Fourier–Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer-Assisted Error Bounds

Abstract

We develop and implement a semi-numerical method for computing high-order Taylor approximations of unstable manifolds for hyperbolic fixed points of compact infinite-dimensional maps. The method can follow folds in the embedding and describes precisely the dynamics on the manifold. In order to ensure the accuracy of our computations in spite of the many truncation and round-off errors, we develop a posteriori error bounds for the approximations. Deliberate control of round-off errors (using interval arithmetic) in conjunction with explicit analytical estimates leads to mathematically rigorous computer-assisted theorems describing precisely the truncation errors for our approximation of the invariant manifold. The method is applied to the Kot-Schaffer model of population dynamics with spatial dispersion.

Appendices

Appendix 1: Computer-Assisted Proof of a Non-trivial Fixed Point for Kot-Schaffer

We discuss an a posteriori scheme for studying the fixed point problem for the Kot-Schaffer mapping $F :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ with logistic nonlinearity as defined in Sect. 2.1. The argument is another application of Theorem 2.2. In order to apply the theorem, we define the mapping $G :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ by

$$\begin{aligned} G(a) = F(a) - a = \mu B a - a - \mu B (c*a*a). \end{aligned}$$

We have that a is a zero of G if and only if a is a fixed point of F. Observe that G is Fréchet differentiable and that for any $a \in \ell _\nu ^1$ the action of the linear operator $DG(a) :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ is given by

$$\begin{aligned} DG(a) h = \mu B h - h - 2 \mu B (c * a * h). \end{aligned}$$

The associated truncated mapping $G^N :{\mathbb {R}}^{N+1} \rightarrow {\mathbb {R}}^{N+1}$ is

$$\begin{aligned} G^N(\bar{a}^N) = \mu B^N \bar{a}^N - \bar{a}^N - \mu B^N (\bar{c}^N * \bar{a}^N * \bar{a}^N)^N. \end{aligned}$$

(49)

(the truncated convolution maps are as defined in Sect. 3.3).

Suppose that $\bar{a}^N \in {\mathbb {R}}^{N+1}$ is an approximate solution of $G^N = 0$. (In practice, $\bar{a}^N$ may be the output of a numerical computation via Newton’s method.). The derivative of $G^N$ at $\bar{a}^N$ is the $N+1 \times N+1$ matrix which acts on $h^N \in {\mathbb {R}}^{N+1}$ by

$$\begin{aligned} DG^N(\bar{a}^N)h^N = \mu B^Nh^N - \text{ Id }_{N+1}h^N - 2 \mu B^N (c^N * \bar{a}^N * h^N)^N. \end{aligned}$$

The matrix representation of this linear transformation is computed column by column in the usual way, i.e. by considering the action on the basis vectors $h^N = e_j \in {\mathbb {R}}^{N+1}$.

Let $A^N$ be an approximate inverse for $DG^N(\bar{a}^N)$ (possibly obtained numerically). Define the linear operators $A^{\dagger }, A \in B(\ell _\nu ^1)$ by

$$\begin{aligned} (A^{\dagger }h)_n := \left\{ \begin{array}{lll} (DG^N(\bar{a}^N) h^N)_n &{} &{} 0 \le n \le N \\ -h_n &{} &{} n \ge N+1 \end{array} \right. \end{aligned}$$

(50)

and

$$\begin{aligned} (A h)_n := \left\{ \begin{array}{lll} (A^N h^N)_n &{} &{} 0 \le n \le N \\ -h_n &{} &{} n \ge N+1 \end{array} \right. \end{aligned}$$

(51)

for all $h \in \ell _\nu ^1$. Note that

$$\begin{aligned} \Vert A^{\dagger }\Vert _{B(\ell _\nu ^1)} \le \max \left( \Vert DG^N(\bar{a}^N)\Vert _{B(\ell _\nu ^1)}, 1 \right) , \quad \text{ and } \quad \Vert A\Vert _{B(\ell _\nu ^1)} \le \max \left( \Vert A^N\Vert _{B(\ell _\nu ^1)}, 1 \right) . \end{aligned}$$

Both of the operators A and $A^{\dagger }$ are invertible, and hence injective, as each is the Cartesian product of a finite dimensional invertible matrix and (minus) the identity in the tail components. The following Lemma gives sufficient conditions for the existence of a fixed point for Kot-Schaffer. The proof of the Theorem follows by estimates similar to those discussed in detail in Sect. 4.4.

Lemma 5.1

(A posteriori existence of a fixed point for Kot-Schaffer) Suppose that $\mu > 0$, $\nu > 1$ and $b, c \in \ell _\nu ^1$ are fixed parameters for the Kot-Schaffer mapping $F :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ defined by Eq. (5). Let $N \in {\mathbb {N}}$, $G^N$ be as defined in Eq. (49), $\bar{a}^N \in {\mathbb {R}}^{N+1}$, $A^N$ be an $N+1 \times N+1$ invertible matrix, and $A \in B(\ell _\nu ^1)$ be the linear operator defined in Eq. (51). Let $b^N, c^N$ denote, respectively, the projections of b, c into ${\mathbb {R}}^{N+1}$ Let $B^N$ denote the $N+1 \times N+1$ diagonal matrix of $b^N$, and $B \in B(\ell _\nu ^1)$ denote the linear operator defined by $(B h)_n = b_n h_n$ for all $n \in {\mathbb {N}}$. Let $r_3 \in {\mathbb {R}}$ have that

$$\begin{aligned} \sup _{n \ge N+1} |b_n| \le r_3. \end{aligned}$$

Let

$$\begin{aligned} \Vert c^\infty \Vert _\nu ^1 = \sum _{n = N+1}^\infty |c_n| \nu ^n. \end{aligned}$$

