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Cusp Catastrophe plot using ParametricPlot3D

Posted 4 months ago
4 Replies
9 Total Likes

Hello, I've been trying to replicate a cusp catastrophe 3D plot I found in Strogatz's book, nonlinear dyamics and chaos (p. 79).

For reference, this is a nonlinear 1D ODE: dx/dt = r x (1 - x/k) - x^2/(1 + x^2)

I verified that the model undergoes saddle-node bifurcations by playing around with the value of the paremeters r and k.

 Plot[r x - (r x^2)/k - x^2/(1 + x^2), {x, 1, 10}], {k, 1, 
  10.}, {r, .01, 2}]

And I managed to replicate the bifurcation diagram by expressing r and k as parametric functions of x. This is achieved by differencing the ODE w.r.t. to x, substituting the values back to the ODE and solving for the parameters, to get 2 parametric equations (r and k) as functions of x.

The book does the same, so I'm surre there's no mistake in this:

r = (2 x^3)/(1 + x^2)^2 k=(2 x^3)/(x^2 - 1)

ParametricPlot[{ (2 x^3)/(x^2 - 1), (2 x^3)/(1 + x^2)^2}, {x, 1.01, 
  40}, AspectRatio -> 1.01/1, AxesLabel -> {"k", "r"}, 
 AxesOrigin -> {0, 0}]

So far so good, I plotted r-k for different values of x and I think I plotted the bifurcation diagram correctly.

When I added x to the above plot as another dimension, I was expecting to get a 3D stability diagram in the x,r,k space (fig. 3.7.6 in book). However, what I get is the same 2D plot as above, only in a 3D space.

ParametricPlot3D[{(2 x^3)/(1 + x^2)^2, (2 x^3)/(x^2 - 1), x}, {x, 1, 
  10}, PlotRange -> {0, 12}]

So, with the parametricPlot3D what I think I get is a projection of the 3D space on r-k but the actual values of x are missing.

Then, following this demonstration, I tried to plot it using the ContourPlot3D, however what I get makes no sense to me.

G[r_, k_, x_] := r  x (1 - x/k) - x^2/(1 + x^2)
ContourPlot3D[Evaluate[D[G[r, k, x], x]],
    {r, 0.1, 1}, {k, 1, 70}, {x, 1.01, 40}, 
    Axes -> True, ContourStyle -> {EdgeForm[]}, 
 AxesLabel -> TraditionalForm /@ {r, k, x}]

I get three flat surfaces on the r-k-x space that I don't even know what they represent. It is not quite clear to me why I get these flat curves instead of something resembling at least the 3D counterpart of the 2D bifurcation diagram.

4 Replies

Another way, highlighting the bifurcation:

 Evaluate@{r /. First@Solve[r x (1 - x/k) - x^2/(1 + x^2) == 0, r], k, x},
 {k, 0, 10}, {x, 0 + 2 $MachineEpsilon, k - 20 $MachineEpsilon}, 
 PlotPoints -> 75, MaxRecursion -> 3, 
 Method -> {"BoundaryOffset" -> False}, BoxRatios -> {1, 1, 1}, 
 PlotRange -> {{0, 1}, Automatic, Automatic},
 MeshFunctions -> {Function[{r, k, x, u, v}, 
    Last@Normalize@D[r x (1 - x/k) - x^2/(1 + x^2), {{r, k, x}}] // 
     Evaluate]}, Mesh -> {{0}}, PlotStyle -> Opacity[0.7], 
 MeshStyle -> Thick, AxesLabel -> {r, k, x}, ViewPoint -> Above]

enter image description here

Posted 4 months ago

Hello, I believe I was working the same problem from Storgatz's class on the model of an insect outbreak. I plotted the cusp as follows:

xprime[x_, r_, k_] := r x (1 - x/k) - x^2/(1 + x^2);

ContourPlot3D[xprime[x, r, k] == 0, {r, 0, 1}, {k, 0, 10}, {x, 0, 10}]

This resulted in the contour plot which matched the visualization from the book (may have to rotate it bit).

Cusp contour plot


The surface in either demo or Strogatz’s book is the contour for r-h-x space in terms of the fixed point equation, So you only need G[r,k,x] at zero.


Posted 4 months ago

Thank you both for your answers!

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