bifurcation diagrams

A bifurcation diagram is a visual tool used in mathematics and physics to show how a system's behavior changes as a parameter varies. It typically plots possible long-term states (like stable outcomes) of a system against different values of a variable. As the parameter changes, the diagram reveals points where the system's behavior transforms suddenly—these are bifurcations—such as shifting from steady to oscillatory patterns. This helps us understand how complex systems can evolve and respond to changes, highlighting critical thresholds where small tweaks lead to different outcomes.