 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.
 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(tt)bN(tt)^{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.
 Hysteresis and Jumps
a) Study the following model numerically,


e^{0.1u} cos4u 0.01u^{3}+1.2*u + A(t) 
 (2)  

 (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 timedependence of the solution change qualitatively when you
increase the amplitude A in steps from A_{0} = 1 to A_{0} = 12?
Make sure that your solution is numerically resolved. Plot in the same
graph A(t) and use it to identify ranges of A_{0} 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 A_{0} and A_{0} is scanned over
the range covered by A(t) in your simulation.
 Perturbed PitchFork Bifurcation
In class we considered the equation
as the general form of perturbing the pitchfork bifurcation.
Often, instead the equation
¶_{t} y = ny  y^{3} + a+ by^{2} 
 (5) 

is considered. It makes the fact more apparent that a special feature
of the pitchfork bifurcation is the missing of all even terms in
x.
 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?
 Use (5) to show that among the perturbations of the
pitchfork 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 pitchfork bifurcation
An example of an experimentally investigated perturbed pitchfork bifurcation
can be found in a paper by Aitta, Ahlers, and Cannell, Phys. Rev. Lett. 54 (1985) 673.
 Nongeneric Scaling for Period
Strogatz: 4.3.10.
 Superconducting Josephson Junction
Read Ch.4.6 in Strogatz and do the problems: 4.6.1, 4.6.2, 4.6.3.