Parameterization of Invariant Manifolds for Periodic Orbits (II): A Posteriori Analysis and Computer Assisted Error Bounds

Abstract

In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier–Taylor parameterizations of local stable/unstable manifolds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical methods developed in Castelli et al. (SIAM J Appl Dyn Syst 14(1):132–167, 2015) in order to obtain a high order Fourier–Taylor series expansion of the parameterization. The main result of the present work is an a-posteriori theorem which provides mathematically rigorous error bounds. The hypotheses of the theorem are checked with computer assistance. The argument relies on a sequence of preliminary computer assisted proofs where we validate the numerical approximation of the periodic orbit, its stable/unstable normal bundles, and the jets of the manifold to some desired order M. We illustrate our method by implementing validated computations for two dimensional manifolds in the Lorenz equations in $\mathbb {R}^3$ and a three dimensional manifold of a suspension bridge equation in $\mathbb {R}^4$.

A Rigorous Enclosure of the Coefficients for the Suspension Bridge Problem

Denote by denote by $\alpha =(\alpha _1,\alpha _2)$ the multi-indices $\alpha \in \mathbb {N}^2$. The superscript $\alpha ^i$ labels different $\alpha$’s. Following the scheme proposed in Sect. 3, the rigorous enclosure of the coefficients $a_\alpha (w)$ of the parameterization with $2\le |\alpha |\le \tilde{N}$ are computed.

Recalling (48), we aim at solving $F_{\alpha ,m}^{(j)} =\left( \frac{2 \pi \mathbf{i}m}{2 T} + \alpha \cdot \varLambda \right) a_{\alpha ,m}^{(j)} - \left( f \circ P \right) ^{(j)}_{\alpha ,m}=0$, where

$$\begin{aligned} \left( \begin{array}{ccc} \left( f \circ P \right) ^{(1)}_{\alpha ,m} \\ \left( f \circ P \right) ^{(2)}_{\alpha ,m} \\ \left( f \circ P \right) ^{(3)}_{\alpha ,m} \\ \left( f \circ P \right) ^{(4)}_{\alpha ,m} \end{array}\right) =\left( \begin{array}{ccc} a_{\alpha ,m}^{(2)}\\ a_{\alpha ,m}^{(3)}\\ a_{\alpha ,m}^{(4)}\\ - 120 a_{\alpha ,m}^{(1)} - 154 a_{\alpha ,m}^{(2)} - 71 a_{\alpha ,m}^{(3)} - 14 a_{\alpha ,m}^{(4)} {\displaystyle {\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ \alpha ^i\ge 0 \end{array}}} } \left( a_{\alpha ^1}^{(1)} * a_{\alpha ^2}^{(1)} *a_{\alpha ^3}^{(1)} \right) _m \end{array} \right) , \end{aligned}$$

(84)

with $2 \le |\alpha | \le \tilde{N}$ and $m \in \mathbb {Z}$. The derivative of F (with respect to $a_{\alpha ,m}$, $|\alpha |\ge 2$) at the point a acts on an element v as

$$\begin{aligned} \left( DF(a) v\right) _{\alpha ,m}{=}\left( \frac{2 \pi \mathbf{i}m}{2 T} {+} \alpha \cdot \lambda \right) v_{\alpha ,m} {+}\left( \begin{array}{ccc} \displaystyle - v_{\alpha ,m}^{(2)}\\ \displaystyle - v_{\alpha ,m}^{(3)}\\ \displaystyle - v_{\alpha ,m}^{(4)}\\ \displaystyle 120 v_{\alpha ,m}^{(1)} +154 v_{\alpha ,m}^{(2)} +71v_{\alpha ,m}^{(3)}+14v_{\alpha ,m}^{(4)} + 3 (a^2*v)_{\alpha ,m} \end{array} \right) , \end{aligned}$$

(85)

where we used the notation

$$\begin{aligned} a^2_\alpha ={\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2=\alpha \\ |\alpha ^i|\ge 0 \end{array}}} a^{(1)}_{\alpha ^1} * a^{(1)}_{\alpha ^2} \quad \text {and} \quad (a^2*v)_{\alpha ,m} = {\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ \alpha ^i \ge 0, |\alpha ^3|\ge 2 \end{array}}} \left( a_{\alpha ^1}^{(1)}* a_{\alpha ^2}^{(1)}*v_{\alpha ^3}^{(1)}\right) _m. \end{aligned}$$

