Hopf bifurcation

In the study of dynamical systems, a Hopf bifurcation (also known as a Poincaré–Andronov–Hopf bifurcation) is a transition where a system shifts from a steady equilibrium to a periodic, self-sustaining oscillation. This transition occurs when a continuous change in a control parameter—such as temperature, electrical voltage, or fluid flow rate—reaches a specific threshold. Below this threshold, small disturbances to the system decay over time, and the system returns to its stable resting state. Once the parameter passes the threshold, the steady state becomes unstable, and the system begins to oscillate rhythmically. The system’s state then traces a repeating, closed loop in its