In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point. A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf. Overview Supercritical and subcritical Hopf bifurcations The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical. The normal form of a Hopf bifurcation is: where z, b are both complex and λ is a parameter. Write: The number α is called the first Lyapunov coefficient. If α is negative then there is a stable limit cycle for λ > 0: where The bifurcation is then called supercritical. If α is positive then there is an unstable limit cycle for λ < 0. The bifurcation is called subcritical. Intuition The normal form of the supercritical Hopf bifurcation can be expressed intuitively in polar coordinates, where is the instantaneous amplitude of the oscillation and is its instantaneous angular position. The angular velocity is fixed. When , the differential equation for has an unstable fixed point at and a stable fixed point at . The system thus describes a stable circular limit cycle with radius and angular velocity . When then is the only fixed point and it is stable. In that case, the system describes a spiral that converges to the origin. Cartesian coordinates The polar coordinates can be transformed into Cartesian coordinates by writing and . Differentiating and with respect to time yields the differential equations, and Subcritical case The normal form of the subcritical Hopf is obtained by negating the sign of , which reverses the stability of the fixed points in . For the limit cycle is now unstable and the origin is stable. Example Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment), the Hodgkin–Huxley model for nerve membrane potential, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator, Classical electromagnetism. Hopf