We now choose the operator A and $A^\dag$. Recall that $A^\dag$ approximates the derivative $DF({\bar{a}})$ (where ${\bar{a}}$ is an approximate solution of $F(a)=0$) while A approximates the inverse of $DF({\bar{a}})$. We have some freedom in defining these operators but it is advisable that the composition $AA^\dag :X\rightarrow X$ acts as the identity out of a certain finite dimensional subspace of X. Arguing as in (69), define a finite dimensional part of $A^\dag$ as the exact derivative of $DF^{(M)}({\bar{a}})$, while the action of the derivative on the infinite dimensional complement is only approximated. In (69) we chose to approximate the derivative with the diagonal action of $\left( \frac{2\pi \mathbf{i}m}{2T}+\alpha \cdot \varLambda \right)$ because this term is growing with m, hence, it is asymptotically dominant. However, besides those terms that are growing, it is advisable to consider in $A^\dag$ also those terms that are big. In our case the vector field has a cubic nonlinearity and the first few Fourier coefficients $\gamma _m$ of the 2T-periodic orbit $\gamma (t)$ are

$$\begin{aligned} \gamma _0=0,\ \gamma _{\pm 2}\approx 0.178\pm 13.34 \mathbf{i},\ \gamma _{\pm 6}=-0.17\pm 0.064\mathbf{i}, \ \gamma _{\pm 1,\pm 3,\pm 4\pm 5}\equiv 0. \end{aligned}$$

It follows that the first few Fourier coefficients of $a^2_\mathbf{0}$ are the following ( $\mathbf{0}=(0,0)$):

$$\begin{aligned} ( a^2_\mathbf{0})_0\approx 356.31,\quad (a^2_\mathbf{0})_{\pm 4}\approx -176.40\pm 9.52\mathbf{i}\quad (a^2_\mathbf{0})_{\pm 1,2,3}\approx 0. \end{aligned}$$

Therefore, the cubic term alone produces contributions in the derivative as big as multiplication by 1100.

We decide to include these contributions (3 times the multiplication by $(a^2_\mathbf{0})_0$, $(a^2_\mathbf{0})_{\pm 4}$), together with the linear term $120 v_{\alpha ,m}^{(1)}, 154 v_{\alpha ,m}^{(2)}, 71v_{\alpha ,m}^{(3)}$ in the definition of $A^\dag$. In practice, define $A^\dag$ so that

$$\begin{aligned} \left( (A^\dag _{i_1,i_2})_{j_1,j_2} d \right) _m = \left\{ \begin{array}{rl} \displaystyle \left( (DF_{i_1,i_2}^{(M)})_{j_1,j_2} d_F \right) _m, &{} |m| < M,\\ \displaystyle \delta _{i_1,i_2} \delta _{j_1,j_2} \left( \frac{2 \pi \mathbf{i}m}{2 T} + \alpha (i_1) \cdot \varLambda \right) d_m, &{} |m| \ge M, \end{array} \right. \end{aligned}$$

(86)

and for any i ranging on the set of possible $\alpha$’s, we augment the action of the operators $(A^\dag _{i,i})_{4,j}$, $j=1,2,3$ with the multiplication by the infinite dimensional tridiagonal matrix or the diagonal matrix as depicted in Fig. 6 where

$$\begin{aligned} d_0=120+3(a^2_\mathbf{0})_0,\quad d_4=3(a^2_\mathbf{0})_4,\quad d_{-4}=3(a^2_\mathbf{0})_{-4}, \quad f_0=154,\quad g_0=71. \end{aligned}$$

Similarly, we define the operator A as

$$\begin{aligned} \left( (A_{i_1,i_2})_{j_1,j_2} c \right) _m = \left\{ \begin{array}{rl} \displaystyle \left( (A_{i_1,i_2}^{(M)})_{j_1,j_2} c_F \right) _m, &{} |m| < M,\\ \displaystyle \delta _{i_1,i_2} \delta _{j_1,j_2} \left( \frac{1}{\frac{2 \pi \mathbf{i}m}{2 T} + \alpha (i_1) \cdot \varLambda } \right) c_m, &{} |m| \ge M, \end{array}\right. \end{aligned}$$

