Computing (Un)stable Manifolds with Validated Error Bounds: Non-resonant and Resonant Spectra

Stable and unstable manifolds are fundamental building blocks for understanding the global dynamics of nonlinear differential equations. Since closed-form analytic expressions for stable/unstable manifolds are rarely available, considerable effort goes into developing numerical techniques for their approximation, see e.g., Krauskopf et al. (2005), Haro and da la Llave (2006), Beyn and Kless (1998) and Castelli et al. (2015) and the references therein. One powerful tool for studying invariant manifolds (stable, unstable, strongly (un)stable and even more general ones) is the parameterization method of Cabré et al. (2003a, b, 2005). The parameterization method is based on formulating certain operator equations (invariance equations) which simultaneously describe both the dynamics on the manifold and its embedding. The method has been implemented numerically to study a variety of problems involving stable and unstable manifolds of equilibria and fixed points (Mireles-James 2013; Mireles-James and Mischaikow 2013; Mireles James and Lomelí 2010; Haro 2011; van den Berg et al. 2011), stable/unstable manifolds of periodic orbits for differential equations (Guillamon and Huguet 2009; Huguet and de la Llave 2013; Castelli et al. 2015), quasi-periodic invariant sets in dynamical systems (Haro and da la Llave 2006) and more recently in order to simultaneously compute invariant manifolds with their unknown dynamics (Canadell and Haro 2014; Haro et al. 2014), to mention just a few examples.

The last reference is a book which provides many other examples and much fuller discussion of the literature.

In addition to facilitating efficient numerical computations, the functional analytic framework of the parameterization method also provides a natural setting for a posteriori analysis of errors. The works of van den Berg et al. (2011), Mireles-James and Mischaikow (2013) and Mireles-James (2015) exploit this a posteriori analysis and implement mathematically rigorous numerical validation methods for the stable and unstable manifolds. The term “validation” here expresses the fact that the computations provide explicit bounds on all approximation errors involved.

Approximate parametrizations are often computed by substituting a power series ansatz into the invariance equation and deriving a sequence of homological equations for the power series coefficients. These homological equations are then solved recursively to any desired order. In this paper we employ an alternative methodology for solving the invariance equation. We recast the infinite system of homological equations as a nonlinear zero finding problem on a Banach space of geometrically decaying sequences, and we implement a parametrized Newton–Kantorovich argument in the style of Yamamoto (1998).

There are three advantages to using a Newton method. First, we note the works of van den Berg et al. (2011), Mireles-James and Mischaikow (2013) and Mireles-James (2015) assume that certain non-resonance conditions between the eigenvalues are satisfied. The main goal of the present work is to weaken this assumption. We build on the theory of Cabré et al. (2003a, b, 2005) and develop computer-assisted methods for rigorous error bounding even in the face of a resonance. More precisely we develop validation schemes which apply at any co-dimension one resonance between the eigenvalues. The formulation of the resonant as well as non-resonant zero finding problems on an infinite sequence space unifies the presentation and implementation as well as the necessary a posteriori analysis.

Second, the zero finding methodology based on a numerical Newton method for the truncated problem leads to improved numerical performance even in the case where validated numerics are not desired. The Newton iteration can always be started from the linear approximation of the manifold by its eigenvectors; however, one has the option of improving the convergence by starting the iteration from a polynomial obtained by solving a few of the lower-order homological equations recursively. Once the iteration begins the order of the polynomial approximation is roughly doubled at each step. The freedom one has in choosing the lengths of the eigenvectors is exploited in order to guarantee that the Taylor coefficients decay at the desired rate. A similar numerical Newton scheme for computing invariant manifolds in discrete time dynamical systems (without rigorous validation) was used in Mireles James and Lomelí (2010). Indeed this kind of Newton scheme is always needed when the parameterization method is applied in Fourier space, see again Haro et al. (2014) and the references therein.

Third, existing rigorous continuation methods (van den Berg et al. 2010; Breden et al. 2013) are directly applicable to our novel approach, thus putting the rigorous computation of branches of connecting orbits within direct reach. Of course, from the viewpoint of continuation, all three advantages are crucial. Indeed, while continuing along one-parameter families of (un)stable manifolds resonances are encountered generically and are thus unavoidable.

Our use of parameterized Newton–Kantorovich arguments is guided by the work of a number of authors on the so-called method of radii polynomials. This method provides a tool kit for solution and continuation of zero finding problems in infinite dimensions. In this method one derives certain polynomials whose coefficients encode information about the approximate solution, the choice of approximate inverse of the derivative, the local regularity properties of the problem and the choice of Banach space on which to work. Once these givens are fixed the roots of the radii polynomials yield not only existence and uniqueness results, but also tight error estimates and isolation bounds for the problem. Radii polynomial methods have been applied successfully to a number of problems in dynamical systems, partial differential equations and delay equations, and we refer the interested reader to Day et al. (2007), Breden et al. (2013), van den Berg et al. (2010), van den Berg et al. (2011), Hungria et al. (2016) and van den Berg et al. (2015) for more discussion and references.

When there is a resonance between the stable eigenvalues, the conjugacy map P as constructed above is no longer analytic and cannot be expressed as a convergent power series. One way to resolve this obstacle is to change the flow \(\theta ^{\prime }=h(\theta )\) in parameter space to a nonlinear “normal form,” in such a way that the conjugacy is again analytic. In this paper we consider two types of resonances in particular, namely the co-dimension one resonances. The first type of resonance is a single “regular” resonance \(\tilde{k}\cdot \lambda = \lambda _{\tilde{\imath }} \in \mathbb {R}\) for some \(1 \le \tilde{\imath }\le d\) and \(\tilde{k}\in \mathbb {N}^d\) with \(\sum _{i=1}^d \tilde{k}_i \ge 2\), and no other resonances. The second type is a double real eigenvalue (\(\tilde{k}=e_i\) for some \(i \ne \tilde{\imath }\)) with geometric multiplicity one, and no other resonances. We note that a double eigenvalue is not a resonance in the strict sense, but we nevertheless use this “uniform” terminology in the current paper. These are the only co-dimension one resonances; hence, they are the types that are encountered generically in one-parameter continuation. For this reason we restrict our attention to these resonance types as our examples. We note that a completely analogous approach works for resonances of higher co-dimension, but here omit the details.

In order to illustrate the application of our methods we discuss three example problems in detail. First we analyze the stable manifold of the origin in the well-known Lorenz equations. We use this model system to scrutinize our method in the non-resonant case and show how the structure of the vector field is directly reflected in the bounds used for validation. Second we tune the parameter in the Lorenz system to obtain double stable eigenvalues at the origin to showcase our method in this context.

