Hopf bifurcation

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. 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. 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: Write: b = α + i β . {displaystyle b=alpha +ieta .,} The number α is called the first Lyapunov coefficient. Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment), the Hodgkin–Huxley model for nerve membrane, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor, and the Brusselator.