(87)

and, at first, we augment the action of $(A_{i,i})_{4,j}$, $j=1,2,3$, for any i ranging on the set of possible $\alpha$’s, with the multiplication by the infinite dimensional tridiagonal matrix as depicted in Fig. 7 where

$$\begin{aligned} e_0^{(m)}=-\frac{d_0}{\mu _m^2},\quad e_{-4}^{(m)}=-\frac{d_{-4}}{\mu _{m+4}\mu _m},\quad e_{4}^{(m)}=-\frac{d_{4}}{\mu _{m-4}\mu _m}, \quad h_0^{(m)}=-\frac{f_0}{\mu _m^2},\quad \ell _0^{(m)}=-\frac{g_0}{\mu _m^2} \end{aligned}$$

and

$$\begin{aligned} \mu _m=\frac{2\mathbf{i}\pi }{2T}m+\alpha (i)\cdot \varLambda . \end{aligned}$$

Fig. 6

Structure of the components of the operator $A^\dag _{i,i}$

Fig. 7

Structure of the components of the operator $A_{i,i}$. The infinite dimensional diagonal and tridiagonal terms are defined so that the composition $AA^\dag$ acts as the identity out of $\varPi ^{(M+4)}X$

Let us now compute the action of $A^\dag$ on $v\in X$. The finite dimensional part results in

$$\begin{aligned} \Big ( \left( A^\dag v \right) _\alpha \Big )_F=\left( \begin{array}{c} (DF^{(M)} v)_F^{(1)}\\ (DF^{(M)} v)_F^{(2)}\\ (DF^{(M)} v)_F^{(3)}\\ (DF^{(M)} v)_F^{(4)}+\epsilon \end{array}\right) . \end{aligned}$$

where $\epsilon$ denotes the multiplication of $d_4$ and $d_{-4}$ times $v^{(1)}_{\alpha ,m}$ where $m=-M-3,\dots , -M$ and $m=M,\dots , M+3$ respectively.

Instead, for $|m|\ge M$

$$\begin{aligned} \left( A^\dag v \right) _{\alpha ,m} = \left( \begin{array}{c} \mu _mv_{\alpha ,m}^{(1)}\\ \mu _mv_{\alpha ,m}^{(2)}\\ \mu _mv_{\alpha ,m}^{(3)}\\ d_4v_{\alpha ,m-4}^{(1)}+d_0v_{\alpha ,m}^{(1)}+d_{-4}v_{\alpha ,m+4}^{(1)} +f_0v_{\alpha ,m}^{(2)}+g_0v_{\alpha ,m}^{(3)}+\mu _mv_{\alpha ,m}^{(4)}\\ \end{array}\right) . \end{aligned}$$

Then, let us apply the operator A to $A^\dag (v)$. For any $\alpha$ and $|m|\ge M+4$,

$$\begin{aligned}&[A(A^\dag (v))]_{\alpha ,m}\\&\quad {=}\left( \begin{array}{c} \frac{1}{\mu _m}(\mu _mv_{\alpha ,m}^{(1)})\\ \frac{1}{\mu _m}(\mu _mv_{\alpha ,m}^{(2)})\\ \frac{1}{\mu _m}(\mu _mv_{\alpha ,m}^{(3)})\\ e_4^{(m)}(\mu _{m-4}v^{(1)}_{\alpha ,m-4})+e_0^{(m)}(\mu _mv^{(1)}_{\alpha ,m}) +e_{-4}^{(m)}(\mu _{m+4}v^{(1)}_{\alpha ,m+4})+h_0^{(m)}\mu _mv^{(2)}_{\alpha ,m} +\ell _0^{(m)}\mu _mv_{\alpha ,m}^{(3)}\\ +\frac{1}{\mu _m}\Big (d_4v_{\alpha ,m-4}^{(1)}+d_0v_{\alpha ,m}^{(1)} +d_{-4}v_{\alpha ,m+4}^{1}+f_0v_{\alpha ,m}^{(2)}+g_0v_{\alpha ,m}^{(3)} +\mu _mv_{\alpha ,m}^{(4)}\Big )\\ \end{array}\right) . \end{aligned}$$

By definition of $e_0,e_4,e_{-4},h_0,\ell _0$, we have $[A(A^\dag (v))]_{\alpha ,m}=v_{\alpha ,m}$, that is $AA^\dag$ acts as the identity on the infinite dimensional subspace $(I-\varPi ^{(M+4)})X$. On the contrary, we can not guarantee that $AA^\dag$ is close to the identity in the finite dimensional space $\varPi ^{(M+4)}(X)$, because of the out of diagonal terms. Thus, we compute a numerical inverse of the restriction of $A^\dag$ on $\varPi ^{(M+4)}(X)$, and we append the result in the construction of A. In practice, the matrices $( A^{(M)}_{i_1,i_2})_{j_1,j_2}$ are replaced by slightly larger matrices $( A^{(M+4)}_{i_1,i_2})_{j_1,j_2}$. In conclusion, the structure of the operator A is the same as the one depicted in Fig. 7 with $M+4$ instead of M.

