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My question stems from exercise 4.3.3 in Murdock's book "Pertubations: Theory and Methods".

I am asked in the following:

Consider the problem $y''+y=\epsilon y^2$ $y(0)=\alpha$, $y'(0)=0$. Draw energy and phase plane portraits.

How can I use mathematica to rdraw these? I mean $\epsilon$ and $\alpha$ aren't given, do I need to choose numerical values for these?

The potnetial energy is $V(y) = y^2/2-\epsilon y^3/3$.

I am also asked to determine conditions on $\alpha$ and $\epsilon$ under which the solutions are periodic.

Alan
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    Possible duplicate of (http://mathematica.stackexchange.com/questions/14160/plotting-a-phase-portrait) and (http://mathematica.stackexchange.com/questions/28611/phase-portraits-and-streamplot). – Chris K Aug 25 '16 at 17:40
  • @ChrisK I don't see it as a duplicate of any of them, in my question there are parameters $\alpha$ and $\epsilon$ which don't have a specific numerical value, but a continuum of possible values. – Alan Aug 25 '16 at 18:00
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    I think you'll need to pick a value for ϵ in order to plot the phase portrait; α is in the initial conditions, which don't contribute to the phase portrait so that doesn't matter. – Chris K Aug 25 '16 at 18:20
  • Multiply the equation by y' and integrate to obtain surfaces of constant energy, which can be plotted using Plot. By the way, this looks like a physics or math problem, not a Mathematica problem. – bbgodfrey Aug 25 '16 at 22:51

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