Breakdown of Heteroclinic Connections in the Analytic Hopf-Zero Singularity: Rigorous Computation of the Stokes Constant

Abstract

Consider analytic generic unfoldings of the three- dimensional conservative Hopf-zero singularity. Under open conditions on the parameters determining the singularity, the unfolding possesses two saddle-foci when the unfolding parameter is small enough. One of them has one-dimensional stable manifold and two-dimensional unstable manifold, whereas the other one has one- dimensional unstable manifold and two-dimensional stable manifold. Baldomá et al. (J Dyn Differ Equ 25(2):335–392, 2013) gave an asymptotic formula for the distance between the one-dimensional invariant manifolds in a suitable transverse section. This distance is exponentially small with respect to the perturbative parameter, and it depends on what is usually called a Stokes constant. The nonvanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. However, up to now there do not exist analytic techniques to check that condition. In this paper we provide a method for obtaining accurate rigorous computer-assisted bounds for the Stokes constant. We apply it to two concrete unfoldings of the Hopf-zero singularity, obtaining a computer-assisted proof that the constant is nonzero.

1 Introduction

One of the fundamental questions in dynamical systems is to assess whether a given model possesses chaotic dynamics or not. In particular, one would like to prove whether the model has a hyperbolic invariant set whose dynamics is conjugated to the symbolic dynamics of the usual Bernouilli shift by means of the construction of a Smale horseshoe. Since the pioneering works by Smale and Shilnikov, it is well known that the construction of such invariant sets may be attained by analyzing the stable and unstable invariant manifolds of hyperbolic invariant objects (critical points, periodic orbits, invariant tori) and their intersections.

Such analysis can be done by classical perturbative techniques such as (suitable versions of) Melnikov theory (Melnikov 1963) or by means of computer assisted proofs (Capiński and Zgliczyński 2017, 2018). However, there are settings where Melnikov theory nor “direct” computer-assisted proofs (that is, rigorous computation of the invariant manifolds) cannot be applied. For instance, in the so-called exponentially small splitting of separatrices setting. That is, on models which depend on a small parameter and where the distance between the stable and unstable invariant manifolds is exponentially small with respect to this parameter.

This phenomenon of exponentially small splitting of separatrices often appears in analytic systems with different time scales, which couple fast rotation with slow hyperbolic motion. Example of such settings is nearly integrable Hamiltonian systems at resonances, near the identity area preserving maps or local bifurcations in Hamiltonian, reversible or volume-preserving settings. In such settings, one needs more sophisticated techniques rather than Melnikov theory to analyze the distance between the stable and unstable invariant manifolds. Most of the results in the area follow the seminal approach proposed by in Lazutkin (2003) (there are though other approaches such as Treschev (1997)). Using these techniques, one can provide an asymptotic formula for the distance between the invariant manifolds, with respect to the perturbation parameter. If we denote by $\varepsilon$ the small parameter, the distance is usually of the form

$$\begin{aligned} d=d(\varepsilon )\sim \Theta \varepsilon ^\alpha e^{\frac{a}{\varepsilon ^\beta }}\qquad \text {as}\qquad \varepsilon \rightarrow 0 \end{aligned}$$

for some constants $\Theta$, $\alpha$, a and $\beta$. In most of the settings, the constants $\alpha$, a and $\beta$ have explicit formulas and can be “easily” computed for given models. However, the constant $\Theta$ is of radically different nature and much harder to compute. Indeed, the constants $\alpha$, a and $\beta$ depend on certain first-order terms of the model, whereas $\Theta$, which we refer to as the Stokes constant, depends in a nontrivial way on the “whole jet” of the considered model. Note that it is crucial to know whether $\Theta$ vanishes or not, since its vanishing makes the whole first order between the invariant manifolds vanish and, consequently, chaos cannot be guaranteed in the system.

The purpose of this paper is to provide (computer-assisted) methods to check, in given models, that the Stokes constant does not vanish. Moreover, our method provides a rigorous accurate computation of this constant. To show the main ideas of the method and avoid technicalities, we focus on the simplest setting where this method can be implemented: the breakdown of a one-dimensional heteroclinic connection for generic analytic unfoldings of the volume-preserving Hopf-zero singularity.

This problem was analyzed in Baldomá and Seara (2008); Baldomá et al. (2013). In these papers and the companions (Baldomá et al. 2018a, b, 2020), the authors prove that, in generic unfoldings of an open set of Hopf-zero singularities, one can encounter Shilnikov chaos Sil’nikov (1970). The fundamental difficulty in these models is to prove that the one-dimensional and two-dimensional heteroclinic manifolds connecting two saddle-foci in a suitable truncated normal form of the unfolding, break down when one considers the whole vector field. These breakdowns, which are exponentially small, plus some additional generic conditions lead to existence of chaotic motions.

Remark 1.1

A bifurcation with very similar behavior to that of the conservative Hopf-zero singularity is the Hamiltonian Hopf-zero singularity where a critical point of a 2 degree of freedom Hamiltonian system has a pair of elliptic eigenvalues and a pair of 0 eigenvalues forming a Jordan block [see for instance Gelfreich and Lerman 2014]. In generic unfoldings, the 0 eigenvalues become a pair of small real eigenvalues and therefore the critical point becomes a saddle-center. In this setting, one can analyze the one-dimensional invariant manifolds of the critical point and obtain an asymptotic formula for their distance (in a suitable section). This distance is exponentially with respect to the perturbative parameter. Then, to prove that they indeed do not intersect, one has to show that a certain Stokes constant is not zero as in the Hopf-zero conservative singularity. The methods presented in this paper can be adapted to this other setting. The Hamiltonian Hopf-zero singularity appears in many physical models, for instance in the Restricted Planar 3 Body Problem [see Baldomá et al. 2021a, b]. It also plays an important role in the breakdown of small amplitude breathers for the Klein–Gordon equation (albeit in an infinite-dimensional setting), see Segur and Kruskal (1987); Gomide et al. (2021). We plan to provide a computer-assisted proof of the Stokes constant to guarantee the nonexistence of small breathers in given Klein–Gordon equations in a future work.

In this paper, we provide a method to compute the Stokes constant associated to the breakdown of the one-dimensional heteroclinic connection in analytic unfoldings of the conservative Hopf-zero singularity.

Let us first explain the Hopf-zero singularity and state the main results about the breakdown of its one-dimensional heteroclinic connection obtained in Baldomá and Seara (2008); Baldomá et al. (2013).

1.1 Hopf-Zero Singularity and Its Unfoldings

The Hopf-zero singularity takes place on a vector field $X^*:{\mathbb {R}}^3\rightarrow {\mathbb {R}}^3$, which has the origin as a critical point, and such that the eigenvalues of the linear part at this point are 0, $\pm i\alpha ^*$, for some $\alpha ^*\ne 0$. Hence, after a linear change of variables, we can assume that the linear part of this vector field at the origin is

$$\begin{aligned} DX^*(0,0,0)=\left( \begin{array}{ccc}0 &{}\alpha ^* &{}0\\ alpha^* &{}0 &{}0\\ 0 &{}0 &{}0\end{array}\right) . \end{aligned}$$

We assume that $X^*$ is analytic. Since $DX^*(0,0,0)$ has zero trace, it is reasonable to study it in the context of analytic conservative vector fields (see Broer and Vegter 1984 for the analysis of this singularity in the ${\mathcal {C}}^\infty$ class). In this case, the generic singularity can be met by a generic linear family depending on one parameter, and so it has codimension one.

We study generic analytic families $X_{\mu }$ of conservative vector fields on ${\mathbb {R}}^3$ depending on a parameter $\mu \in {\mathbb {R}}$, such that $X_{0}=X^*$, the vector field described above.

Following Guckenheimer (1981) and Guckenheimer and Holmes (1990), after some changes of variables, we can write $X_{\mu }$ in its normal form up to order two, namely

(1)

Note that the coefficients $\beta _1$, $\gamma _2$ and $\alpha _3$ depend exclusively on the vector field $X^*$.

From now on, we will assume that $X^*$ and its unfolding $X_\mu$ satisfy the following generic conditions:

$$\begin{aligned} \beta _1\ne 0, \qquad \gamma _0\ne 0. \end{aligned}$$

(2)

Depending on the other coefficients $\alpha _i$ and $\gamma _i$, one obtains different qualitative behaviors for the orbits of the vector field $X_{\mu }$. We consider $\mu$ satisfying

$$\begin{aligned} \beta _1 \gamma _0\mu >0 . \end{aligned}$$

(3)

In fact, redefining the parameters $\mu$ and the variable ${\bar{z}}$, one can achieve

$$\begin{aligned} \beta _1>0,\qquad \gamma _0=1, \end{aligned}$$

(4)

and consequently the open set defined by (3) is now

$$\begin{aligned} \mu >0. \end{aligned}$$

(5)

Moreover, dividing the variables ${\bar{x}},{\bar{y}}$ and ${\bar{z}}$ by $\sqrt{\beta _1}$, and scaling time by $\sqrt{\beta _1}$, redefining the coefficients and denoting $\alpha _0=\alpha ^*/\sqrt{\beta _1}$, we can assume that $\beta _1=1$, and therefore system (1) becomes

(6)

We denote by $X_{\mu }^2$, usually called the normal form of second order, the vector field obtained considering the terms of (6) up to order two. Therefore, one has

$$\begin{aligned} X_{\mu }=X_{\mu }^2+F_{\mu }^2, \quad \text{ where } \ F_{\mu }^2({\bar{x}},{\bar{y}},{\bar{z}})= {\mathcal {O}}_{3}({\bar{x}},{\bar{y}},{\bar{z}},\mu ). \end{aligned}$$

It can be easily seen that system (6) has two critical points at distance ${\mathcal {O}}(\sqrt{\mu })$ to the origin. Therefore, we scale the variables and parameters so that the critical points are ${\mathcal {O}}(1)$ and not ${\mathcal {O}}(\sqrt{\mu })$. That is, we define the new parameter $\delta =\sqrt{\mu }$, and the new variables $x=\delta ^{-1}{\bar{x}}$, $y=\delta ^{-1}{\bar{y}}$, $z=\delta ^{-1}{\bar{z}}$ and $t=\delta {\bar{t}}$. Then, renaming the coefficients $b=\gamma _2$, $c=\alpha _3$, system (6) becomes

(7)

where f, g and h are real analytic functions of order three in all their variables, $\delta >0$ is a small parameter and $\alpha (\delta ^2)=\alpha _0+\alpha _2\delta ^2$.

Remark 1.2

Without loss of generality, we can assume that $\alpha _0$ and c are both positive constants. In particular, for $\delta$ small enough, $\alpha (\delta ^2)$ will be also positive.

Observe that if we do not consider the higher-order terms (that is, $f=g=h=0$), we obtain the unperturbed system

(8)

The next lemma gives the main properties of this system.

Lemma 1.3

(Baldomá et al. (2013)) For any value of $\delta >0$, the unperturbed system (8) has the following properties:

• 1. It possesses two hyperbolic fixed points $S_{\pm }^0=(0,0,\pm 1)$ which are of saddle-focus type with eigenvalues ${\mp } 1+|\frac{\alpha }{\delta }\pm c|i$, ${\mp } 1-|\frac{\alpha }{\delta }\pm c|i$, and $\pm 2$.

• 2. The one-dimensional unstable manifold of $S_{+}^0$ and the one-dimensional stable manifold of $S_{-}^0$ coincide along the heteroclinic connection $\{(0,0,z):\, -1<z<1\}.$ The time parameterization of this heteroclinic connection is given by

  $$\begin{aligned} \Upsilon _0(t)=(0,0,z_0(t))=(0,0,-\tanh t), \end{aligned}$$

  if we require $\Upsilon _0(0)=(0,0,0).$

Their 2-dimensional stable/unstable manifolds also coincide, but we will not deal with this problem in this paper.

The critical points given in Lemma 1.3 are persistent for system (7) for small values of $\delta >0$. Below we summarize some properties of system (7).

Lemma 1.4

(Baldomá et al. (2013)) If $\delta >0$ is small enough, system (7) has two fixed points $S_{\pm }(\delta )$ of saddle-focus type,

$$\begin{aligned} S_{\pm }(\delta )=(x_\pm (\delta ), y_\pm (\delta ),z_\pm (\delta )), \end{aligned}$$

with

$$\begin{aligned} x_\pm (\delta )={\mathcal {O}}(\delta ^2),\quad y_\pm (\delta )={\mathcal {O}}(\delta ^2),\quad z_\pm (\delta )=\pm 1+{\mathcal {O}}(\delta ). \end{aligned}$$

The point $S_+(\delta )$ has a one-dimensional unstable manifold and a two-dimensional stable one. Conversely, $S_-(\delta )$ has a one-dimensional stable manifold and a two-dimensional unstable one.

Moreover, there are no other fixed points of (7) in the closed ball $B(\delta ^{-1/3}).$

The theorem proven in Baldomá et al. (2013) is the following.

Theorem 1.5

(Baldomá et al. (2013)) Consider system (7), with $\delta >0$ small enough. Then, there exists a constant $C^*$, such that the distance $d^{\textrm{u},\textrm{s}}$ between the one-dimensional stable manifold of $S_-(\delta )$ and the one-dimensional unstable manifold of $S_+(\delta )$, when they meet the plane $z=0$, is given by

$$\begin{aligned} d^{\textrm{u},\textrm{s}}=\delta ^{-2}e^{-\frac{\alpha _0\pi }{2\delta }}e^{\frac{\pi }{2}(\alpha _0h_0+c)}\left( C^*+{\mathcal {O}}\left( \frac{1}{\log (1/\delta )}\right) \right) , \end{aligned}$$

where $\alpha _0=\alpha (0)$, and $h_0=-\lim _{z\rightarrow 0} z^{-3} h(0,0,z,0,0)$.