The bound ${{\varvec{Z}}}^{(\mathbf{1})}$. As in (75), let $\tilde{\mathcal {A}}^2_\alpha$ be the matrix with components $(\tilde{\mathcal {A}}^2_{\alpha })(m,n)=({\bar{a}}^2_{\alpha })_{m-n}$. The matrices ${\overline{\varGamma }}(s,t)$ used in the computation of the $Z^{(1)}$ bound, are of the following form

$$\begin{aligned} \begin{array}{r l l} s_j=1 \qquad &{} s_\alpha =t_\alpha &{} t_j=2\quad {\overline{\varGamma }}(s,t)=- {\tilde{I}}\\ s_j=2 \qquad &{} s_\alpha =t_\alpha &{} t_j=3\quad {\overline{\varGamma }}(s,t)=-{\tilde{I}}\\ s_j=3 \qquad &{} s_\alpha =t_\alpha &{} t_j=4\quad {\overline{\varGamma }}(s,t)=-{\tilde{I}}\\ s_j=4 \qquad &{} s_\alpha =t_\alpha &{} t_j=1\quad {\overline{\varGamma }}(s,t)=3(\tilde{\mathcal {A}}^2_\mathbf{0})^*\\ &{}&{}t_j=4\quad {\overline{\varGamma }}(s,t)=14 {\tilde{I}}\\ &{} s_\alpha >t_\alpha &{} t_j=1\quad {\overline{\varGamma }}(s,t)=3\tilde{\mathcal {A}}^2_{s_\alpha -t_\alpha } \end{array} \end{aligned}$$

where $(\tilde{\mathcal {A}}^2_\mathbf{0})^*$ is the same as $\tilde{\mathcal {A}}^2_\mathbf{0}$ after replacing $({\bar{a}}^2_\mathbf{0})_0=({\bar{a}}^2_\mathbf{0})_{\pm 4}=0$. That is one of the consequences of considering the tridiagonal action in $(A^\dag _{i,i})_{4,1}$. The other main consequence is that the linear terms $120 {\tilde{I}}$, $154{\tilde{I}}$ and $71{\tilde{I}}$ vanish.

The bounds ${{\varvec{Z}}}^{(\mathbf{2})}$ and ${{\varvec{Z}}}^{(\mathbf{3})}$. Because of the cubic nonlinearity, besides the $Z^{(2)}$ bound we also have the $Z^{(3)}$ bound. Indeed

$$\begin{aligned}&{\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ |\alpha ^1|\ge 0,|\alpha ^2|\ge 0,|\alpha ^3|\ge 2 \end{array}}}\left( {\bar{a}}_{\alpha ^1}^{(1)}+ru_{\alpha ^1}\right) *\left( {\bar{a}}_{\alpha ^2}^{(1)} +ru_{\alpha ^2}\right) *\left( rv_{\alpha ^3}^{(1)}\right) \\&\quad =r{\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2=\alpha \\ |\alpha ^1|\ge 0,|\alpha ^2|\ge 2 \end{array}}}({\bar{a}}^2)_{\alpha ^1}*v^{(1)}_{\alpha ^2} +r^2{\mathop {\mathop {\mathop {\sum }\limits _{\alpha ^1+\alpha ^2+\alpha ^3=\alpha }} \limits _{|\alpha ^1|,|\alpha ^2|\ge 0}}\limits _{|\alpha ^3|\ge 2}}\left( {\bar{a}}_{\alpha ^1}^{(1)}*u_{\alpha ^2}+{\bar{a}}_{\alpha ^2}^{(1)}*u_{\alpha ^1}\right) *v_{\alpha ^3}+r^3\\&\quad {\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ |\alpha ^i|\ge 2 \end{array}}} u_{\alpha ^1}*u_{\alpha ^2}*v_{\alpha ^3} \end{aligned}$$

