Bifurcation diagram

The behavior of a system can vary greatly depending on the model parameters. A value of a model parameter at which the fixed points of the system suddenly change is called a bifurcation point.

A bifurcation diagram shows how an ODEs fixed points depend on the parameters to the system.

Consider for example the system $\dot x = f(x;h)$ where $f(x;h) = 3x-x^2-h$. This ODE might be used to model, for example, the population of fish in a farm, where $x$ is the number of fish and $h$ is the harvesting rate. The fixed points at which $f(x_*;h)=0$ are

The bifurcation diagram below shows the fixed points as a function of $h$.

The bifurcation point is at $h=\frac94$, where there is only one fixed point.