Thread: Some simple DiffEQ problems.

The differential equation dx/dt = (1/10)*x*(10-x)-h models a logistic population with with harvesting at a rate h. Determine the dependence of the critical points on the paramater h, and then construct a bifurcation diagram.


Consider the differential equation dx/dt = kx-x^3. (a) If k<=0, show that the only critical value c=0 of x is stable. (b) If k>0, show that the critical point c=0 is now unstable, but that the critical points c= +/- sqrt(k) are stable. Thus the qualitative nature of the solutions changes at k=0 as the parameter k increases, and so k=0 is a bifurcation point for the differential equation with parameter k. The plot of all points of the form (k,c) where c is a critical point of the equation is the "pitchfork diagram" (here the book shows a c vs. k plot of a parabola that opens to the right)



Any help?

Let's write the first DE as...

(1)

... and suppose to know the 'initial value' . It is important to establish the sign of second member of (1) and to do that we have to factorize it as follows...

(2)

... where...



(3)

No we have three possibilities...

1)

In this case is everywhere negative and is...



2)

In this case is everywhere negative with the only exception of where is null. The situation is the same as 1) unless you have and in this case you have the constant solution

3)

In this case is negative for or , is null for or and is positive for . We have three possibilities...

3a)

3b)

3b)

Kind regards

Originally Posted by qtpipi

and then construct a bifurcation diagram.
Any help?

Hey, you into programming? Mathematica is a big help with this stuff. No I don't work for them; just know a good thing when I see it. You know what a bifurcation diagram is right? It illustrates the dependence of the critical points on the parameter. So just calculate the zeros of the right side as a function of h and then just plot the results. If you're interested, try interpreting the code below. It first finds the roots, selects only the real ones, then arranges them in a suitable order to connect the points and draw a smooth plot. Now, modify my code to construct the pitch-fork bifurcation for the second one.

Also, in the news yesterday was an article about tipping points in global warming:

Is There a Climate-Change Tipping Point? - TIME

Here's something fun: Explain what that has to do with this thread.

Code:

Clear[h]
myList = 
  Table[myRoots = x /. Solve[1/10 x (10 - x) - h == 0, x];
   {h, #} & /@ myRoots,
   {h, -5, 5, 0.1}];
myRealRoots = 
  Flatten[Select[myList, Element[#[[2]], Reals] &], 1];

curve = FindCurvePath[myRealRoots]
ListLinePlot[myRealRoots[[curve[[1]]]]]