Hence $(Z^{(2)})_\alpha ^{(j)}=(Z^{(3)})_\alpha ^{(j)}=0$, for $j=1,2,3$, and

$$\begin{aligned} (Z^{(2)})^{(4)}_\alpha =2{\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ |\alpha ^1|\ge 0,|\alpha ^2|,|\alpha ^3|\ge 2 \end{array}}}\Vert {\bar{a}}_{\alpha ^1}^{(1)}\Vert _{1,\nu },\quad (Z^{(3)})_\alpha ^{(4)}={\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ |\alpha ^i| \ge 2 \end{array}}}1 \end{aligned}$$

1.1 A.1 Extra Coefficients, $\tilde{{{\varvec{N}}}}<|{\varvec{\alpha }}|\le {{\varvec{N}}}$

Once the enclosure of the function $a_\alpha (w)$ for $|\alpha |<\tilde{N}$ is computed, following the approach of Sect. 3.3, the coefficients $a_\alpha (w)$ for $\tilde{N}< |\alpha |\le N$ can be computed layer by layer. In the case under analysis, the value of N required by the proof is pretty big. As already stated, we do not compute all the coefficients one-by-one for any $|\alpha |$ up to N, rather for $|\alpha |$ big enough a uniform bound is employed. More precisely, for a choice of $N^*$, $3\tilde{N}<N^*<N$, the functions $a_\alpha$ are one-by-one enclosed for any $\tilde{N}<|\alpha |\le N^*$. Then uniform bounds provide the enclosure for all the remaining $a_\alpha$.

The case $|{\varvec{\alpha }}|\le {{\varvec{N}}}^{*}$. In the unknown $a_\alpha$, the function $F_\alpha$ is the same as in (84). The nonlinearity is decomposed into

$$\begin{aligned} \sum _{\alpha ^1+\alpha ^2+\alpha ^3=\alpha } a^{(1)}_{\alpha ^1}*a^{(1)}_{\alpha ^2}*a^{(1)}_{\alpha ^3} ={\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ |\alpha ^{i}|<|\alpha | \end{array}}}a^{(1)}_{\alpha ^1}*a^{(1)}_{\alpha ^2}*a^{(1)}_{\alpha ^3} +3a_\alpha ^{(1)}*(a^2_\mathbf{0}). \end{aligned}$$

Since ${\bar{a}}={\bar{a}}_\alpha =0$, it follows that

$$\begin{aligned} (F({\bar{a}}))_\alpha =\left( 0,0,0,{\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ |\alpha ^i|<|\alpha | \end{array}}}\left( a_{\alpha ^1}^{(1)} * a_{\alpha ^2}^{(1)} *a_{\alpha ^3}^{(1)} \right) \right) ^T \end{aligned}$$

The definition of ${{\varvec{A}}}_{{\varvec{\alpha }},{\varvec{\alpha }}}$. Let us first write $A^\dag _{\alpha ,\alpha }$ explicitly as

$$\begin{aligned} \left( (A^\dag _{\alpha ,\alpha })_{j_1,j_2} d \right) _m = \left\{ \begin{array}{rl} \displaystyle \left( \left( DF_{\alpha ,\alpha }^{(M)}\right) _{j_1,j_2} d_F \right) _m, &{} |m| < M,\\ \displaystyle \delta _{j_1,j_2} \left( \frac{2 \pi \mathbf{i}m}{2 T} + \alpha \cdot \varLambda \right) d_m, &{} |m| \ge M, \end{array} \right. \end{aligned}$$

(88)

and, as done before, we augment the operators $(A^\dag _{\alpha ,\alpha })_{4,j_2}$, $j_2=1,2,3$ with the tridiagonal and diagonal operators depicted in Fig/. 6. Here $DF_{\alpha ,\alpha }^{(M)}$ is the derivative of $F_\alpha ^{(M)}$ with respect to $a_\alpha ^{(M)}$ and it is given by

$$\begin{aligned} DF_{\alpha ,\alpha }^{(M)}=\left( \begin{array}{cccc} \mu _\alpha ^{(M)} &{}-I^{(M)} &{} 0 &{}0\\ 0 &{}\mu _\alpha ^{(M)} &{}-I^{(M)} &{} 0 \\ 0&{}0&{}\mu _\alpha ^{(M)} &{}-I^{(M)} \\ 120I^{(m)}+3({\mathcal {A}}^2_\mathbf{0})^{(M)}&{}154 I^{(M)}&{}71 I^{(M)}&{} \mu _\alpha ^{(M)}+14I^{(M)} \end{array} \right) , \end{aligned}$$

where $\mu _\alpha ^{(M)}$ is the $(2M-1)\times (2M-1)$ diagonal matrix with $\frac{2 \pi \mathbf{i}m}{2 T} + \alpha \cdot \varLambda$ on the diagonal, $|m|<M$, $I^{(M)}$ is the $(2M-1)\times (2M-1)$ identity matrix and $({\mathcal {A}}^2_\mathbf{0})^{(M)}$ is the $(2M-1)\times (2M-1)$ matrix representing the action of the convolution $a^2_\mathbf{0}*x$ on $X^{(M)}$.

