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Linear Systems equillibrium

  1. Apr 7, 2006 #1
    Hey, I'm trying to find two linear functions, f(r) and h(r) for the following system:

    dp/dt = A*f(r)

    dr/dt = -B*h(r)

    where A and B are constants greater than zero. I'm trying to find linear functions that will tend to equillibrium, and also where

    limit df(r)/dt = 0
    r->0

    I have been trying various linear functions and have been unable to come up with a solution. Is there a solution? and if so, what would be one? It's probalby something simple that I'm overlooking. Any help would be appreciated. Thanks.
     
  2. jcsd
  3. Apr 8, 2006 #2
    The linear (homogeneous) functions in one variable, 'r' in this case, have the general form:

    [tex]f(r) \ = \ \alpha\cdot r,\quad g(r) \ = \ \beta\cdot r, \qquad \alpha,\beta\in\mathbb{R}[/tex]

    So this system can be trivially solved, since the variables are separated. On one hand, you have:

    [tex]\dot{r}\ = \ -B\beta\, r[/tex]

    so

    [tex]r(t) \ = \ r_0e^{-B\beta t}[/tex]

    The condition for stability is obviously [tex]\beta > 0[/tex].

    On the other hand:

    [tex]\dot{p}\quad = \quad A\alpha\, r \quad = \quad A\alpha r_0 e^{-B\beta t}[/tex]

    Then,

    [tex]p(t) \quad = \quad \gamma \ - \ \frac{A\alpha r_0}{B\beta} e^{-B\beta t}[/tex]

    where [tex]\gamma[/tex] is an integration constant. This means that, as long as [tex]\beta > 0[/tex], your system will be stable, with equilibrium point [tex]p = \gamma, r = 0[/tex].
     
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