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Math Help - bifurcations

  1. #1
    Newbie
    Joined
    May 2009
    Posts
    7

    bifurcations

    think i am on a role for not realising the obvious here...

    i have a equationdy/dt= (1-y)(y^2-&)

    &=lamda for the moment if i refer to it as that.

    i need to find all birfucation points and sketchthe diagram. and also expand this by expanding f(y,&) usining taylors series.)


    i have found critical points p1=(1,1)

    and p2=(1/3,-1/3)

    then found when fy <0 &<1

    and fy>0 &>1
    when y=1


    when y= -1/3


    i then dont know what i am doin please may you help
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    First solve for the equilibrium points:

    -y^3+y^2+y\lambda-\lambda=0

    I get y=1 and y=\pm \sqrt{\lambda}

    So when \lambda<0 we have 1 (real) equilibrium point, when \lambda=0 we have two equilibrium points (a bifurcation occurs), when \lambda>0 another bifurcation occurs creating 3 equilibrium points.

    How about I just show you the Mathematica code and the resulting plots for \lambda=4 and you try to figure it out:

    Code:
    myLambda = 4; 
    sol1 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /. 
          \[Lambda] -> myLambda, y[0] == 4}, y, {x, 0, 5}]; 
    sol2 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /. 
          \[Lambda] -> myLambda, y[0] == 0.82}, y, {x, 0, 5}]; 
    sol3 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /. 
          \[Lambda] -> myLambda, y[0] == -4}, y, {x, 0, 5}]; 
    sol4 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /. 
          \[Lambda] -> myLambda, y[0] == 1.1}, y, {x, 0, 5}]; 
    Show[{Plot[Evaluate[y[x] /. First[sol1]], {x, 0, 5}, 
        PlotRange -> {{0, 5}, {-5, 5}}], Plot[Evaluate[y[x] /. First[sol2]], 
        {x, 0, 5}, PlotRange -> {{0, 5}, {-5, 5}}], 
       Plot[Evaluate[y[x] /. First[sol3]], {x, 0, 5}, 
        PlotRange -> {{0, 5}, {-5, 5}}], Plot[Evaluate[y[x] /. First[sol4]], 
        {x, 0, 5}, PlotRange -> {{0, 5}, {-5, 5}}], 
       Graphics[{Dashed, Line[{{-5, 1}, {5, 1}}]}], 
       Graphics[{Dashed, Line[{{-5, Sqrt[myLambda]}, {5, Sqrt[myLambda]}}]}], 
       Graphics[{Dashed, Line[{{-5, -Sqrt[myLambda]}, {5, -Sqrt[myLambda]}}]}]}]
    Attached Thumbnails Attached Thumbnails bifurcations-cubic-dif-eqn.jpg  
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