Math 21b, Spring 2017. Worksheet 26.
Problem 1.
1. Find a first order nonlinear system in R2such that the x- and y-nullclines are y=nand x=n,
respectively, for all integers n.
2. Given a 3×3matrix Aand a vector v0∈R3, find a first order system of differential equations
where v0is a stationary point and Ais J(v0).
Solution. (1) One example is dx
dt = sin(2πy),dy
dt = sin(2πx).
(2) The system x0(t) = A(x(t)−v0) is one possibility.
Problem 2. Use the phase space approach to analyze the second order differential equation
d2x
dt2= (x−1)dx
dt .
Solution. We expand the equation to phase space as:
x0=y, y0= (x−1)y.
The nullclines are y= 0 and (x−1)y= 0. Every point on the line y= 0 is stationary and this corresponds
to the fact that x≡afor a∈Rsolves our differential equation. We compute
J=0 1
y x −1
So
J(a, 0) = 0 1
0a−1
which has eigenvales 0,(a−1). So the stationary points to the left of x= 1 are stable while those to the
right are unstable.