Define the constants

$$\begin{aligned} \kappa _n^1 := \max _{N+1 \le k \le 2N-n} \frac{| (c^N * \bar{a}^N )_{n+k} |}{2 \nu ^k}, \quad \kappa _n^2 := \max _{N+1 \le k \le 2N+n} \frac{| (c^N * \bar{a}^N)_{k-n} |}{2 \nu ^k} \end{aligned}$$

as in Lemma 3.3. Let

$$\begin{aligned} Y_0:= & {} \Vert A^N G^N(\bar{a}^N)\Vert + 2 \mu \sum _{n=N+1}^{3N} | b_n| |(c^N * \bar{a}^N * \bar{a}^N)_n|\nu ^n \nonumber \\&+\,\mu (\Vert A^N B^N \Vert + r_3) \Vert \bar{a}\Vert ^2 \Vert c^\infty \Vert \end{aligned}$$

(52)

$$\begin{aligned} Z_0:= & {} \Vert \text{ Id }_{N+1} - A^N DG^N(\bar{a}^N) \Vert ,\\ Z_1:= & {} 4 |\mu | \sum _{n=0}^N \sum _{k=0}^N |a^N_{nk}| |b_k| (\kappa _k^1 + \kappa _k^2)\, \nu ^n \nonumber \\&+\,|\mu |\Vert A^N B^N \Vert \Vert \bar{a} \Vert _\nu ^1 \Vert c^\infty \Vert _\nu ^1\nonumber \\&+ \, |\mu | r_3 \left( 1 + 2\Vert c\Vert _\nu ^1 \Vert \bar{a} \Vert _\nu ^1 \right) ,\nonumber \end{aligned}$$

(53)

and

$$\begin{aligned} Z_2 := 2 |\mu | \, \Vert c\Vert _\nu ^1 \Vert AB \Vert . \end{aligned}$$

If $r > 0$ is a number with

$$\begin{aligned} p(r) := Z_2 r^2 - (1 - Z_0 - Z_1) r + Y_0 \le 0, \end{aligned}$$

then there exists a unique fixed point $a^* \in B_r(\bar{a})$ of the Kot-Schaffer mapping.

Appendix 2: CAP Results for Non-trivial Fixed Points of the Kot-Schaffer map

For parameters of the Kot-Schaffer map as discussed in Sect. 2.1.1, we choose to work in the sequence space $\ell _\nu ^1$ with $\nu = 1.1$. We project to order $N = 26$ cosine modes (i.e. projecting the problem into ${\mathbb {R}}^{27}$) as discussed in Sect. 5. We run a numerical Newton method in order to find an approximate fixed point. The IntLab program kotSchaffer_validateFP.m is then used in order to check the conditions of Theorem 5.1. We obtain the interval constants

$$\begin{aligned} Y_0= & {} 2.3 \times 10^{-15},\\ Z_0= & {} 10^{-13},\\ Z_1= & {} 1.15 \times 10^{-7}, \end{aligned}$$

and

$$\begin{aligned} Z_2 = 6. \end{aligned}$$

From these constants, we obtain that there is a fixed point $a^*$ of the Kot-Schaffer map with

$$\begin{aligned} a^* \in B_{r_0}(\bar{a}) \subset \ell _\nu ^1, \end{aligned}$$

with $r_0 = 7.6 \times 10^{-15}$. Moreover the fixed point is unique in a ball of radius at least $R_0 = 0.169$. The reader interested in the details of the proof can run the IntLab program a_KotPaperProofs_firstOrderData.m from [22] for more details.

Appendix 3: Computer-Assisted Analysis of the Eigenvalue/Eigenvector Problem for the Kot-Schaffer Map

In this section, we discuss a numerical validation scheme for the Kot-Schaffer eigenvalue/eigenvector problem. We focus on the case of a real eigenvalue, as this is the case of interest in main body of the paper. Let $F :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ be the Kot-Schaffer mapping given in Eq. (5).

Assumption C

(a priori fixed point) Assume that $a^* \in \ell _\nu ^1$ is a fixed point for F and that $a^*$ admits the decomposition

$$\begin{aligned} a^* = \bar{a} + a^\infty , \end{aligned}$$

where $\bar{a}, a^\infty \in \ell _\nu ^1$ and $\bar{a}_n = 0$ for $n \ge N+1$. Let $\bar{a}^N \in {\mathbb {R}}^{N+1}$ denote the projection of $\bar{a}$. We do not assume that the $N+1$ leading coefficients of $a^\infty$ are zero. Rather we assume that there is an $r_0 \ge 0$ so that $\Vert a^\infty \Vert _\nu ^1 \le r_0$. (Computer- assisted validation of such a decomposition for $a^*$ is obtained using the techniques discussed in Appendix 1).

