Hopf bifurcation

Remarks

Under these conditions, a family of periodic orbits branches off the equilibrium at $\mu = 0$, with periods close to $2\pi / \beta(0)$. If moreover a certain coefficient $\ell_1$, the first Lyapunov coefficient, is nonzero, the orbits exist for $\mu$ on one side of $0$ only and have amplitude of order $\sqrt{\abs{\mu}}$. The bifurcation is supercritical when $\ell_1 < 0$; in the plane the orbits are then stable limit cycles. It is subcritical when $\ell_1 > 0$; the orbits are then unstable.

Example: $x' = \mu x - y - x(x^2 + y^2)$, $y' = x + \mu y - y(x^2 + y^2)$. The Jacobian at $0$ is $\begin{pmatrix} \mu & -1 \\ 1 & \mu \end{pmatrix}$, with eigenvalues $\mu \pm i$. So $\alpha(\mu) = \mu$, $\beta(\mu) = 1$, and all three clauses hold. In polar coordinates the system is $r' = \mu r - r^3$, $\theta' = 1$, and the circle $r = \sqrt{\mu}$ is a stable limit cycle for $\mu > 0$.

Non-example: $x' = \mu^2 x - y$, $y' = x + \mu^2 y$. The eigenvalues are $\mu^2 \pm i$, so $\alpha(\mu) = \mu^2$ and $\alpha'(0) = 0$; clause 3 fails.