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Math Help - Differntial Equations

  1. #1
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    Differntial Equations

    ODE:
    x' = -3x^2 - x + 2

    X[0] is given.

    What values will yield equilibrium. I got those values 2/3 and -1.

    Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?

    Another Q : Let x[0] > 0 and if we solve it by Forward Euler, what should be the max stepsize?

    Any help is appreciated!

    Thanks
    Robert
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by robert123
    ODE:
    x' = -3x^2 - x + 2

    X[0] is given.

    What values will yield equilibrium. I got those values 2/3 and -1.

    Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?
    Put y=x-2/3, then the first of the equilibria corresponds to
    y=0. Now rewriting the DE in terms of y gives:

    <br />
\frac{dy}{dt}=-3y(y+\frac{5}{3})<br />

    And if y is small:

    <br />
\frac{dy}{dt}\approx -5y<br />

    So we see for small perturbations about y=0 the DE gives a
    rate of change of y in the direction restoring y
    towards zero (+ve perturbation gives -ve rate of change
    and vice versa), so y=0 and therefore x=2/3 is a stable equilibrium.

    RonL
    Last edited by CaptainBlack; February 27th 2006 at 05:10 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by robert123
    ODE:
    x' = -3x^2 - x + 2

    X[0] is given.

    What values will yield equilibrium. I got those values 2/3 and -1.

    Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?
    Now lets look at the other equilibrium point.

    <br />
\frac{dx}{dt}=-3(x-2/3)(x+1)<br />

    So let y=x+1, then:

    <br />
\frac{dy}{dt}=-3(y-\frac{5}{3})y<br />
,

    or for small perturbations y is very small and so:

    <br />
\frac{dy}{dt}\approx 5y<br />
.

    So the rate of change of y is in the same sence as y, which gives us an unstable equilibrium.

    RonL
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by robert123
    ODE:
    x' = -3x^2 - x + 2

    X[0] is given.

    What values will yield equilibrium. I got those values 2/3 and -1.

    Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?

    Another Q : Let x[0] > 0 and if we solve it by Forward Euler, what should be the max stepsize?
    The answer to this second question depend on what you want the
    answer for. If we are just interested in convergence to the equilibrium
    without oscillations then a step size of:

    <br />
h_{max}=|(x_0-2/3)/ x_0'|<br />

    will suffice. If we want to reproduce the trajectory with reasonable
    verisimilitude we would use a h<<h_{max}, a typical rule of thumb that
    has worked reasonably well for control problems is to use

    h=h_{max}/3.

    RonL
    Last edited by CaptainBlack; February 27th 2006 at 10:14 AM.
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  5. #5
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    Thanks!

    Hey ,

    Thanks alot for you great help !

    Robert
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