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Nonlinear ODE system

  1. Jun 17, 2010 #1
    Given the ODE system:
    v' = u(u2-1)
    u' = v-u

    Define w=u2+v2. Compute w'.
    Find the largest radius R for which u2+v2<R so that the if the solution curve (u,v) is inside that circle the solution tends to (0,0) as t--> +[tex]\infty[/tex]


    Any guidance would be appriciated !
     
  2. jcsd
  3. Jun 17, 2010 #2
    First, use the chain rule of differentiation and the expressions for [itex]u'[/itex] and [itex]u'[/itex] to find [itex]w'[/itex]. Please show us the result of your work.
     
  4. Jun 17, 2010 #3
    sorry I forgot to mention I only had difficulty with the second part of the question.

    W'= 2uu'+2vv'=2u[v-u]+2v[u(u2-1)]=2vu-2u2+2vu3-2uv
    W' = -2u2+2vu3

    I think (but perhaps I'm mistaken) W is suppose to be a lyoponouv function and I'm suppose to find the radius R in which W is monotically decreasing thus proving that the origin is stable fixed point (so every solution tends to the origin) in the said circle.

    so my problem is the second part, how to find the radius R in which W is monotically decreasing, if what I've written earlier is even correct..

    thank you for your reply :-)
     
  5. Jun 17, 2010 #4
    What is the condition so W(t) would monotonically decrease?
     
  6. Jun 17, 2010 #5
    W'=-2u2+2vu3<0

    v>0, u<0 no problem
    v<0, u>0 no problem

    u,v both positive or both negitive are problematic because W' can be positive in those regions, no ?
     
  7. Jun 17, 2010 #6
    This is not correct.
     
  8. Jun 17, 2010 #7
    Could you please elaborate ?
     
  9. Jun 17, 2010 #8
    Simplify the inequality you got to get a simpler relation.
     
  10. Jun 17, 2010 #9
    You can factorize the inequality you got. Then use the following rule:

    [tex]
    A B < 0 \Leftrightarrow \left[\begin{array}{l}
    \left\{\begin{array}{l}
    A > 0 \\

    B < 0
    \end{array}\right. \\

    \left\{\begin{array}{l}
    A < 0 \\

    B > 0
    \end{array}\right.
    \end{array}\right.
    [/tex]
     
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