We outline the proof of the Hopf bifurcation theorem, emphasizing the key steps and techniques involved.

Setup: Consider $\dot{\mathbf{x}} = \mathbf{F}_\mu(\mathbf{x})$ with $\mathbf{F}_0(0) = 0$ and linearization having eigenvalues $\lambda = \pm i\omega_0$ at $\mu = 0$. We assume the transversality condition $\alpha'(0) > 0$ (eigenvalues cross to the right half-plane as $\mu$ increases).

Step 1: Complexification and change of variables

Convert to complex notation. Let $z = x + iy$ represent the plane, so the system becomes $\dot{z} = \lambda z + g(z, \bar{z}, \mu)$ where $g$ contains nonlinear terms. The goal is to transform this into polar coordinates and analyze the radial dynamics.

Step 2: Center manifold reduction

For $\mu$ near 0, the center manifold is two-dimensional (the plane itself in this case). The key is to eliminate non-resonant terms through near-identity changes of variables, leaving only the essential dynamics.

Step 3: Normal form computation

Using the method of normal forms, perform successive near-identity transformations to eliminate as many nonlinear terms as possible. The transformed system, in polar coordinates $(r, \theta)$, takes the form:

$\dot{r} = \alpha(\mu) r + c(\mu) r^3 + O(r^5)$ $\dot{\theta} = \omega(\mu) + O(r^2)$

where $\alpha(0) = 0$, $\alpha'(0) > 0$, and $c(0) = \ell_1$ is the first Lyapunov coefficient.

Step 4: Analysis of the radial equation

For small $|\mu|$, the radial equation $\dot{r} = \alpha(\mu) r + \ell_1 r^3 + O(r^5)$ determines stability:

• If $\mu < 0$ (so $\alpha(\mu) < 0$): The origin $r = 0$ is stable; small perturbations decay exponentially.

• If $\mu > 0$ (so $\alpha(\mu) > 0$): The origin becomes unstable. If $\ell_1 < 0$ (supercritical), there exists a stable periodic orbit at:

$r^*(\mu) = \sqrt{\frac{-\alpha(\mu)}{\ell_1}} = \sqrt{\frac{-\alpha'(0)\mu}{\ell_1}} + O(\mu)$

The orbit is stable because $\frac{d}{dr}\left(\alpha r + \ell_1 r^3\right)\bigg|_{r=r^*} = \alpha + 3\ell_1(r^*)^2 = \alpha - 2\alpha = -\alpha < 0$ for $\mu > 0$.

Step 5: Existence and smoothness

The implicit function theorem guarantees that the periodic orbit $r^*(\mu)$ extends to a smooth family for small $\mu > 0$. The period is approximately:

$T(\mu) = \frac{2\pi}{\omega(\mu)} \approx \frac{2\pi}{\omega_0}$

which varies smoothly with $\mu$.

Step 6: Return to original coordinates

The periodic orbit in polar coordinates corresponds to a limit cycle in the original $(x, y)$ coordinates. The inverse transformations (from normal form back to original system) preserve the existence and stability of the limit cycle, completing the proof.

Conclusion: Under the non-degeneracy and transversality conditions, a Hopf bifurcation creates a branch of periodic orbits bifurcating from the fixed point. The stability of the limit cycle is determined by the sign of the first Lyapunov coefficient $\ell_1$.