Math 130 Final Quiz 3
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Math 130 Jekel
Final Quiz 3
March 22, 2023
Question 1 (4 points). Below is a bifurcation diagram for a function f(a,θ). The horizontal axis represents the parameter a and the vertical axis shows θ from 0 to 2π . The curves are where f(a,θ) = 0, and the shaded region is where ∂f/∂θ > 0.
(a) Mark the bifurcations in the picture above and identify what type of bifurcation they are.
(b) For a = 1, draw a phase diagram on the circle for θ˙ = f(a,θ). (Following the usual convention, θ should start on the right side and go counterclockwise.)
Question 2 (2 points). Consider the equation θ˙ = 1 + 2sin(θ − T/6). Let θ(t) be a solution with θ(0) = 0.
❼ Does θ(t) move counterclockwise or clockwise?
❼ Does θ(t) approach a limit as t → ∞ , or is it periodic?
❼ If it approaches a limit, what is the limit?
❼ If it is periodic, write down an integral that would compute the period (but don’t evaluate the integral).
Question 3 (2 points). Consider the equation θ˙ = sin(θ) − 2cos(3θ)−4. Let θ(t) be a solution with θ(0) = 0. Answer the same bullet points as in Question 2.
Question 4 (2 points). Consider the equation y˙ = f(a,y) where f(a,y) = ey − ay . There is only one value of (a,y) where a bifurcation could occur. Find it.
2023-03-25