A simple nondegenerate Hopf point creates a local periodic branch

Because $\pm i{\omega }_0$ and every other eigenvalue of $A_0$ are nonzero, $A_0$ is invertible. First set $y=x-x_0$ and $\widehat{f}(y,\lambda )=f(x_0+y,\lambda )$, so $\widehat{f}(0,0)=0$. The data in Hopf spectral and multilinear normal-form data are those of this field: $A=A_0$, $B=D_x^{2}f(x_0,0)$, $C=D_x^{3}f(x_0,0)$, and $p,q$ are eigenvectors of this same $A_0$ with both declared normalizations. Apply [F1] to obtain the unique nearby equilibrium $\xi (\lambda )$ of the original field. In the parameter-dependent translated state $v=x-\xi (\lambda )$, the field is $f(v+\xi (\lambda ),\lambda )$ and vanishes at $v=0$ for each fixed parameter; its critical spatial jets agree with the preceding data, since $\xi (0)=x_0$ and $\dot{\lambda }=0$. Apply [F2] to obtain $\zeta (\lambda )$ with ${\beta }^{\mathrm{\prime }}(0)\mathrm{\ne }0$, and apply [F1] to $\mathrm{Re}\zeta (\lambda )-\widehat{\beta }=0$ in solved variable $\lambda$. This makes $\widehat{\beta }$ a valid local parameter, which we relabel $\beta$.

By [F3], every sufficiently small periodic orbit is on the parameterized center graph. Restricting to a fixed parameter and using [F4], its reduced dynamics have $\dot{r}=r(\beta +{\omega }_0{\mathrm{\ell }}_1{r}^2)+O(r^4+\mathrm{\mid }\beta \mathrm{\mid }{r}^3)$ and $\dot{\theta }={\omega }_0+O(\mathrm{\mid }\beta \mathrm{\mid }+r^2)$, with the first-derivative bounds supplied by [F4]. In particular $\dot{\theta }>0$ after shrinking the neighborhood.

Reparameterize by the strictly increasing angle from step 1.2 and write $dr\mathrm{/}d\theta =\rho (r,\theta ,\beta ):=\dot{r}\mathrm{/}\dot{\theta }$. The bounds in [F4] give uniformly in $\theta$ that $\rho =(r\mathrm{/}{\omega }_0)(\beta +{\omega }_0{\mathrm{\ell }}_1{r}^2)+O({\beta }^{2}r+\mathrm{\mid }\beta \mathrm{\mid }{r}^3+r^4)$ and ${\mathrm{\partial }}_r\rho =(\beta +3{\omega }_0{\mathrm{\ell }}_1{r}^2)\mathrm{/}{\omega }_0+O({\beta }^2+\mathrm{\mid }\beta \mathrm{\mid }{r}^2+r^3)$. On $r>0$ the radial field is jointly $C^1$ in $(\theta ,r,\beta )$ by the finite-regularity coordinate construction in [F4]. Applying [F6] to $(\theta ,r,\beta {)}^{\mathrm{\prime }}=(1,\rho (\theta ,r,\beta ),0)$ gives the required common local solution and first variations, with finite composition over $[0,2\pi ]$. For small $r_0>0$ and $\mathrm{\mid }\beta \mathrm{\mid }$, while $r\in [r_0\mathrm{/}2,2r_0]$ the bound $\mathrm{\mid }\rho \mathrm{/}r\mathrm{\mid }\le C(\mathrm{\mid }\beta \mathrm{\mid }+4r_0^2)$ makes the change in $\log r$ over $[0,2\pi ]$ smaller than $\log 2$. Thus no first exit from that interval occurs; compact continuation in [F6] supplies the full angular interval and positivity, and ${\mathrm{\partial }}_{\theta }\log r=\rho \mathrm{/}r=O(\mathrm{\mid }\beta \mathrm{\mid }+r^2)$; integrating over the bounded interval $[0,2\pi ]$ gives $r(\theta ;r_0)=r_0+O(\mathrm{\mid }\beta \mathrm{\mid }{r}_0+r_0^3)$. Substituting this bound into $P_{\beta }(r_0)-r_0={\int }_0^{2\pi }\rho (r(\theta ;r_0),\theta ,\beta )d\theta$ gives $P_{\beta }(r_0)-r_0=\frac{2\pi }{{\omega }_0}{r}_0(\beta +{\omega }_0{\mathrm{\ell }}_1{r}_0^2)+O({\beta }^2{r}_0+\mathrm{\mid }\beta \mathrm{\mid }{r}_0^3+r_0^4)$. The variational derivative $Q=\mathrm{\partial }r\mathrm{/}\mathrm{\partial }{r}_0$ solves ${\mathrm{\partial }}_{\theta }Q=({\mathrm{\partial }}_r\rho )Q$, $Q(0)=1$, so $Q(2\pi )=\exp ({\int }_0^{2\pi }{\mathrm{\partial }}_r\rho d\theta )$. The same bound on $r(\theta ;r_0)$ in the derivative expansion yields $P_{\beta }^{\mathrm{\prime }}(r_0)=1+\frac{2\pi }{{\omega }_0}(\beta +3{\omega }_0{\mathrm{\ell }}_1{r}_0^2)+O({\beta }^2+\mathrm{\mid }\beta \mathrm{\mid }{r}_0^2+r_0^3)$. The factor $r_0$ is the persistent equilibrium.