In Baldomá et al. (2013), it was proven that the constant $C^*$ comes from the so-called inner equation and that, generically, it does not vanish. However, for a given model is usually very hard to prove analytically whether the associated $C^*$ vanishes or not. In this paper, we provide a rigorous (computer-assisted) method to check whether it vanishes and to compute its value.

1.2 The Inner Equation

One of the key parts of the proof of Theorem 1.5 is to analyze an inner equation. This equation provides the Stokes constant $C^*$, and it was obtained and analyzed in Baldomá and Seara (2008). To obtain it,

we perform the change of coordinates $(\phi ,\varphi ,\eta )=C_{\delta }(x,y,z)$ given by

$$\begin{aligned} \phi =\delta (x+iy), \qquad \varphi = \delta (x-iy),\qquad \eta = \delta z, \qquad \tau =\frac{t-i\pi /2}{\delta }. \end{aligned}$$

(9)

Applying this change to system (7), one obtains

$$\begin{aligned} \frac{d\phi }{d\tau }= & {} \big (-\alpha i-\eta \big )\phi +{\tilde{F}}_1(\phi ,\varphi ,\eta ,\delta ), \nonumber \\ \frac{d\varphi }{d\tau }= & {} \big (\alpha i-\eta \big )\varphi +{\tilde{F}}_2(\phi ,\varphi ,\eta ,\delta ), \\ \frac{d\eta }{d\tau }= & {} -\delta ^2 + b\phi \varphi +\eta ^2 + {\tilde{H}}(\phi ,\varphi ,\eta ,\delta ), \nonumber \end{aligned}$$

(10)

where

$$\begin{aligned} {\tilde{F}}_1(\phi ,\varphi , \eta ,\delta )= & {} f(C_{\delta }^{-1}(\phi ,\varphi ,\eta ),\delta )+ig(C_{\delta }^{-1}(\phi ,\varphi ,\eta ),\delta ),\\ {\tilde{F}}_2(\phi ,\varphi , \eta ,\delta )= & {} f(C_{\delta }^{-1}(\phi ,\varphi ,\eta ),\delta )-ig(C_{\delta }^{-1}(\phi ,\varphi ,\eta ),\delta ),\\ {\tilde{H}}(\phi ,\varphi , \eta ,\delta )= & {} h(C_{\delta }^{-1}(\phi ,\varphi ,\eta ),\delta ). \end{aligned}$$

The inner equation comes from (10) taking $\delta =0$. Defining $F_i(\phi ,\varphi , \eta )={\tilde{F}}_i(\phi ,\varphi , \eta ,0)$ and $H(\phi ,\varphi ,\eta )={\tilde{H}}(\phi ,\varphi ,\eta ,0)$ and, for technical reasons, performing the change $\eta =-s^{-1}$, we get

$$\begin{aligned} \frac{d\phi }{d\tau }= & {} -\left( \alpha i- \frac{1}{s}\right) + F_1\left( \phi ,\varphi ,-s^{-1}\right) , \nonumber \\ \frac{d\varphi }{d\tau }= & {} \left( \alpha i+ \frac{1}{s}\right) + F_2\left( \phi ,\varphi ,-s^{-1}\right) , \\ \frac{ds}{d\tau }= & {} 1+s^2\left( b \phi \varphi + H\left( \phi ,\varphi ,-s^{-1}\right) \right) .\nonumber \end{aligned}$$

(11)

Remark 1.6

Even if the purpose of this paper is not the proof of Theorem 1.5, let us explain why the inner equation plays a fundamental role in the study of the difference between the stable and unstable manifolds of the points $S_\pm (\delta )$. One of the key points in the study of exponentially small splitting is to obtain good parameterizations $(x^{u,s}(t),y^{u,s}(t),z^{u,s}(t))$ of these manifolds. As system (7) is a small perturbation of system (8) and for this system the points $S_\pm (0)=(0,0,\pm 1)$ have an heteroclinic connection given by $(0,0,-\tanh {t})$, it is natural to look for these manifolds as a perturbation of it. However, to detect the exponentially small splitting, the proof in Baldomá et al. (2013) requires to obtain these parameterizations in a complex domain which reaches a neighborhood of order $\delta$ of the singularities $t=\pm i\frac{\pi }{2}$ of the unperturbed heteroclinic connection, that is when $t\mp \frac{i \pi }{2}={\mathcal {O}}(\delta )$. Roughly speaking, in Baldomá et al. (2013), it is shown that

$$\begin{aligned} x^{u,s}(t),y^{u,s}(t)\simeq \frac{\delta ^2}{(t-\frac{i\pi }{2})^3}, \quad z^{u,s}(t)\simeq \frac{ \delta }{(t-\frac{i\pi }{2})^2}, \end{aligned}$$

and observed that, for $t\mp \frac{i \pi }{2}\sim \delta$, one has:

$$\begin{aligned} x^{u,s}(t),y^{u,s}(t)\simeq \frac{1}{\delta }, \quad z^{u,s}(t)\simeq \frac{1 }{\delta }. \end{aligned}$$

Therefore, these parameterizations are not close to the unperturbed heteroclinic connection anymore, which behaves as

$$\begin{aligned} (0,0,\tanh {t})\sim \left( 0,0,\frac{1 }{\delta }\right) . \end{aligned}$$

To obtain a new approximation of the invariant manifolds near the singularities one performs the change of variables $\tau =\delta ^{1}(t-\frac{i\pi }{2})$ and, to work with bounded solutions, one also scales the functions by $\delta$. This is the reason of the change of variables (9) and the inner equation (11) is set to give the first order of the invariant manifolds in these new variables.

We reparameterize time so that Eq. (11) becomes a nonautonomous two-dimensional equation with time s,

$$\begin{aligned} \begin{aligned} \phi '&= \frac{-\left( \alpha i- \frac{1}{s}\right) \phi + F_1(\phi ,\varphi ,-s^{-1})}{1+s^2(b \phi \varphi + H(\phi ,\varphi ,-s^{-1}))}, \\ \varphi '&= \frac{\left( \alpha i+ \frac{1}{s}\right) \varphi + F_2(\phi ,\varphi ,-s^{-1})}{1+s^2(b \phi \varphi + H(\phi ,\varphi ,-s^{-1}))}, \end{aligned} \end{aligned}$$

(12)

with $'=\frac{d}{ds}$.

To analyze this system, we separate its linear terms from the nonlinear ones. Indeed, defining

$$\begin{aligned} {\mathcal {A}}(s)= \begin{pmatrix} -i\alpha +\frac{1}{s}&{}0\\ 0&{}i\alpha +\frac{1}{s} \end{pmatrix}, \end{aligned}$$

(13)

and

$$\begin{aligned}{} & {} {\mathcal {S}}(\phi ,\varphi ,s)=\nonumber \\{} & {} \begin{pmatrix} \displaystyle \frac{\left( \alpha i- \frac{1}{s}\right) \phi s^2\big (b \phi \varphi + H(\phi ,\varphi ,-s^{-1})\big )+F_1(\phi ,\varphi ,-s^{-1}) }{1+s^2(b \phi \varphi + H(\phi ,\varphi ,-s^{-1}))} \\ \displaystyle \frac{-\left( \alpha i+ \frac{1}{s}\right) \varphi s^2 \big (b \phi \varphi + H(\phi ,\varphi ,-s^{-1})\big )+F_2(\phi ,\varphi ,-s^{-1}) }{1+s^2(b \phi \varphi + H(\phi ,\varphi ,-s^{-1}))} \end{pmatrix}, \end{aligned}$$

(14)

Equation (12) can be expressed as

$$\begin{aligned} \begin{pmatrix} \frac{d\phi }{ds}\\ \frac{d\varphi }{ds}\end{pmatrix}={\mathcal {A}}(s) \begin{pmatrix}\phi \\ \varphi \end{pmatrix}+{\mathcal {S}}(\phi ,\varphi ,s). \end{aligned}$$

(15)

From now on, we will refer to (15) as the inner equation.

For $s\in {\mathbb {C}}$, we shall write $\Re s$ and $\Im s$ for its real and imaginary part, respectively. Following (Baldomá and Seara 2008), we define the inner domains as

$$\begin{aligned} {\mathcal {D}}^-_\rho = \left\{ s \in {\mathbb {C}}: |\Im s|\ge -\tan \beta \Re s -\rho , \; \Re s\le 0 \right\} , \qquad {\mathcal {D}}^+_\rho =\{s:-s\in {\mathcal {D}}^-\} \end{aligned}$$

(16)

for some $\rho >0$.

Theorem 1.7

(Baldomá and Seara (2008)) If $\rho$ is big enough, the inner equation has two solutions $\psi ^\pm =(\phi ^{\pm },\varphi ^{\pm })$ defined in ${\mathcal {D}}^\pm _\rho$ satisfying the asymptotic condition

$$\begin{aligned} \lim _{\Re s\rightarrow \pm \infty }\psi ^\pm (s)=0. \end{aligned}$$

(17)

Moreover its difference satisfies that, for $s\in {\mathcal {D}}^+_\rho \cap {\mathcal {D}}^-_\rho \cap \{\Im s<0\}$

$$\begin{aligned} \Delta \psi (s) = \psi ^{+}(s)-\psi ^-(s) = s e^{-i \alpha (s - h_0\log s )}\left[ \left( \begin{array}{c}\Theta \\ 0\end{array}\right) +{\mathcal {O}}\left( \frac{1}{|s|}\right) \right] . \end{aligned}$$

(18)

In addition $\Theta \ne 0$ if and only if $\Delta \psi \not \equiv 0$.

Note that the Stokes constant $\Theta \in {\mathbb {C}}$ can be defined as

$$\begin{aligned} \Theta =\lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha ( s - h_0\log s)}\Delta \phi (s). \end{aligned}$$

(19)

Later, in Baldomá et al. (2013) the authors prove that, if $C^*$ is the constant introduced in Theorem 1.5, then

$$\begin{aligned} C^*=|\Theta |. \end{aligned}$$

However there is no closed formula for $\Theta$, which depends on the full jet of the nonlinear terms in (15). Our strategy to compute $\Theta$ is to perform a computer-assisted proof.

2 Rigorous Computation of the Stokes Constant

We propose a method to compute the Stokes constant $\Theta$ relying on rigorous, computer-assisted, interval arithmetic-based validation. The method takes advantage from the constructive method for proving Theorem 1.7 based on fixed point arguments, and we strongly believe that it can be applied to other settings as, for instance, the classical rapidly forced pendulum and close to the identity area preserving maps.

The method we propose to compute the Stokes constant $\Theta$ is divided into two parts.

• Part 1: We provide an algorithm to give an explicit $\rho _*>0$ such that the existence of the solutions $\psi ^{\pm }$ of the inner equation, in the domain $\{\Re s=0, \, \Im s \le -\rho _*\}$ is guaranteed. The algorithm is based in giving explicit bounds (which depend on the nonlinear terms ${\mathcal {S}}$ of the inner equation, see (14)) of all the constants involved in the fixed point argument. We believe that this algorithm can be generalized to other situations where the proof of the existence of the corresponding solutions of the inner equation relies on fixed point arguments. In the case of the Hopf-zero singularity, by Theorem 1.7, if one can check (using rigorous computer computations) that $\Delta \psi (-i\rho ^*)=\psi ^+(-i\rho ^*)-\psi ^-(-i\rho ^*) \ne 0$ one can ensure that $\Theta \ne 0$.

• Part 2: Using that $\Delta \psi (s)$ is defined for $s\in \{\Re s=0,\, \Im s\le -\rho _*\}$ with $\rho _*$ given in Part 1, we give a method which provides rigorous accurate estimates for $\Theta$. We give an algorithm to compute $\rho _{0}\ge \rho _*$ such that, for all $\rho \ge \rho _0$, the Stokes constant and $\Delta \phi (-i\rho )$ satisfies the relation

  $$\begin{aligned} \Theta = i \rho ^{-1} \Delta \phi (-i\rho )e^{\alpha \left( \rho -i h_0 \log \rho - h_0\frac{\pi }{2}\right) } (1+g(\rho )), \end{aligned}$$

  (20)

  with $|g(\rho )|<1$. By (18), we know that $|g(\rho )|$ is of order ${\mathcal {O}}(\rho ^{-1})$. We provide explicit upper bounds for it. Part 2 also relies on evaluating $\Delta \psi (-i\rho )$ but takes more advantage on the fixed point argument techniques used to prove formula (18) in Theorem 1.7. A similar formula to (20) for $\Theta$ can be deduced in other settings such as the rapidly forced pendulum and close to the identity area preserving maps. We should be able to adapt our method to a plethora of different situations.

In Sect. 2.1, we show the theoretical framework we use to design the method. In particular, the functional setting needed for the fixed point argument. It is divided in Sects. 2.1.1 and 2.1.2 which deal with Part 1 and Part 2, respectively. In Sect. 2.2, we follow the theoretical approach given in the previous sections and compute all the necessary constants to implement the method. After that, in Sect. 2.3 we write the precise algorithm, pointing out all the constants that need to be computed to find $\Theta$. In Sect. 3, we apply our method to two examples. Finally, in Sect. 4, we explain how to improve the accuracy in the computation of the Stokes constant in one of the examples considered in Sect. 3.

