  We prove that a class of superlinear indefinite problems with homogeneous
Neumann boundary conditions admits an arbitrarily high number of positive
solutions, provided that the parameters of the problem are adequately chosen.
The sign-changing weight in front of the nonlinearity is taken to be piecewise
constant, which allows to perform a sharp phase-plane analysis, firstly to
study the sets of points reached at the end of the regions where the weight is
negative, and then to connect such sets through the flow in the positive part.
Moreover, we study how the number of solutions depends on the amplitude of the
region in which the weight is positive, using the latter as the main
bifurcation parameter and constructing the corresponding global bifurcation
diagrams.
