We briefly introduce the setup of the continuation problems under consideration in this paper. Additional details can be found in~\cite{QJBcontinuation}. Consider a polynomial vector field of the form 

\begin{equation} \label{eq:h_vector_field}
  \left\{ \begin{array}{l} \dot{u} = h(u,\lambda),\\
 g(\lambda)=0, \\
  u \text{ is $2\pi$-periodic},
  \end{array}\right.
\end{equation}

where $h:\R^n \times \R^m \to \R^n$ is a polynomial vector field and $g:\R^m \to \R^{m'}$ is a polynomial mapping. The maps $h$ and $g$ depend on the problem under study. For the standard \emph{pseudo-arclength} continuation of periodic orbits, $m=2$, $m'=0$, $\lambda=(\tau,\mu) \in \R^2$, $h=\tilde{f}:\R^n \times \R^{2} \to \R^{n}$ is defined in \eqref{e:prelim} and $g \equiv 0$. For the continuation of periodic orbits passing through a Hopf bifurcation, $m=3+n$, $m'=n$, $\lambda=(\tau, a, \y,\mu) \in \R^{3+n}$, $h=\bar{f}:\R^{n} \times \R^{3+n} \to \R^{n}$ is defined in \eqref{eq:bar_f_defn} and $g(\lambda)=f(\y,\mu)$.

\begin{remark}
For the pseudo-arclength continuation, $g \equiv 0$ and therefore two equations are ``missing'' to balance the variable $\lambda \in \R^2$. In this case a phase condition and a continuation equation are appended. For the desingularized Hopf problem, three equations are missing as $\lambda \in \R^{3+n}$ is variable and $g(\lambda)=0 \in \R^n$. A phase condition, a continuation equation and an amplitude equation are then introduced.
\end{remark}

\subsection{Formulation in Fourier space}
\label{s:fourier}

We write the Fourier expansion of a $2\pi$-periodic function $u =(u_1,\dots,u_n) : \R \to \R^n$ as

\begin{equation}\label{e:fourier}
u(t)=\sum_{k\in\mathbb{Z}}(v)_{k} e^{\imag k t }, \qquad (v)_{k}=(v_1,\dots,v_n)_k\in \mathbb{C}^n.
\end{equation}

The differential equation $\dot{u} = h(u,\lambda) \in \R^n$ then transforms in Fourier space to

\begin{equation}\label{e:defF}
\left(F_i(x)\right)_k \bydef \imag k  (v_i)_{k} -
  (\widehat{h}_i (v,\lambda))_k =0 \qquad \text{for } 1\leq i \leq n, k\in \Z,
\end{equation}

where $\widehat{h}_i$ is the polynomial $h_i$, but with multiplications interpreted as convolutions, denoted

\[
(\tilde{v} \tilde{v}')_k \bydef \sum_{k' \in \Z}  \tilde{v}_{k'} \tilde{v}'_{k-k'}.
\]

To fix a Banach space in which we will apply contraction arguments, 
we introduce the $\nu$-norm ($\nu \geq 1$) on $\C^\Z$ as

\begin{equation}\label{norm_l1nu}
\|\tilde{v}\|_\nu \bydef \sum_{k \in \Z}  |\tilde{v}_k| \nu^{|k|} ,
\end{equation}

with corresponding Banach space 
$\ell^1_\nu \bydef \{ \tilde{v} \in \C^\Z \,:\, \|\tilde{v}\| _\nu <\infty\}$.
The space of variables $x=(v,\lambda)$ is then $X=X_\nu \bydef (\ell^1_\nu)^n \times \C^m$.
On the components of $x=(x_1,\dots,x_{m+n})=(v_1,\dots,v_n,\lambda_1,\dots,\lambda_m)$

we define the norm
\[
  \|x_j\| \bydef \begin{cases}
  \|x_j\|_\nu \qquad &\text{for }1\leq j\leq n,\\
  |x_j| & \text{for }n+1\leq j \leq n+m,
  \end{cases}
\]
leading to the product norm
\begin{equation}\label{e:maxnorm}
	\|x\|_X \bydef \max_{1\leq j \leq m+n} \| x_j\|.
\end{equation}	
We introduce the time derivative on $\ell^1_\nu$ by
\[
  (\imag \K  \tilde{v})_k \bydef \imag k \tilde{v}_k, 
  \qquad \text{for } k \in \Z,
\]
which we extend to $X$ via $\imag \K x = (\imag \K v_1,\dots,\imag \K v_n,0,\dots,0)$.