2.1 Scheme of the Method: Theoretical Approach

2.1.1 Existence Domain of the Solutions of the Inner Equation

We analyze the solutions $\psi ^\pm =(\phi ^\pm ,\varphi ^\pm )$ of Eq. (15) in the inner domains ${\mathcal {D}}^\pm _\rho$ introduced in (16). To prove the existence of the solutions $\psi ^\pm$, we set up a fixed point argument. From now on, we use subindices 1 and 2 to denote the two components of all vectors and operators.

Note that the right hand side of Eq. (15) has a linear part plus higher-order terms (which will be treated as perturbation). We consider a fundamental matrix M(s) associated to the matrix ${\mathcal {A}}$ in (13) given by

$$\begin{aligned} M(s)=s \begin{pmatrix} e^{-i\alpha s}&{}0\\ 0&{}e^{i\alpha s} \end{pmatrix}, \end{aligned}$$

(21)

and we define also the integral operators

$$\begin{aligned} {\mathcal {B}}^{\pm }(h)=\begin{pmatrix}{\mathcal {B}}_1^{\pm }(h)\\ {\mathcal {B}}_2^{\pm }(h)\end{pmatrix}=M(s)\int _{\pm \infty }^0 M(s+t)^{-1}h(s+t)dt. \end{aligned}$$

(22)

Then, the solutions $\psi ^\pm$ of Eq. (15) satisfying the asymptotic conditions (17) must be also solutions of the integral equation

$$\begin{aligned} \psi ^\pm ={\mathcal {B}}^\pm ({\mathcal {S}}(\psi ,s)). \end{aligned}$$

Therefore, we look for fixed points of the operators

$$\begin{aligned} {\mathcal {F}}^\pm (\psi )={\mathcal {B}}^\pm ({\mathcal {S}}(\psi ,s)). \end{aligned}$$

(23)

We define the Banach spaces

$$\begin{aligned} {\mathcal {X}}_\nu ^{\pm }=\left\{ h: {\mathcal {D}}^{\pm }_\rho \rightarrow {\mathbb {C}}: \text {analytic}, \Vert h\Vert _\nu <\infty \right\} \Vert h\Vert _\nu =\sup _{s\in {\mathcal {D}}^{\pm }_\rho }|s^\nu h(s)|.\nonumber \\ \end{aligned}$$

(24)

Then, we obtain fixed points of the operators ${\mathcal {F}}^\pm$ in the Banach spaces ${\mathcal {X}}_\nu \times {\mathcal {X}}_\nu$ with the norm

$$\begin{aligned} \Vert (\phi ,\varphi )\Vert _{\nu }= \textrm{max}\{\Vert \phi \Vert _\nu , \Vert \varphi \Vert _\nu \}, \end{aligned}$$

for some $\nu$ to be chosen.

In Baldomá and Seara (2008), it is proven that the operators ${\mathcal {F}}^{\pm }$ are contractive operators in some ball of ${\mathcal {X}}_3\times {\mathcal {X}}_3$ if $\rho \ge \rho ^*$ is big enough. Consequently the existence of solutions $\psi ^\pm$ of Eq. (15) in the domains ${\mathcal {D}}^{\pm }_\rho$ is guaranteed. However, we want to be explicit in the estimates to compute the smallest $\rho _*$ such that one can prove that ${\mathcal {F}}^{\pm }$ are contractive operators.

To this end, we need to control the dependence on $\rho$ of the Lipschitz constant of the operators ${\mathcal {F}}^{\pm }$. Let us explain briefly the procedure, which is performed only for the − case being the $+$ case analogous.

• In Sect. 2.2.1, we provide explicit bounds for the norm of the linear operator ${\mathcal {B}}^-$ in (22).

• In Sect. 2.2.2, we define a set of constants depending on the nonlinear terms ${\mathcal {S}}$ (see (14)) of the inner equation.

• We deal with the bounds of the first iteration, ${\mathcal {F}}^+(0)$ in Sect. 2.2.3. We conclude that it belongs to a closed ball of ${\mathcal {X}}_3 \times {\mathcal {X}}_3$ if $\rho \ge \rho _*^1$ where $\rho _*^1$ is determined by the constants in the previous step. The radius of the ball $M_0(\rho )/2$ is fully determined also by the previous constants.

• In Sect. 2.2.4, we provide explicit bounds of the derivative of the nonlinear operator ${\mathcal {S}}$ and consequently of its Lipschitz constant, which depends on $\rho$. These computations hold true for values of $\rho \ge \rho _*^2\ge \rho _1^*$ with $\rho _*^2$ satisfying some explicit conditions.

• In Sect. 2.2.5, for $\rho \ge \rho _*^2$, we compute the Lipschitz constant $L(\rho )$ of ${\mathcal {F}}^-$ in the closed ball of ${\mathcal {X}}_3 \times {\mathcal {X}}_3$ of radius $M_0(\rho )$.

• In Sect. 2.2.6, we set $\rho _*\ge \rho _*^2$ for the existence result. We choose $\rho ^*$ such that $L(\rho _*)\le \frac{1}{2}$. Then, since

  $$\begin{aligned} \Vert \psi ^-\Vert _3 \le \Vert {\mathcal {F}}^-(0) \Vert _3 + \Vert {\mathcal {F}}^-(\psi ^-) - {\mathcal {F}}^-(0) \Vert _3 \le \frac{M_0(\rho )}{2} + L(\rho ) \Vert \psi ^-\Vert _3 \le M_0(\rho ), \end{aligned}$$

  the fixed point theorem ensures the existence of a fixed point $\psi ^-$ satisfying $\Vert \psi ^- \Vert _3 \le M_0(\rho )$ for $\rho \ge \rho ^*$.

• Finally, we compute $\Delta \phi (-i\rho _*)$ by computed-assisted proofs techniques. This completes the Part 1 of the algorithm since $\Delta \phi (-i\rho _*)\ne 0$ implies $\Theta \ne 0$.

All the steps described above are written with all the detailed constants in Sect. 2.3.

2.1.2 Rigourous Computation of the Stokes Constant

In this section we describe a method to compute rigorously the Stokes constant $\Theta$ defined in (19) (Part 2 of the algorithm). The method is based in the alternative formula for $\Theta$ proposed in (20):

$$\begin{aligned} \Theta = i\frac{e^{\alpha \left( \rho -ih_0\log \rho -h_0\frac{\pi }{2}\right) }\Delta \phi (-i\rho )}{\rho } (1+ g(\rho )), \qquad \lim _{\rho \rightarrow \infty } g(\rho )=0. \end{aligned}$$

(25)

Let us to explain how this formula is derived. The key point is to analyze the difference

$$\begin{aligned} \Delta \psi (s)=\psi ^+(s)-\psi ^-(s) \end{aligned}$$

as a solution of a linear equation on the vertical axis $\Im s\in (-\infty , -\rho _*)$ where $\rho _*$ is provided by the method explained in Sect. 2.1.1. Indeed $\Delta \psi =(\Delta \phi , \Delta \varphi )$ satisfies the equation

$$\begin{aligned} \begin{pmatrix}\Delta \phi '\\ \Delta \varphi '\end{pmatrix}= \left( {\mathcal {A}}(s)+{\mathcal {K}}(s)\right) \begin{pmatrix}\Delta \phi \\ \Delta \varphi \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {K}}(s)=\int _0^1D{\mathcal {S}}\left( \psi ^-(s)+t(\psi ^+(s)-\psi ^-(s)),s\right) dt, \end{aligned}$$

(26)

and ${\mathcal {S}}$ is given in (14). We look for the linear terms of lower order in $s^{-1}$ of ${\mathcal {S}}$. Indeed, we have that

$$\begin{aligned} {\mathcal {S}}(\psi ,s)= \frac{1}{s} \left( \begin{array}{c} -\alpha i h_0\phi \\ \alpha ih_0\varphi \end{array}\right) +{\widetilde{{\mathcal {S}}}}(\psi ,s), \end{aligned}$$

with ${\widetilde{{\mathcal {S}}}}(\psi ,s)={\mathcal {O}}(|s|^{-2})$ when $\psi \in {\mathcal {X}}_3$ (see (24)). Then, $\Delta \psi$ satisfies the equation

$$\begin{aligned} \begin{pmatrix}\Delta \phi '\\ \Delta \varphi '\end{pmatrix}= \left( {\widetilde{{\mathcal {A}}}}(s)+{\widetilde{{\mathcal {K}}}}(s)\right) \begin{pmatrix}\Delta \phi \\ \Delta \varphi \end{pmatrix}, \end{aligned}$$

(27)

where

$$\begin{aligned} {\widetilde{{\mathcal {A}}}}(s) = \begin{pmatrix} -i\alpha +\frac{1}{s} -i\alpha \frac{h_0}{s}&{}0\\ 0&{}i\alpha +\frac{1}{s} +i\alpha \frac{h_0}{s} \end{pmatrix}, \end{aligned}$$

(28)

and

$$\begin{aligned} {\widetilde{{\mathcal {K}}}}(s)=\int _0^1D{\widetilde{{\mathcal {S}}}}\left( \psi ^-(s)+t(\psi ^+(s)-\psi ^-(s)),s\right) dt. \end{aligned}$$

(29)

A fundamental matrix for the linear system $z'={\widetilde{A}}(s) z$ is

$$\begin{aligned} \begin{pmatrix} s e^{-i\alpha ( s+h_0\log s)} &{} 0 \\ 0 &{} s e^{i\alpha ( s+h_0\log s)} \end{pmatrix}. \end{aligned}$$

Therefore, any solution of system (27) can be expressed as

$$\begin{aligned} \begin{pmatrix}\Delta \phi \\ \Delta \varphi \end{pmatrix} = \begin{pmatrix} \displaystyle se^{-i\alpha (s+h_0\log s)} \left[ \kappa _0 + \int _{-i\rho }^s\frac{e^{i\alpha ( t+h_0\log t)}}{t}\left( {\widetilde{{\mathcal {K}}}}_{11}\Delta \phi +{\widetilde{{\mathcal {K}}}}_{12}\Delta \varphi \right) dt \right] \\ \displaystyle se^{i\alpha (s+h_0\log s)} \left[ \kappa _1 + \int _{-i\rho }^s\frac{e^{-i\alpha ( t+h_0\log t)}}{t}\left( \widetilde{ {\mathcal {K}}}_{21}\Delta \phi +{\widetilde{{\mathcal {K}}}}_{22}\Delta \varphi \right) dt \right] \end{pmatrix}, \end{aligned}$$

with $\kappa _0,\kappa _1$ two constants.

Since $|\psi ^{\pm }|\le M_0(\rho ) |s|^{-3}$, $\Delta \psi$ goes to 0 as $\Im s\rightarrow -\infty$ and, therefore,

$$\begin{aligned} \kappa _1=-\int _{-i\rho }^{-i\infty } \frac{e^{-i\alpha ( t+h_0\log t)}}{t}\left( \widetilde{ {\mathcal {K}}}_{21}\Delta \phi +{\widetilde{{\mathcal {K}}}}_{22}\Delta \varphi \right) dt. \end{aligned}$$

Then, we deduce that the difference $\Delta \psi (s)$ is a fixed point of the equation

$$\begin{aligned} \begin{aligned} \Delta \psi (s)&= \Delta \psi ^0(s)+{\mathcal {G}}(\Delta \psi ) (s), \ \text{ where } \\ \Delta \psi ^0(s)&=\left( \begin{array}{c} s e^{-i\alpha ( s+h_0\log s)}\kappa _0\\ 0\end{array}\right) , \end{aligned} \end{aligned}$$

(30)

with $\kappa _0$ a constant depending on $\rho$ and ${\mathcal {G}}$ is the linear operator

$$\begin{aligned} {\mathcal {G}}(\Delta \psi )=\begin{pmatrix} \displaystyle se^{-i\alpha (s+h_0\log s)}\int _{-i\rho }^s\frac{e^{i\alpha ( t+h_0\log t)}}{t}\left( {\widetilde{{\mathcal {K}}}}_{11}\Delta \phi +{\widetilde{{\mathcal {K}}}}_{12}\Delta \varphi \right) dt\\ \displaystyle se^{i\alpha (s+h_0\log s)}\int _{-i\infty }^s\frac{e^{-i\alpha ( t+h_0\log t)}}{t}\left( \widetilde{ {\mathcal {K}}}_{21}\Delta \phi +{\widetilde{{\mathcal {K}}}}_{22}\Delta \varphi \right) dt \end{pmatrix}. \end{aligned}$$

(31)

By construction, $\kappa _0$ is defined as

$$\begin{aligned} \kappa _0=\kappa _0(\rho )=i\frac{e^{\alpha \left( \rho -ih_0\log \rho -h_0\frac{\pi }{2}\right) }\Delta \phi (-i\rho )}{\rho }. \end{aligned}$$

(32)

Using (19) and (30), we have

$$\begin{aligned} \begin{aligned} \Theta&=\lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha ( s + h_0\log s)}\Delta \phi (s)\\&=\kappa _0\left( 1+\kappa _0^{-1}\lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha (s + h_0\log s)}{\mathcal {G}}_1\left( \Delta \psi (s)\right) \right) . \end{aligned} \end{aligned}$$

(33)

We use equality (33) to obtain formula (25) of $\Theta$. To bound $|g(\rho )|$ in formula  (25), we need to control the linear operator $s^{-1}e^{i\alpha (s + h_0\log s)}{\mathcal {G}}_1$. To this end, we consider a norm with exponential weights,

$$\begin{aligned} \Vert \psi \Vert =\textrm{max}\left\{ \textrm{max}_{s\in E} \left| s^{-1}e^{i\alpha (s+h_0\log s)}\phi \right| ,\textrm{max}_{s\in E} \left| e^{i\alpha ( s+h_0\log s) }\varphi \right| \right\} , \end{aligned}$$

(34)

with $E=\{\Re s=0, \, \Im s\in (-\rho _*,-\infty )\}$.

