Math 5110/6830
Homework 4.2
1. Find fixed points, their stability, and if a parameter is involved draw a
bifurcation diagram.
a)
pt+1 = p2t − qt
qt+1 = pt qt − 100pt
b)
xn+1 = rxn − x2n + xn
with r > 0
2. (dV 2.4.16) Consider the Romeo and Juliet model that we discussed in
class in a “feeling preserving” mode when aR + pJ = 1 and aJ + pR = 1.
a) Why is this mode called the “feeling preserving”
b)Write down the system of equations for the fixed points (in matrix form,
as we did in class). Show that in the “feeling preserving” mode the determinant
of this system is equal to zero.
c) Show that the two eigenvalues of the Jacobian matrix are
λ1 = 1,
λ2 = aR + aJ − 1.
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