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Math 134 Exam 1
Summer 2023
1. Draw the flow diagrams. Indicate the stable, unstable, and half-stable critical points.
(a) (8 points)
(b) (8 points)
Final answer must be drawn on the circle. But I’d suggest graphing y = 2 sin(2θ) and y = 1 first on the interval [0, 2π] to see what the correct critical points are.
2. Draw the bifurcation diagrams, and give the names of each.
(a) (12 points) 
(b) (12 points) 
3. A nonlinear RLC circuit has the current equation 
If we divide
by I0 and multiply by C, we can rewrite the ODE in terms of the unitless quantity 
(a) (5 points) Set 
and rewrite the ODE in terms of derivatives w.r.t. τ .
(b) (5 points) Solve for T such that the equation is of the form 
(c) (5 points) What are ε and γ in terms of R, L,C, I0 , k?
(d) (5 points) Explain when the above second order equation can be approximated by a first order equation, in terms of physical parameters R, L,C, I0 , k.
4. Suppose that γ = 2/1 in the previous equation so that 
(a) (4 points) Set v = dτ/du, and find the phase plane system, by finding equations for both dτ/du and dτ/dv in terms of u and v.
(b) (4 points) Graph the solution to the first-order equation 
in the phase plane. (The uv-plane).
(c) (4 points) Sketch the solution to the second order ODE 
with initial conditions u(0) = −1, u′ (0) = 0 in the same phase-plane graph above.
(d) (4 points) Indicate the regions Tfast and Tslow in your phase-plane graph above. Give an explicit expression for Tslow in terms of what you solved for in the previous question.
(e) (4 points) Explain (in terms of your system from part a)), why the solution to the second order ODE in part c) must always move quickly in the Tfast region.
5. (a) (5 points) Show that dt/dx = r + x2 has a saddle-node bifurcation at r = 0. (Hint: write as r − (−x2).)
(b) (5 points) For r > 0, set up and evaluate an integral to find the time T (r) that it takes for a solution x(t) to travel from x = −1 to x = 1, when dt/dx = r + x2 .
(c) (5 points) Find 
T (r).
(d) (5 points) Explain why part c) is consistent with the flow diagram for r = 0.