Observe that (30) can be rewritten

$$\begin{aligned} (\text {Id}-{\mathcal {G}})(\Delta \psi )=\Delta \psi ^0. \end{aligned}$$

Now we see that $\text {Id}-{\mathcal {G}}$ is an invertible operator. Indeed, in Baldomá and Seara (2008), it was proven that for $\rho$ big enough

$$\begin{aligned} \Vert {\mathcal {G}}_1(\Phi )\Vert \le A_1(\rho ) \Vert \Phi \Vert , \qquad \Vert {\mathcal {G}}_2(\Phi )\Vert \le A_2(\rho )\Vert \Phi \Vert , \end{aligned}$$

(35)

with $0<A(\rho ):=\textrm{max}\{A_1(\rho ),A_2(\rho )\}<1$. Moreover, Baldomá and Seara (2008) also shows that

$$\begin{aligned} \lim _{\rho \rightarrow \infty } A(\rho )=0. \end{aligned}$$

These estimates imply that, if $\rho$ is big enough, $\text {Id}-{\mathcal {G}}$ is invertible and therefore $\Delta \psi =(\text {Id}-{\mathcal {G}})^{-1}(\Delta \psi ^0)$. Moreover,

$$\begin{aligned} \Vert \Delta \psi \Vert \le \frac{1}{1-A(\rho )}\Vert \Delta \psi ^0\Vert = \frac{|\kappa _0(\rho )|}{1-A(\rho )}, \end{aligned}$$

(36)

and this inequality directly gives

$$\begin{aligned} \left| \lim _{\Im s \rightarrow -\infty } s^{-1} e^{i\alpha (s+h_0\log s)} {\mathcal {G}}_1(\Delta \psi (s)) \right| \le A_1(\rho ) \frac{|\kappa _0(\rho )|}{1-A(\rho )}. \end{aligned}$$

Therefore, from (33), we can conclude that the Stokes constant $\Theta$, which is independent of $\rho$, can be computed as

$$\begin{aligned} \Theta =\kappa _0(\rho ) (1+g(\rho )), \end{aligned}$$

for any $\rho$ big enough, where $\kappa _0$ is given in (32) and g satisfies

$$\begin{aligned} |g(\rho )|\le {\overline{M}}(\rho ):=\frac{A_1(\rho ) }{1- A(\rho )}. \end{aligned}$$

(37)

Since $A(\rho ),A_1(\rho )$ go to zero as $\rho \rightarrow \infty$, the same happens for ${\overline{M}}(\rho )$. Then (25) is proven.

Notice that the relative error to approximate $\Theta$ by $\kappa _0$ is

$$\begin{aligned} \frac{ \left| \Theta - \kappa _0(\rho )\right| }{|\kappa _0(\rho )|} \le {\overline{M}}(\rho ). \end{aligned}$$

As a consequence,

$$\begin{aligned} |\Theta | \in \left[ |\kappa _0(\rho )| (1-{\overline{M}}(\rho )), |\kappa _0(\rho )| (1+{\overline{M}}(\rho ))\right] . \end{aligned}$$

In Sect. 2.2, the procedure described above is implemented:

• Following the fixed point argument in Baldomá and Seara (2008), in Sect. 2.2.7 we give a explicit formula for $A(\rho )=\textrm{max}\{A_1(\rho ),A_2(\rho )\}$ in (35) for $\rho \ge \rho _*$, where $\rho ^*$ is the constant given by Part 1.

• In Sect. 2.2.8, we set $\rho _0\ge \rho _*$ such that ${\overline{M}}(\rho )<1$ for $\rho \ge \rho _0$.

2.2 Computing the Stokes Constant: Method

In this section we are going to give explicit expressions for all the constant involved in the method explained in the previous section.

2.2.1 The Linear Operator ${\mathcal {B}}^-$

Lemma 2.1

Consider the linear operator ${\mathcal {B}}^-$ defined in (22).

• 1. When $\nu >1$, the linear operator ${\mathcal {B}}^-:{\mathcal {X}}_{\nu }\times {\mathcal {X}}_{\nu }\rightarrow {\mathcal {X}}_{\nu -1}\times {\mathcal {X}}_{\nu -1}$ is continuous and

  $$\begin{aligned} \Vert {\mathcal {B}}^-(\psi )\Vert _{\nu -1} \le B_{\nu +1} \Vert \psi \Vert _\nu , \end{aligned}$$

  where

  $$\begin{aligned} \begin{aligned} B_{m}&= \frac{\pi }{2} \frac{(m-3))!!}{(m-2)!!} \qquad{} & {} \text {if } m \text { is even},\\ B_m&= \frac{(m-3)!!}{(m-2)!!} \qquad{} & {} \text { if } m \text { is odd}. \end{aligned} \end{aligned}$$

  (38)

• 2. When $\nu >0$, the linear operator ${\mathcal {B}}^-:{\mathcal {X}}_{\nu }\times {\mathcal {X}}_{\nu }\rightarrow {\mathcal {X}}_{\nu }\times {\mathcal {X}}_{\nu }$ is continuous and, for all $0<\gamma \le \beta$ (see (16)),

  $$\begin{aligned} \Vert {\mathcal {B}}^-(\psi )\Vert _\nu \le \frac{1}{\alpha \sin \gamma (\cos \gamma )^{\nu +1}}\Vert \psi \Vert _\nu . \end{aligned}$$

  Define $\gamma _*\in (0,\frac{\pi }{2})$ such that $\sin ^2 \gamma _* =\frac{1}{\nu +2}$. If $\gamma _* \le \beta$,

  $$\begin{aligned} \Vert {\mathcal {B}}^-(\psi )\Vert _\nu \le C_{\nu } \Vert \psi \Vert _\nu \qquad \text { where }\qquad C_{\nu }=\frac{(\nu +2)^{\frac{\nu +2}{2}}}{\alpha (\nu +1)^{\frac{\nu +1}{2}}}. \end{aligned}$$

  (39)

This lemma is proven in “Appendix A”.

From now on we choose $\beta$, the angle in the definition (16) of ${\mathcal {D}}^-_{\rho }$, be such that $6 \sin \beta ^2=1$. Then for all $\nu \ge 4$, the optimal value $\gamma _*$ in second item of Lemma 2.1 satisfies that $\gamma _*\le \beta$ and the optimal bound (39) will be used throughout the paper.

We emphasize that, if $s\in {\mathcal {D}}^{-}_\rho$, one has that $|s|\ge \rho$. Recall that we are looking for $\rho _*$ the minimum value for $\rho$ to ensure that the inner equation has a solution $\psi ^-$ defined in ${\mathcal {D}}^-_\rho$. Since we need $\rho ^{-1}_*$ to be small, we start by assuming that $\rho _*\ge 2$. We will change this value along the proof.

2.2.2 Explicit Constants for the Inner Equation

We consider the $\textrm{max}$ norm $|(x,y,z)|=\textrm{max}\{|x|,|y|,|z|\}$. Let $a_3=\lim _{z\rightarrow 0}z^{-3}F_1(0,0,z)$, $h_0=\lim _{z\rightarrow 0}z^{-3}H(0,0,z)$ and $C_{F}^0,C_{H}^0,{\overline{C}}_H^0$ be such that for $|z|\le \frac{1}{2}$,

$$\begin{aligned} \begin{aligned} \left| \Delta F_1(z)\right| = \left| F_1(0,0,z)+a_3z^3\right| \le C_{F}^0|z|^4, \\ \left| \Delta F_2(z)\right| =\left| F_2(0,0,z)+\overline{a_3}z^3\right| \le C_{F}^0|z|^4,\\ \left| \Delta H(z)\right| =\left| H(0,0,z)+h_0z^3\right| \le C_{H}^0|z|^4,\\ \left| H(0,0,z)\right| \le {\overline{C}}_H^0 |z|^3. \end{aligned} \end{aligned}$$

(40)

We also introduce $C_F, C_F^{\phi ,\varphi }$, $C_H,C_{H}^{\phi ,\varphi }$ such that, for $|(x,y)|\le |z|$,

$$\begin{aligned} \begin{aligned} \left| H(x,y,z)\right|&\le C_H|(x,y,z)|^3 \le C_H|z|^3 \\ \left| F_{1,2}(x,y,z)\right|&\le C_F|(x,y,z)|^3 \le C_F|z|^3, \\ \left| \partial _x F_{1,2}(x,y,z) \right|&\le C_F^{\phi } |(x,y,z)|^2 \le C_F^{\phi }|z|^2, \\ \left| \partial _y F_{1,2}(x,y,z) \right|&\le C_F^{\varphi } |(x,y,z)|^2 \le C_F^{\varphi }|z|^2, \\ \left| \partial _x H(x,y,z) \right|&\le C_H^{\phi } |(x,y,z)|^2 \le C_H^{\phi }|z|^2, \\ \left| \partial _yH(x,y,z) \right|&\le C_H^{\varphi } |(x,y,z)|^2 \le C_H^{\varphi }|z|^2 . \end{aligned} \end{aligned}$$

(41)

As a consequence, setting

$$\begin{aligned} {\overline{C}}_H= C_H^\phi + C_H^\varphi \quad \text { and }\quad {\overline{C}}_F= C_F^\phi + C_F^\varphi , \end{aligned}$$

(42)

we have

$$\begin{aligned} \begin{aligned} \left| H(x,y,z) + h_0z^3\right|&\le C_H^0 |z|^4 + {\overline{C}}_H |(x,y)| |z|^2,\\ \left| F_1(x,y,z) +a_3z^3\right|&\le C_F^0 |z|^4 + {\overline{C}}_F |(x,y)| |z|^2,\\ \left| F_2(x,y,z) +\overline{a_3}z^3\right|&\le C_F^0 |z|^4 + {\overline{C}}_F |(x,y)| |z|^2. \end{aligned} \end{aligned}$$

2.2.3 Bounds for the Norm of the First Iteration

The second step in the proof consists on studying ${\mathcal {F}}^-(0)(s)={\mathcal {B}}^- ({\mathcal {S}}(0,s))$, where ${\mathcal {F}}^-$ is the operator introduced in (23).

Lemma 2.2

Chose any $\rho _*^1 >\textrm{max}\{2, {\overline{C}}_H^0\}$, take $\rho \ge \rho _*^1$ and define

$$\begin{aligned} {\mathcal {C}}_0(\rho )= \frac{C_{F}^0 + |a_3|{\overline{C}}_H^0}{1- \frac{|{\overline{C}}_H^0|}{\rho }}. \end{aligned}$$

Then ${\mathcal {F}}^-(0)\in {\mathcal {X}}_3\times {\mathcal {X}}_3$ and

$$\begin{aligned} \Vert {\mathcal {F}}^-(0)\Vert _3 \le \frac{11 |a_3|}{3\alpha }+ B_5 {\mathcal {C}}_0(\rho ). \end{aligned}$$

Proof

By (14), it is clear that

$$\begin{aligned} {\mathcal {S}}_1(0,s) - \frac{a_3}{s^3}= \frac{F_1(0,0,-s^{-1})}{1+ s^2 H(0,0,-s^{-1})}-\frac{a_3}{s^3}= \frac{\Delta F_1 (-s^{-1})-\frac{a_3 H(0,0,-s^{-1})}{s}}{1+s^2 H(0,0,-s^{-1})}, \end{aligned}$$

and therefore

$$\begin{aligned} \left| {\mathcal {S}}_1(0,s)-\frac{a_3}{s^3}\right| \le \frac{1}{|s|^4} \frac{C_{F}^0 + |a_3|{\overline{C}}_H^0}{1- \frac{{\overline{C}}_H^0}{ \rho }}. \end{aligned}$$

An analogous bound works for ${\mathcal {S}}_2(0,s)$ and therefore

$$\begin{aligned} \Vert {\mathcal {S}}(0,s)- s^{-3} (a_3,\overline{a_3})\Vert _4 \le \frac{C_{F}^0 + |a_3|{\overline{C}}_H^0}{1- \frac{|{\overline{C}}_H^0|}{\rho }}={\mathcal {C}}_0(\rho ). \end{aligned}$$

(43)

We introduce ${\mathcal {S}}_0(s)= s^{-3} (a_3,\overline{a_3})$. We have that

$$\begin{aligned} {\mathcal {B}}_1^-({\mathcal {S}}_0(s)) = a_3 s \int _{-\infty }^0 \frac{e^{i\alpha t}}{(s+t)^4} \, dt = \frac{a_3}{i \alpha s^3 } + \frac{4sa_3}{i\alpha }\int _{-\infty }^0 \frac{e^{i\alpha t}}{(s+t)^5 }\, dt . \end{aligned}$$

(44)

Notice that, for $s\in {\mathcal {D}}^-_\rho$ and $t\in {\mathbb {R}}$, $|s+t|^2 \ge |s|^2 + t^2$. Then, using also Lemma A.1 (see “Appendix A”),

$$\begin{aligned} \left| s \int _{-\infty }^0 \frac{e^{i\alpha t}}{(s+t)^5 }\, dt \right| \le \frac{1}{|s|^3} \int _{-\infty }^0 \frac{1}{\big (t^2 + 1\big )^{5/2}} = \frac{2}{3 |s|^3}. \end{aligned}$$

Using this last bound and formula (44), we obtain

$$\begin{aligned} \left| {\mathcal {B}}_1^-({\mathcal {S}}_0(s)) \right| \le \frac{1}{|s|^3} \left( \frac{|a_3|}{\alpha } + \frac{8|a_3|}{3 \alpha }\right) \le \frac{11 |a_3|}{3 \alpha |s|^3}. \end{aligned}$$

To finish we notice that, from (43) and the first item of Corollary 2.1,

$$\begin{aligned} \left\| {\mathcal {F}}^-_1(0)\Vert _3\le \Vert {\mathcal {B}}^-_1({\mathcal {S}}_0)\right\| _3 + \left\| {\mathcal {B}}^-_1 ({\mathcal {S}}(0,\cdot )- {\mathcal {S}}_0)\right\| _3\le \frac{11|a_3|}{3\alpha }+{\mathcal {C}}_0(\rho ) B_{5}. \end{aligned}$$

Analogous computations lead to the same estimate for $\Vert {\mathcal {F}}_2^-(0)\Vert _3$.

