kov Glycolysis model (described in next):

$$u'=-u+av+u^2v$$
$$v'=b-av-u^2v$$

for $$a,b>0$$. Now if we do $$a=0$$ we keep the limit cycle but now we got to get a new trapping region in order to demonstrate that the nonlinear ODE system preserve all the proporties and $$b$$ is between $$b*<b<1$$. And I can assure you that $$b*=0.90016$$ which meand that $$a$$ known as the phophofructoquinase once is eliminated from the glycolytic cycle $$b$$ which I don't know what kind of nature element is takes those values in order to keep the cycle functional.

That's the reason why we degenerated the nonlinear ODE system and the necesity to find a trapping region as Strogartz et el do in their book when they talk about the limit cycle of the glycolysis.

 

[epenguin]

Trouble is you have with your talk of trapping region not told what you are looking for and asking that I can understand. It looks as if it might be "the condition (b value) for existence of a limit cycle".

Ax limit cycle is a trapping region. If you are looking for a limit cycle, or something about it, please say so. If instead you are looking for a trapping region that contains the limit cycle, or contains the s.p., there is an infinite number of them, so you have to add some condition to what is demanded like "the smallest right triangle that contains the limit cycle" or something.