bifurcation diagram (mathematical)

A bifurcation diagram is a visual tool that shows how a system's behavior changes as a parameter varies. It plots the possible long-term states (like steady points or cycles) against different values of the parameter. When a parameter reaches certain critical values, the diagram reveals sudden shifts—called bifurcations—where the system's behavior fundamentally changes, such as shifting from stable to chaotic. Essentially, it helps visualize the points at which small changes in conditions cause significant qualitative differences in system dynamics, illustrating how complex patterns emerge from simple rules.