Math 130 Final Quiz 2
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Math 130 Jekel
Final Quiz 2
March 22, 2023
Question 1 (5 points). Let f(a,y) = y(y − a)[1 − a(1 + y2 )].
(a) Graph the region f(a,y) ≥ 0 in the ay-plane.
(b) Make a bifurcation diagram for y˙ = f(a,y), using dashed lines for the unstable equilibria and solid lines
for the stable equilibria.
(c) Identify the bifurcations visually, label them, and label what type of bifurcation they are.
Question 2 (3 points). Let f(a,y) = log(1 + y2 ) − ay . The equation y˙ = f(a,y) has a bifurcation at (a* ,y* ) = (0, 0). Use the Taylor expansion technique to determine what type of bifurcation it is. You may use the expansion log(1 + x) = x − 
+ 
− ...
Question 3 (2 points). Consider an equation y˙ = f(a,y). Suppose that
❼ At a = 0, the equation has a stable equilibrium at y = 0 and no other equilibria.
❼ At a = 1, the equation has stable equilibria at y = −1 and +1 and an unstable equilibrium at y = 0
and no other equilibria.
What could the bifurcation diagram of y˙ = f(a,y) look like for a between 0 and 1? There are many different possibilities that could occur for different f . Please draw two possible bifurcation diagrams that satisfy the conditions above and are qualitatively different.
2023-03-25