$\square$

2.2.4 The Lipschitz Constant of ${\mathcal {S}}$

Let

$$\begin{aligned} M_0(\rho )= \frac{22 |a_3|}{3\alpha } + 2 B_5 {\mathcal {C}}_0(\rho ), \end{aligned}$$

(45)

in such a way that $2 \Vert {\mathcal {F}}^-(0)\Vert _3 \le M_0(\rho )$.

Lemma 2.3

Assume that $\Vert \phi \Vert _3,\Vert \varphi \Vert _3 \le M_0(\rho )$ and take $\rho \ge \rho _*^2$ being $\rho _*^2\ge \rho _*^1$ such that

$$\begin{aligned} \min \left\{ 1- \frac{b M_0^2(\rho _*^2)}{(\rho _*^2)^4} - \frac{C_H}{(\rho _*^2)}, 1 - \frac{M_0(\rho _*^2)}{(\rho _*^2)^2}\right\} >0. \end{aligned}$$

(46)

Then

$$\begin{aligned} \begin{aligned} \big |\partial _\phi {\mathcal {S}}_1(\psi ,s) \big | , \big |\partial _\varphi {\mathcal {S}}_2(\psi ,s) \big |&\le \frac{M_{11}^1(\rho )}{|s|} + \frac{M_{11}^2(\rho )}{|s|^2}+ \frac{M_{11}^3(\rho )}{|s|^3}+ \frac{M_{11}^4(\rho )}{|s|^4}, \\ \big |\partial _\varphi {\mathcal {S}}_1(\psi ,s)|, |\partial _\phi {\mathcal {S}}_2(\psi ,s) \big |&\le \frac{M_{12}^2(\rho )}{|s|^2}+ \frac{M_{12}^3(\rho )}{|s|^3}+ \frac{M_{12}^4(\rho )}{|s|^4}, \end{aligned} \end{aligned}$$

with

$$\begin{aligned} M_{11}^1(\rho ) ={} & {} \frac{\alpha |h_0|}{1 - \frac{bM_0^2}{\rho ^4}-\frac{C_H}{\rho }},\nonumber \\ M_{11}^2(\rho ) ={} & {} \frac{|h_0| + \alpha C_H^0+ C_{F}^\phi }{1 - \frac{bM_0^2}{\rho ^4}-\frac{C_H}{\rho }},\nonumber \\ M_{11}^3(\rho ) ={} & {} \frac{1 }{ \left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2} \left[ M_0\alpha C_{H}^\phi +C_F C_H^\phi + \left( \alpha {\overline{C}}_H M_0 + C_H^0 \right) \right. \nonumber \\ {}{} & {} \left. \quad \left( 1 -\frac{b M_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) \right] ,\nonumber \\ M_{11}^4 (\rho ) ={} & {} \frac{M_0}{ \left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2}\nonumber \\{} & {} \cdot \left[ b (C_F + \alpha M_0) + C_H^\phi + \left( \alpha b M_0 +{\overline{C}}_H+\frac{bM_0}{\rho }\right) \right. \nonumber \\ {}{} & {} \left. \quad \left( 1 -\frac{b M_0^2}{\rho ^4} - \frac{C_H}{\rho } \right) + \frac{bM_0}{\rho } \right] \nonumber \\ M_{12}^2(\rho ) ={} & {} \frac{C_{F}^\varphi }{1 - \frac{bM_0^2}{\rho ^4}-\frac{C_H}{\rho }},\nonumber \\ M_{12}^3(\rho ) ={} & {} \frac{M_0\alpha C_{H}^\varphi +C_F C_H^\varphi }{ \left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2},\nonumber \\ M_{12}^4 (\rho ) ={} & {} \frac{M_0}{ \left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2} \left( b (C_F + \alpha M_0) + C_H^\varphi + \frac{bM_0}{\rho } \right) . \end{aligned}$$

(47)

Proof

Notice that $\rho _*^2\ge \rho _*^1$ and therefore, Lemma 2.2 can be applied for $\rho \ge \rho _*^2$. Moreover, if $s\in {\mathcal {D}}^-_\rho$,

$$\begin{aligned} |\psi (s)| \le \frac{M_0(\rho )}{|s|^3} \le \frac{1}{|s|}, \end{aligned}$$

so that the bounds in (41) can also be used.

We start with $\partial _{\phi }{\mathcal {S}}$. We introduce

$$\begin{aligned} \begin{aligned} S_1(\psi ,s)&=\frac{\partial _{\phi } F_1(\psi ,-s^{-1})}{1+ s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )} + \frac{F_1(\psi ,-s^{-1}) s^2 \big (b \varphi + \partial _{\phi } H(\psi ,-s^{-1})\big )}{\left( 1+ s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )\right) ^2 }, \\ S_2(\psi ,s)&= \frac{\left( \alpha i -\frac{1}{s}\right) s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )}{1 + s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )}- \frac{\left( \alpha i -\frac{1}{s}\right) \phi s^2 \big (b \varphi + \partial _{\phi } H(\psi ,-s^{-1})\big )}{\left( 1+ s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )\right) ^2 }. \end{aligned} \end{aligned}$$

Straightforward computations lead us to

$$\begin{aligned} \partial _\phi {\mathcal {S}}_1 = S_1 + S_2. \end{aligned}$$

When $\Vert \phi \Vert _3,\Vert \varphi \Vert _3 \le M_0(\rho )$,

$$\begin{aligned} |S_1(\psi ,s)|\le{} & {} \frac{1}{|s|^2 } \frac{C_F^\phi }{1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }} +\frac{1}{|s|^3}\frac{C_F \left( C_H^\phi + b \frac{M_0}{|s|}\right) }{\left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2} \\ \le{} & {} \frac{1}{|s|^2 } \frac{C_F^\phi }{1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }} + \frac{1}{|s|^3} \frac{C_F C_H^\phi }{\left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2}\\ {}{} & {} \quad + \frac{1}{|s|^4} \frac{bM_0 C_F}{\left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2}, \\ |S_2(\psi ,s)| \le{} & {} \frac{1}{|s|}\frac{\left( \alpha + \frac{1}{|s|} \right) \left( |h_0| + \frac{C_H^0}{|s|} + \frac{{\overline{C}}_HM_0}{|s|^2}+ b \frac{M_0^2}{|s|^3} \right) }{1 - \frac{bM_0^2}{\rho ^4}-\frac{C_H}{\rho }}\\ {}{} & {} \quad + \frac{M_0}{|s|}\frac{\left( \alpha + \frac{1}{|s|} \right) \left( \frac{C_H^\phi }{|s|^2} + \frac{bM_0}{|s|^3} \right) }{ \left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2}\\ ={} & {} \frac{1}{1 - \frac{bM_0^2}{\rho ^4}-\frac{C_H}{\rho }} \left( \frac{\alpha |h_0|}{|s|} + \frac{|h_0| + \alpha C_H^0}{|s|^2} + \frac{\alpha {\overline{C}}_H M_0 + C_H^0}{|s|^3}\right. \\ {}{} & {} \left. \quad + \frac{\alpha b M_0^2 + {\overline{C}}_H M_0}{|s|^4} + \frac{bM_0^2}{|s|^5}\right) \\ {}{} & {} + \frac{1}{ \left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2} \left( \frac{M_0 \alpha C_H^\phi }{|s|^3} + \frac{\alpha b M_0^2 + M_0 C_{H}^\phi }{|s|^4} + \frac{b M_0^2}{|s|^5}\right) . \end{aligned}$$

Therefore we have that

$$\begin{aligned} \big |\partial _\phi {\mathcal {S}}_1(\psi ,s) \big | \le \frac{M_{11}^1(\rho )}{|s|} + \frac{M_{11}^2(\rho )}{|s|^2}+ \frac{M_{11}^3(\rho )}{|s|^3}+ \frac{M_{11}^4(\rho )}{|s|^4}, \end{aligned}$$

where $M_{11}^k$ are the constants introduced in the lemma.

We now compute a bound for $\partial _{\varphi } {\mathcal {S}}$. As for $\partial _{\phi }{\mathcal {S}}$, we define

$$\begin{aligned} \begin{aligned} S_1(\psi ,s)&=\frac{\partial _{\varphi } F_1(\psi ,-s^{-1})}{1+ s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )} + \frac{F_1(\psi ,-s^{-1}) s^2 \big (b \phi + \partial _{\varphi } H(\psi ,-s^{-1})\big )}{\left( 1+ s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )\right) ^2 }, \\ S_2(\psi ,s)&= -\frac{\left( \alpha i -\frac{1}{s}\right) \phi s^2 \big (b \phi + \partial _{\varphi } H(\psi ,-s^{-1})\big )}{\left( 1+ s^2 \big (b\phi \varphi + H(\psi ,-s^{-1})\big )\right) ^2 }. \end{aligned} \end{aligned}$$

and we notice that

$$\begin{aligned} \partial _\varphi {\mathcal {S}}_1 = S_1 + S_2. \end{aligned}$$

We have that, if $\Vert \phi \Vert _3,\Vert \varphi \Vert _3 \le M_0(\rho )$,

$$\begin{aligned} |S_1(\psi ,s)| \le{} & {} \frac{1}{|s|^2 } \frac{C_F^\varphi }{1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }} + \frac{1}{|s|^3} \frac{C_F C_H^\varphi }{\left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2}\\ {}{} & {} \quad + \frac{1}{|s|^4} \frac{bM_0 C_F}{\left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2}, \\ |S_2(\psi ,s)| \le{} & {} \frac{1}{ \left( 1 - \frac{bM_0^2}{\rho ^4} - \frac{C_H}{\rho }\right) ^2} \left( \frac{M_0 \alpha C_H^\varphi }{|s|^3} + \frac{\alpha b M_0^2 + M_0 C_{H}^\varphi }{|s|^4} + \frac{b M_0^2}{|s|^5}\right) . \end{aligned}$$

Then

$$\begin{aligned} \big |\partial _\varphi {\mathcal {S}}_1(\psi ,s) \big | \le \frac{M_{12}^2(\rho )}{|s|^2}+ \frac{M_{12}^3(\rho )}{|s|^3}+ \frac{M_{12}^4(\rho )}{|s|^4}, \end{aligned}$$

with the constants $M_{12}^k$ defined in the lemma.

Since the bounds for $F_1,\partial _\phi F_1,\partial _\varphi F_1$ are the same as for $F_2,\partial _\phi F_2,\partial _\varphi F_2$ and using the symmetry in the definition of ${\mathcal {S}}$, we have that

$$\begin{aligned} \begin{aligned} |\partial _\phi {\mathcal {S}}_2(\psi ,s)|&\le \frac{M_{12}^2(\rho )}{|s|^2}+ \frac{M_{12}^3(\rho )}{|s|^3}+ \frac{M_{12}^4(\rho )}{|s|^4}, \\ \big |\partial _\varphi {\mathcal {S}}_2(\psi ,s) \big |&\le \frac{M_{11}^1(\rho )}{|s|} + \frac{M_{11}^2(\rho )}{|s|^2}+ \frac{M_{11}^3(\rho )}{|s|^3}+ \frac{M_{11}^4(\rho )}{|s|^4}. \end{aligned} \end{aligned}$$

$\square$

As a corollary, we obtain the following.