Suppose now that $\bar{\lambda } \in {\mathbb {R}}$ and $\bar{\xi }^N \in {\mathbb {R}}^{N+1}$ are an approximate (possibly numerically computed) eigenvalue/eigenvector pair for ${\textit{DF}}^N(\bar{a}^N)$, i.e. suppose that

$$\begin{aligned} DF^N(\bar{a}^N) \bar{\xi }^N - \bar{\lambda } \bar{\xi }^N \approx 0. \end{aligned}$$

Our goal is to prove that there exist $\lambda ^* \in {\mathbb {R}}$, and $\xi ^* \in \ell _\nu ^1$ having $|\bar{\lambda }_0 - \lambda ^*|, \Vert \bar{\xi } - \xi ^* \Vert _\nu ^1$ small and that

$$\begin{aligned} {\textit{DF}}(a^*) \xi ^* - \lambda ^* \xi ^* = 0. \end{aligned}$$

(54)

In order to isolate a unique solution, we choose an arbitrary $s \in {\mathbb {R}}$, $s \ne 0$ and impose the constraint

$$\begin{aligned} \sum _{n=0}^N \xi _n = s. \end{aligned}$$

(55)

This constraint is somewhat arbitrary and could be replaced with any condition which fixes the length of $\xi$.

In order to formalize the problem, we define the Banach space

$$\begin{aligned} {\mathcal {Y}} := {\mathbb {R}} \times \ell _\nu ^1, \end{aligned}$$

which we endow with the norm

$$\begin{aligned} \Vert (\lambda , \xi )\Vert _{{\mathcal {Y}}} = |\lambda | + \Vert \xi \Vert _\nu ^. \end{aligned}$$

We denote a general element $y \in {\mathcal {Y}}$ as a sequence indexed from $-1, 0, 1, 2, \ldots$, i.e.

$$\begin{aligned} y = \{y_n\}_{n = -1}^\infty = \{y_{-1}, y_0, y_1, \ldots \}, \end{aligned}$$

and think of the norm as being

$$\begin{aligned} \Vert y\Vert _{{\mathcal {Y}}} = |y_{-1}| + \sum _{n= 0}^{\infty } |y_n| \nu ^n \end{aligned}$$

With $s \ne 0$ fixed, we define the mapping $E :{\mathcal {Y}} \rightarrow {\mathcal {Y}}$ by

$$\begin{aligned} E(\lambda , \xi )_n := \left\{ \begin{array}{lll} s - \sum _{n=0}^N \xi _n &{} &{} n = -1 \\ \\ \left[ {\textit{DF}}(a^*)\xi \right] _n - \lambda \xi _n &{} &{} n \ge 0 \end{array} \right. , \end{aligned}$$

(56)

where

$$\begin{aligned} {\textit{DF}}(a)h = \mu {\mathcal {B}} h - 2 \mu {\mathcal {B}} (c * a * h). \end{aligned}$$

Note that E is a nonlinear (quadratic) mapping, due to the appearance of the product of the unknowns $\lambda$ and $\xi$.

The truncated system $E^N :{\mathbb {R}}^{N+2} \rightarrow {\mathbb {R}}^{N+2}$ is given by

$$\begin{aligned}&E^N(\lambda , \xi ^N)_n := \left\{ \begin{array}{lll} s - \sum _{n=0}^N \xi _n &{} &{} n = -1 \\ \\ {\textit{DF}}^N(\bar{a}^N) \xi ^N - \lambda \xi ^N &{} &{} 0 \le n \le N \end{array} \right. \nonumber \\&= \left\{ \begin{array}{lll} s - \sum _{n=0}^N \xi _n &{} &{} n = -1 \\ \left[ (\mu B^N - 2 \mu M^N) \xi ^N \right] _n - \lambda \xi ^N_n &{} &{} 0 \le n \le N \end{array} \right. \end{aligned}$$

(57)

where $M^N$ is the $N+1 \times N+1$ matrix defined linear operator

$$\begin{aligned} (c^N * \bar{a}^N * h^N)^N = M^N h^N, \quad \quad \quad \text {for all} \, h^N \in {\mathbb {R}}^{N+1}. \end{aligned}$$

(58)

and $B^N$ is the $N+1 \times N+1$ matrix diagonal matrix with the entries of $b^N$ in the diagonal and zeros elsewhere. ($M^N$ is computed column by column by considering $h^N = e_j$ the j-th basis vector in ${\mathbb {R}}^{N+1}$ for each j.)

The Fréchet derivative of E is given by

$$\begin{aligned}{}[DE(\lambda , \xi )h]_n = \left\{ \begin{array}{lll} - \sum _{n=0}^N h_n &{} &{} n=-1 \\ \mu b_n h_n - 2 \mu b_n (c* a^* * h)_n - \lambda h_n - \xi _n h_{-1} &{} &{} n \ge 0 \end{array} \right. , \end{aligned}$$

and the derivative of the truncated mapping is given by the $N+2 \times N+2$ matrix

$$\begin{aligned} DE^N(\lambda , \xi ^N) = \left( \begin{array}{cc} 0 &{} -\mathbf {1}_N \\ -\xi ^N &{} \mu B^N - 2\mu B^N M^N - \lambda \text{ Id }_N \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} \mathbf {1}_N = (1, \ldots , 1)^T \quad (1, \ldots , 1)\in {\mathbb {R}}^{N+1}, \end{aligned}$$

i.e. this is the row vector of length $N+1$ with ones in all entries.

