Interdisciplinary Nonlinear Dynamics (438)

Problem Set 3

For Discussion Section October 19

1. Hysteresis and Jumps

   a) Study the following model numerically,

   
   

   du dt = e0.3u sin2u + e-0.1u + A(t) (1) A(t) = sin(wt). (2)

   Use as initial condition u(t = 0) = 0 and run the solution to t_max = 200. Choose w = 0.05 to mimic a slow change of the "forcing" A. How does the time-dependence of the solution change qualitatively when you increase the amplitude A in steps from A = 1 to A = 23? Make sure that your solution is numerically resolved. Plot in the same graph A(t) and use it to identify ranges of A for which the solution exhibits hysteresis.
   b) Interpret the result in terms of the bifurcations that the solutions to (1) undergo when A(t) is replaced by a constant A₀ and A₀ is scanned over the range covered by A(t) in your simulation.

2. Perturbed Transcritical Bifurcation
   

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

   
   

   ¶t x = mx - x2 + h. (3)

   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 m and h.

3. Perturbed Pitch-Fork Bifurcation
   

   In class we considered the equation

   
   

   ¶t x = mx - x3 + h (4)

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

   
   

   ¶t y = ny - y3 + a+ by2 (5)

   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 (5) into the form (4). What is the connection between the coefficients n, a, b, and m and h?
   2. Use (5) 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 (4)? 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

4. Non-generic Scaling for Period

   Strogatz: 4.3.10.

5. Superconducting Josephson Junction

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