Hopf Bifurcation

We give a complete proof for the planar case, using polar coordinates, an amplitude-equation reduction, and the Poincaré–Bendixson theorem as the topological tool that promotes an annulus of trapped trajectories into an actual periodic orbit. The proof is organized in five steps.

Step 1: Reduction to polar form. Fix $\mu$ near $\mu_c$ and write $u = r\cos\theta$, $v = r\sin\theta$. A direct computation (chain rule, using $\dot u = \dot r\cos\theta - r\dot\theta \sin\theta$, etc., and solving the resulting linear system for $\dot r,\dot\theta$) converts the system into

valid in a neighborhood of the origin uniform for $\mu$ near $\mu_c$. This is a standard computation: the cubic terms $f^{(3)}, g^{(3)}$ contribute to $\dot r$ only through their component in the radial direction, and rotational averaging over $\theta \in [0,2\pi)$ (since $\dot\theta \approx \omega_c \neq 0$, $\theta$ sweeps a full circle in time $\approx 2\pi/\omega_c$, much faster than $r$ changes) shows that only specific combinations of the third-order Taylor coefficients survive; carrying this out yields precisely $a(\mu_c) = \ell_1$, the formula stated in (H3). (This averaging/normal-form computation is classical; we take the formula for $\ell_1$ as given and focus the proof on how it controls dynamics, which is the logical heart of the theorem.) Because $\omega(\mu_c) = \omega_c > 0$ and $\omega$ is continuous, we may shrink the neighborhood of $\mu_c$ so that $\omega(\mu) \ge \omega_c/2 > 0$ throughout; hence $\dot\theta > 0$ always in this range, and $\theta$ is a genuine angular (monotonic) variable. The radial equation

decouples from $\theta$ at leading order and entirely controls the existence and stability of periodic orbits, since $\dot\theta$ never vanishes: any periodic orbit near the origin must be a circle $r = r_0(\mu)$ traversed with monotonically increasing $\theta$, hence corresponds exactly to a nonzero equilibrium $r_0 > 0$ of the truncated radial equation (up to the $O(r^5)$ correction, handled in Step 4).

Step 2: Analysis of the truncated radial equation. Consider first the truncated equation $\dot r = \alpha r + a r^3$ with $a = a(\mu_c) = \ell_1 \ne 0$ by (H3), and treat $\alpha = \alpha(\mu)$ as the bifurcation parameter (justified by (H2): $\alpha'(\mu_c)\ne 0$ means $\alpha$ is itself a valid, monotonic local reparametrization of $\mu$ near $\mu_c$, by the inverse function theorem).

• If $\ell_1 < 0$: for $\alpha \le 0$, $\dot r = \alpha r + \ell_1 r^3 < 0$ for all $r>0$ (both terms strictly negative), so $r=0$ is the only nonnegative equilibrium and it is asymptotically stable. For $\alpha > 0$, $\dot r = r(\alpha + \ell_1 r^2) = 0$ at $r_0 = \sqrt{-\alpha/\ell_1} > 0$; since $\ell_1 < 0$, $d/dr[\alpha r + \ell_1 r^3]$ at $r_0$ equals $\alpha + 3\ell_1 r_0^2 = \alpha - 3\alpha = -2\alpha < 0$, so $r_0$ is an asymptotically stable equilibrium of the radial flow, while $r=0$ is now unstable ($\dot r \approx \alpha r > 0$ for small $r$).

• If $\ell_1 > 0$: symmetric argument with signs reversed — for $\alpha \ge 0$, $r=0$ is the only equilibrium and it is unstable for $\alpha>0$; for $\alpha<0$ an unstable equilibrium $r_0 = \sqrt{-\alpha/\ell_1}$ exists, and $r=0$ is stable there (a locally stable focus with an unstable circle around it).

This proves parts (a) and the amplitude formula in (b) and (c) for the truncated system, since $r_0^2 = -\alpha/\ell_1$, and by (H2), $\alpha = \alpha'(\mu_c)(\mu - \mu_c) + O((\mu-\mu_c)^2)$, giving

matching the claimed formula.

