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?

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


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 (http://www.mathhelpforum.com/math-he...es/redface.gif) Van Der Pol equation.

Quote:

---

Originally Posted by Rebesques

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


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 (:o) 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?

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!

Quote:

---

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?

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." :p:eek::D

Quote:

---

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." :p:eek::D

---

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?

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.

Quote:

---

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

Quote:

---

phase portrait

---

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