Finite dimensional numerical approximations are found in the truncated space
\[
  X_K \bydef \{ x=(v,\lambda) \in X : (v_i)_k=0 \text{ for all } |k|>K, 1 \leq i \leq n \}
\]
for some $K \in \N$.
The space $X_K$ can be identified with $\C^{m+n(2K+1)}$ and it will be convenient notationally to introduce the following bilinear form on $X_K$:

\begin{equation}\label{e:bilinear}
  \langle x, x' \rangle \bydef \sum_{i=1}^{n} \sum_{k=-K}^K (v_i)_k (v'_i)_k  
  + \sum_{j=1}^m \lambda_j \lambda'_j .
\end{equation}

In order to recover a real-valued solution we will check a posteriori that  $(v_i)_{-k} = \overline{(v_i)_k}$ using equivariance of the problem under the conjugation symmetry defined below.
\begin{definition}
The \emph{conjugate} $x^* = (v^*,\lambda^*)$ of $x=(v,\lambda) \in X$ is given by 
\[
  \lambda^*_j = \overline{\lambda_j} ~~\text{for } 1\leq j \leq m, 
  \qquad 
  (v^*_i)_k = \overline{(v_i)_{-k}} ~~\text{for } 1\leq i \leq n, k \in \Z.
\]
The set of conjugate symmetric elements is denoted by $\XS = \{ x \in X : x^*=x\}$, and $\XS_K = \XS \cap X_K$.
\end{definition}
We note that both $F$ and $g$ are equivariant under the conjugation symmetry: $F(x^*)=F(x)^*$ and $g(\lambda^*)=g(\lambda)^*$. 
Note that the equivariance of  $g$ follows from the fact that it is a real polynomial mapping. 
Furthermore,
$ \langle x, x' \rangle \in \R$ for $x, x' \in \XS_K$.

To set up the rigorous continuation framework, let us assume we have two points $\hx_0 = (\hv_0,\hl_0) \in \XS_K$ and $\hx_1 = (\hv_1,\hl_1) \in \XS_K$ which each represent an approximate solution
of
\begin{equation}\label{e:Fgsystem}
  \left\{
  \begin{array}{l}
	  F(x)=0, \\
	  g(\lambda)=0 .
  \end{array}
  \right.
\end{equation}

We define the interpolation
\[
  \hx_s = (\hv_s, \hl_s)  \bydef (1-s) \hx_0 + s\hx_1
  \qquad\text{for  } s\in [0,1].
\]
To introduce the phase condition we define 
$q^\phase = (q^\phase_v,0) \in \XS_K$
with 
\begin{equation}\label{e:choiceofphaseq}
  (q^\phase_v)_j  =  \overline{ \imag \K (\hv_{\frac{1}{2}})_j} \qquad\text{for }
  |k| \leq K, 1\leq j \leq n.
\end{equation}
We choose the phase condition
\[
  G^\phase(x) \bydef   
  \bigl\langle q^\phase , x \bigr\rangle  
  
  
  =0 .
\] 
This phase equation does not depend on $s$, which deviates slightly from the one in~\cite{QJBcontinuation}. Having an $s$-dependent phase condition is convenient when validating  long stretches of a solution branch (essentially, it assists in ``gluing'' short pieces into a long smooth curve). Here we are interesting in verifying a solution branch near a bifurcation point, hence $s$-dependence is not necessary and merely complicates the algebra and notation.


\begin{remark}\label{r:dependence}
We note that $G^\phase$ depends linearly on $x$.

Furthermore, $G^\phase$ only depends on the Fourier coefficients with indices $|k| \leq K$. We will encounter slightly more general dependence for more general ``phase'' equations throughout, namely we allow for affine linear dependence on  $x$ and affine linear dependence on $s$. For this purpose we introduce the notation
\begin{equation}\label{e:generalG}
  \G^{\pp_0,\pp_1}_{\yy_0,\yy_1}(x,s) \bydef 
  \bigl\langle (1-s) \pp_0 + s \pp_1 , x \bigr\rangle  - \bigl[(1-s)
    \yy_0  + s \yy_1  \bigr],
\end{equation}
for $\pp_0,\pp_1\in \mathcal{S}_K$ and $\yy_0,\yy_1 \in \R$. 
In this notation $G^\phase(x)=\G^{q^\phase,q^\phase}_{0,0}(x,s)$.

