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Possible to find a gradient system for this?

  1. Feb 21, 2013 #1
    1. The problem statement, all variables and given/known data

    Does

    x' = xex2tanh(x+y) + (1/2)ex2sech2(x+y)
    y' = (1/2)ex2sech2(x+y)

    contain a limit cycle? Possibly-relevant theorem below.

    2. Relevant equations

    Theorem. Closed orbits are impossible in gradient systems.

    Definition. If x' = -ΔV for some cont. diff'ble, single-valued scalar func. V(x), then such a system is called a gradient system with potential function V.

    3. The attempt at a solution

    So the problem looks like a setup to apply this theorem .... but I can't figure out how. I've started at it for like 1 hr.

    Thanks in advance.
     
  2. jcsd
  3. Feb 22, 2013 #2

    haruspex

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    You mean -∇V, right?
     
  4. Feb 22, 2013 #3

    haruspex

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    Since you want that to look like -∂V/∂y, what does it suggest for V?
     
  5. Feb 22, 2013 #4
    That V(x,y) should contain ∫-∂V/∂y dy somewhere
     
  6. Feb 22, 2013 #5
    But I don't know how to integrate that. I can't even find it on an integral chart.
     
  7. Feb 22, 2013 #6

    haruspex

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    Remember, you're only trying to integrate that wrt y. You should find sech2 on the chart. If not, what's the integral of sec2?
     
  8. Feb 22, 2013 #7
    Ah, okay. So I know, for one thing, that V(x,y) somehow involves (1/2)xex2tanh(x+y). And ....... I see it now. If V(x,y) = (1/2)xex2tanh(x+y), then ∂V/∂x = xex2tanh(x+y) + (1/2)xex2sech2(x+y). But, I want <-∂V/∂x, -∂V/∂y> = < xex2tanh(x+y) + (1/2)ex2sech2(x+y), (1/2)ex2sech2(x+y)>.
     
    Last edited: Feb 22, 2013
  9. Feb 22, 2013 #8
    Ah, wait. I could do

    V(x,y) = (1/2)ex2tanh(-(x+y)). Then

    -∂V/∂x = -[xex2 * tanh(-(x+y)) + (1/2)ex2 * -sech2(-(x+y))]
    = xex2 * tanh(x+y) + (1/2)ex2 * sech2(x+y)

    since sech is even and tanh is odd.
     
  10. Feb 22, 2013 #9

    haruspex

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    That's it.
     
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