Let $(\bar{\lambda }, \bar{\xi }_N)$ be an approximate solution of $E^N = 0$. Assume that $DE^N(\bar{\lambda }, \bar{\xi }^N)$ is an invertible matrix and let $A^N$ be an approximate inverse. Define the linear operators $A, A^\dagger \in B({\mathcal {Y}})$ by

$$\begin{aligned} (A^{\dagger }h)_n := \left\{ \begin{array}{lll} \left[ DE^N(\bar{\lambda }, \bar{\xi }^N) h \right] _n &{} &{} -1 \le n \le N \\ - \lambda h_n &{} &{} n \ge N+1 \end{array} \right. \end{aligned}$$

(59)

and

$$\begin{aligned} (A h)_n := \left\{ \begin{array}{lll} \left[ A^N h \right] _n &{} &{} -1 \le n \le N \\ - \frac{h_n}{\lambda } &{} &{} n \ge N+1 \end{array} \right. \end{aligned}$$

(60)

for all $h = \{h_{-1}, h_0, h_1, \ldots \} \in {\mathcal {Y}}$. Note that $A, A^{\dagger }$ are injective. We have the following a posteriori validation lemma for the eigenvalue/eigenvector problem, whose proof follows by estimates similar to those carried out in detail in Sect. 4.4.

Lemma 5.2

(A posteriori existence of an eigenvalue/eigenvector pair for the Kot-Schaffer map) Suppose that $\mu > 0$, $\nu > 1$ and $b, c \in \ell _\nu ^1$ are fixed parameters for the Kot-Schaffer mapping $F :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ defined by Eq. (5). Suppose in addition that $a^*$ is a fixed point of F satisfying Assumption C above. Let $N \in {\mathbb {N}}$, $\bar{a}^N \in {\mathbb {R}}^{N+1}$, $a^\infty \in \ell _\nu ^1$, and $r_0 > 0$ be as in Assumption C. Let $E^N$ be as defined in Eq. (57), $\bar{\xi }^N \in {\mathbb {R}}^{N+2}$, $A^N$ be a $N+2 \times N+2$ invertible matrix, $\bar{\lambda } \in {\mathbb {C}}$, $M^N$ be the $N+1 \times N+1$ matrix defined in Eq. (58), and $A \in B({\mathcal {Y}})$ be the linear operator defined in Eq. (60). Let $b^N, c^N$ denote, respectively, the projections of b, c into ${\mathbb {R}}^{N+1}$, and let $B^N$ denote the diagonal matrix of $b^N$. Let $r_3 \in {\mathbb {R}}$ be any bound of the form

$$\begin{aligned} \sup _{n \ge N+1} |b_n| \le r_3. \end{aligned}$$

Let

$$\begin{aligned} \Vert c^\infty \Vert _\nu ^1 = \sum _{n = N+1}^\infty |c_n| \nu ^n. \end{aligned}$$

Define the constants

$$\begin{aligned} \kappa _n^1 := \max _{N+1 \le k \le 2N-n} \frac{| (c^N * \bar{a}^N )_{n+k} |}{2 \nu ^k}, \quad \kappa _n^2 := \max _{N+1 \le k \le 2N+n} \frac{| (c^N * \bar{a}^N)_{k-n} |}{2 \nu ^k} \end{aligned}$$

as in Lemma 3.3. Let

$$\begin{aligned} Y_0:= & {} \Vert A^N E^N(\bar{\lambda }, \bar{\xi }^N) \Vert _{{\mathcal {Y}}} + \frac{4 \mu }{|\bar{\lambda }|} \sum _{n = N+1}^{3N} |b_n| | (c^N * \bar{a}^N * \bar{\xi }^N)_n| \nu ^n\\&+\, 2 \mu \Vert \bar{\xi } \Vert _\nu ^1 \left( \Vert A^N B^N \Vert _{B({\mathcal {Y}})} + \frac{r_3}{|\bar{\lambda }|} \right) (\Vert c^N\Vert _\nu ^1 r_0 + \Vert c^\infty \Vert _\nu ^1 \Vert a^*\Vert _\nu ^1 ) ,\\ Z_0:= & {} \left\| \text{ Id }_{N+2} - A^N D E^N(\bar{\lambda }, \bar{\xi }^N) \right\| _{B({\mathcal {Y}})},\\ Z_1:= & {} 4\mu \sum _{k=0}^N |a_k^{-1}||b_k| (\kappa _k^1 + \kappa _k^2) \\&+\,\, 4 \mu \sum _{n=0}^N \sum _{k=0}^N |a_k^n||b_k| (\kappa _k^1 + \kappa _k^2) \nu ^n \\&+\, 2 \mu \Vert A^N B^N \Vert _{B({\mathcal {Y}})} \Vert c^N \Vert _\nu ^1 r_0 \\&+\, 2 \mu \Vert A^N B^N \Vert _{B({\mathcal {Y}})} \Vert c^\infty \Vert _\nu ^1 \Vert a^* \Vert _\nu ^1 \\&+\, \mu \frac{r_3}{|\bar{\lambda }|} (1 + 2\Vert c\Vert _\nu ^1 \Vert a^*\Vert _\nu ^1) \end{aligned}$$

and

$$\begin{aligned} Z_2 := \Vert A \Vert _{B({\mathcal {Y}})}. \end{aligned}$$

If $r > 0$ has that

$$\begin{aligned} p(r) := Z_2 r^2 - (1 - Z_0 - Z_1) r + Y_0 \le 0, \end{aligned}$$

then there exists a unique $(\lambda ^*, \xi ^*) \in B_r(\bar{\lambda }, \bar{\xi }) \subset {\mathcal {Y}}$ so that $E(\lambda ^*, \xi ^*) = 0$.