Following the approach of van den Berg et al. (2015) we prove the existence of a heteroclinic connection between the hexagon and ground states. The proof is based on rigorous numerics for a boundary value problem, where the validated manifolds are used to formulate the boundary conditions. We settle a case left open in van den Berg et al. (2015) due to the presence of resonances in the stable eigenvalues at the origin. In Fig. 1 we depict the verified connecting orbit of (1) as well as the corresponding stationary transition layer between hexagonal spots and the uniform state of (2).

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We remark that an interesting future extension of this work would be to apply the ideas developed here to the validated computation of local stable/unstable manifolds of periodic orbits. The parameterization method has already been adapted to this context. See Guillamon and Huguet (2009), Huguet and de la Llave (2013) and Castelli et al. (2015) for more details and numerical examples. Using the techniques of the present work it should be possible to validate these computations even in the presence of a resonance between the Floquet multipliers. This will be the subject of a future study.

Finally we remark that the references mentioned in this introductory discussion are far from exhaustive, and a comprehensive overview of the literature is beyond the scope of the present work. In recent years a number of authors have developed numerical validation procedures which provide mathematically rigorous a posteriori error bounds on approximations of invariant manifolds associated to various kinds of invariant sets. We refer the interested reader to CAPD (2015), Capinksi and Simo (2012), Johnson and Tucker (2011), Wittig et al. (2010) and Wittig (2011) for fuller discussion of methods other than those presented here.

The outline of the paper is as follows. Section 2 is dedicated to the description of the general setup of our approach. In Sect. 3 we give more details on how we derive the zero finding problem. In Sect. 4 we transform it to an equivalent local fixed point problem to be solved by a parametrized Newton–Kantorovich-type argument. In Sect. 5 we illustrate the performance of our method with the three examples described above. The code implementing these examples can be found at the webpage Code page (2015).

In this section we derive the zero finding problem on the space of geometrically decaying series coefficients whose solution corresponds to a solution P of (8) via (7). The functional analytic setup is close to the one utilized in Hungria et al. (2016) with the main difference lying in the convolution structure.

We now can make the correspondence between zeros of f and parametrizations of local stable manifolds more precise.

We aim to show that T is contraction on a small ball around an approximate zero \(\widehat{x}=\iota \widehat{x}_F \in \mathcal {X}\). It follows from Lemma 4.5 that if \(\widehat{x}\) is symmetric (\(\widehat{x}^* = \widehat{x}\)) or almost symmetric, then the unique fixed point of T in the ball is a symmetric zero of f (provided A is injective).

The crux of this definition is that the bounds Y on the residue and Z on the derivative of T can be constructed explicitly, see the examples in Sect. 5. We note that the inclusion of the scalar r in (55) trivially scales the bounds on DT by a factor r. We include this factor here to keep the notation compatible with earlier papers (going back to Yamamoto 1998). The radius r of the ball is not fixed a priori and the \(l_0+n\) radii polynomials p(r) are used in the following parametrized version of the Newton–Kantorovich theorem.

A proof of this lemma can be found, e.g., in Yamamoto (1998), Day et al. (2007) and Hungria et al. (2016).

To derive the bounds (54) and (55) good analytical control on the operator f and its derivative Df at \(\widehat{x}\) is essential. In Sects. 5.1 and 5.2 we illustrate the “mechanics” involved in the derivation of these bounds.

In this section we consider three applications to illustrate the performance of our method. We start with the well-known Lorenz equations and compute a 2D local stable manifold at the classical parameter values yielding non-resonant eigenvalues. Subsequently, we consider non-standard parameter values in the Lorenz system to investigate the specific issues in the double eigenvalue case. Finally we consider the case of regular resonant eigenvalues in a system of ODEs originating from a pattern formation model. We compute (un)stable manifolds that serve as ingredient for a connecting orbit computation in this system.

5.1 Local Manifolds in the Lorenz System

We consider the Lorenz differential equations (Lorenz 1963)

$$\begin{aligned} \dot{u} = g(u) = \begin{pmatrix} \sigma (u_2-u_1)\\ \rho u_1-u_2-u_1u_3\\ u_1u_2-\beta u_3 \end{pmatrix}. \end{aligned}$$

(59)

In Sect. 5.1.1 we choose the classical parameter values \(\beta = \frac{8}{3}\) and \(\sigma = 10\) and \(\rho = 28\) and describe the validation process including the derivation of the necessary bounds for the 2D local stable manifold of the origin (local Lorenz manifold) in some detail. In particular, we investigate how to choose the various computational parameters involved in the validation process taking different objectives into account. In Sect. 5.1.2 we use the fact that we have explicit formulas for the stable eigenvalues at the origin to tune the parameters \(\beta ,\rho ,\sigma \) such that there are double eigenvalues at the origin. We explain the validation analysis in this context, which serves as preparation for the regular resonant case.

5.1.1 Detailed Analysis for the Local Lorenz Manifold: Non-resonant Eigenvalues

The dimension of the manifold is \(d = 2\), and the phase space dimension is \(n = 3\). Denote the stable eigenvalues by \(\lambda _{1,2}\) and the corresponding eigenvectors by \(\xi _{1,2}\). As a matter of fact \(\lambda _{1,2}\in {\mathbb {R}}\). Hence, we search for a map \(P: \mathbb {R}^{2}\supset \mathbb {B}_{\nu }\rightarrow \mathbb {R}^{3}\) together with \(\nu = (\nu _1,\nu _2)\) fulfilling (8), with g given by the Lorenz vector field (59), and the linear constraints (9). As we are in the non-resonant case we can choose \(\nu = \hat{\nu }\) as discussed in Sect. 2.5. We make the power series ansatz (7) with \(a_{k} = (a_{k_1 k_2}^{1},a_{k_1 k_2}^{2},a_{k_1 k_2}^{3})\in \mathbb {R}^{3}\). Note that the coefficients \(c_{k} = (c_{k_1k_2}^{1},c_{k_1k_2}^{2},c_{k_1k_2}^{3})\) in the expansion (18) are

$$\begin{aligned} c_{k_1 k_2} = \begin{pmatrix} \sigma \left( a_{k_1 k_2}^{2}-a_{k_1 k_2}^{1}\right) \\ \rho a_{k_1 k_2}^{1} - a_{k_1 k_2}^{2} - (a^{1}*a^{3})_{k_1 k_2}\\ (a^{1}*a^{2})_{k_1 k_2} - \beta a^{3}_{k_1 k_2} \end{pmatrix} \end{aligned}$$

(60)

with \(*\) defined in (36). Together with \(p = 0\in \mathbb {R}^3\) this completes the ingredients for \(f^{\text {nonres}}\) defined in (39).

Recalling the definition of the radii polynomials in (56b), we notice that a necessary condition for finding a radius r fulfilling \(p^{i}(r)<0\) for \(i=1,2,3\) is that the components of \(Z_{1}\) derived in (65) be smaller than one. The parameters that are under our direct control are \(m = (m_1,m_2)\) and \(\nu = (\nu _1,\nu _2)\) or \(\Vert \xi _{1,2}\Vert \), respectively. Note that varying \(\nu _{1,2}\) is in the following precise sense equivalent to varying \(\xi _{1,2}\), which implies that we may as well fix either \(\nu \) or \(\xi \).