Corollary 2.4

If $\psi ,\psi '\in B(M_0(\rho ))$ with $\rho \ge \rho _*^2$ as in Lemma 2.3. Then, there exist functions $\Delta {\mathcal {S}}_j$, $j=1\ldots 4$, such that

$$\begin{aligned} {\mathcal {S}}(\psi ,s)-{\mathcal {S}}(\psi ',s) = \sum _{j=1}^4 \Delta {\mathcal {S}}_j(\psi ,s)- \Delta {\mathcal {S}}_j(\psi ',s), \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Vert \Delta {\mathcal {S}}_1(\psi ,s)-\Delta {\mathcal {S}}_1(\psi ',s) \Vert _4&\le M_{11}^1 (\rho ) \Vert \psi - \psi '\Vert _3,\\ \Vert \Delta {\mathcal {S}}_j(\psi ,s)-\Delta {\mathcal {S}}_j(\psi ',s) \Vert _{3+j}&\le (M_{11}^j (\rho ) + M_{12}^j(\rho ))\Vert \psi - \psi '\Vert _3. \end{aligned} \end{aligned}$$

As a consequence

$$\begin{aligned} \begin{aligned} |{\mathcal {S}}(\psi ,s)-{\mathcal {S}}(\psi ',s)| \le&|\psi (s) - \psi '(s)| \left( \frac{M_{11}^1(\rho )}{|s|} + \frac{M_{11}^2(\rho )+ M_{12}^2(\rho )}{|s|^2} \right. \\ {}&+ \left. \frac{M_{11}^3(\rho )+ M_{12}^3(\rho )}{|s|^3}+ \frac{M_{11}^4(\rho )+M_{12}^4(\rho )}{|s|^4} \right) . \end{aligned} \end{aligned}$$

Proof

Indeed:

$$\begin{aligned} \begin{aligned} {\mathcal {S}}_1(\psi ,s)-{\mathcal {S}}_1(\psi ',s)=&(\phi - \phi ') \int _{0}^1 \partial _{\phi } {\mathcal {S}}_1 (\psi ' + \lambda (\psi -\psi ')) \, d\lambda \\ {}&+ (\varphi - \varphi ') \int _{0}^1 \partial _{\varphi } {\mathcal {S}}_1 (\psi ' + \lambda (\psi -\psi ')) \, d\lambda \\ =&(\phi - \phi ') (S_1^\phi + S_2^\phi + S_3^\phi + S_4^\phi ) + (\varphi - \varphi ') (S_2^\varphi + S_3^\varphi + S_4^\varphi ), \end{aligned} \end{aligned}$$

with $S_{j}^{\phi ,\varphi }\in {\mathcal {X}}_j$ and

$$\begin{aligned} \Vert S_{j}^{\phi }\Vert _j \le M_{11}^j(\rho ), \qquad \Vert S_{j}^{\varphi }\Vert _j \le M_{12}^j(\rho ). \end{aligned}$$

In analogous way, we decompose ${\mathcal {S}}_2(\psi )-{\mathcal {S}}_2(\psi ')$, and by symmetry, we obtain that

$$\begin{aligned} {\mathcal {S}}(\psi ,s)-{\mathcal {S}}(\psi ',s)= \left( \begin{array}{cc} S_{11}(s) &{} S_{12}(s) \\ S_{21}(s) &{} S_{22}(s) \end{array}\right) \left( \begin{array}{c} \phi - \phi ' \\ \varphi - \varphi ' \end{array}\right) , \end{aligned}$$

(48)

with

$$\begin{aligned} \begin{aligned} |S_{11}(s)|, |S_{22}(s)|&\le \frac{M_{11}^1(\rho )}{|s|} + \frac{M_{11}^2(\rho )}{|s|^2}+ \frac{M_{11}^3(\rho )}{|s|^3}+ \frac{M_{11}^4(\rho )}{|s|^4}, \\ |S_{12}(s)|, |S_{21}(s)|&\le \frac{M_{12}^2(\rho )}{|s|^2}+ \frac{M_{12}^3(\rho )}{|s|^3}+ \frac{M_{12}^4(\rho )}{|s|^4}. \end{aligned} \end{aligned}$$

Namely, $S_{ij}$ can be decomposed as a sum of functions belonging to the adequate ${\mathcal {X}}_{k}$. Therefore, taking the supremmum norm in (48), we get the result. $\square$

Remark 2.5

Notice that, for some concrete functions $F_{1,2}$ and H the general bounds in that we have used for them and their derivatives (see (41)) may not be sharp. To improve the estimates for $|\partial _{\phi ,\varphi } {\mathcal {S}}|$ in Lemma 2.3, we need to track some terms of the functions $F_{12},H$. Indeed, instead of (41) we can use bounds of the derivatives of the form

$$\begin{aligned} \begin{aligned} \big |\partial _x F_{1,2}(x,y,z) \big |&\le c_{F}^\phi |z|^2 + K_{F}^\phi |(x,y)||z|, \\ \big |\partial _y F_{1,2}(x,y,z) \big |&\le c_{F}^\varphi |z|^2 + K_{F}^\varphi |(x,y)||z|, \\ \big |\partial _x H(x,y,z) \big |&\le c_{H}^\phi |z|^2 + K_{H}^\phi |(x,y)||z|, \\ \big |\partial _y H(x,y,z) \big |&\le c_{H}^\varphi |z|^2 + K_{H}^\varphi |(x,y)||z|, \end{aligned} \end{aligned}$$

which, together with (40), imply

$$\begin{aligned} \begin{aligned} \big |F_{1,2}(x,y,z) +a_3z^3 \big |&\le C_F^0 |z|^4 + {\bar{C}}_F |(x,y)||z|^2 + {\bar{K}}_F|(x,y)|^2 |z|, \\ \big |H(x,y,z) +h_0z^3 \big |&\le C_H^0 |z|^4 + {\bar{C}}_H |(x,y)||z|^2 + {\bar{K}}_H|(x,y)|^2 |z|, \end{aligned} \end{aligned}$$

with ${\bar{C}}_F=c_F^\phi + c_F^{\varphi }$, ${\bar{C}}_H=c_H^\phi + c_H^{\varphi }$, ${\bar{K}}_F=K_F^\phi + K_F^{\varphi }$, ${\bar{K}}_H=K_H^\phi + K_H^{\varphi }$. If necessary, since $|(x,y)|\le |z|$, we can also use

$$\begin{aligned} \begin{aligned} \big |F_{1,2}(x,y,z) +a_3z^3 \big |&\le C_F^0 |z|^4 + {\tilde{C}}_F |(x,y)||z|^2, \\ \big |H(x,y,z) +h_0z^3 + a_4 z^4 + a_5 z^5 \big |&\le {\bar{c}}_H^0 |z|^6 + {\tilde{C}}_H |(x,y)||z|^2. \end{aligned} \end{aligned}$$

where

$$\begin{aligned} {\tilde{C}}_F={\bar{C}}_F+ {\bar{K}}_F,\qquad {\tilde{C}}_H={\bar{C}}_H+ {\bar{K}}_H. \end{aligned}$$

It is clear that, taking

$$\begin{aligned} \begin{aligned} C_F&= |a_3| + \frac{C_F^0}{\rho } + \frac{{\tilde{C}}_F M_0}{\rho ^2}, \qquad C_F^{\phi ,\varphi } = c_{F}^{\phi ,\varphi } + \frac{K_F^{\phi ,\varphi }M_0}{\rho ^2}, \\ {C}_H&= |a_4| + \frac{|a_5|}{\rho } + \frac{{\bar{c}}_H^0}{\rho ^2} + \frac{{\tilde{C}}_H M_0}{\rho }, \qquad C_H^{\phi ,\varphi } = c_{H}^{\phi ,\varphi } + \frac{K_H^{\phi ,\varphi }M_0}{\rho ^2}, \end{aligned} \end{aligned}$$

we can get a more accurate bound for $|\partial _{\phi ,\varphi } {\mathcal {S}}|$. In fact, we can just change the definition of $M_{ij}^k$ by changing the value of $C_F, C_F^{\phi ,\varphi }, C_H^{\phi ,\varphi }$ by their new value.

2.2.5 The Lipschitz Constant of ${\mathcal {F}}^-$

Now we are going to compute the Lipschitz constant of the operator ${\mathcal {F}}^-$ in (23).

Lemma 2.6

Take $\rho \ge \rho _*^2$ as in Lemma 2.3. The operator ${\mathcal {F}}:B(M_0) \rightarrow {\mathcal {X}}_3\times {\mathcal {X}}_3$ is Lipschitz with Lipschitz constant $L(\rho )=\min \{L_1(\rho ),L_2(\rho )\}$ with

$$\begin{aligned} \begin{aligned} L_1(\rho )=\,&C_4\frac{M_{11}^1(\rho )}{\rho }+B_{6} \frac{M_{11}^2(\rho )+ M_{12}^2 (\rho )}{\rho } + B_{7} \frac{M_{11}^3(\rho )+ M_{12}^3 (\rho )}{\rho ^2} \\ {}&+B_{8} \frac{M_{11}^4(\rho )+ M_{12}^4 (\rho )}{\rho ^3}, \\ L_2(\rho )=\,&C_4 \frac{M_{11}^1(\rho )}{\rho }+ C_5 \frac{M_{11}^2(\rho )+ M_{12}^2 (\rho )}{\rho ^2} + C_6 \frac{M_{11}^3(\rho )+ M_{12}^3 (\rho )}{\rho ^3} \\ {}&+C_7 \frac{M_{11}^4(\rho )+ M_{12}^4 (\rho )}{\rho ^4}. \end{aligned} \end{aligned}$$

(49)

where $B_\nu$ and $C_\nu$ are the constants introduced in (38) and (39), respectively.

Proof

We apply the second item of Lemma 2.1 to $\Delta {\mathcal {S}}_1(\psi ,s)-\Delta {\mathcal {S}}_1(\psi ',s)$ and we obtain that

$$\begin{aligned}{} & {} \Vert {\mathcal {B}}^-(\Delta {\mathcal {S}}_1(\psi ,s)-\Delta {\mathcal {S}}_1(\psi ',s))\Vert _4\le \nonumber \\{} & {} C_4M_{11}^1(\rho ) \Vert \phi - \phi '\Vert _3. \end{aligned}$$

(50)

Now we apply the first item of Lemma 2.1 to $\Delta {\mathcal {S}}_j(\psi ,s)-\Delta {\mathcal {S}}_j(\psi ',s)$ and we obtain

$$\begin{aligned}{} & {} \Vert {\mathcal {B}}^-\left( \Delta {\mathcal {S}}_j(\psi ,s)-\Delta {\mathcal {S}}_j(\psi ',s)\right) \Vert _{2+j}\nonumber \\{} & {} \le B_{j+4}(M_{11}^j(\rho ) + M_{12}^j(\rho ))\Vert \phi - \phi '\Vert _3. \end{aligned}$$

(51)

Then, we get $L_1(\rho )$ adding the results in (50) and (51). Furthermore, applying the second item of Corollary 2.1, we obtain $L_2(\rho )$ using that

$$\begin{aligned} \Vert {\mathcal {B}}^-\left( \Delta {\mathcal {S}}_j(\psi ,s)-\Delta {\mathcal {S}}_j(\psi ',s)\right) \Vert _{3+j}\le C_{j+2}(M_{11}^j(\rho ) + M_{12}^j(\rho ))\Vert \phi - \phi '\Vert _3, \end{aligned}$$

$\square$

Remark 2.7

Notice that $B_{m}$ is decreasing with respect to m, but $C_{m}$ is increasing. It is not difficult to check that when $\rho \ge C_7/B_8\ge 3^9 \cdot 32 /(2^{12}\cdot 5\pi ) \sim 9.7895$ then $L_{1}(\rho )\ge L_{2}(\rho )$. This fact will be used in Sect. 3.1 and 3.2.

2.2.6 Setting $\rho _*$ for the Existence Result

We choose $\rho _*\ge \rho _*^2$ satisfying

$$\begin{aligned} L(\rho _*)\le \frac{1}{2}, \end{aligned}$$

so that Lemma 2.3 can be applied for $\rho \ge \rho _*$. Then, the operator ${\mathcal {F}}^- : B(M_0) \rightarrow B(M_0)$ is contractive. Indeed,

$$\begin{aligned} \Vert {\mathcal {F}}^-(\psi )\Vert _{3} \le \Vert {\mathcal {F}}^-(0)\Vert _{3} + \Vert {\mathcal {F}}^-(\psi )-{\mathcal {F}}^-(0)\Vert _{3} = \frac{M_0}{2} + LM_0\le M_0, \end{aligned}$$

provided $L\le \frac{1}{2}$. Therefore, the operator has a fixed point $\psi ^-$ defined in ${\mathcal {D}}^{-}_{\rho ^*}$ (see (16)) and therefore satisfies

$$\begin{aligned} |\phi ^-(s)|,|\varphi ^-(s)|\le \frac{M_0}{|s|^3}. \end{aligned}$$

(52)

2.2.7 Explicit Bounds for the Norm of the Linear Operator ${\mathcal {G}}$

The next lemma gives estimates for the linear operator ${\mathcal {G}}$ defined in (31) with respect to the norm introduced in (34).