For $r>0$, divide step 2.1 by $(2\pi \mathrm{/}{\omega }_0)r$ and call the result $H(r,\beta )$. Then $H(r,\beta )=\beta +{\omega }_0{\mathrm{\ell }}_1{r}^2+O({\beta }^2+\mathrm{\mid }\beta \mathrm{\mid }{r}^2+r^3)$. Differentiating this quotient using both expansions in step 2.1 gives ${\mathrm{\partial }}_{r}H(r,\beta )=2{\omega }_0{\mathrm{\ell }}_{1}r+O({\beta }^2\mathrm{/}r+\mathrm{\mid }\beta \mathrm{\mid }r+r^2)$. If $\beta {\mathrm{\ell }}_1>0$, then $\mathrm{\mid }\beta +{\omega }_0{\mathrm{\ell }}_1{r}^2\mathrm{\mid }=\mathrm{\mid }\beta \mathrm{\mid }+{\omega }_0\mathrm{\mid }{\mathrm{\ell }}_1\mathrm{\mid }{r}^2$; the error is bounded by $C({\beta }^2+\mathrm{\mid }\beta \mathrm{\mid }{r}^2+r^3)$, which is at most an arbitrarily small multiple of $\mathrm{\mid }\beta \mathrm{\mid }+r^2$ after shrinking $r,\mathrm{\mid }\beta \mathrm{\mid }$. Thus there is no positive root. If $\beta =0$, then $H(r,0)={\omega }_0{\mathrm{\ell }}_1{r}^2+O(r^3)$, whose leading term dominates for all sufficiently small $r>0$, so there is no nonconstant periodic orbit at the threshold either. If $\beta {\mathrm{\ell }}_1<0$, choose $0<c_{-}<1\mathrm{/}\sqrt{{\omega }_0\mathrm{\mid }{\mathrm{\ell }}_1\mathrm{\mid }}<c_{+}$. At $r=c_{-}\sqrt{\mathrm{\mid }\beta \mathrm{\mid }}$ and $r=c_{+}\sqrt{\mathrm{\mid }\beta \mathrm{\mid }}$ the leading terms have opposite signs and the error is $O(\mathrm{\mid }\beta {\mathrm{\mid }}^{3\mathrm{/}2})$, so [F9] gives a root between them. Every root lies between fixed such multiples of $\sqrt{\mathrm{\mid }\beta \mathrm{\mid }}$: below a sufficiently small multiple, $\mathrm{\mid }\beta \mathrm{\mid }$ dominates the error; above a sufficiently large multiple, ${\omega }_0\mathrm{\mid }{\mathrm{\ell }}_1\mathrm{\mid }{r}^2$ dominates both $\mathrm{\mid }\beta \mathrm{\mid }$ and the error. On this annulus, ${\beta }^2\mathrm{/}r+\mathrm{\mid }\beta \mathrm{\mid }r+r^2=o(r)$ uniformly as $\beta \to 0$, so ${\mathrm{\partial }}_{r}H$ has the sign of ${\mathrm{\ell }}_1$ and the root is unique. Since this derivative is nonzero, [F1] with $k=1$ makes the radius a $C^1$ function of $\beta$ on the punctured existence side, using the joint $C^1$ return map from [F6].