Appendix 4: CAP Results for an Unstable Eigenvalue/Eigenvector Pair for the Kot-Schaffer Map

We take parameters for the Kot-Schaffer map as in Sect. 2.1.1 and let $\bar{a}^N \in {\mathbb {R}}^{27}$ be as in Appendix 2. In Appendix 2, we showed that the fixed point validation for these data resulted in a fixed point $a^* \in \ell _\nu ^1$ with $\nu = 1.1$,

$$\begin{aligned} a_* = \bar{a}^N + a^\infty , \quad \quad \Vert a^\infty \Vert \le r_0, \end{aligned}$$

and $r_0 = 7.6 \times 10^{-15}$. We compute an unstable eigenvalue/eigenvector pair for $DF^N(\bar{a}^N)$ resulting in the numerical approximation of the eigenvalue

$$\begin{aligned} \bar{\lambda }= -1.688296535873959. \end{aligned}$$

We choose a scaling of 1.5 for the eigenvector as discussed in Appendix 3, and fix $s = -1.76841177760678$ as discussed in Sect. 1. The IntLab program kotSchaffer_validateEig.m is used in order to check the conditions of Theorem 5.2, and we obtain that

$$\begin{aligned} Y_0 \le 10^{-13}, \quad \quad Z_0 \le 8.2 \times 10^{-14}, \quad \quad Z_1 \le 6.8 \times 10^{-8}, \quad \quad \text{ and } \quad Z_2 \le 18.9. \end{aligned}$$

This results in a validated error bound of

$$\begin{aligned} r_1 \le 3.087 \times 10^{-13}. \end{aligned}$$

The reader interested in the details of the implementation will find them in the program a_KotPaperProofs_firstOrderData.m from [22] for more details.

Appendix 5: A Spectral Perturbation Lemma

The spectrum of an eventually diagonal linear operator on $\ell _\nu ^1$ [recall the definition of Eq. (22)] is determined by considering the spectrum of its matrix part in conjunction with the (diagonal) entries in the tail. The spectrum of the matrix can often be computed using validated numerical methods. The spectrum of the tail can be read off by hand in the case that the diagonal entries decay to zero rapidly enough. Many of the operators we encounter in applications (especially in this paper) can be viewed as small perturbations of eventually diagonal linear operators, and we would like to use the fact that we understand precisely the spectrum of the unperturbed (eventually diagonal) linear operator to understand the spectrum after a small perturbation.

Lemma 5.3 is an example of the kinds of results one can obtain. More general results of this kind can be given, but this simple lemma is sufficient for the needs of the present work. We include the elementary proof for the sake of completeness.

Lemma 5.3

With $N \in {\mathbb {N}}$ let $A^N$ be an $N+1 \times N+1$ matrix. Assume that $A^N$ is diagonalizable with

$$\begin{aligned} A^N = Q \Sigma Q^{-1}, \quad \Sigma = \left( \begin{array}{cccc} \lambda _0 &{} 0 &{} \ldots &{} 0 \\ 0 &{} \lambda _1 &{} \ldots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \ldots &{} \lambda _{N} \end{array} \right) . \end{aligned}$$

Suppose that $|\lambda _0| > 1$ and that

$$\begin{aligned} |\lambda _N| \le \ldots \le |\lambda _1| < 1, \end{aligned}$$

so that $\lambda _0$ is the only unstable eigenvalue of $A^N$. Let $\{a_n\}_{n = N+1}^\infty$ be a sequence of real numbers with

$$\begin{aligned} a_n \rightarrow 0, \quad \quad \quad \text {as} \quad \quad \quad n\rightarrow \infty , \end{aligned}$$

and

$$\begin{aligned} | a_n | \le |a_{N+1}| < |\lambda _N|, \quad \quad \quad \text {for all} \quad \quad \quad n \ge N+1. \end{aligned}$$

Let $A :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ be the eventually diagonal operator defined by

$$\begin{aligned} (A h)_n := \left\{ \begin{array}{lll} [A^N h^N]_n &{} &{} \text{ for } \,\, 0 \le n \le N \\ &{} &{} \\ a_n h_n &{} &{} \text{ for } \,\, n \ge N+1 \end{array} \right. \end{aligned}$$

with $h \in \ell _{\nu }^1$. Suppose that

$$\begin{aligned} \Vert Q\Vert \Vert Q^{-1}\Vert \max \left( (|\lambda _0|-1)^{-1}, (1 - |\lambda _1|)^{-1} \right) \le C. \end{aligned}$$

Let $\tilde{A} :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ be a bounded linear operator and suppose that

$$\begin{aligned} \Vert \tilde{A} \Vert C < 1. \end{aligned}$$

Then

$$\begin{aligned} B = A + \tilde{A}, \end{aligned}$$

has exactly one unstable eigenvalue.

Proof

We study the homotopy given by

$$\begin{aligned} H(s) = s A + (1-s)B = A - (s-1) \tilde{A}. \end{aligned}$$

The proof is in two steps. In the first, we prove that: If each operator H(s) has no eigenvalues on the unit circle during the homotopy (i.e. “no eigenvalues cross the unit circle”) and A has exactly one unstable eigenvalue, this implies that each H(s) has exactly one unstable eigenvalue. In the second step of the proof, we show that the hypotheses of the lemma are sufficient to guarantee that indeed each H(s) has no eigenvalue on the unit circle.

