# Thread: help with qualitative analysis of ODEs

1. ## help with qualitative analysis of ODEs

Please can someone explain the following ideas to me?

Definition: the stationary point of an autonomous equation dy/dt = f(y) is defined by: F (y0) = 0 (so the points y such that the function is zero?)

y0 is a therefore a solution of the problem:

dy/dt = f(y), y(0) = y0

dont understand what this is saying?

A simple observation is that if f(x)>0 then dx/dt>0
and if f(x)<0 then dx/dt<0

why???

Definition: A stationary point is said to be asymptotically stable if for all dy/dt sufficiently close to y0, the solution of the problem dy/dt=f(y), y(0)=dy/dt

satisfies lim y(t) = y0 (as t tends to infinity)

Really have no idea what this is saying? And what is the purpose of qualitative analysis?

2. So, you are asking "what's the point in investigating chaos dynamics"?

Ok some facts. When you solve f(y)=0, you come up with denegerate solutions y(t)=y0. These solutions are important, because they give us information about the flow associated with the DE - the behaviour of non-degenerate solutions.

As you say, if f(y)>0, then dy/dt>0; So y(t) is increasing, and so the limit of (t,y(t)) when t->t0 exists (maybe infinite, depends on the domain for t.) If y0 is asymptotically stable, then (t,y(t)) will tend to (t0,y0); Isn't it nice to know all solutions tend to a fixed one as time passes? Warms my heart. Too bad "stable points" are really rare :P

If you need to see some examples, a classic one is the logistic equation, and here's some nice stuff concerning its chaotic behaviour.

Actually, I personally enjoy the behaviour of the sexier () Van Der Pol equation.

3. Originally Posted by Rebesques
So, you are asking "what's the point in investigating chaos dynamics"?

Ok some facts. When you solve f(y)=0, you come up with denegerate solutions y(t)=y0. These solutions are important, because they give us information about the flow associated with the DE - the behaviour of non-degenerate solutions.

As you say, if f(y)>0, then dy/dt>0; So y(t) is increasing, and so the limit of (t,y(t)) when t->t0 exists (maybe infinite, depends on the domain for t.) If y0 is asymptotically stable, then (t,y(t)) will tend to (t0,y0); Isn't it nice to know all solutions tend to a fixed one as time passes? Warms my heart. Too bad "stable points" are really rare :P

If you need to see some examples, a classic one is the logistic equation, and here's some nice stuff concerning its chaotic behaviour.

Actually, I personally enjoy the behaviour of the sexier () Van Der Pol equation.

I don't follow exactly. Shouldn't y be a function of y rather than t? Or is that just for autonomous equations?
How would you apply what you said if for example y' = 2 - y ?
Obviously the stationary point is y=2 but then what?

4. Just for autonomous ofcourse. We got the valuable help of the Poincare theorems.

Now for y'=y-2, t>0. Since y(t)=2 is a stationary point, for y>2 you see that the solutions flow away from the stationary point, towards infinity. This is because they cannot intersect y=2 and are increasing. For y<2, they will be decrasing to minus infinity. Draw a graph!

5. Originally Posted by Rebesques
Just for autonomous ofcourse. We got the valuable help of the Poincare theorems.

Now for y'=y-2, t>0. Since y(t)=2 is a stationary point, for y>2 you see that the solutions flow away from the stationary point, towards infinity. This is because they cannot intersect y=2 and are increasing. For y<2, they will be decrasing to minus infinity. Draw a graph!

I drew a graph. But i'm not sure if it's right. How should it look? If you can describe it?

6. Well, I suck at drawing plots, so I 'll describe it.

Consider the half-plane, t>0. Draw a bold line at y=2. Then, loads of wavey graphs, non-intersecting one another, not crossing the line y=2, and the ones above it going to infinity, while the ones below the line tend to minus infinity.

Wow... I wonder if i can describe the Mona Lisa this way? "Dull chick, no boobs."

7. Originally Posted by Rebesques
Well, I suck at drawing plots, so I 'll describe it.

Consider the half-plane, t>0. Draw a bold line at y=2. Then, loads of wavey graphs, non-intersecting one another, not crossing the line y=2, and the ones above it going to infinity, while the ones below the line tend to minus infinity.

Wow... I wonder if i can describe the Mona Lisa this way? "Dull chick, no boobs."

Ok maybe i've got this wrong so do you have y on the horizontal axis and f(y) on the vertical? Hence making y=2 a vertical line. Then you can draw in little lines indicating the gradients for different values of y? Then somehow you connect the lines?

8. Not really, that graph contains t and y(t). If you want to draw the flows, yes y=2 is vertical, and the gradients flow away from it. Check the webpages I mentioned for examples.

9. Originally Posted by Rebesques
Not really, that graph contains t and y(t). If you want to draw the flows, yes y=2 is vertical, and the gradients flow away from it. Check the webpages I mentioned for examples.
Ah so you're drawing a logistic map? I'm attempting a phase portrait, still not sure if im doing it right

10. phase portrait

Call it what you like, just do it Coordinates are (y,y').