Remark 5.1

From the scaling operation analyzed in Lemma 2.2, we notice that

$$\begin{aligned} \Vert \mu a^j \Vert _{\nu } = \Vert a^j\Vert _{\mu \nu }, \qquad j=1,2,3. \end{aligned}$$

This implies that we should either vary the decay rates (domain radius) \(\nu \) or the (eigenvector) scalings \(\mu \) to affect \(\Vert \hat{a}^{j}\Vert _{\nu }\), \(j = 1,2,3\) in \(Z_1\). Below we fix \(\nu = (1,1)\), as this is numerically the most stable choice.

Let us describe the dependence of the coefficients of \(p^i(r)\) on these computational parameters in more detail by deriving explicit formulas for the radii polynomials.

Derivation of the bounds: details To define the radii polynomials specified in (56b) we first need to compute the bounds \(Y^j\) and \(Z^j\) \((j = 1,2,3)\) defined in (54) and (55). Assume we have already calculated an approximate zero \(\hat{x} = (\hat{a}^{1},\hat{a}^{2},\hat{a}^{3}) = \iota \hat{x}_{F}\) where \(\iota \) is defined in (49), and set \(A_{m} \approx (Df^{m}(\hat{x}_{F}))^{-1}\). We start by noting that \((f(\hat{x}))_{k} = 0\) for \(k\notin \mathcal {I}_{2m}\), since g(u) is quadratic. Using this and the fact that \(T(\hat{x})-\hat{x} = Af(\hat{x})\) we set \(y_{k}^j = (|(A_mf^{m}(\hat{x}))_{k}^j|)\) for \(k\in \mathcal {I}_{m}\), \(y_{k}^j = \frac{1}{k_1|\lambda _1|+k_2|\lambda _2|}((|f^{2m}(\hat{x}))_{k}^j|)\) for \(k\in \mathcal {I}_{2m}\setminus \mathcal {I}_m\) and \(y_{k} = 0\in \mathbb {R}^{3}\) for \(k\notin \mathcal {I}_{2m}\). The bounds \(Y^j\) \((j = 1,2,3)\) are then obtained by computing the finite sums \(\Vert y^{j}\Vert _{\nu }\).

To derive the bounds \(Z^{j}(r)\) defined in (55) we use the splitting (53). Let \(v,w\in B_{1}(0)\). We start by deriving an expansion

$$\begin{aligned} (Df(\hat{x}+rv)rw- A^{\dag }rw)_{k} = z_{k,1}r + z_{k,2}r^{2}\qquad \text {with } z_{k,1},z_{k,2}\in \mathbb {R}^3. \end{aligned}$$

(61)

The explicit expressions for z are listed in Table 1.

Table 1

Expansion coefficients for \((Df(\bar{x}+rv)rw- A^{\dag }rw)_{k}\) in (61)

	\(k\in \mathcal {I}_{m}\)	\(k\notin \mathcal {I}_{m}\)
\(z_{k,1}\)	\(\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}\)	\(\begin{pmatrix}\sigma (w^{2}_{k}-w^{1}_{k})\\ \rho w^{1}_{k}-w^{2}_{k}-((\hat{a}^{1}*w^{3})_{k}+(\hat{a}^{3}*w^{1})_{k})\\ (\hat{a}^{1}*w^{2})_{k}+(\hat{a}^{2}*w^{1})_{k}-\beta w^{3}_{k}\end{pmatrix}\)
\(z_{k,2}\)	\( \begin{pmatrix}0\\ -((w^{1}*v^{3})_{k}+(w^{3}*v^{1})_{k})\\ (w^{1}*v^{2})_{k}+(w^{2}*v^{1})_{k}\end{pmatrix}\)	\(\begin{pmatrix}0\\ -((w^{1}*v^{3})_{k}+(w^{3}*v^{1})_{k})\\ (w^{1}*v^{2})_{k}+(w^{2}*v^{1})_{k}\end{pmatrix}\)

Our next goal is to compute a bound

$$\begin{aligned} |(DT(\hat{x}+rv)rw)_{k}^j|\le Z^j_{k,1}r + Z^j_{k,2}r^2. \end{aligned}$$

(62)

The first term to estimate is \((\text {Id }-AA^{\dag })w\). We introduce the notation \(A^{\dag }_m \mathop {=}\limits ^{\text{ def }}Df^m(\widehat{x}_F)\), and by using \(|w^{j}_k|\le \frac{1}{\nu ^k}\) we compute, by a direct computation, \(\epsilon _{k}\in \mathbb {R}^{3}\), \(k\in {\mathcal {I}}_m\) such that

$$\begin{aligned} |((\text {Id }-AA^{\dag })w)_{k}^j| \le (|\text {Id }-A_mA^{\dag }_m| \, |w_F|)_k^j \le \epsilon _{k}^j \qquad \text {for all }k\in {\mathcal {I}}_m, \end{aligned}$$

(63)

where absolute values are to be understood component-wise. Furthermore, for the tail terms we have \(((\text {Id }-AA^{\dag })w)_{k} =0\in \mathbb {R}^{3}\) for \(k\notin {\mathcal {I}}_m\) by construction. To accommodate matrix multiplication we collect all estimates of \(z_{k,2}\) for \(k\in \mathcal {I}_m\) in a vector. Namely, we define the vector \(\chi _{2} \in \mathbb {R}^{3n_m}\) with \(n_m = (m_1+1)(m_2+1)\) such that

$$\begin{aligned} |z_{k,2}^j|\le \chi ^{3(k_2(m_1+1)+k_1)+j}_2 \qquad \text {for } 0\le k_1 \le m_1, 0\le k_2 \le m_2, \quad j = 1,2,3. \end{aligned}$$

(64)

To be explicit, we use the estimate

$$\begin{aligned} \chi ^{3(k_2(m_1+1)+k_1)+1}_2 = 0,\quad \chi ^{3(k_2(m_1+1)+k_1)+2}_2 =\frac{2}{\nu ^k},\quad \chi ^{3(k_2(m_1+1)+k_1)+3}_2 = \frac{2}{\nu ^k}. \end{aligned}$$

Then applying the definition of A given in (51), and interpreting \(A_m\) as an \(3n_m \times 3n_m\) matrix, yields the values for \(Z_{k,1}\) and \(Z_{k,2}\) summarized in Table 2.

