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Differential equation bifurcation, how to find equilibrium points?

  1. Sep 23, 2013 #1
    1. The problem statement, all variables and given/known data

    What are the bifurcation values for the equation:

    dy/dt = y^3 +ay^2


    2. Relevant equations



    3. The attempt at a solution

    Equilibrium solutions:
    y^3 + ay^2 = 0
    ==> y^2 (y + a) = 0
    ==> y = 0 (double root), or y = -a.

    a = 0 is the sole bifurcation point, since

    a < 0 ==> two equilibrium solutions
    a = 0 ==> one equilibrium solution
    a > 0 ==> two equilibrium solutions.

    my question is how can you tell that a < 0 has two equilibrium solutions and a=0 has one and a>0 has two again?
     
  2. jcsd
  3. Sep 23, 2013 #2

    epenguin

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    Homework Helper
    Gold Member

    What is an equilibrium solution? It is a point where dy/dt = 0. When you are exactly at such a point the system will no longer move from there, that is called equilibrium. Your algebra tells you there is only one such point (I.e. value of y) when a = 0, and two otherwise.

    You should also look at in what direction the y moves when it is not at an equilibrium point to start to understand what this is for.
     
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