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Math Help - help with qualitative analysis of ODEs

  1. #1
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    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?
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  2. #2
    Super Member Rebesques's Avatar
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    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.
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  3. #3
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    Quote Originally Posted by Rebesques View Post
    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?
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  4. #4
    Super Member Rebesques's Avatar
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    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!
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    Quote Originally Posted by Rebesques View Post
    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?
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  6. #6
    Super Member Rebesques's Avatar
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    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."
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  7. #7
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    Quote Originally Posted by Rebesques View Post
    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?
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  8. #8
    Super Member Rebesques's Avatar
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    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.
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  9. #9
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    Quote Originally Posted by Rebesques View Post
    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
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  10. #10
    Super Member Rebesques's Avatar
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    phase portrait

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