The operator $A_{\alpha ,\alpha }$ is of the same shape as done previously

$$\begin{aligned} \left( (A_{\alpha ,\alpha })_{j_1,j_2} c \right) _m = \left\{ \begin{array}{rl} \displaystyle \left( \left( A_{\alpha ,\alpha }^{(M+4)}\right) _{j_1,j_2} c_F \right) _m, &{} |m| < M+4,\\ \displaystyle \delta _{j_1,j_2} \left( \frac{1}{\frac{2 \pi \mathbf{i}m}{2 T} + \alpha \cdot \varLambda } \right) c_m, &{} |m| \ge M+4, \end{array} \right. \end{aligned}$$

(89)

with the tridiagonal and diagonal elements of Fig. 7 appended to $(A_{\alpha ,\alpha })_{4,j2}$, $j_2=1,2,3$. As before, we increase M to $M+4$ to ensure that $AA^\dag =I$ on $I-\varPi ^{(M+4)}X$ and $A_{\alpha ,\alpha }^{(M+4)}$ is a numerical inverse of $(A^\dag _{\alpha ,\alpha })^{(M+4)}$.

Construction of the bounds ${{\varvec{Y}}}$ and ${{\varvec{Z}}}({{\varvec{r}}})$.

$$\begin{aligned} Y_\alpha \ge |||A_{\alpha ,\alpha }||| \left[ 0,0,0,\left\| {\mathop {\sum }_{\begin{array}{c} \alpha ^1+\alpha ^2+\alpha ^3=\alpha \\ |\alpha ^i|<|\alpha | \end{array}}}\left( a_{\alpha ^1}^{(1)} * a_{\alpha ^2}^{(1)} *a_{\alpha ^3}^{(1)} \right) \right\| _\nu \right] ^T. \end{aligned}$$

For any $\alpha$, let $R_\alpha =I-A_{\alpha ,\alpha }^{(M+4)}(A_{\alpha ,\alpha }^\dag )^{(M+4)}$ the residual that occurs when multiplying $A^\dag$ with the approximative inverse A and define

$$\begin{aligned} \left( Z^{(0)}_\alpha \right) ^{(j)}\ge \sum _{j_1}||| (R_\alpha )_{j,j_1}|||. \end{aligned}$$

More precisely, for any $j=1,\dots ,4$ we have the $Z^{(1)}$ bound

$$\begin{aligned} \begin{array}{rl} Z^{(1)}_{\alpha ,j}=&{}|||(A_{\alpha ,\alpha })_{j,1}{\tilde{I}} |||+|||(A_{\alpha ,\alpha })_{j,2}{\tilde{I}} |||+|||(A_{\alpha ,\alpha })_{j,3}{\tilde{I}} |||+|||(A_{\alpha ,\alpha })_{j,4} 14{\tilde{I}} |||\\ &{}+3|||(A_{\alpha ,\alpha })_{j,4}(\tilde{\mathcal {A}}^2_\mathbf{0})^* |||+3|||(A_{\alpha ,\alpha })_{j,4}|||\epsilon _\mathbf{0,2}\\ \end{array} \end{aligned}$$

(90)

where $\Vert (\bar{a}^2)_0-(a^2)_0\Vert _\nu \le \epsilon _\mathbf{{0},2}$.

