Interdisciplinary Nonlinear Dynamics (438)

Problem Set 2

For Discussion Section October 23

1. Vector Fields and Bifurcations

   Do 3.1.4, 3.2.2, 3.4.3, 3.4.11, 3.4.12, 3.4.15, 3.4.16.
   

2. Population Growth with Eggs II
   

   Consider again the dynamics of a population of animals that lay eggs and investigate numerically

   
   

   dN dt = -N(t)-N(t)2+aN(t-t)-bN(t-t)2. (1)

   Explore what happens to the oscillating solution you found in the last homework if you keep increasing the delay t from t = 3 in not too large steps to at least t = 15, with a = 17 and b = 3.7 fixed.

3. Hysteresis and Jumps

   a) Study the following model numerically,

   
   

   du dt = e0.1u cos4u -0.01u3+1.2*u + A(t) (2) A(t) = A0sin(wt). (3)

   Use as initial condition u(t = 0) = 0 and run the solution to t_max = 400. Choose w = 0.04 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 A₀ = 1 to A₀ = 12? Make sure that your solution is numerically resolved. Plot in the same graph A(t) and use it to identify ranges of A₀ for which the solution exhibits hysteresis.
   b) Interpret the result in terms of the bifurcations that the solutions to (2) undergo when A(t) is replaced by a constant A₀ and A₀ is scanned over the range covered by A(t) in your simulation.

4. 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

   An example of an experimentally investigated perturbed pitch-fork bifurcation can be found in a paper by Aitta, Ahlers, and Cannell, Phys. Rev. Lett. 54 (1985) 673.

5. Non-generic Scaling for Period

   Strogatz: 4.3.10.

6. Superconducting Josephson Junction

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