Step 3: From a radial equilibrium to a genuine periodic orbit — Poincaré–Bendixson. So far $r_0$ is an equilibrium only of the truncated radial ODE; we must show the full planar system (with the $O(r^5)$ tail restored) genuinely has a periodic orbit near $r=r_0$, not merely an approximate one. This is where the Poincaré–Bendixson Theorem enters as the essential topological tool. Suppose $\ell_1 < 0$ and $\alpha > 0$ small (the supercritical, cycle-exists-for-$\alpha>0$ case; the subcritical case is handled by reversing time, since reversing $t \mapsto -t$ turns a stable limit cycle of $\dot{\mathbf x}=f$ into an unstable one of $\dot{\mathbf x}=-f$ and swaps the sign of $\ell_1$'s role). We exhibit a trapping annulus $\Omega = \{ (r,\theta) : r_1 \le r \le r_2 \}$ with $0 < r_1 < r_0 < r_2$, both radii $O(\sqrt\alpha)$, such that the vector field points strictly inward on both boundary circles. Choose $r_1 = r_0(1-\epsilon)$, $r_2 = r_0(1+\epsilon)$ for small fixed $\epsilon>0$. On the truncated equation, $\dot r(r_1) = \alpha r_1 + \ell_1 r_1^3 > 0$ and $\dot r(r_2) < 0$ (immediate from Step 2's monotonicity computation, since $r_1 < r_0 < r_2$ straddle the unique positive zero of the cubic $\alpha r + \ell_1 r^3$, which is increasing then decreasing... more precisely $\alpha r+\ell_1 r^3$ is positive on $(0,r_0)$ and negative on $(r_0,\infty)$ when $\ell_1<0,\alpha>0$, verified directly by factoring $r(\alpha+\ell_1 r^2)$). Both $|\dot r(r_1)|$ and $|\dot r(r_2)|$ scale like $\alpha \cdot r_0 \cdot \epsilon \sim \alpha^{3/2}\epsilon$ (using $r_0 = \sqrt{-\alpha/\ell_1} = \Theta(\sqrt\alpha)$), a positive quantity of definite sign bounded away from zero relative to $\epsilon$, while the neglected $O(r^5)$ terms contribute at most $C r_0^5 = O(\alpha^{5/2})$, which is strictly smaller in $\alpha$-order for $\alpha$ sufficiently small. Hence for $\alpha$ small enough, the sign of $\dot r$ on $r=r_1$ and $r=r_2$ is unchanged by the tail: $\dot r > 0$ on $r=r_1$ and $\dot r< 0$ on $r = r_2$, for all $\theta$. Thus the closed annulus $\Omega$ is positively invariant: any trajectory starting in $\Omega$ stays in $\Omega$ for all future time (it cannot cross either boundary circle, since the vector field points inward there, and $\dot\theta \ge \omega_c/2>0$ prevents any escape through the "sides," there being none — $\Omega$ is an annulus with the two circles as its entire boundary). Moreover $\Omega$ contains no equilibrium of the full system: the only equilibrium of the planar system near the origin in this range is the origin itself (a hyperbolic unstable focus since $\alpha>0$, by the Hartman–Grobman theorem applied to the linearization, or directly since $\dot r = \alpha r + O(r^3) > 0$ near $0$), and the origin lies outside $\Omega$ since $r_1 > 0$. By the Poincaré–Bendixson theorem, a compact positively invariant region containing no equilibria contains at least one periodic orbit. Applying this to $\Omega$: $\Omega$ is compact (closed and bounded in $\mathbb R^2$), positively invariant, and equilibrium-free, so it contains a periodic orbit $\Gamma$. Since $\epsilon$ was arbitrary, we may shrink $\epsilon \to 0$, and by continuity of the flow in $\epsilon$ (or, more precisely, uniqueness of the limit cycle established in Step 4) the orbit $\Gamma$ converges to the circle $r=r_0$ as $\epsilon \to 0$, giving amplitude $r_0 = \sqrt{-\alpha/\ell_1} + O(\alpha)$ as claimed.

Step 4: Uniqueness and stability of the cycle. Uniqueness within $\Omega$, and hence a precise identification of $\Gamma$ with amplitude $r_0(1+O(\epsilon))$, follows from monotonicity of the Poincaré return map. Define the Poincaré section $\Sigma = \{\theta = 0, r>0\}$ (a valid transversal since $\dot\theta > 0$ everywhere in the annulus). The return map $P: \Sigma \to \Sigma$, $r \mapsto P(r)$, is well-defined for $r \in [r_1,r_2]$ (since $\Omega$ is invariant and trajectories starting on $\Sigma \cap \Omega$ return to $\Sigma$ after time $\approx 2\pi/\omega_c$, by $\dot\theta$ bounded away from $0$), and is smooth because the flow is smooth and $\Sigma$ is transversal. Differentiating the radial equation along the flow shows $P'(r) = \exp\!\left(\int_0^{2\pi/\omega(\mu)} \frac{\partial}{\partial r}\big[\alpha + a r^2 + O(r^4)\big]\, r \, dt \Big/ r \right)$ — more directly, for the truncated system, the displacement function $d(r) = P(r) - r$ satisfies $d(r_0)=0$ and, from Step 2's computation, $d'(r_0) \propto -2\alpha < 0$ strictly. A strictly negative derivative at an isolated zero means $d(r)$ changes sign from $+$ to $-$ as $r$ increases through $r_0$, so $r_0$ is the unique zero of $d$ in a neighborhood, hence the unique fixed point of $P$ in $\Omega$ for $\epsilon$ small, hence $\Gamma$ is the unique periodic orbit in $\Omega$. Because $d'(r_0)<0$, the fixed point $r_0$ of $P$ is attracting: $|P(r)-r_0| < |r-r_0|$ for $r$ near $r_0$, $r\ne r_0$, which is precisely the statement that $\Gamma$ is an orbitally stable limit cycle. This proves the stability claim in (c) for the supercritical case; the subcritical case ($\ell_1>0$) follows by the time-reversal argument noted in Step 3, which turns attracting into repelling.

Step 5: Smooth dependence on $\mu$. Finally, since $d(r,\mu) = P(r,\mu) - r$ is smooth in both variables (smooth dependence of solutions of ODEs on parameters and initial conditions, together with smoothness of $f$), $d(0,\mu_c) = 0$ and $\partial d/\partial r(0,\mu_c) = 0$ (both from the vanishing of $\alpha(\mu_c)$), while $\partial^2 d/\partial r^2$ and $\partial d/\partial \mu$ are nonzero by (H2)–(H3); the implicit function theorem applied to $d(r,\mu)/r$ (which extends smoothly through $r=0$ by evenness of $\dot r$ in $r$ at leading orders) produces a smooth curve $\mu = \mu(r)$ of periodic orbits for $r$ near $0$, giving the full one-parameter family asserted in (b), with the stated leading-order formula. $\blacksquare$