We note that \[ 
\G^{\pp_0,\pp_1}_{\yy_0,\yy_1}(x^*,s)=  \overline{\G^{\pp_0,\pp_1}_{\yy_0,\yy_1}(x,s)}.\]
\end{remark} 

\begin{remark}

In case $h=\bar{f}:\R^{3+n} \times \R^{n} \to \R^{n}$ is the desingularized Hopf problem defined in \eqref{eq:bar_f_defn}, an extra phase-like condition is appended to the system, namely the ($s$-independent) amplitude equation
\[
G^\ameq(x) \bydef   \bigl\langle q^\ameq , x \bigr\rangle  - 1 =   \G^{q^\ameq,q^\ameq}_{1,1}(x,s)  =0,
\]

where 
$q^\ameq = (q^\ameq_v,0) \in \XS_K$,
with 


\[
  (q^\ameq_v)_i =  \overline{\K^2 (\hv_{\frac{1}{2}})_i} \qquad\text{for }
  |k| \leq K, 1\leq i \leq n.
\]
See Section~\ref{s:hopf} for more details and a motivation for this choice. 
\end{remark}


In an analogous manner we introduce an $s$-dependent continuation equation $G^\cont_s(x)=0$. Indeed, we determine numerically ``predictors'' $\dot{\hx}_\s \in \XS_K$ of the tangent direction of the solution curve (for the problem including the phase condition) at $\hx_\s$ for $\s=0,1$. 
We set 
\[
  q^\cont_{\s} = \overline{\dot{\hx}_{\s}} ,
\]

so that $q^\cont_{\s} \in \XS_K$, and we define 

\begin{align}
G^\cont_s(x) &\bydef \bigl\langle (1-s) q^\cont_0 + s q^\cont_1 , x \bigr\rangle - \bigl[ (1-s) \langle  q^\cont_0 , \hx_0 \rangle + s \langle q^\cont_1 , \hx_1 \rangle \bigr] \nonumber \\
&= \G^{q^\cont_0,q^\cont_1}_{\langle  q^\cont_0 , \hx_0  \rangle ,\langle  q^\cont_1 ,  \hx_1 \rangle }(x,s) \label{e:continuationequation},
\end{align}
hence the dependence of $G^\cont$ on $x$ and $s$ is  as described in Remark~\ref{r:dependence}. 

The full set of ``algebraic'' equations is

\begin{equation*}
G_s(x) =
\begin{bmatrix}  G^\phase(x)  \\ G^\cont_s(x) \end{bmatrix} 


\qquad\text{or}\qquad
G_s(x) =
\begin{bmatrix}  G^\phase(x) \\ G^\ameq(x) \\ g(\lambda) \\G^\cont_s(x)  \end{bmatrix} ,
\end{equation*}

for the pseudo-arclength continuation and the desingularized Hopf problem, respectively. The general zero finding problem for continuation is 

\begin{equation}\label{e:defHs}
H_s(x) \bydef\begin{bmatrix} F(x) \\ G_s(x) \end{bmatrix} = 0 .
\end{equation}

Clearly, conjugate symmetric zeros of $H_s$ correspond to periodic orbits of~\eqref{e:initial}, provided $\tau=\lambda_1 \neq 0$.

We define the associated fixed point operator
\begin{equation}\label{e:defTs}
T_s(x) \bydef x-A_s H_s(x), \qquad T_s:X \to X, \qquad s\in [0,1].
\end{equation}
Here $A_s$ is an injective map that approximates the inverse of the Jacobian $D_x H_s(\hx_s)$. We do not elaborate on the choice for $A_s$, which is discussed in detail in~\cite[Section~8.2]{QJBcontinuation}. 
For the current discussion it suffices to say that $A_s=(1-s)A_0 + s A_1$,
and $A_{\s}$,  $\s=0,1$ are approximate inverses of the Jacobians at the end points $\hx_\s$. Each linear operator $A_{\s}$, $\s=0,1$ is
made up from a $(m+n(2K+1))\times(m+n(2K+1))$ matrix and a diagonal infinite tail.
In particular, let $\Pi_K$ denote the natural projection of $X$ onto $X_K$, then
the block structure (finite matrix and infinite tail) of $A_{\s}$ is characterized by

