ASSIGNMENT
(1) Show that
x· = x3
blows up (escapes to +∞ in a finite time) starting from x(0) = x0 > 0 by computing the solution explicitly.
(2) For the following:
(a) x· = 1 + r/2x + x2,
(b) x· = r - cosh(x),
(c) x· = 3r + x - ln(x + 1),
(d) x· = ½x - x/x+1 - r + 2,
Find a transformation putting each of the f(r, x) in normal form for a saddle-node bifurcation in a vicinity of the bifurcation point.
(3) For x· = x(r - ex/2),
(a) Sketch each qualitatively different vector field as r is varied.
(b) Find the bifurcation point, and show that it is a trancritical bifurcation.
(c) Sketch the bifurcation diagram in the x∗, r plane (the curves f(r, x∗) = 0 containing critical points).
(d) Bring the equation into normal form in a vicinity of the bifurcation point.
(4) Consider the model,
x· = (1 - x) - x/a + x.
(a) Show that the number of critical points depends on the values of a. Classify the stability of the critical points.
(b) Determine the type of bifurcation that occurs.
(c) Reduce the equation to a normal form (Hint: use Taylor expansion with respect to the model parameter AND x).
Questions 2a and 2c.