ESAM 438

Problem Set 3

For Discussion Section October 18

1. Perturbed Transcritical Bifurcation
   

   To unfold the transcritical bifurcation consider adding a perturbation h to obtain

   
   

   ¶t x = mx - x2 + h. (1)

   Determine all possible bifurcation diagrams that are obtained as m or h are varied, respectively. Try to sketch the solution surface x = x(m,h). By projecting onto the (m,h)-plane, determine the complete phase diagram, which shows how many solutions are obtained for any combination of the two parameters x and h.

2. Perturbed Pitch-Fork Bifurcation
   

   In class we considered the equation

   
   

   ¶t x = mx - x3 + h (2)

   as the general form of perturbing the pitch-fork bifurcation. Often, instead the equation

   
   

   ¶t y = ny - y3 + a+ by2 (3)

   is considered. It makes the fact more apparent that a special feature of the pitch-fork bifurcation is the missing of all even terms in x.

   1. Introduce a suitable variable transformation form relating y and x as well as n and m that brings (3) into the form (2). What is the connection between the coefficients n, a, b, and m and h?
   2. Use (3) to show that among the perturbations of the pitch-fork bifurcation are also diagrams of the form shown below.

      Can such a bifurcation diagram also be obtained in (2)? If so, how do the parameters m and h have to be varied to get it?

   Figure 1: One possible perturbation of the pitch-fork bifurcation

3. Non-generic Scaling for Period

   Strogatz: 4.3.10.

4. Superconducting Josephson Junction

   Read Ch.4.6 in Strogatz and do the problems: 4.6.1, 4.6.2, 4.6.3.