Consider the following system depending on two parameters:
\( \dot{x} = x^4 + ax + b. \)
(1) For each value of a, find the value of b when a bifurcation occurs (we regard a as fixed and
vary b). Sketch the graph showing the dependence of that value b on a in the (a, b) plane.
(2) Sketch the phase portrait of the system for the values a, b in each of the regions separated
by the above graph.
(3) Sketch all qualitatively different bifurcation diagrams in the (a, x) plane that occur as we
vary b. Indicate all bifurcations and stability of branches of the diagram.
(4) Sketch all qualitatively different bifurcation diagrams in the (b, x) plane that occur as we
vary a. Indicate all bifurcations and stability of branches of the diagram.