$A_{\s} \Pi_K = \Pi_K A_{\s}$ and $ A_{\s} (I -\Pi_K) =  (I -\Pi_K) A_{\s}$,
while the diagonal tail is given by
\[
(I -\Pi_K) A_{\s} x = (I-\Pi_K)  (-\imag \K^{-1} v_1, \dots ,-\imag \K^{-1} v_n,
0,\dots , 0),
\] 
with $(\K^{-1} \tilde{v})_k \bydef k^{-1} \tilde{v}_k$ for any $k \neq 0$.
With regards to conjugation symmetry, the choice of $A_{\s}$ is such that
$A_{\s} x^* = (A_{\s} x)^*$, hence $T_s(x^*)=T_s(x)^*$ for all $s\in[0,1]$. 

Let $B_r(x) \bydef \{ x' \in X : \|x-x'\|_X \leq r \}$, then the ``tube'' around the numerical line segment $\{\hx_s: s\in [0,1]\}$
is given by

\begin{equation}\label{e:tube}
  \mathcal{C}_r \bydef \bigcup_{s \in [0,1]} B_r(\hx_s).
\end{equation}
In~\cite[Sections~6 and~8]{QJBcontinuation} explicitly computable bounds 
$Y=(Y_1,\dots, Y_{m+n})$ and $Z(r)=(Z_1,\dots,Z_{m+n})(r)$ are derived, that satisfy

\begin{subequations}
\label{e:boundsYZ}
\begin{alignat}{1}
Y_j &\geq \max_{s\in [0,1]}\| (T_s(\hx_s) - \hx_s )_j \| ,\\
Z_j(r) &\geq \max_{s\in [0,1]}\sup_{b, c\in B_1(0)}\|[  D_x T_s(\hx_s+r b)r c]_j \|,
\label{e:boundZ}
\end{alignat}
\end{subequations}

for $j=1,\dots,m+n$.
The following theorem, which is itself based on the uniform contraction principle (see for example~\cite{BLM,breden-vanicat,GLPmulti} for similar results), is the crux of rigorously verified continuation.
\begin{theorem}[Theorems~3.1~and~4.2 in~\cite{QJBcontinuation}]\label{thm:radii_bound}
Assume $Y$ and $Z(r)$ satisfy~\eqref{e:boundsYZ}.
Assume moreover that $A_s$ is injective for all $s\in[0,1]$. 
If there exists an $\hr>0$ such that 

\begin{equation}\label{e:radpoly}
  Y_j+Z_j(\hr)-\hr < 0 
\qquad \text{for all } j=1,\dots, m+n,
\end{equation}

then $T_s$ is a contraction on $B_{\hr}(\hx_s)$  for every $s$ in $[0,1]$. 
The fixed points $\sol{x}(s)$ of $T_s$ in $B_{\widehat{r}}(x_s)$
are conjugate symmetric and form 
a continuous parametrized curve $\sol{x}:[0,1]\rightarrow X$ in $\mathcal{C}_{\hr}$, such that $H_s(\sol{x}(s))=0$ for every $s\in [0,1]$.
\end{theorem}

\begin{remark}\label{r:injectivity}
Injectivity of $A_s$ follows by a computational check on the finite part, since invertibility of its diagonal tail is trivial to establish. For the implementation of the $Z$-bound chosen in \cite{QJBcontinuation}, the former computational check is in fact implied by $Z(\hr)<\hr$, see~\cite[Section~8.5]{QJBcontinuation}.
Since $\sol{x}(s) \in B_r(\hx_s)$, it follows from the inequalities~\eqref{e:radpoly} and the definition of the $Z$-bound~\eqref{e:boundZ} that
\[
  \| I - A_s D_x H_s(\sol{x}(s)) \|_{B(X)} <1,
\]

where $\| \cdot \|_{B(X)}$ is the bounded linear operator norm on $X$. Hence $D_x H_s(\sol{x}(s))$ is injective.


\end{remark}
{"config": "arxiv", "file": "2006.13373/setup.tex"}