Table 2

Expansion coefficients for \(Z_{k}(r) = Z_{k,1}r + Z_{k,2}r^2\)

	\(k\in \mathcal {I}_{m}\)	\(k\notin \mathcal {I}_{m}\)
\(Z_{k,1}\)	\(\epsilon _{k}\)	\(\frac{1}{k_1|\lambda _1|+k_2|\lambda _2|}\begin{pmatrix}\sigma (|w|^{2}_{k}+|w|^{1}_{k})\\ \rho |w|^{1}_{k}+|w|^{2}_{k}+((|\hat{a}|^{1}*|w|^{3})_{k}+(|\hat{a}|^{3}*|w|^{1})_{k})\\ (|\hat{a}|^{1}*|w|^{2})_{k}+(|\hat{a}|^{2}*|w|^{1})_{k}+\beta |w|^{3}_{k}\end{pmatrix}\)
\(Z_{k,2}\)	\( (|A_{m}| \chi _{2})_{k}\)	\(\frac{1}{k_1|\lambda _1|+k_2|\lambda _2|}\begin{pmatrix}0\\ ((|w|^{1}*|v|^{3})_{k}+(|w|^{3}*|v|^{1})_{k})\\ (|w|^{1}*|v|^{2})_{k}+(|w|^{2}*|v|^{1})_{k}\end{pmatrix}\)

All absolute values are to be understood component-wise. The values of \(\epsilon _k\) are defined in (63)

Finally, by estimating the finite part (\(k\in {\mathcal {I}}_m\)) and the tail part (\(k \notin {\mathcal {I}}_m\)) separately, we can compute \(Z^j(r) = Z_{1}^j r+Z_{2}^j r^{2}\) by

$$\begin{aligned} Z_{1}= & {} \displaystyle \sum _{k\in \mathcal {I}_{m}}\epsilon _{k}\nu ^{k} + \frac{1}{\min \bigl \{(m_1+1)|\lambda _1|,(m_2+1)|\lambda _2|\bigr \}} \begin{pmatrix} 2|\sigma |\\ |\rho |+1+\Vert \hat{a}^1\Vert _{\nu }+\Vert \hat{a}^{3}\Vert _{\nu }\\ |\beta | + \Vert \hat{a}^1\Vert _{\nu }+\Vert \hat{a}^{2}\Vert _{\nu } \end{pmatrix} \end{aligned}$$

(65a)

$$\begin{aligned} Z_{2}= & {} \displaystyle \sum _{k\in \mathcal {I}_{m}}(|A_m| \chi _{2})_{k}\nu ^{k} + \frac{1}{\min \bigl \{(m_1+1)|\lambda _1|,(m_2+1)|\lambda _2|\bigr \}} \begin{pmatrix} 0\\ 2\\ 2 \end{pmatrix}, \end{aligned}$$

(65b)

where we have used the Banach algebra convolution estimate in \(W^{\nu }\) to bound the tail sums. This completes the ingredients for (56b).

The main influence of \(\nu \) on these estimates is via the terms \(\Vert \hat{a}^{i}\Vert _{\nu }\) \((i = 1,2,3)\). On the other hand these norms are also controlled by the length of the stable eigenvectors \(\xi _{1,2}\) appearing in definition (39), and also, albeit weakly, by \(m = (m_1,m_2)\). Let us analyze this interplay.

Choice of m and \(\Vert \xi _{1,2}\Vert \) The considerations that lead to the settings of the computational parameters depend on the goals in the application at hand. One might, for example, be interested in uniformly maximizing the image in phase space of the parametrization, or one might want to capture a particular point in phase space in the image, i.e., maximize the image in a certain direction. In the following we collect several observations for either task. Let us begin with a fundamental restriction; namely, the components \(Z^{j}_1\) in (65) need to be smaller than one. This can be used as a first feasibility check for the validation.

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This motivates the following procedure (recall that we have fixed \(\nu =(1,1)\), see Remark 5.1):

1. 1.

   Choose an order \(m = (m_1,m_2)\). Compute \(\hat{a}\) with \(\Vert \xi _{1,2}\Vert = 1.\)

    
2. 2.

   Attempt to check the conditions of Lemma 4.6 using the formulas derived for (54) and (56b).

    
3. 3.

   • In case of failure rescale \(\xi _{1,2} = \mu _{1,2}\xi _{1,2}\) and \(\hat{a} = \mu \hat{a}\) with \(0<\mu _i< 1\) \((i = 1,2)\) and repeat the second step.

   • In case of success rescale \(\xi _{1,2} = \mu _{1,2}\xi _{1,2}\) and \(\hat{a} = \mu \hat{a}\) with \(\mu _i>0\) \((i = 1,2)\) chosen according to the maximization objective and repeat the second step until stop at failure.

    

In this procedure there remain two open choices, namely of \(m = (m_1,m_2)\) and \(\mu = (\mu _1,\mu _2)\). Let us review two options. More thorough discussion of the choice of scalings, including algorithms and implementations, can be found in Breden et al. (2015).

1. 1.

   \(m_1 = m_2\) and \(\mu _1 = \mu _2>1\): uniformly maximizing the image

   We consider the two cases \(m = (5,5)\) and \(m = (15,15)\), see Fig. 2. First, for \(m = (5,5)\), the validation succeeds with \(\Vert \xi _{1,2}\Vert = 1\) with a radius of \(r \approx 10^{-7}\). Recall that by Remark 4.2 the validation radius r can be seen as an accuracy measure. We thus consider smaller radius r as higher accuracy. After conducting 5 rescalings with factors \(\mu _1 = \mu _2 = \frac{7}{6}\) to uniformly maximize the image, we fail to validate. The accuracy r we obtain after 4 rescalings decreased to \(10^{-5}\) (for fixed m the (uniform estimate on the) accuracy naturally decreases when increasing the domain). Second, for \(m = (15,15)\) we also succeed to validate for \(\Vert \xi _{1,2}\Vert = 1\) but with smaller uniform error bound \(r \approx 10^{-15}\). We are able to rescale 15 times with the same factors. This increases \(\Vert \xi _{1,2}\Vert \) to 10.09 and increases the validation radius to \(10^{-2}\).

    
2. 2.

   Fast-slow choice of m and \(\mu \): maximizing the image in the slow direction

   Next we consider the cases \(m_1\ne m_2\) and/or \(\mu _1\ne \mu _2\), see Fig. 3. We recall that \(|\lambda _{1}|\approx 10|\lambda _2|\), hence we refer to \(\lambda _{1/2}\) as the fast/slow eigenvalue. For most orbits the dynamics close to the origin is dominated by the slow direction. This is, for example, of interest when computing connecting orbits that approach the equilibrium along the slow direction. Capturing a large portion of the slow direction can thus be desirable. We choose \(m_2 \ge m_1\) and \(\mu _2>1>\mu _1\). First we choose \(m_1 = m_2 = 15\) . We succeed to validate for \(\Vert \xi _{1,2}\Vert = 1\) with a radius \(r\approx 10^{-15}\). Let \(\mu _1 = \frac{6}{7}\) and \(\mu _2 = \frac{7}{6}\). We obtain 14 successful rescalings with gradually decreasing accuracy (\(r \approx 10^{-3}\) after 14 rescalings). If we choose \(m_1 = 5\) and \(m_2 = 15\) we observe qualitatively different behavior of the validation radii.