Every sufficiently small periodic trajectory lies on the center graph by [F3], and [F4] gives $\dot{\theta }>0$, so it meets the ray $\theta =0$ and returns after some positive integer number of turns. Reparameterize the radial equation by $\theta$; $dr\mathrm{/}d\theta =\dot{r}\mathrm{/}\dot{\theta }$ is a $C^1$ scalar nonautonomous ODE on $r>0$, so two solutions cannot cross by uniqueness from the augmented $C^1$ flow [F6]. Therefore the one-turn radius map is strictly increasing. If a periodic trajectory met the section in a $k$-cycle with $P_{\beta }^k(r)=r$, strict increase forces $P_{\beta }(r)=r$; indeed $P_{\beta }(r)>r$ would imply $P_{\beta }^k(r)>r$, and $P_{\beta }(r)<r$ would imply $P_{\beta }^k(r)<r$. Thus the unique positive fixed point from step 3.1 describes the unique local periodic orbit up to phase, and none exists on the opposite side.

At the root, step 3.1 gives $r^2=-\beta \mathrm{/}({\omega }_0{\mathrm{\ell }}_1)+O(\mathrm{\mid }\beta {\mathrm{\mid }}^{3\mathrm{/}2})$. The angular speed in step 1.2 is ${\omega }_0+O(\mathrm{\mid }\beta \mathrm{\mid }+r^2)$, so the return period is $2\pi \mathrm{/}{\omega }_0+O(\mathrm{\mid }\beta \mathrm{\mid })$. Differentiating the return map at this fixed radius yields $P_{\beta }^{\mathrm{\prime }}(r)=1+\frac{2\pi }{{\omega }_0}(\beta +3{\omega }_0{\mathrm{\ell }}_1{r}^2)+O(\mathrm{\mid }\beta {\mathrm{\mid }}^{3\mathrm{/}2})=1-\frac{4\pi }{{\omega }_0}\beta +O(\mathrm{\mid }\beta {\mathrm{\mid }}^{3\mathrm{/}2})$. Thus this nontrivial radial multiplier is inside the unit circle for $\beta >0$ and outside it for $\beta <0$, when $\beta$ is sufficiently small.

The full monodromy matrix along the periodic orbit converges to $e^{(2\pi \mathrm{/}{\omega }_0)A_0}$. Indeed its variational coefficient $D_{x}f(x(t),\lambda )$ converges uniformly to $A_0$; subtracting the two fundamental-matrix integral equations and iterating the resulting bound gives uniform convergence on the bounded period interval. The limiting matrix has the critical multiplier $1$ of algebraic multiplicity two, and all remaining multipliers $e^{2\pi {\nu }_j\mathrm{/}{\omega }_0}$ have modulus different from one. By [F5], exactly two multipliers remain near $1$ and the others converge with multiplicity to the hyperbolic multipliers. One of the two is exactly $1$ because ${\dot{x}}_{\ast }(t)$ solves the variational equation and is periodic; the other is the radial multiplier of step 4.2. For a section $\mathrm{\Sigma }$ transverse to $v={\dot{x}}_{\ast }(0)$, differentiating $P(y)={\varphi }_{\tau (y)}(y)$ gives $DP=\pi M{\mathrm{\mid }}_{T\mathrm{\Sigma }}$, where $M$ is the monodromy matrix and $\pi$ is projection along $v$ onto $T\mathrm{\Sigma }$. Thus $DP$ represents the induced map on the quotient by $\mathrm{span}\{v\}$, so its eigenvalues are exactly the non-phase Floquet multipliers.

On the branch $\beta \mathrm{\ne }0$, step 4.2 makes the radial multiplier different from one, and the other multipliers are off the unit circle by step 5.1. The return-map derivative has exactly the non-phase multipliers: if $v={\dot{x}}_{\ast }(0)$ and $\pi$ projects onto the section tangent along $v$, differentiating $P(y)={\varphi }_{\tau (y)}(y)$ gives $DP=\pi M{\mathrm{\mid }}_{T\mathrm{\Sigma }}$, which represents the map induced by $M$ on the quotient by $\mathrm{span}\{v\}$. Thus the return map is hyperbolic. If all $\mathrm{Re}{\nu }_j<0$ and $\beta >0$, every return multiplier is inside the unit circle, so [F8] gives orbital asymptotic stability. If $\beta <0$, the radial multiplier is outside; if any $\mathrm{Re}{\nu }_j>0$, a transverse multiplier is outside. In either case [F8] gives orbital instability. Steps 3.1–4.1 prove local existence, side, and uniqueness up to phase; the arguments use only the local center graph and return section, so assert no global completeness.