a bifurcation diagram

Instead of asking 'what does the system do for this one parameter value?', step back and ask the whole-story question: 'how do the steady states and cycles change as I sweep the parameter across its entire range?' A bifurcation diagram is the single picture that answers it — a map of every long-term state of the system plotted against the parameter, so you can see at a glance where states appear, disappear, swap stability, or give birth to oscillations.

You build it by putting the parameter (say r) on the horizontal axis and a representative coordinate of the long-term state (a fixed-point value, or the amplitude of a cycle) on the vertical axis. For each r you mark every equilibrium and cycle, drawing stable ones as solid curves and unstable ones as dashed curves by convention. The result is a set of branches: a saddle-node shows up as a fold where a solid and a dashed branch meet and turn back; a transcritical shows up as two lines crossing and swapping solid/dashed; a pitchfork shows up as one branch splitting into three; a Hopf shows up as a branch from which a cycle's amplitude opens up like a parabola. The bifurcation points are exactly where branches are created, destroyed, or change stability.

The diagram is the master summary of a parametrized system's qualitative behaviour: it compresses an infinite family of phase portraits into one readable chart and shows the route from simple to complex. For the logistic map the bifurcation diagram is famous in its own right — a single fixed point splits to a 2-cycle, then a 4-cycle, then 8, in the period-doubling cascade, finally dissolving into the speckled bands of chaos, the whole road to chaos visible in one image. Reading a bifurcation diagram is reading the system's life story as a function of its control knob.