   Starting with a success at \(\Vert \xi _{1,2}\Vert = 1\) and radius \(r\approx 10^{-8}\) the accuracy increases to \(10^{-11}\) in the first 8 rescalings with the factors \(\mu _1 = \frac{6}{7}\) and \(\mu _2 = \frac{7}{6}\) until the norms of \(\Vert \xi _{1,2}\Vert \) “align” with the choice of m. Then the accuracy decreases to \(10^{-3}\) after 18 rescalings.

    

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The above considerations can serve as a starting point for more elaborate future investigations. One might, for example, devise an optimization scheme in which one takes not only the radius r as an unknown in the radii polynomials but also considers \(\nu \) or \(\Vert \xi _{1,2}\Vert \), respectively, as variables.

5.1.2 Analysis for the Local Lorenz Manifold: Double Eigenvalues

In order to analyze the situation for double eigenvalues as discussed in Sect. 2.3, we choose for the parameters in the Lorenz system (59) the relation \(\rho =1+(\sigma +1)^2/(4\sigma )\), leading to double eigenvalues \(\lambda =-(\sigma +1)/2\). Using this data we set up the operator \(f^{\text {double}}\) specified in (40). Note that the (generalized) eigenvectors \(\xi _{1,2}\) fulfilling (23) can be computed explicitly in this model case. For the general case we refer to methods developed in Alefeld and Spreuer (1986) and Rump (2001). It will be subject of future work to discuss their applicability in the current context. Let us now discuss the influence of the choice of parameters m and \(\nu \) (or \(\Vert \xi _{1,2}\Vert \)) we first define the radii polynomials as we did in 5.1.1.

To compute the bounds \(Y^j\) \((j = 1,2,3)\) defined in (54) and \(Z^j\) \((j = 1,2,3)\) defined in (55) we follow the same strategy as in Sect. 5.1.1. First the derivation of the Y-bounds is exactly analogous. In deriving \(z_{k,i}(r)\), \(i=1,2\) fulfilling in the analogue of (61) we notice that the only difference from the formulas in Table 1 is induced by the additional linear term \((k_1+1)a_{\overline{k}}\), with \(\overline{k}\mathop {=}\limits ^{\text{ def }}(k_1+1,k_2-1)\) in \(f^{\text {double}}\) for \(|k|\ge 2\) and \(k_2>0\). For \(k_2=0\) one should read \(a_{\overline{k}}\equiv 0\). The resulting differences in \(z_{k,1}\) can be read off in Table 3.

Table 3

Expansion coefficients for \((Df(\hat{x}+rv)rw- A^{\dag }rw)_{k}\) for the double eigenvalue case, using the standard Kronecker \(\delta \) symbol

	\(k\in \mathcal {I}_{m}\)	\(k\notin \mathcal {I}_{m}\)
\(z_{k,1}\)	\(\delta _{k_1m_1}(m_1+1)w_{\overline{k}}\)	\(\begin{pmatrix}\sigma (w^{2}_{k}-w^{1}_{k})\\ \rho w^{1}_{k}-w^{2}_{k}-((\hat{a}^{1}*w^{3})_{k}+(\hat{a}^{3}*w^{1})_{k})\\ (\hat{a}^{1}*w^{2})_{k}+(\hat{a}^{2}*w^{1})_{k}-\beta w^{3}_{k}\end{pmatrix} + (k_1+1)w_{\overline{k}}\)

To obtain \(Z_{k}(r)\) fulfilling the equivalent of (62) we obtain, in addition to \(\epsilon _{k}\) from (63) and \(\chi _{2}\) from (64), a vector \(\chi _{1}\in {\mathbb {R}}^{3n_m}\) such that

$$\begin{aligned} |z_{k,1}^{j}| \le \chi _{1}^{3(k_2(m_1+1)+k_1)+j}\qquad j = 1,2,3. \end{aligned}$$

To be precise, we set \(\chi _1^{3(k_2(m_1+1)+m_1)+j} = (m_1+1)\nu _1^{-(m_1+1)}\nu _{2}^{-(k_2-1)}\) for all \(1 \le k_2 \le m_2\) and \(j=1,2,3\), whereas all other components of \(\chi _1\) vanish. To obtain the analogue of \(Z_1\) in (65) we need to bound the sum

$$\begin{aligned} \displaystyle \sum _{k\notin {\mathcal {I}}_m}\frac{k_1+1}{|\lambda _1|(k_1+k_2)}|w_{\overline{k}}^j|\nu _1^{k_1}\nu _2^{k_2} = \frac{\nu _2}{\nu _1}\displaystyle \sum _{k\notin {\mathcal {I}}_m} \frac{k_1+1}{|\lambda _1|(k_1+k_2)}|w_{\overline{k}}^j|\nu _1^{k_1+1}\nu _2^{k_2-1}. \end{aligned}$$

The following lemma can be used to control the factor \(\frac{k_1+1}{|\lambda _1|(k_1+k_2)}\) uniformly for \(k\notin {\mathcal {I}}_m\). We formulate it in this more general form as it will be reused for more general terms of this type in analyzing resonant eigenvalues in Sect. 5.2.

Lemma 5.1

Let \(m\in \mathbb {N}^{d}\) with \(m_{\tilde{\imath }} \ge 1\) for an \(\tilde{\imath }\in \{1,\ldots ,d\}\). Let

$$\begin{aligned} Q_{\tilde{\imath }}\mathop {=}\limits ^{\text{ def }}\min \{ |\lambda _i|(m_i+1) : i \ne \tilde{\imath }\}. \end{aligned}$$

Then we have, with \(\mathcal {I}_{m} = \{ k \in \mathbb {N}^d : k \preceq m \}\),

$$\begin{aligned} \max _{k\in \mathbb {N}^d\setminus \mathcal {I}_{m}} \frac{k_{\tilde{\imath }}+1}{|\lambda _1|k_1+\ldots +|\lambda _d|k_d} \le \max \left\{ \frac{m_{\tilde{\imath }}+2}{|\lambda _{\tilde{\imath }}| (m_{\tilde{\imath }}+1) } \,,\, \frac{m_{\tilde{\imath }}+1}{Q_{\tilde{\imath }}+|\lambda _{\tilde{\imath }}| m_{\tilde{\imath }}} \,,\, \frac{1}{Q_{\tilde{\imath }}} \right\} . \end{aligned}$$

(66)

Proof 5.1

For any \(k \in \mathcal {I}_m\) there is at least one \(j \in \{1,\ldots ,d\}\) such that \(k_j \ge m_j+1\). We distinguish two cases:

