2.8.3, 2.8.4, 2.8.5
: Sample implementations of the Euler method for Maple, and some comments about
doing to in Matlab, are given in the resources for lecture 6; these are easily
modified to implement the improved Euler and Runge-Kutta methods.
3.1.3 : Also use appropriate software, such as Matlab or Maple,
to plot some representative numerical solutions of x(t) for three values of r
below, at and above the bifurcation value.
3.2.4 : As in 3.1.3, plot some representative numerical solutions
below, at and above the bifurcation.
10.2.4 : Notice that
the bifurcation sequence occurs for negative values of r.
10.2.6 : Try to find
stable fixed points and period-doubling bifurcation sequences for various ranges
of r; you could hand in one orbit diagram for a restricted r-range, showing a
single bifurcation sequence, and another for a larger range of values of r
showing several sequences.
10.3.9 : You should
look through the different parts of 10.3.7 and 10.3.8 for instructions and
hints. This is quite a long problem; we can discuss it in office hours on
Wednesday, February 26 (AQ 5016).
10.5.3 : You can just
draw a few cobweb diagrams by hand, and discuss the stability of the fixed point(s); this should be fairly quick.
10.5.4 : Recall that a
periodic window in the orbit diagram corresponds to a stable periodic orbit,
which we showed would have negative Liapunov exponent; but we already know the
Liapunov exponent for the tent map, for any initial conditions...
6.8.2, 6.8.3, 6.8.4 : These questions should be fairly quick; you
just need to find the fixed points and nullclines, and identify the general
direction of the vector field in the different regions of the phase plane.
It is sufficient if you give a rough sketch of the nullclines and direction
field and give the answer for the index.
6.4.7 : There are many different possible cases here. To help you organize them: you can assume that k1/G1 < k2/G2, and consider the different cases as N0 varies. You do not need to consider degenerate cases (where some parameters or groups are equal).
Back to Course Web Page