Step 1: This part of the argument is based on the well-known semi-continuity properties of separated parts of the spectrum (below we cite precisely the property needed for the argument). Let $\Sigma _s$ denote the spectrum of H(s). Note that

$$\begin{aligned} \Vert H(s)\Vert \le \Vert A\Vert + \Vert \tilde{A} \Vert < \infty , \end{aligned}$$

Then for $|\lambda | > \Vert A \Vert + \Vert \tilde{A} \Vert$ we have that

$$\begin{aligned} H(s) - \lambda \text{ Id } = -\lambda \left( \text{ Id } - \frac{1}{\lambda }H(s)\right) , \end{aligned}$$

with $\Vert H(s)/\lambda \Vert < 1$. It follows by the Neumann theorem that the operator is invertible, i.e. that $|\lambda | > \Vert A \Vert + \Vert \tilde{A} \Vert$ is not in the spectrum of H(s). More precisely, we have that

$$\begin{aligned} \Sigma _s \subset \{ z \in {\mathbb {C}} : \, |z| < \Vert A \Vert + \Vert \tilde{A} \Vert \}, \end{aligned}$$

for each $s \in [0, 1]$.

Since each H(s) is a compact linear operator, its spectrum accumulates only at the origin in ${\mathbb {C}}$. It follows that for each $s \in [0,1]$, the unstable spectrum of H(s) consists of only finitely many unstable eigenvalues of finite multiplicity.

Now assume that for all $s \in [0, 1]$, the operator H(s) has no eigenvalues on the unit circle. We claim that this implies that there is an $r > 0$ so that the stable spectrum of H(s) is interior to the circle of radius $1 - r$ and so that the unstable spectrum of H(s) is in the complement of the circle of radius $1 + r$. To put it another way: The spectrum of H(s) is uniformly bounded away from the unit circle throughout the homotopy.

To see this, note that the assumption that H(s) has no eigenvalues on the unit circle means that for all $\theta \in [0, 2 \pi ]$ the operator $H(s) - \hbox {e}^{i \theta } \text{ Id }$ is boundedly invertible. Since $[0, 2 \pi ]$ is compact, there is a $C > 0$ so that

$$\begin{aligned} \Vert (H(s) - \hbox {e}^{i \theta } \text{ Id })^{-1}\Vert \le C, \end{aligned}$$

for all $\theta \in [0, 2\pi ]$.

Choose $r < 1/C$ and let $\epsilon \in [-r, r]$. Let $M(s, \theta ) = (H(s) - \hbox {e}^{i \theta } \text{ Id })^{-1}$, and write

$$\begin{aligned} H(s) - (1+\epsilon ) \hbox {e}^{i\theta }\text{ Id } = (H(s) - \hbox {e}^{i \theta } \text{ Id }) \left[ \text{ Id } - \epsilon M(s, \theta ) \hbox {e}^{i \theta } \text{ Id } \right] . \end{aligned}$$

Since

$$\begin{aligned} \Vert \epsilon M(s, \theta ) \hbox {e}^{i \theta } \text{ Id } \Vert \le r \Vert M(s, \theta )\Vert < 1, \end{aligned}$$

we have that $\text{ Id } - \epsilon M(s, \theta ) \hbox {e}^{i \theta } \text{ Id }$ is boundedly invertible by the Neumann series argument. Then

$$\begin{aligned} \left( H(s) - (1+\epsilon ) \hbox {e}^{i\theta }\text{ Id }\right) ^{-1} = \left[ \text{ Id } - \epsilon M(s, \theta ) \hbox {e}^{i \theta } \text{ Id } \right] ^{-1} M(s, \theta ), \end{aligned}$$

is a bounded linear operator for all $s \in [0,1]$ and $\epsilon \in [-r, r]$. It follows that for each $s \in [0,1]$, the spectrum of H(s) is bounded away from the unit circle by a distance of r.

Since the spectrum is uniformly bounded away from the unit circle, there are simple closed curves $\gamma _1, \gamma _2 \subset {\mathbb {C}}$ so that:

• (i) for each $s \in [0, 1]$, the unstable eigenvalues of H(s) are enclosed by $\gamma _1$,

• (ii) for each $s \in [0,1]$, the stable eigenvalues of H(s) are enclosed by $\gamma _2$,

• (iii) the curve $\gamma _1$ is contained in the complement of the unit disk and is a nonzero distance from the unit circle,

• (iv) the curve $\gamma _2$ is contained in the interior of the unit disk and is a nonzero distance from the unit circle.

Then observe that for each $s \in [0,1]$ the curves $\gamma _1, \gamma _2$ and the linear operator H(s) satisfy the hypotheses of Theorem 3.16 of Part IV, Section 4 of the book of Kato [23].

By applying this stability theorem, we have that for each $\hat{s} \in [0,1]$, there is a $\delta _{\hat{s}}$ so that for all $s \in (\hat{s}-\delta _{\hat{s}}, \hat{s} + \delta _{\hat{s}})$ the dimension of the unstable eigenspace of H(s) is constant. By compactness of [0, 1], we obtain finitely many $s_1, \ldots , s_N$, and $\delta _1, \ldots , \delta _N$ so that the dimension of the unstable eigenspace of H(s) is constant on each of the intervals $(s_j - \delta _j, s_j + \delta _j)$ and $[0, 1] \subset \bigcup _{j=1}^N (s_j - \delta _j, s_j + \delta _j)$. Then the dimension of the unstable eigenspace is constant in the intersection of any two intervals, i.e. it cannot change from one subinterval to the next. But this gives that unstable dimension is constant throughout the entire homotopy. Since $H(1) = A$ has one-dimensional unstable eigenspace, it follows that H(s) has one-dimensional eigenspace for each $s \in [0,1]$.

