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Describing anEquilibrium Solution which is neither Stable or unstable?

  1. May 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Discuss the stability of the equilibrium solutions

    2. Relevant equations
    dy/dx = cos^2(y) between 0<=y<=2pi

    3. The attempt at a solution
    Found the equilibrium solutions to be pi/2 and 3pi/2.
    Rough graphed y(x) which was a y vs x graph with horizontal lines (Eq Solu) at pi/2 and 3pi/2.
    Then added the curves y(0)=0 and y(0)=pi which are graphs approaching Eq Sols pi and 3pi/2 respectively.

    Now, I'm not sure how to describe the equilibrium solutions. The graph is neither stable or unstable but just keeps repeating in the given domain. How would I describe this type of behaviour? Thanks
    Last edited: May 12, 2013
  2. jcsd
  3. May 12, 2013 #2


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    Repeating is not the important feature, it's whether a solution starting near your equilibrium solution will flow into or out of the equilibrium solution. If you start with an initial condition a little below y=pi/2, what does it do? What about a little above?
  4. May 13, 2013 #3


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    If, as you appear to be saying, the solutions near the equilibrium solution tend neither toward it nor away from it but circulate around it, then the equilibrium solution is a "center": there are periodic solutions in its vicinity.
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