Uniform bound for ${{\varvec{N}}}^*<|{\varvec{\alpha }}|\le {{\varvec{N}}}$. It remains to compute rigorous enclosures for $a_\alpha$ for all $N^*<|\alpha |\le N$. The idea is to solve the system $\{ F_\alpha =0\}_{N^*<|\alpha |\le N}$ in the unknowns $\{a_\alpha \}_{N^*<|\alpha |\le N}$. The operators $A^\dag$ and A are constructed as diagonal operators in $\alpha$ so that any polynomial $p_\alpha$ depends on the operator $A_{\alpha ,\alpha }$. Also, the “numerical approximation” is taken to be zero for all $\alpha$. In order to define a unique polynomial that provides the enclosure for any $\alpha$, uniform bounds on $Y_\alpha$, $Z_\alpha$ are sought, together with a uniform bound of $|||A_{\alpha ,\alpha }|||$. Let us briefly discuss how to define a uniform bound for $|||A_{\alpha , \alpha }|||$. The crucial point is to bound $|||A_{\alpha ,\alpha }^{M+4}|||$, where $A_{\alpha ,\alpha }^{(M+4)}$ is an approximate inverse of $(A^\dag _{\alpha ,\alpha })^{(M+4)}$. For the system (83), we have

$$\begin{aligned} \left( A^\dag _{\alpha ,\alpha }\right) ^{(M+4)}=\left( \begin{array}{cccc} \mu _\alpha ^{(M+4)} &{}-I^{(M)} &{} 0 &{}0\\ 0 &{}\mu _\alpha ^{(M+4)} &{}-I^{(M)} &{} 0 \\ 0&{}0&{}\mu _\alpha ^{(M+4)} &{}-I^{(M)} \\ C_{4,1}&{}C_{4,2}&{}C_{4,3}&{} \mu _\alpha ^{(M+4)}+14I^{(M)} \end{array}\right) , \end{aligned}$$

where $C_{4,1}$ is the $2(M+4)-1 \times 2(M+4)-1$ matrix obtained by enlarging $120I^{(M)}+3({\mathcal {A}}^2_\mathbf{0})^{(M)}$ with the terms on the 3 diagonals $d_0, d_4, d_{-4}$, $C_{4,2}=154 I^{(M+4)}$, $C_{4,3}=71 I^{(M+4)}$.

Write $(A^\dag _{\alpha ,\alpha })^{(M+4)}=P+B_\alpha$ where

$$\begin{aligned} P=\left( \begin{array}{cccc} 0 &{}-I^{(M)} &{} 0 &{}0\\ 0 &{}0 &{}-I^{(M)} &{} 0 \\ 0&{}0 &{}0 &{}-I^{(M)} \\ 0&{}0&{}0&{} 14I^{(M)} \end{array}\right) \quad \text {and} \quad B_\alpha =\left( \begin{array}{cccc} \mu _\alpha ^{(M+4)} &{}0 &{} 0 &{}0\\ 0 &{}\mu _\alpha ^{(M+4)} &{}0 &{} 0 \\ 0&{}0&{}\mu _\alpha ^{(M+4)} &{}0 \\ C_{4,1}&{}C_{4,2}&{}C_{4,3}&{} \mu _\alpha ^{(M+4)} \end{array}\right) . \end{aligned}$$

Now, we define $A_{\alpha ,\alpha }^{(M+4)}$ as

$$\begin{aligned} A_{\alpha ,\alpha }^{(M+4)}={\tilde{B}}_\alpha -{\tilde{B}}_\alpha P{\tilde{B}}_\alpha , \end{aligned}$$

where

$$\begin{aligned} {\tilde{B}}_\alpha =\left( \begin{array}{cccc} \mu _\alpha ^{-1} &{}0 &{} 0 &{}0\\ 0 &{}\mu _\alpha ^{-1} &{}0 &{} 0 \\ 0&{}0&{}\mu _\alpha ^{-1} &{}0 \\ -\mu _\alpha ^{-1}C_{4,1}\mu _\alpha ^{-1}&{}-\mu _\alpha ^{-1}C_{4,2}\mu _\alpha ^{-1}&{}-\mu _\alpha ^{-1}C_{4,3}\mu _\alpha ^{-1}&{} \mu _\alpha ^{-1} \end{array}\right) ,\quad \mu _\alpha ^{-1}=\left( \mu _\alpha ^{(M+4)}\right) ^{-1}. \end{aligned}$$

It follows that $B_\alpha {\tilde{B}}_\alpha =I$ and that $I-A_{\alpha ,\alpha }^{(M+4)}(A_{\alpha ,\alpha }^\dag )^{(M+4)}={\tilde{B}}_\alpha P{\tilde{B}}_\alpha P$. For the definition of $Z^0$, a uniform bound of the latest product is needed for any $|\alpha |>N^*$. Since ${\tilde{B}}_\alpha$ is component wise decreasing in $|\alpha |$, that is $| {\tilde{B}}_{\alpha '} | <\max | {\tilde{B}}_{\alpha } |$ if $|\alpha '|=|\alpha |+1$, a bound is obtained by computing the expression for all the $\alpha$’s with $|\alpha |=N^*$.

