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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
    y(0)=0
    y(0)=pi/2
    y(0)=pi


    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

    Dick

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    Science Advisor
    Homework Helper

    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

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    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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