Hopf bifurcation

In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, 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 as a parameter crosses a threshold value. Under reasonably generic assumptions about the dynamical system, the fixed point becomes a small-amplitude limit cycle as the parameter changes. 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 the following time-dependent differential equation: where z, b are both complex and λ is a real 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. 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. The polar coordinates can be transformed into Cartesian coordinates by writing and .