- 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
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.
= -N(t)-N(t)2+aN(t-t)-bN(t-t)2. ||(1)|
- Hysteresis and Jumps
a) Study the following model numerically,
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.
e0.1u cos4u -0.01u3+1.2*u + A(t) ||(2)|
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.
- Perturbed Pitch-Fork Bifurcation
In class we considered the equation
as the general form of perturbing the pitch-fork bifurcation.
Often, instead the equation
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
- 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
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.
- Non-generic Scaling for Period
- Superconducting Josephson Junction
Read Ch.4.6 in Strogatz and do the problems: 4.6.1, 4.6.2, 4.6.3.