Families of dynamical systems

 

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Notes:
• Right now this applet doesn't work with Safari on a Mac. I'll try to fix it.
• To see the long term behavior check "Skip first 100 iterates." If no cobweb plot appears, then there is either an attracting fixed point (in which case you'll see a single red dot on the x-axis) or the orbit goes off the screen. Checking this box enables you to see attracting periodic points.
• You can make your own examples using the input field. The function is f(x) and the slider runs from kmin to kmax.

Quadratic family: f(x)=x2+k
    a. Tangent bifurcation at k=0.25
    b. Period doubling cascade (decreasing values of k):
        • fixed point for k>-0.75
        • becomes period 2 orbit at k=-0.75
        • becomes period 4 orbit at k=-1.23
        • becomes period 8 orbit at k=-1.365
        • becomes period 16 orbit at k=-1.39
        • becomes period 32 orbit at k=-1.401

Cubic family: f(x)=x3+kx
    a. Pitchfork bifurcation at k=1
       • repelling fixed point for k>1
       • three fixed points (two repelling, one attracting) for k<1
    b. Period doubling cascade (decreasing values of k):
        • fixed point for k>-1
        • becomes period 2 orbit at k=-2.23
        • becomes period 4 orbit at k=-2.29
        • becomes period 8 orbit at k=-2.30

Logistic family: f(x)=kx(1-x)
    a. Period doubling cascade (increasing values of k):
        • fixed point for k<3
        • becomes period 2 orbit at k=3
        • becomes period 4 orbit at k=1+sqrt(6) (approx. 3.45)
        • becomes period 8 orbit at k=3.54
        • becomes period 16 orbit at k=3.56
        • becomes period 32 orbit at k=3.57
    b. Period doubling cascade (increasing values of k):
        • period 3 orbit created at k=3.8284
        • becomes period 6 orbit at k=3.841
        • becomes period 12 orbit at k=3.8477
        • becomes period 24 orbit at k=3.8495
    c. Period doubling cascade (increasing values of k):
        • period 5 orbit created at k=3.7388
        • becomes period 10 orbit at k=3.7413
        • becomes period 20 orbit at k=3.7428

Tent family:
        f(x)=2kx when x<1/2
        f(x)=k(2-2x) otherwise

Baker's family: f(x)=2k(x-floor(2x))

Created by Dave Richeson using GeoGebra.