Step 2: Now let $h \in \ell _\nu ^1$ and note that

$$\begin{aligned} (A - \hbox {e}^{i \theta } \text {Id})^{-1}h = {\left\{ \begin{array}{ll} \left[ Q (\Sigma - \hbox {e}^{i \theta } \text {Id})^{-1} Q^{-1} h^N \right] _n &{} 0 \le n \le N \\ (a_n - \hbox {e}^{i \theta })^{-1} h_n &{} n \ge N+1 \end{array}\right. }, \end{aligned}$$

where

$$\begin{aligned} (\Sigma - \hbox {e}^{i \theta } \text {Id})^{-1} = \left( \begin{array}{lll} (\lambda _0 - \hbox {e}^{i \theta })^{-1} &{} \ldots &{} 0 \\ \vdots &{} \ddots &{} \vdots \\ 0 &{} \ldots &{} (\lambda _N - \hbox {e}^{i \theta })^{-1} \end{array} \right) . \end{aligned}$$

Then

$$\begin{aligned} \left\| \left( A - \hbox {e}^{i \theta } \text {Id}\right) ^{-1} \right\| \le \Vert Q \Vert \Vert Q^{-1} \Vert \max \left( (|\lambda _0|-1)^{-1}, (1 - |\lambda _1|)^{-1} \right) \le C. \end{aligned}$$

We now write

$$\begin{aligned} M = (A - \hbox {e}^{i \theta } \text {Id})^{-1}, \end{aligned}$$

and consider invertibility of the operator

$$\begin{aligned} H(s) - \hbox {e}^{i \theta }\text {Id}= & {} A - (1 -s)\tilde{A} - \hbox {e}^{i \theta } \text {Id} \end{aligned}$$

(61)

$$\begin{aligned}= & {} (A - \hbox {e}^{i \theta } \text {Id}) \left[ \text {Id} - M(1- s) \tilde{A} \right] . \end{aligned}$$

(62)

By hypothesis, we have that

$$\begin{aligned} \Vert M (1-s)\tilde{A} \Vert \le \Vert M \Vert \Vert \tilde{A}\Vert \le C \Vert \tilde{A}\Vert < 1, \end{aligned}$$

so that $\left[ \text {Id} - M(1- s)\tilde{A} \right]$ is boundedly invertible. From this, we obtain that $H(s) - \hbox {e}^{i \theta }\text {Id}$ is boundedly invertible for each $s \in [0,1]$, i.e. H(s) has no eigenvalues on the unit circle. By the argument of step one, $B = H(1)$ has exactly one unstable eigenvalue as desired. $\square$

Appendix 6: Computer-Assisted Proof that the Unstable Spectrum is Exactly One Eigenvalue

We consider again the Kot-Schaffer map $F :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ given by Eq. (5) with parameters $\mu > 0$ and $b, c \in \ell _\nu ^1$ as discussed in Sect. 2.1.1. Let $a_*, \xi _* \in \ell _\nu ^1$ and $\lambda _* \in {\mathbb {R}}$ be as in the Appendices 2 and 4. In particular, we consider projection dimension $N = 26$ and have

$$\begin{aligned} a_* = \bar{a} + a^\infty , \end{aligned}$$

where $\pi _N(\bar{a}) = \bar{a}^N$ and $\bar{a}_n = 0$ for $n \ge N+1$ and

$$\begin{aligned} \Vert a^\infty \Vert _\nu ^1 \le r_0. \end{aligned}$$

Using the notation of Appendix 4, let

$$\begin{aligned} A^N = DF^N(\bar{a}^N), \end{aligned}$$

and define the linear operator $A :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ by

$$\begin{aligned} (Ah)_n = {\left\{ \begin{array}{ll} (A^N h^N)_n &{} 0 \le n \le N \\ b_n h_n &{} n \ge N+1. \end{array}\right. } \end{aligned}$$

Computing rigorous enclosures of the eigenvalues and eigenvectors of $A^N$ (using IntLab) gives

$$\begin{aligned} \lambda _0 \in [-1.68829653587397, -1.68829653587393],\\ \lambda _1 \in [-0.74018685544601, -0.74018685544599], \end{aligned}$$

so that

$$\begin{aligned} \max ((|\lambda _0|-1)^{-1}, (1 - |\lambda _{1}|)^{-1}) \le 3.84892. \end{aligned}$$

Moreover we report that

$$\begin{aligned} \lambda _N \in [ -0.91030349913583, -0.91030349913582] \times 10^{-7}, \end{aligned}$$

and that $|\lambda _N| \le |\lambda _j| \le |\lambda _1| < 1$. Then we check that

$$\begin{aligned} r_3 = |b_{N+1}| = \mu 2^{-(N+1)} \in [0.74505805969238, 0.74505805969239] \times 10^{-8}, \end{aligned}$$

which verifies the hypothesis that

$$\begin{aligned} |b_{N+1}| < |\lambda _{N}|, \end{aligned}$$

as required by Lemma 5.3. Since $b_n = \mu 2^{-n}$, we see that these are strictly decreasing. We also compute (using IntLab) interval enclosures of Q and $Q^{-1}$, and using these, we check that

$$\begin{aligned} C = 2.5 \times 10^{3}, \end{aligned}$$

is a number satisfying the hypotheses of Lemma 5.3.

