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.

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.