• \(k_{\tilde{\imath }}\ge m_{\tilde{\imath }}+1\). Then

  $$\begin{aligned} \frac{k_{\tilde{\imath }}+1}{|\lambda _1|k_1+\ldots +|\lambda _d|k_d} \le \frac{k_{\tilde{\imath }}+1}{|\lambda _{\tilde{\imath }}| k_{\tilde{\imath }} } \le \frac{m_{\tilde{\imath }}+2}{|\lambda _{\tilde{\imath }}| (m_{\tilde{\imath }}+1)}. \end{aligned}$$
• \(0\le k_{\tilde{\imath }}\le m_{\tilde{\imath }}\). We estimate

  $$\begin{aligned} \frac{k_{\tilde{\imath }}+1}{|\lambda _1|k_1+\ldots +|\lambda _d|k_d} \le \frac{k_{\tilde{\imath }}+1}{Q_{\tilde{\imath }}+|\lambda _{\tilde{\imath }}| k_{\tilde{\imath }}} \le \max \left\{ \frac{m_{\tilde{\imath }}+1}{Q_{\tilde{\imath }}+|\lambda _{\tilde{\imath }}| m_{\tilde{\imath }}}, \frac{1}{Q_{\tilde{\imath }}} \right\} . \end{aligned}$$

The proof follows from combining these estimates. \(\square \)

Following the same steps as in Sect. 5.1.1 we obtain \(Z_{k,1}\) as given in Table 4, while \(Z_{k,2}\) remain unaltered from Table 2.

Table 4

Expansion coefficients for \(Z_{k}(r) = Z_{k,1}r + Z_{k,2}r^2\)

	\(k\in \mathcal {I}_{m}\)	\(k\notin \mathcal {I}_{m}\)
\(Z_{k,1}\)	\(\epsilon _{k} + (|A_m|\chi _1)_k \)	\(\frac{1}{|\lambda _1|(k_1+k_2)}\begin{pmatrix}\sigma (|w|^{2}_{k}+|w|^{1}_{k})\\ \rho |w|^{1}_{k}+|w|^{2}_{k}+((|\hat{a}|^{1}*|w|^{3})_{k}+(|\hat{a}|^{3}*|w|^{1})_{k})\\ (|\hat{a}|^{1}*|w|^{2})_{k}+(|\hat{a}|^{2}*|w|^{1})_{k}+\beta |w|^{3}_{k}\end{pmatrix} +\frac{k_1+1}{|\lambda _1|(k_1+k_2)} \begin{pmatrix}|w|^{1}_{\overline{k}}\\ |w|^{2}_{\overline{k}}\\ |w|^{3}_{\overline{k}}\end{pmatrix}\)

The absolute value \(|A_m|\) is to be understood component-wise

Hence, one finds (in all the cases discussed below) that \(Z_1\) is given by

$$\begin{aligned} Z_{1}= & {} \displaystyle \sum _{k\in \mathcal {I}_{m}}Z_{k,1}\nu ^{k} + \frac{1}{|\lambda _1|\min \{m_1+1,m_2+1\}}\begin{pmatrix} 2|\sigma |\\ |\rho |+1+\Vert \hat{a}^1\Vert _{\nu }+\Vert \hat{a}^{3}\Vert _{\nu }\\ |\beta | + \Vert \hat{a}^1\Vert _{\nu }+\Vert \hat{a}^{2}\Vert _{\nu } \end{pmatrix} \\&+ \frac{\nu _2}{\nu _1}\frac{m_1+2}{|\lambda _1|(m_1+1)}. \end{aligned}$$

Recalling the necessary condition \(Z_1^j\le 1\) from above, we point out that the term \(\frac{\nu _2}{\nu _1}\frac{m_1+2}{|\lambda _1|(m_1+1)}\) is crucial. In particular, we certainly need \(\frac{\nu _2}{\nu _1 |\lambda _1|} <1\). Therefore, in our analysis of the computational parameters we fix \(\Vert \xi _{1,2}\Vert \) and vary \(\nu _{1,2}\).

Remark 5.2

The condition from Lemma 2.7 reads \(\ell _0\bigl (\hat{\nu }_{1},\frac{\hat{\nu }_{2}}{|\lambda _1|}\bigr ) \le \nu _{1}\). However, the necessary condition \(\frac{\nu _2}{\nu _1 |\lambda _1|} <1\) implies that \(\ell _0\bigl (\nu _{1},\frac{\nu _{2}}{|\lambda _1|}\bigr ) = \nu _1\), hence in practice one simply takes \(\hat{\nu }= \nu \).

Let us now consider different choices of parameters \(\beta ,\rho ,\sigma \). We fix \(\beta = \frac{8}{3}\) and consider \(\sigma = -3,-5,-11,-21\) with \(\rho = \frac{4}{3},\frac{9}{5},\frac{36}{11},\frac{121}{21}\). This corresponds to double eigenvalues \(\lambda _1 = 1,2,5,10\). Thus we compute 2D unstable manifolds (simply replacing g by \(-g\) everywhere in the analysis). Note that this does not necessitate an adaptation of the formulas for the radii polynomials. Table 5 offers a comparison of the validation success in terms of the decay rate \(\nu \).

Table 5

For fixed \(m = (7,6)\) and \(\Vert \xi _{1,2}\Vert = \frac{1}{50}\), we give the corresponding validation radii (if there was one; failure is denoted by \(-\)) on the different domain sizes corresponding to the different decay rates \(\nu \)

	\(\lambda _1 = 1\)	\(\lambda _1 = 2\)	\(\lambda _1= 5\)	\(\lambda _1 = 10 \)
\(\nu = (7,0.5)\)	\( 2.61\times 10^{-8}\)	\(2.58\times 10^{-12}\)	\(1.53\times 10^{-13}\)	\(1.38\times 10^{-14}\)
\(\nu = (7,1)\)	\( - \)	\(2.06\times 10^{-11}\)	\(4.96\times 10^{-13}\)	\(2.40\times 10^{-14}\)
\(\nu = (10,2)\)	\( - \)	\(-\)	\(2.66\times 10^{-10}\)	\(8.11\times 10^{-12}\)
\(\nu = (12,3)\)	\( - \)	\(-\)	\(-\)	\(2.84\times 10^{-10}\)

Thus we see that the bigger we choose the magnitude of the eigenvalue, the bigger is the domain of convergence of the parametrization that we are able to validate. Moreover, the smaller the eigenvalue, the bigger the ratio \(\nu _1/\nu _2\) needs to be. This reflects the crucial role of the term \(\frac{\nu _2}{\nu _1}\frac{m_1+2}{|\lambda _1|(m_1+1)}\).

