Visual summary of the succession of period-doubling produced as a control parameter is changed.

This is a depiction of the solution to a dynamical system as one or more parameters vary. Typically, the horizontal axis has the parameter and the vertical axis has some aspect of the solution, such as, the norm of the solution, the maximum and/or minimum values of one of the state variables, the frequency of a solution, or the average of one of the state variables.

a visual summary of the succession of period-doubling produced as r increases

a diagram produced (by iteration) in which bifurcation is a pronounced characteristic

A graph that shows the critical points where bifurcation occurs, and the possible solutions that exist at that point.

In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria or fixed points) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable fixed points with a solid line and unstable fixed points with a dotted line.