The first Lyapunov coefficient is the frequency-normalized radial cubic Hopf coefficient

Because $A$ has no zero eigenvalue, $D_{x}f(0,0)=A$ is invertible. Apply [F3] to $f(x,\lambda )=0$ to obtain the local $C^5$ equilibrium branch $\xi (\lambda )$; translate $x$ by $\xi (\lambda )$, so the translated field vanishes at $x=0$ for each small $\lambda$. Its Jacobian path is $C^4$. By [F4] the critical eigenpair is $C^4$ in $\lambda$ and ${\beta }^{\mathrm{\prime }}(0)=\mathrm{Re}(p^{\ast }{A}^{\mathrm{\prime }}(0)q)\mathrm{\ne }0$. Apply [F3] to $\mathrm{Re}\zeta (\lambda )-\widehat{\beta }=0$ in solved variable $\lambda$; its derivative is nonzero, so $\widehat{\beta }$ is a valid local parameter, which we relabel $\beta$.

At $\beta =0$, parameterize the center graph by $u=p^{\ast }x$. From $p^{\ast }A=i{\omega }_0{p}^{\ast }$ and $A\bar{q}=-i{\omega }_0\bar{q}$ we get $2i{\omega }_0{p}^{\ast }\bar{q}=0$, hence $p^{\ast }\bar{q}=0$; also $p^{\ast }q=1$. Its embedding is $H_0(u,\bar{u})=uq+\bar{u}\bar{q}+h(u,\bar{u})$ with $h$ in the hyperbolic complement. Its reduced equation is $\dot{u}=i{\omega }_{0}u+g_2(u,\bar{u})+g_3(u,\bar{u})+O(\mathrm{\mid }u{\mathrm{\mid }}^4)$, where $g_j$ is homogeneous of degree $j$. The graph and reduced equation have the required derivatives by [F2] and the chain rule [F5].

Make a quadratic coordinate change $u=w+{\sum }_{j+k=2}{c}_{jk}{w}^j{\bar{w}}^k$. Along the linear flow, $D(w^j{\bar{w}}^k)(i{\omega }_{0}w,-i{\omega }_0\bar{w})=(j-k)i{\omega }_0{w}^j{\bar{w}}^k$; substitution therefore changes that coefficient by $-(j-k-1)i{\omega }_0{c}_{jk}$. This divisor is nonzero for every $j+k=2$, so choose $c_{jk}$ to remove all quadratic terms. Use conjugate corrections in the conjugate equation to preserve the real system. A cubic change with the same divisor removes every cubic term except $(j,k)=(2,1)$, whose divisor is zero. These changes are local diffeomorphisms because their derivative at zero is the identity; [F3] applies to their inverse equations. The resulting equation is $\dot{w}=i{\omega }_{0}w+\frac{1}{2}{G}_{21}{w}^2\bar{w}+O(\mathrm{\mid }w{\mathrm{\mid }}^4)$. Composing the graph embedding with the changes gives a full-space center-manifold parameterization $H(w,\bar{w})$.

Write $H(w,\bar{w})=wq+\bar{w}\bar{q}+\frac{1}{2}{h}_{20}{w}^2+h_{11}w\bar{w}+\frac{1}{2}{h}_{02}{\bar{w}}^2+H_3(w,\bar{w})+O(\mathrm{\mid }w{\mathrm{\mid }}^4)$, where $H_3$ is homogeneous cubic. The invariance equation is $DH\dot{w}=f(H)$. With $f(x,0)=Ax+\frac{1}{2}B(x,x)+\frac{1}{6}C(x,x,x)+O(\mathrm{\parallel }x{\mathrm{\parallel }}^4)$ and no quadratic terms in $\dot{w}$, comparison of degree-two coefficients gives $(2i{\omega }_{0}I-A)h_{20}=B(q,q)$ and $A{h}_{11}=-B(q,\bar{q})$. Zero and $2i{\omega }_0$ are not eigenvalues of $A$: its only imaginary-axis eigenvalues are $\pm i{\omega }_0$. Thus both operators are invertible and $h_{20}=(2i{\omega }_{0}I-A{)}^{-1}B(q,q)$, $h_{11}=-A^{-1}B(q,\bar{q})$.

Let $h_{21}$ be the coefficient of $w^2\bar{w}$ in $H_3$. Comparing that monomial in $DH\dot{w}=f(H)$ gives $(i{\omega }_{0}I-A)h_{21}+\frac{1}{2}q{G}_{21}=\frac{1}{2}C(q,q,\bar{q})+B(q,h_{11})+\frac{1}{2}B(\bar{q},h_{20})$. The derivative term on the left has factor $i{\omega }_0$ because $w^2\bar{w}$ evolves with linear frequency $(2-1)i{\omega }_0$. Left multiplication by $p^{\ast }$ annihilates $(i{\omega }_{0}I-A)h_{21}$ and uses $p^{\ast }q=1$, yielding $G_{21}=p^{\ast }\text{\hspace{0.17em}⁣}(C(q,q,\bar{q})+2B(q,h_{11})+B(\bar{q},h_{20}))$. Substituting step 3.1's formulas gives exactly the expression in [F1].

By [F2], the tangent plane of the graph is spanned by $q(\beta ),\bar{q}(\beta )$ from [F4]. The coordinate $z=p(\beta {)}^{\ast }x$ is valid there because $p(\beta {)}^{\ast }q(\beta )=1$ and $p(\beta {)}^{\ast }\bar{q}(\beta )=0$; the latter follows from $p^{\ast }A=\zeta p^{\ast }$, $A\bar{q}=\bar{\zeta }\bar{q}$, and $\zeta \mathrm{\ne }\bar{\zeta }$. Hence the linear center equation is exactly $\dot{z}=(\beta +i\omega (\beta ))z$. The center graph is $C^5$ and the normalized left and right eigenpair are $C^4$ in the parameter. The coordinate changes are obtained by solving finite-dimensional homological equations whose divisors stay nonzero for small $\beta$, so their coefficients have at least $C^2$ parameter dependence. The resonant coefficient is $G_{21}+O(\mathrm{\mid }\beta \mathrm{\mid })$; its change contributes $O(\mathrm{\mid }\beta \mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^3)$ and its $z$-derivative contributes $O(\mathrm{\mid }\beta \mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^2)$. Taylor's theorem for the $C^5$ reduced vector field gives an $O(\mathrm{\mid }z{\mathrm{\mid }}^4)$ remainder with derivative $O(\mathrm{\mid }z{\mathrm{\mid }}^3)$. Combining these terms yields $R=O(\mathrm{\mid }z{\mathrm{\mid }}^4+\mathrm{\mid }\beta \mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^3)$ and $D_{z}R=O(\mathrm{\mid }z{\mathrm{\mid }}^3+\mathrm{\mid }\beta \mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^2)$.

Finally $\mathrm{Re}(G_{21}\mathrm{/}2)={\omega }_0{\mathrm{\ell }}_1$ by [F1]. Thus the real radial cubic coefficient is the stated value; step 5.1 establishes the normal form and its bounds.