5.2 Coexistence of Hexagonal and Trivial Patterns

As an example of an application to a case with regular resonant eigenvalues, we turn our attention to the existence of solutions of (2) exhibiting certain patterns. The asymptotic analysis in Doelman et al. (2003) reduces the problem of finding transition layers between stationary patterns of (2) to connecting orbit problems for the system of ODEs (1). We proceed to analyze (1) along the same lines as in van den Berg et al. (2015). To this end we set \(u_1 = u, u_2 = u^{\prime }, u_3 = v\) and \(u_4 = v^{\prime }\) in (1) and rewrite it as

$$\begin{aligned} {u}^{\prime } = g(u) = \begin{pmatrix} u_2\\ -\frac{\gamma }{4}u_1-\frac{\sqrt{2}}{4}u_3^{2} + \frac{3}{8}u_1^{3}+3u_1u_3^{2}\\ u_4\\ -\gamma u_3 -\frac{\sqrt{2}}{2} u_1u_3 +9u_3^{3} + 3u_1^{2}u_3 \end{pmatrix} \end{aligned}$$

(67)

with \(u = (u_1,u_2,u_3,u_4)^{T}\). The relation between the parameters \(\gamma ,\mu \) and \(\beta \) is given by \(\gamma = \frac{\mu }{\beta ^2}\). We refer to Doelman et al. (2003) for a complete description of the seminal asymptotic reduction of the PDE (2) to the ODE system (67), obtained via a combination of spatial dynamics, bifurcation theory and geometric singular perturbation theory. For further details about the particular form (67) one may consult (van den Berg et al. 2015). We take the same viewpoint as in van den Berg et al. (2015) and investigate by rigorous numerical techniques, for fixed parameter value \(\gamma \), connecting orbits between equilibria of (67). While in van den Berg et al. (2015) the configuration for the coexistence of hexagons and rolls (spots and stripes) is considered, we concern ourselves with the coexistence of the trivial state and hexagonal patterns. In terms of (67) this corresponds to a connecting orbit between the two fixed points \(p_1 = (\sqrt{2}u_3^{*},0,u_3^{*},0)^T\), where \(u_3^{*} = \frac{1+\sqrt{1+60\gamma }}{30}\), and \(p_2 = (0,0,0,0)^T\). As (67) is Hamiltonian, a necessary condition for the connection between \(p_1\) and \(p_2\) to exist is for the equilibria to lie on the same energy level, which is the case for \(\gamma = -\frac{2}{135}\).

The equilibrium \(p_2\) has resonant eigenvalues (\(\lambda _1=2\lambda _2\)). For this reason the case of coexistence of the trivial state and the hexagonal spot pattern was not considered in van den Berg et al. (2015). However, our current approach to validating the (un)stable manifold is well suited for this situation.

To compute the connecting orbit we adapt the approach of van den Berg et al. (2015). Introducing a rescaling factor \(L>0\) we aim at solving

$$\begin{aligned} \left\{ \begin{array}{rl} u^{\prime }(t) &{}= Lg(u(t)) \qquad t\in [0,1]\\ u(0) &{}\in W^{u}_{\text {loc}}(p_1)\\ u(1) &{}\in W^{s}_{\text {loc}}(p_2) \end{array}\right. \end{aligned}$$

(68)

with \(u(t) = (u_1(t),u_2(t),u_3(t),u_4(t))\). The main aim of the current paper is to construct efficient rigorously validated descriptions of the local (un)stable manifolds. As explained in detail in van den Berg et al. (2015), once we have such parametrizations \(P^{u}: \mathbb {R}^{2}\supset \mathbb {B}_{1}^{{sym }}\rightarrow \mathbb {R}^{4}\) of the local unstable manifold of \(p_1\) and \(P^{s}: \mathbb {R}^{2}\supset \mathbb {B}_{1}^{{sym }}\rightarrow \mathbb {R}^{4}\) of the local stable manifold of \(p_2\), (68) can be solved by finding a zero of the operator

$$\begin{aligned} F(\psi ,\phi ,u)(t) = \begin{pmatrix} u_{1}(1) - P^{s}_1(\theta (\psi ))\\ u_{3}(1) - P^{s}_3(\theta (\psi ))\\ u_{4}(1) - P^{s}_4(\theta (\psi ))\\ u(t) - P^{u}(\phi ) - \displaystyle \int _{0}^{t}Lg(u(s))ds \end{pmatrix}, \end{aligned}$$

(69)

where \(\theta (\psi ) \mathop {=}\limits ^{\text{ def }}(\rho \cos (\psi ),\rho \sin (\psi ))\) with some fixed \(\rho < 1\) (playing the role of the phase condition in this otherwise autonomous problem). The fact that the second component \(u_{2}(1) - P^{s}_2(\theta (\psi ))=0\) can be replaced by the a posteriori check of

$$\begin{aligned} \text {sign}(u_{2}(1)) = \text {sign}(P^{s}_{2}(\theta (\psi ))) \end{aligned}$$

is explained in Lemma 2 in van den Berg et al. (2015) and is related to the Hamiltonian nature of the problem. Using the phase condition (i.e., fixing \(\rho \)) and omitting the second component deals on the one hand with the fact that every time shift of a connecting orbit is again a connecting orbit and on the other hand with the fact that the intersection of the two-dimensional unstable and stable manifolds corresponding to the connecting orbit is not transverse in \(\mathbb {R}^4\). In a nutshell, they guarantee the isolation of the zero of F that we set out to find.

To rigorously compute zeros of (69) we choose to discretize the time dependence using a Chebyshev series (other choices, such as splines, are also possible). For details on how this is done in this example we refer the reader to van den Berg et al. (2015) and for the general method to Lessard and Reinhardt (2014). In this paper we focus on the rigorous computation of the maps \(P^{s,u}\) and especially \(P^{s}\), as there we encounter eigenvalue resonances. The validated computation of \(P^{u}\) is conducted analogously to the Lorenz equation explained above and we do not give any further details below. Furthermore, in order to validate zeros of (69) we need rigorous information on \(\Vert DP^{s,u}(\theta )\Vert \) that we obtain in the same way as explained in Remark 3 in van den Berg et al. (2015). Note that by Remark 4.2 the a posteriori bound \(\delta _{s,u}\) corresponds to our validation radius r.

We now delve into the validated computation of the stable manifold of the origin, which has resonant eigenvalues. Explicitly, the eigenvalues of \(Dg(p_2)\) with negative real part are given by \(\lambda _1 = -\sqrt{-\gamma }\) and \(\lambda _2 = -\frac{\sqrt{-\gamma }}{2}\) with corresponding eigenvectors \(\xi _1 = (0,0,-\frac{1}{\sqrt{-\gamma }},1)^T\) and \(\xi _{2} = (-\frac{2}{\sqrt{-\gamma }},1,0,0)^T\). We note that \(2\lambda _2 = \lambda _1\), so condition (5) holds with \(\tilde{\imath }= 1\) and \(\tilde{k} = (0,2)\).