Lemma 2.8

Take $\rho \ge \rho _*$ and let

$$\begin{aligned} \begin{aligned} A_1(\rho )&=\frac{M_{11}^2}{\rho } + \frac{M_{11}^3 + M_{12}^2}{2\rho ^2} + \frac{M_{11}^4+M_{12}^3}{3\rho ^3}+ \frac{M_{12}^4}{4\rho ^4}, \\ A_2(\rho )&=\frac{M_{12}^2}{2\alpha \rho ^2} + \frac{M_{11}^2 + M_{12}^3}{2\alpha \rho ^3} + \frac{M_{11}^3+M_{12}^4}{2 \alpha \rho ^4}+ \frac{M_{11}^4}{2\alpha \rho ^5}, \end{aligned} \end{aligned}$$

(53)

where $M_{ij}=M_{ij}(\rho )$ are the constants introduced in (47). Then, we have that, for s with $\Re s=0$ and $\Im s\le -\rho$,

$$\begin{aligned} \begin{aligned} \left| s^{-1}e^{i\alpha (s+h_0\log s)}{\mathcal {G}}_1(\Delta \psi )\right| \le&A_1(\rho )\Vert \Delta \psi \Vert ,\\ \left| e^{i\alpha (s+h_0\log s)}{\mathcal {G}}_2(\Delta \psi )\right| \le&A_2(\rho )\Vert \Delta \psi \Vert . \end{aligned} \end{aligned}$$

(54)

In particular,

$$\begin{aligned} \Vert {\mathcal {G}}(\Delta \psi )\Vert \le A(\rho )\Vert \Delta \psi \Vert , \end{aligned}$$

(55)

with

$$\begin{aligned} A=A(\rho ) = \textrm{max}\left\{ A_1(\rho ),A_2(\rho )\right\} . \end{aligned}$$

(56)

Proof

In this proof we omit the dependence on $\rho$ of $M_{i,j}^k$. We use Lemma 2.3 to bound ${\widetilde{{\mathcal {K}}}}_{ij}$, the components of the matrix ${\widetilde{{\mathcal {K}}}}$ in (29). By construction, if $\psi \in B(M_0)$,

$$\begin{aligned} \begin{aligned} \left| {\widetilde{{\mathcal {K}}}}_{11}(\psi ,s)\right| ,\left| {\widetilde{{\mathcal {K}}}}_{22}(\psi ,s)\right|&\le \frac{M_{11}^2}{|s|^2} + \frac{M_{11}^3}{|s|^3}+ \frac{M_{11}^4}{|s|^4}, \\ \left| {\widetilde{{\mathcal {K}}}}_{12}(\psi ,s)\right| ,\left| {\widetilde{{\mathcal {K}}}}_{21}(\psi ,s)\right|&\le \frac{M_{12}^2}{|s|^2} + \frac{M_{12}^3}{|s|^3}+ \frac{M_{12}^4}{|s|^4}. \end{aligned} \end{aligned}$$

Then, for the first component,

$$\begin{aligned} \begin{aligned} \left| s^{-1}e^{i\alpha (s+h_0\log s)}{\mathcal {G}}_1(\Delta \psi )\right|&\le \left| \int _{-i\rho }^s\frac{e^{i\alpha ( t+h_0\log t)}}{t}\left( {\widetilde{{\mathcal {K}}}}_{11}\Delta \phi + {\widetilde{{\mathcal {K}}}}_{12}\Delta \varphi \right) dt\right| \\&\le \int _{-i\rho }^s\left( \frac{M_{11}^2}{|t|^2}+\frac{M_{11}^3}{|t|^3}+\frac{M_{11}^4}{|t|^4} \right) \Vert \Delta \phi \Vert \, dt \\&+ \int _{-i \rho }^s \left( \frac{M_{12}^2}{|t|^3}+\frac{M_{12}^3}{|t|^4}+\frac{M_{12}^4}{|t|^5}\right) \Vert \Delta \varphi \Vert dt\\&\le \left[ \frac{M_{11}^2}{\rho }+\frac{M_{11}^3}{2\rho ^2}+\frac{M_{11}^4}{3\rho ^3} \right] \Vert \Delta \phi \Vert \\ {}&+ \left[ \frac{M_{12}^2}{2\rho ^2}+\frac{M_{12}^3}{3 \rho ^3}+\frac{M_{12}^4}{4\rho ^4} \right] \Vert \Delta \varphi \\&\le \left( \frac{M_{11}^2}{\rho } + \frac{M_{11}^3 + M_{12}^2}{2\rho ^2} + \frac{M_{11}^4+M_{12}^3}{3\rho ^3}+ \frac{M_{12}^4}{4\rho ^4} \right) \Vert \Delta \psi \Vert . \end{aligned} \end{aligned}$$

(57)

For the second component, using that $\big |e^{i\alpha h_0\log t}\big | = e^{\alpha h_0\pi /2}$,

$$\begin{aligned} \left| e^{i\alpha (s+h_0\log s)}{\mathcal {G}}_2(\Delta \psi )\right|\le & {} \left| se^{2i\alpha (s+h_0\log s)}\int _{-\infty }^s\frac{e^{-i\alpha ( t+h_0\log t)}}{t} \left( {\widetilde{{\mathcal {K}}}}_{21}\Delta \phi + {\widetilde{{\mathcal {K}}}}_{22}\Delta \varphi \right) dt\right| \\\le & {} \left| s\right| e^{-2\alpha \Im s}\int _{-\infty }^se^{2\alpha \Im t} \left( \left[ \frac{M_{12}^2}{|t|^2}+\frac{M_{12}^3}{|t|^3}+\frac{M_{12}^4}{|t|^4} \right] \Vert \Delta \phi \Vert \right) \, dt \\ {}{} & {} + |s| e^{-2 \alpha \Im s} \int _{-\infty }^s e^{2 \alpha \Im t} \left( \frac{M_{11}^2}{|t|^3}+\frac{M_{11}^3}{|t|^4}+\frac{M_{11}^4}{|t|^5} \Vert \Delta \varphi \Vert \right) \,dt\\\le & {} \left[ \frac{M_{12}^2}{2\alpha \rho ^2}+\frac{M_{12}^3}{2\alpha \rho ^3}+\frac{M_{12}^4}{2\alpha \rho ^4} \right] \Vert \Delta \phi \Vert \\ {}{} & {} + \left[ \frac{M_{11}^2}{2 \alpha \rho ^3}+\frac{M_{11}^3}{2\alpha \rho ^4}+\frac{M_{11}^4}{2\alpha \rho ^5} \right] \Vert \Delta \varphi \Vert \\\le & {} \left( \frac{M_{12}^2}{2\alpha \rho ^2} + \frac{M_{11}^2 + M_{12}^3}{2\alpha \rho ^3} + \frac{M_{11}^3+M_{12}^4}{2 \alpha \rho ^4}+ \frac{M_{11}^4}{2\alpha \rho ^5} \right) \Vert \Delta \psi \Vert . \end{aligned}$$

and the result is proven. $\square$

2.2.8 Computation of the Stokes Constant

Using the estimates of the operator ${\mathcal {G}}$ given in (55), we can provide a rigorous computation of the Stokes constant.

Let $\rho _0\ge \rho _*$ be such that

$$\begin{aligned} A(\rho _0)< \frac{1}{2}. \end{aligned}$$

Then, the constant ${\overline{M}}(\rho _0)$ defined in (37) satisfies

$$\begin{aligned} {\overline{M}}(\rho _0) = \frac{A_1(\rho _0)}{1-A(\rho _0)}<\frac{A(\rho _0)}{1-A(\rho _0)}<1, \end{aligned}$$

(58)

and, as a consequence,

$$\begin{aligned} \Theta \in \left[ \kappa _0(\rho _0)(1-{\overline{M}}(\rho _0)), \kappa _0(\rho _0)(1+{\overline{M}}(\rho _0))\right] . \end{aligned}$$

In the next section we give the precise algorithm which allows, by means of computer rigorous computations, to compute $\Theta$ with a previous known accuracy. This algorithm is applied to two concrete examples in Sects. 3.1 and  3.2.

2.3 Computing the Stokes Constant: Algorithm

We describe the steps needed to obtain the values of $\rho _*$ and $\rho _0$ which guarantees that $\psi ^+,\psi ^-$ are defined for $s\in (-i\rho _*, -i\infty )$ and a good accuracy of $\Theta$.

• Step 1: Compute the constants $a_3,h_0$ and $C_{F}^0,C_{H}^0,{\overline{C}}_H^0$, $C_F^{\phi ,\varphi }$, $C_{H}^{\phi ,\varphi }$ which satisfy (40) and (41) and ${\overline{C}}_H$, ${\overline{C}}_F$ given in (42).

• Step 2: Take $\rho _*^1 \ge \textrm{max}\{2, {\overline{C}}_H^0\}$ and compute, for $\rho \ge \rho _*^1$, the constants ${\mathcal {C}}_0(\rho )$ given in Lemma 2.2 and $M_0(\rho )$ defined in (45).

• Step 3: Choose $\rho _*^2\ge \rho _*^1$ such that (46) is satisfied. Compute also the constants $M^j_{11}(\rho )$, $j=1,2,3,4$ and $M_{12}^j(\rho )$, $j=2,3,4$, in (47), for $\rho \ge \rho _*^2$.

• Step 4: Compute the constants $L_1(\rho )$ and $L_2(\rho )$ in (49) for $\rho \ge \rho _*^2$.

• Step 5: Choose $\rho _*\ge \rho _*^2$ satisfying

  $$\begin{aligned} L(\rho _*)=\min \{L_1(\rho _*),L_2(\rho _*)\}\le \frac{1}{2}. \end{aligned}$$

• Step 6: Take $\rho _*$ and check that the difference

  $$\begin{aligned} \psi ^+(-i\rho _*)-\psi ^-(-i\rho _*)\ne 0. \end{aligned}$$

• Step 7: For $\rho \ge \rho _*$, compute the constants $A_1(\rho )$ and $A_2(\rho )$ in (53) and $A(\rho )$ in (56).

• Step 8: Compute $\rho _0\ge \rho _*$ such that $A(\rho _0)\le 1/2$. Then, compute $\kappa _0(\rho _0)$ in (32) and ${\overline{M}}(\rho _0)$ in (58).

Therefore, the Stokes constant satisfies

$$\begin{aligned} \Theta \in \left[ \kappa _0(\rho _0)(1-{\overline{M}}(\rho _0)), \kappa _0(\rho _0)(1+{\overline{M}}(\rho _0))\right] . \end{aligned}$$

(59)

Remark 2.9

By Theorem 1.7, the first 6 steps allows us to check whether $\Theta \ne 0$ or not.

3 Examples

To illustrate the algorithm, we consider two concrete examples of analytic unfoldings of a Hopf-zero singularity (7) whose corresponding inner equation can be found in (10). In both cases, we prove that the associated constants $\Theta$ do not vanish and give rigorous estimates for them.

3.1 The First Example

As first example, we take

$$\begin{aligned} \alpha =1,\qquad \qquad b=1,\qquad g=h=0\qquad \text {and}\qquad f(X,Y,Z,\delta )=Z^3. \end{aligned}$$

(60)

This corresponds to $F_1(\phi ,\varphi ,s)=-s^{-3}$, $F_2=F_1$ and $H=0$. The inner equation (15) associated to this model is the following

$$\begin{aligned} \begin{pmatrix} \frac{d\phi }{ds}\\ \frac{d\varphi }{ds}\end{pmatrix}={\mathcal {A}}(s) \begin{pmatrix}\phi \\ \varphi \end{pmatrix}+{\mathcal {S}}(\phi ,\varphi ,s), \end{aligned}$$

(61)

with

$$\begin{aligned} {\mathcal {S}}(\phi ,\varphi ,s)=\begin{pmatrix} {\mathcal {S}}_1(\phi ,\varphi ,s)\\ {\mathcal {S}}_2(\phi ,\varphi ,s) \end{pmatrix} = \begin{pmatrix} \displaystyle \frac{\left( i-\frac{1}{s}\right) \varphi \phi ^2 s^2-\frac{1}{s^3}}{1+\varphi \phi s^2}\\ \displaystyle \frac{\left( i+\frac{1}{s}\right) \varphi ^2\phi s^2-\frac{1}{s^3}}{1+\varphi \phi s^2} \end{pmatrix}. \end{aligned}$$

(62)

Now we follow the steps of the algorithm in Sect. 2.3.

Step 1. In this case, $h_0=0$, and moreover, among all the constants defined in Step 1, the only one that is different form 0 is $a_3=1$.

Step 2. In this case, we have that $\rho _*^1=2$ and ${\mathcal {C}}_0=0$ so that $M_0=\frac{22}{3}$ is independent on $\rho$.

Step 3. We have that $\rho _*^2$ has to be such that

$$\begin{aligned} \sqrt{M_0}=\sqrt{\frac{22}{3}} < \rho _*^2. \end{aligned}$$

In addition $M_{ij}^{1, 2, 3}=0$ and

$$\begin{aligned} M_{11}^4(\rho ){} & {} = \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 1 + M_0 + \left( M_0 + \frac{M_0}{\rho }\right) \left( 1 - \frac{M_0^2}{\rho ^4}\right) + \frac{M_0}{\rho }\right) \\ {}{} & {} = \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 1 + M_0\left( 1+ \frac{1}{\rho }\right) \left( 2- \frac{M_0^2}{\rho ^4}\right) \right) ,\\ M_{12}^4(\rho ){} & {} = \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 1 + M_0 \left( 1 + \frac{1}{\rho }\right) \right) . \end{aligned}$$

Step 4 and Step 5. One can check that

$$\begin{aligned} L_1(\rho )=B_8\frac{M_{11}^4(\rho ) + M_{12}^4(\rho )}{\rho ^3}\le \frac{1}{2}, \end{aligned}$$

for $\rho \ge \rho _*=9.7895$. Under this condition, as we claimed in Remark 2.7,

$$\begin{aligned} L(\rho )\le \frac{1}{2}. \end{aligned}$$

Therefore we can guarantee the existence of $\psi ^{\pm }$ for $\rho \ge \rho _*=9.7895$.

