# Interdisciplinary Nonlinear Dynamics (438)

 Fall 2002
 Hermann Riecke

 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

 dNdt = -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,

 dudt
 =
 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 tmax = 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.

File translated from TEX by TTH, version 2.78.
On 17 Oct 2002, 15:23.