MATH 134 UCLA Differential Equations Questions
1. Consider the system
x˙ = (x − h)(r − x).
(a) Draw a stability diagram in the (r, h)-plane. Be sure to indicate the types of
bifurcation curves/points.
(b)Draw phase portraits for each qualitatively different choice of (r, h).
2. For each parameter r, consider the following flow on the circle:
˙θ = cos2
(θ) + r
(a) Find all bifurcation points (r
∗
, θ∗
).
(b) Draw all qualitatively different phase portraits. What type of bifurcation occurs
at the bifurcation points?
(c) Find the period of oscillation when r is small and positive.