Now we write the differential at $a_*$ as

$$\begin{aligned}&[DF(a_*)h]_n\\&\quad = \mu b_n h_n - 2 \mu b_n (c * a_* * h)_n\\&\quad = {\left\{ \begin{array}{ll} \mu b_n h_n - 2 \mu b_n (c^N * \bar{a}^N * h^N)_n - 2 \mu b_n (c^\infty * a_* * h)_n - 2 \mu b_n (c * \bar{a}^N * h^\infty )_n - 2 \mu b_n (c * a^\infty * h)_n &{} 0 \le n \le N\\ \mu b_n h_n - 2\mu b_n (c * a_* * h)_n &{} n \ge N+1 \end{array}\right. }\\&\quad = {\left\{ \begin{array}{ll} [DF(\bar{a}^N)h^N]_n - 2 \mu b_n (c^\infty * a_* * h)_n - 2 \mu (c^N * \bar{a}^N * h^\infty )_n - 2 \mu b_n (c^N * a^\infty * h)_n &{} 0 \le n \le N\\ \mu b_n h_n - 2\mu b_n (c * a_* * h)_n &{} n \ge N+1 \end{array}\right. } \end{aligned}$$

Define the bounded linear operator $\tilde{A} :\ell _\nu ^1 \rightarrow \ell _\nu ^1$ by

$$\begin{aligned} (\tilde{A} h)_n = {\left\{ \begin{array}{ll} - 2 \mu b_n (c^\infty * a_* * h)_n - 2 \mu (c^N * \bar{a}^N * h^\infty )_n - 2 \mu b_n (c^N * a^\infty * h)_n &{} 0 \le n \le N\\ - 2\mu b_n (c * a_* * h)_n &{} n \ge N+1 \end{array}\right. }, \end{aligned}$$

so that

$$\begin{aligned} DF(a_*) = A + \tilde{A}. \end{aligned}$$

We write

$$\begin{aligned} \tilde{A} = \tilde{A}_1 + \tilde{A}_2 + \tilde{A}_3 + \tilde{A}_4, \end{aligned}$$

where $\tilde{A}_1, \tilde{A}_2, \tilde{A}_3, \tilde{A}_4 :\ell _\nu ^1 \rightarrow \ell _\nu$ are defined by

$$\begin{aligned} (\tilde{A}_1 h)_n= & {} {\left\{ \begin{array}{ll} - 2 \mu b_n (c^\infty * a_* * h)_n &{} 0 \le n \le N \\ 0 &{} n \ge N+1 \end{array}\right. },\\ (\tilde{A}_2 h)_n= & {} {\left\{ \begin{array}{ll} - 2 \mu b_n (c^N * \bar{a}^N * h^\infty )_n &{} 0 \le n \le N \\ 0 &{} n \ge N+1 \end{array}\right. },\\ (\tilde{A}_3 h)_n= & {} {\left\{ \begin{array}{ll} - 2 \mu b_n (c^N * a^\infty * h)_n &{} 0 \le n \le N \\ 0 &{} n \ge N+1 \end{array}\right. }, \end{aligned}$$

and

$$\begin{aligned} (\tilde{A}_4 h)_n = {\left\{ \begin{array}{ll} 0 &{} 0 \le n \le N \\ - 2\mu b_n (c * a_* * h)_n &{} n \ge N+1 \end{array}\right. }. \end{aligned}$$

We bound $\Vert \tilde{A}_2\Vert$ using Lemma 3.3. More precisely, we take

$$\begin{aligned} \kappa _n^1 := \max _{N+1 \le k \le 2N-n} \frac{| (c^N * \bar{a}^N )_{n+k} |}{2 \nu ^k}, \quad \kappa _n^2 := \max _{N+1 \le k \le 2N+n} \frac{| (c^N * \bar{a}^N)_{k-n} |}{2 \nu ^k} \end{aligned}$$

as in Lemma 3.3. Then

$$\begin{aligned} \Vert \tilde{A}_2\Vert \le 4 |\mu | \sum _{n=0}^N |b_n| (\kappa _n^1 + \kappa _n^2) \nu ^n. \end{aligned}$$

For the remaining terms, we have that

$$\begin{aligned} \Vert \tilde{A}_1 \Vert \le 2 |\mu | \Vert B a_*\Vert \Vert c^\infty \Vert ,\\ \Vert \tilde{A}_3 \Vert \le 2 |\mu | \Vert B c^N\Vert \Vert a^\infty \Vert , \end{aligned}$$

and

$$\begin{aligned} \Vert \tilde{A}_4\Vert \le 2 |\mu | \Vert c\Vert \Vert a_*\Vert r_3. \end{aligned}$$

Using interval arithmetic, we compute the bound

$$\begin{aligned} \Vert \tilde{A}\Vert\le & {} \Vert \tilde{A}_1\Vert + \Vert \tilde{A}_2\Vert + \Vert \tilde{A}_3\Vert + \Vert \tilde{A}_4\Vert \\\le & {} 9\times 10^{-8}, \end{aligned}$$

so that

$$\begin{aligned} C \Vert \tilde{A}\Vert \le 2.25 \times 10^{-4} < 1. \end{aligned}$$

Then by Lemma 5.3, and the fact that A has exactly one unstable eigenvalue, we conclude that $DF(a_*) = A + \tilde{A}$ has exactly one eigenvalue, that the unstable manifold of $a_*$ is exactly one dimensional, and the computations in the main body of the present work are justified. The reader interested in the details of the proof should see the IntLab program a_KotPaperProofs_firstOrderData.m from [22] for more details.