  A computational and asymptotic analysis of the solutions of Carrier's problem
is presented. The computations reveal a striking and beautiful bifurcation
diagram, with an infinite sequence of alternating pitchfork and fold
bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak
is then applied to construct asymptotic solutions to the problem. This
asymptotic approach explains the bifurcation structure identified numerically,
and its predictions of the bifurcation points are in excellent agreement with
the numerical results. The analysis yields a novel and complete taxonomy of the
solutions to the problem, and demonstrates that a claim of Bender & Orszag is
incorrect.