Similarly, a bound for $|||A_{\alpha ,\alpha }^{(M+4)}|||$ is computed, as explained in the next remark.

Remark 13

Direct computation provides

$$\begin{aligned}&A_{\alpha ,\alpha }^{(M+4)}={\tilde{B}}_\alpha -{\tilde{B}}_\alpha P{\tilde{B}}_\alpha \\&\quad =\left( \begin{array}{cccc} \mu _\alpha ^{-1} &{} \mu _\alpha ^{-1}I^{(M)} \mu _\alpha ^{-1} &{} 0 &{}0\\ 0&{}\mu _\alpha ^{-1} &{}\mu _\alpha ^{-1}I^{(M)} \mu _\alpha ^{-1} &{} 0\\ -\mu _\alpha ^{-1} I^{(M)} \mu _\alpha ^{-1}C_{4,1}\mu _\alpha ^{-1} &{}-\mu _\alpha ^{-1} I^{(M)} \mu _\alpha ^{-1}C_{4,2}\mu _\alpha ^{-1}&{} \mu _\alpha ^{-1}-\mu _\alpha ^{-1} I^{(M)} \mu _\alpha ^{-1}C_{4,3}\mu _\alpha ^{-1}&{}\mu _\alpha ^{-1}I^{(M)} \mu _\alpha ^{-1} \\ X_1 &{} X_2 &{} X_3 &{} X_4 \end{array}\right) \end{aligned}$$

where

$$\begin{aligned} X_1&=-\left( I-\mu _\alpha ^{-1}C_{4,3}\mu _\alpha ^{-1}I^{(M)} -14\mu _\alpha ^{-1}I^{(M)}\right) \mu _\alpha ^{-1}C_{4,1}\mu _\alpha ^{-1}\\ X_2&=-\mu _\alpha ^{-1}C_{4,1}\mu _\alpha ^{-1}I^{(M)}\mu _\alpha ^{-1} -\left( I-\mu _\alpha ^{-1}C_{4,3}\mu _\alpha ^{-1}I^{(M)}-14\mu _\alpha ^{-1} I^{(M)}\right) \mu _\alpha ^{-1}C_{4,2}\mu _\alpha ^{-1}\\ X_3&=-\mu _\alpha ^{-1}C_{4,2}\mu _\alpha ^{-1}I^{(M)}\mu _\alpha ^{-1} -\left( I-\mu _\alpha ^{-1}C_{4,3}\mu _\alpha ^{-1}I^{(M)}-14\mu _\alpha ^{-1} I^{(M)}\right) \mu _\alpha ^{-1}C_{4,3}\mu _\alpha ^{-1}\\ X_4&=-\mu _\alpha ^{-1}C_{4,3}\mu _\alpha ^{-1}I^{(M)}\mu _\alpha ^{-1} +\mu _\alpha ^{-1}-14\mu _\alpha ^{-1}I^{(M)}\mu _\alpha ^{-1} \end{aligned}$$

For any $\alpha$ the operator norm of the operator $\mu _\alpha ^{-1}$ is given by

$$\begin{aligned} |||\mu _\alpha ^{-1}|||=\max _m\left| \frac{1}{2\pi im/{2T}+\alpha \cdot \varLambda }\right| =\left| \frac{1}{\alpha \cdot \varLambda }\right| =\frac{1}{\alpha \cdot |\varLambda |} \end{aligned}$$

Clearly, if $|\alpha '|=N+1$, $\alpha '\cdot |\varLambda |> \min _{|\alpha |=N}\alpha \cdot |\varLambda |$. Then

$$\begin{aligned} |||\mu _{\alpha '}^{-1}|||<\max _{|\alpha |=N} |||\mu _\alpha ^{-1}|||. \end{aligned}$$

According to this remark, the knowledge of $|||\mu _\alpha ^{-1}|||$ with $|\alpha |=N^*$ provides a uniform bound $|||\mu _{\alpha '}^{-1}|||$ for any $|\alpha '|>N^*$. So we have a uniform bound for the operator norm of each of the entries of $A_{\alpha ,\alpha }^{(M+4)}$ that is valid for all $\alpha$ with $|\alpha |>N^*$.