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Harmonic oscillator with slight non-linearity

  1. Jan 5, 2012 #1
    I have an interesting problem I have come across in my research. It results in the differential equation as follows:


    Primes indicate the derivative with respect to [itex]t[/itex]. [itex]\gamma[/itex] and [itex]\omega[/itex] are constants. The non-linearity comes from the first derivative [itex]x'[/itex] which is raised to the power of [itex]\nu[/itex]. [itex]\nu[/itex] is known to be 0.12 but can be between 0 and 1. The cases where [itex]\nu=0[/itex] or [itex]\nu=1[/itex] are easy enough. But how to go about tackling an arbitrary [itex]\nu[/itex]?

    The problem may be made easier by noting that [itex]g(t)=1[/itex] for [itex]t\geq0[/itex] and [itex]0[/itex] for [itex]t<0[/itex].

    Any ideas on how to go about solving it? Numerically or analytically (which would be amazing).

  2. jcsd
  3. Jan 6, 2012 #2
    You gotta' get Mathematica. Here's some Mathematica code:

    Code (Text):

    \[Gamma] = 0.25
    \[Nu] = 0.5;
    \[Omega] = 0.1
    mysol = NDSolve[{Derivative[2][x][t] +
          2*\[Gamma]*Derivative[1][x][t]^\[Nu] + \[Omega]*x[t] ==
         1, x[0] == 1, Derivative[1][x][0] == 1}, x,
       {t, 0, 5}]
    Plot[x[t] /. mysol, {t, 0, 5}]
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