Following the approach described in Sect. 2.4 we choose the nonlinear normal form

$$\begin{aligned} h^{s}(\theta ) = \begin{pmatrix} 2 \lambda _2 \theta _1 + \tau \theta _2^2\\ \lambda _2 \theta _2 \end{pmatrix} \end{aligned}$$

(70)

to describe the dynamics in the (stable) parameter space. The coefficients \(c_k = c_k(a;\gamma )\) in (41) are given by

$$\begin{aligned} c_k = \begin{pmatrix} a_{k}^{2}\\ -\frac{\gamma }{4}a_k^1-\frac{\sqrt{2}}{4}(a^3*a^3)_k + \frac{3}{8}(a^1*a^1 *a^1)_k+3(a^1*a^3*a^3)_k\\ a_k^4\\ -\gamma a_k^3 -\frac{\sqrt{2}}{2} (a^1*a^3)_k +9(a^3 *a^3 *a^3)_k + 3(a^1 *a^1 *a^3)_k \end{pmatrix}. \end{aligned}$$

This completes the ingredients for (41) in the particular case of (67). Note that the condition in Lemma 2.7c) reads \(\ell _0\bigl (\theta _{1},\frac{\tau \theta _2^{2}}{|\lambda _{1}|}\bigr ) \le \nu _{1}\), which we easily check explicitly.

Derivation of the bounds To define the radii polynomials specified in (56b) we first need to compute the bounds Y and Z defined in (54) and (55). Let \(f = f^{\text {regres}}\) as specified in (41) where \(\zeta = \frac{1}{\sqrt{1-\gamma }} (0,0,-\sqrt{-\gamma },1)^T \in \ker (A^T)\), with A defined in (29). Finally, let an approximate solution \(\hat{x} = \iota \hat{x}_{F}\) with \(f^m(\hat{x}_{F})\approx 0\) be given, and let \(A_m \approx (Df^{m}(\hat{x}_{F}))^{-1}\). Concerning the bounds Y we (again) note that \(f(\hat{x})_{k} = 0\) for \(k\notin \mathcal {I}_{3m}\), since g is a cubic nonlinearity. Therefore the construction of Y fulfilling (54) is analogous to the Lorenz case in Sect. 5.1. Again as in Sect. 5.1, to obtain \(Z^j(r) = Z^j_{1}r+Z^j_{2}r^2 + Z^j_3 r^3\) in (55) we first compute \(z_{k}(r) = z_{k,1}r+z_{k,2}r^2+z_{k,3}r^3\) fulfilling the analogue of (61). The result is summarized in Table 6 in “Appendix 1.” Note that \(z_{-1}(r) = 0\), as \(f_{-1}\) is linear in \(a_{\tilde{k}}\) and \(\tilde{k}\preceq m\) for our choice of m.

The main deviation from the non-resonant case is given by the off-diagonal linear terms introduced by the resonance. Note that this is analogous to the double eigenvalue case, but with the additional difference given by the presence of the additional unknown \(\tau \). To expand the bounds

$$\begin{aligned}&|(DT(\hat{x}+rv)rw)_k^j| \le Z^j_{k,1}r + Z^j_{k,2}r^2 + Z^j_{k,3} r^3 \\&\quad \text {for } k \in {\mathcal {I}}_m= \{0\} \sqcup \{k\in \mathbb {N}^2 : k \preceq m\}, \end{aligned}$$

we first find bounds \(\epsilon _k\) for \(k \in \mathcal {I}_{m}\) such that

$$\begin{aligned} |((\text {Id }-AA^{\dag })w)_{k}^j| \le \epsilon _{k}^j \qquad j = 1,2,3,4. \end{aligned}$$

(71)

Note that \(((\text {Id }-AA^{\dag })w)_{k} = 0\in \mathbb {R}^{4}\) for \(k\notin \mathcal {I}_{m}\) by construction. Furthermore we construct, for \(i = 1,2,3\), the vector \(\chi _{i} \in \mathbb {R}^{4n_m+1}\) with \(n_m = (m_1+1)(m_2+1)\) such that

$$\begin{aligned} |z_{k,i}^j|\le \chi _i^{1+4(k_2(m_1+1)+k_1)+j} \qquad j = 1,2,3,4. \end{aligned}$$

(72)

Note that \(\chi _{i}^{1} = 0\) for \(i = 1,2,3\), since \(f_{-1}\) is linear. Then, applying the definition of A given in (51) yields the values for \(Z_{k,1}\), \(Z_{k,2}\) and \(Z_{k,3}\) summarized in Table 7 in “Appendix 1.” Summing up \(Z_{k,i}\) component-wise while splitting into a finite part and infinite tail yields Z(r). The result is shown in Table 8 in “Appendix 1.” An important ingredient in computing \(Z_1\) is the control of terms of the form

$$\begin{aligned} \begin{aligned}&\hat{\tau }\displaystyle \sum _{k\notin \mathcal {I}_{m}}\frac{k_{\tilde{\imath }}}{|\lambda _1|k_1 + |\lambda _2|k_2}|w_{\underline{k}}|\nu _1^{k_1}\nu _2^{k_2}, \qquad \text {where } \underline{k}\mathop {=}\limits ^{\text{ def }}(k_1+1,k_2-2). \end{aligned} \end{aligned}$$

(73)

We use Lemma 5.1 to bound the factor \(\frac{k_{\tilde{\imath }}}{2|\lambda _2|k_1 + |\lambda _2|k_2}\) uniformly for \(k\notin {\mathcal {I}}_m\). In the specific problem under consideration we are in luck, since \(\hat{\tau } \approx 0\), hence the contribution from the term (73) is small. This stems from the fact that the exact solution is \(\tau = 0\) as we derive analytically in “Appendix 2.”

However, the strength of our method is that one does not need to determine the coefficients of the normal form beforehand, as they are part of the overall set of unknowns for the nonlinear problem.

Numerical implementation The implementation of the validation of an approximate zero of (69) can be found at the webpage Code page (2015). There, a complete instruction on how to run the codes can be found.

We list the main parameters for the computations, which were chosen after numerical experimentation. As time rescaling parameter we choose \(L = 43.2034\) in (68). For both the stable and unstable manifold we choose as domain radius in (6) \(\nu = (1,1)\).

• Parametrization order: unstable \(m = (15,15)\), stable \(m = (15,15)\).

• Scaling of eigenvectors: unstable \(\Vert \xi _1\Vert _2 = 0.005\), \(\Vert \xi _2\Vert _2 = 0.02\) stable \(\Vert \xi _1\Vert _2 = 0.0085\), \(\Vert \xi _2\Vert _2 = 0.0214\)

The Y-bounds are explained above and the Z-bounds which we use to define the radii polynomials from (56) are summarized in Table 8. Using them to check the conditions of Lemma 4.6 we obtain the following validation radii:

$$\begin{aligned} \begin{aligned} \text {unstable manifold:}\quad r = 1.0823\times 10^{-12}, \\ \text {stable manifold:} \quad r = 5.2217\times 10^{-12}. \end{aligned} \end{aligned}$$

Open image in new window

For the validation of the connection we use the approach of van den Berg et al. (2015) in conjunction with the implementation of van den Berg and Sheombarsing (2015). The validated solution profiles are shown in Fig. 4.