A bifurcation occurs when a small change in a control parameter changes the qualitative structure of a dynamical system: equilibria can appear, disappear or change stability.

Consider the one-dimensional system $$\dot x=\mu-x^2.$$ Equilibria satisfy $$x^2=\mu.$$ For $\mu<0$, there are no real equilibria. For $\mu>0$, two appear: $$x_\pm=\pm\sqrt\mu.$$ The derivative of the vector field is $f'(x)=-2x$. Thus $x_+=+\sqrt\mu$ is stable and $x_-=-\sqrt\mu$ unstable. At $\mu=0$, the pair is created or destroyed in a saddle-node bifurcation.

Other common local bifurcations include transcritical and pitchfork bifurcations, where equilibria exchange stability or symmetry produces new branches, and Hopf bifurcations, where an equilibrium can give rise to a periodic orbit.

A bifurcation is not simply a large numerical response. The topology of long-term behavior changes. Plotting equilibrium branches against the control parameter produces a bifurcation diagram that shows where these qualitative changes occur.

Bifurcation reasoning is useful whenever nonlinear feedback produces multiple operating states or oscillations: mechanical buckling, fluid instabilities, electronic oscillators, population dynamics and control systems all exhibit the same mathematical structures.