Step 6. Now it only remains to compute

$$\begin{aligned} \psi ^+(-i\rho _*)-\psi ^-(-i\rho _*)\ne 0. \end{aligned}$$

By means of rigorous computer computations, which are discussed in more detail in “Appendix B,” we obtain that there exists a

$$\begin{aligned} \rho _* \in [15.99999965, 16.00000035] , \end{aligned}$$

(63)

for which

$$\begin{aligned}&\psi ^{+}\left( -i\rho _{*}\right) -\psi ^{-}\left( -i\rho _{*}\right) \\&=\left( \begin{array}{c} \Delta \phi \left( -i\rho _{*}\right) \\ \Delta \varphi \left( -i\rho _{*}\right) \end{array} \right) \nonumber \\&\in \left( \begin{array}{c} {[}-4.50096\cdot 10^{-10}, 4.50096\cdot 10^{-10}]-[1.88812\cdot 10^{-6},1.88897\cdot 10^{-6}]i\\ {[}-3.85539\cdot 10^{-10}, 3.85539\cdot 10^{-10}]+[-4.01544\cdot 10^{-10}, 3.4832\cdot 10^{-10}]i \end{array} \right) . \nonumber \end{aligned}$$

(64)

Therefore, the Stokes constant associated to the first example (60) does not vanish.

Now we follow Step 7. and Step 8. to provide rigorous accurate estimates for it.

Step 7. The constants $A_1$ and $A_2$ in (53) are

$$\begin{aligned} A_1(\rho )= \frac{M_{11}^4(\rho )}{3\rho ^3}+ \frac{M_{12}^4(\rho )}{4\rho ^4}, \qquad A_2(\rho )=\frac{M_{12}^4(\rho )}{2 \rho ^4}+ \frac{M_{11}^4(\rho )}{2\rho ^5}, \end{aligned}$$

(65)

which give the constant $A(\rho )=\textrm{max}\{A_1(\rho ),A_2(\rho )\}$ in (56). We obtain

$$\begin{aligned} A_1(\rho _*)&\in [0.010155523, 0.010155525], \\ A_2(\rho _*)&\in [0.0009597786, 0.0009597788], \\ A(\rho _*)&= A_1(\rho _*) < 1/2. \end{aligned}$$

Step 8. One can choose $\rho _0=\rho _*$. Then, by (32), (58) and (59), one obtains

$$\begin{aligned} \Theta \in [1.0378681, 1.0598665]+[-0.000253, 0.000253]i. \end{aligned}$$

(66)

We can see that the accuracy of the computation is roughly $2\cdot 10^{-2}$.

Remark 3.1

The computation suggests that the Stokes constant $\Theta \in {\mathbb {R}}$. Indeed, this fact can be proved for this example by considering ${\widehat{\psi }}^{\pm } (r) =({\widehat{\phi }}^{\pm }(r) , {\widehat{\varphi }}^{\pm }(r))= \psi ^{\pm }(i r )$, $r\in (-\infty , -\rho _0]$ that satisfies the real differential equation

$$\begin{aligned} \frac{d}{dr} {\widehat{\psi }} = \begin{pmatrix} -1 + \frac{1}{r} &{} 0 \\ 0 &{} 1+ \frac{1}{r}\end{pmatrix} {\widehat{\psi }} + \begin{pmatrix} \displaystyle -\frac{\left( 1-\frac{1}{r}\right) {\widehat{\varphi }}{\widehat{\phi }}^2 r^2-\frac{1}{r^3}}{1-{\widehat{\varphi }}{\widehat{\phi }} r^2}\\ \displaystyle - \frac{\left( 1+\frac{1}{r}\right) {\widehat{\varphi }}^2\widehat{\phi }r^2-\frac{1}{r^3}}{1-{\widehat{\varphi }}{\widehat{\phi }} r^2} \end{pmatrix} \end{aligned}$$

along with the real boundary conditions $\lim _{r\rightarrow -\infty } {\widehat{\psi }}^{\pm }(r) = 0$. Therefore ${\widehat{\psi }}^{\pm }$ are real functions and so their difference is.

As we will see in the next example, the fact that $\Theta \in {\mathbb {R}}$ is not generic and depends strongly on the symmetries of the inner equation.

3.2 The Second Example

The second example breaks the reversibility. It consists in taking $\alpha =b=1$, $g=h=0$ and $f(X,Y,Z,\delta )=Z^3+ 2XYZ$ which corresponds to

$$\begin{aligned} F_1(\phi ,\varphi ,s) = F_2(\phi ,\varphi ,s)= -\frac{1}{s^3}+ \frac{i}{s} (\phi ^2 -\varphi ^2),\qquad H=0, \end{aligned}$$

and then, the inner equation associated to this unfolding is:

$$\begin{aligned} \begin{pmatrix} \frac{d\phi }{ds}\\ \frac{d\varphi }{ds}\end{pmatrix}={\mathcal {A}}(s) \begin{pmatrix}\phi \\ \varphi \end{pmatrix}+{\mathcal {S}}(\phi ,\varphi ,s), \end{aligned}$$

(67)

with ${\mathcal {S}}$ defined as

$$\begin{aligned} {\mathcal {S}}(\phi ,\varphi ,s)=\begin{pmatrix} {\mathcal {S}}_1(\phi ,\varphi ,s)\\ {\mathcal {S}}_2(\phi ,\varphi ,s) \end{pmatrix} = \begin{pmatrix} \displaystyle \frac{\left( i-\frac{1}{s}\right) \varphi \phi ^2 s^2-\frac{1}{s^3}+\frac{i}{2s} (\phi ^2 - \varphi ^2)}{1+\varphi \phi s^2}\\ \displaystyle \frac{\left( i+\frac{1}{s}\right) \varphi ^2\phi s^2-\frac{1}{s^3}+ \frac{i}{2s}(\phi ^2 - \varphi ^2)}{1+\varphi \phi s^2} \end{pmatrix}. \end{aligned}$$

Step 1. We have that all the constants are zero except $a_3=1$, $C_F=2$, $C_F^\phi =C_F^\varphi =1$.

Step 2. As for the first example $\rho _*^1=2$ and ${\mathcal {C}}_0=0$ so that $M_0=\frac{22}{3}$.

Step 3. We have that $\rho _*^2$ has to be such that

$$\begin{aligned} \sqrt{M_0}=\sqrt{\frac{22}{3}} < \rho _*^2. \end{aligned}$$

In addition $M_{ij}^{1, 3}(\rho )=0$ being

$$\begin{aligned} M_{11}^2(\rho ){} & {} = M_{12}^2 (\rho )= \frac{2}{1- \frac{M_0^2}{\rho ^4}}\\ M_{11}^4(\rho ){} & {} = \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 2 + M_0 + \left( M_0 + \frac{M_0}{\rho }\right) \left( 1 - \frac{M_0^2}{\rho ^4}\right) + \frac{M_0}{\rho }\right) \\ {}{} & {} = \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 2 + M_0\left( 1+ \frac{1}{\rho }\right) \left( 2- \frac{M_0^2}{\rho ^4}\right) \right) \\ M_{12}^4(\rho ){} & {} = \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 2+ M_0 \left( 1 + \frac{1}{\rho }\right) \right) . \end{aligned}$$

Step 4 and Step 5. One can check that

$$\begin{aligned} L_1(\rho )=B_6 \frac{M_{11}^2(\rho )+ M_{12}^2(\rho )}{\rho } + B_8\frac{M_{11}^4(\rho ) + M_{12}^4(\rho )}{\rho ^3}\le \frac{1}{2} \end{aligned}$$

for $\rho \ge \rho _* \ge 9.7895$. Therefore, under this condition, using Remark 2.7 as for Example 1, we can guarantee that $L(\rho )\ge 1/2$ and, then, the existence of $\psi ^{\pm }$.

Remark 3.2

For Example 2, we can obtain more accurate estimates by computing directly the derivatives $\partial _{\phi ,\varphi } {\mathcal {S}}$. Indeed, performing straightforward computations, we obtain that $M_{ij}^1(\rho )=M_{ij}^2 (\rho ) = M_{ij}^3(\rho )=0$ and

$$\begin{aligned} \begin{aligned} M_{11}^4(\rho )&= \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 3 + M_0 \left( 1+ \frac{1}{\rho }\right) \left( 2- \frac{M_0^2}{\rho ^4}\right) \right) \\ M_{12}^4(\rho )&= \frac{M_0}{\left( 1 - \frac{M_0^2}{\rho ^4}\right) ^2} \left( 3 + M_0 \left( 1 + \frac{1}{\rho }\right) \right) . \end{aligned} \end{aligned}$$

(68)

Therefore

$$\begin{aligned} L_1(\rho )= B_8 \frac{M_{11}^4(\rho ) + M_{12}^4(\rho )}{\rho ^3}= \frac{5\pi }{32} \frac{M_{11}^4(\rho ) + M_{12}^4(\rho )}{\rho ^3}. \end{aligned}$$

Since we need to be as precise as possible, we use the constants $M_{ij}^k(\rho )$ defined in Remark 3.2 instead of the constants provided by the general method.

Step 6. We compute $\psi ^+(-i\rho _*)-\psi ^-(-i\rho _*)$. By means of rigorous computer computations, we obtain that there exists a $\rho _*$

$$\begin{aligned} \rho _*\in [15.99999965, 16.00000035], \end{aligned}$$

(69)

for which

$$\begin{aligned}&\psi ^{+}\left( -i\rho _{*}\right) -\psi ^{-}\left( -i\rho _{*}\right) \nonumber \\&=\left( \begin{array}{c} \Delta \phi \left( -i\rho _{*}\right) \\ \Delta \varphi \left( -i\rho _{*}\right) \end{array} \right) \\&\in \left( \begin{array}{c} {[}8.63066\cdot 10^{-9}, 9.53086\cdot 10^{-9}]-[ 1.88812\cdot 10^{-6},1.88897\cdot 10^{-6}]i\\ {[}-4.20777\cdot 10^{-10}, 3.50313\cdot 10^{-10}]+[-4.01721\cdot 10^{-10}, 3.48156\cdot 10^{-10}]i \end{array} \right) .\nonumber \end{aligned}$$

(70)

This implies that $\Theta \ne 0$. Now we perform the last two steps in the algorithm.

Step 7. For Example 2, we have that

$$\begin{aligned} A_1(\rho )=\frac{M_{11}^4(\rho )}{3\rho ^3}+ \frac{M_{12}^4(\rho )}{4\rho ^4}, \qquad A_2(\rho )=\frac{M_{12}^4(\rho )}{2 \rho ^4}+ \frac{M_{11}^4(\rho )}{2 \rho ^5}, \end{aligned}$$

and $A(\rho )=\textrm{max}\left\{ A_1(\rho ),A_2(\rho )\right\}$, with $M_{11}^4(\rho ),M_{12}^4(\rho )$ defined in (68). We obtain

$$\begin{aligned} A_1(\rho _*)&\in [0.0114071016, 0.0114071033], \\ A_2(\rho _*)&\in [0.00066984056, 0.00066984069], \\ A(\rho _*)&=A_1(\rho _*) < 1/2. \end{aligned}$$

Step 8. By means of rigorous computer validation, for $\rho _0=\rho _*$ using (32), (58) and (59) we obtain

$$\begin{aligned} \Theta \in [1.036525, 1.0612062]+[0.004738, 0.005355]i. \end{aligned}$$

We can see that the accuracy of the computation is roughly $2.5\cdot 10^{-1}$. Note that Step 8 implies that the Stokes constant has both nonzero real and imaginary part (see Remark 3.1).

4 Improving the Computation of the Stokes Constant

In this section, we give an improvement of the Steps 7 and 8 in Sect. 2.3 to obtain accurate estimates for the Stokes constant $\Theta$. We explain this improvement for the Example 1 given in Sect. 3.1 but the method we present is general and can be applied to any system.

Recall that, using (19) and (30),

$$\begin{aligned} \begin{aligned} \Theta&=\lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha s}\Delta \phi (s)=\kappa _0+\lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha s}{\mathcal {G}}_1\left( \Delta \psi (s)\right) \\&= \kappa _0+\lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha s}{\mathcal {G}}_1\left( \Delta \psi _0(s)\right) + \lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha s}{\mathcal {G}}_1\left( {\mathcal {G}}(\Delta \psi )(s)\right) . \end{aligned} \end{aligned}$$

Therefore, by (54), (55) and (36), the remainder

$$\begin{aligned} {\mathcal {E}}_{\Theta }=\Theta -\kappa _0-\lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha s}{\mathcal {G}}_1\left( \Delta \psi _0(s)\right) , \end{aligned}$$

satisfies

$$\begin{aligned} \begin{aligned} \left| {\mathcal {E}}_{\Theta }\right|&\le \sup _s\left| s^{-1}e^{i\alpha s}{\mathcal {G}}_1\left( {\mathcal {G}}(\Delta \psi )(s)\right) \right| \le A_1 \Vert {\mathcal {G}}(\Delta \psi )\Vert \le A_1 A \Vert \Delta \psi \Vert \\&\le A_1A \frac{|\kappa _0|}{1-A}. \end{aligned} \end{aligned}$$

where $A=A(\rho )=\textrm{max}\{A_1,A_2\}$ and $A_1, A_2$ are given in (65).

Using (31) and the fact that in Example 1, $h_0=0$ and therefore $\widetilde{{\mathcal {K}}}={\mathcal {K}}$,

$$\begin{aligned} \lim _{\Im s\rightarrow -\infty }s^{-1}e^{i\alpha s}{\mathcal {G}}_1\left( \Delta \psi _0(s)\right){} & {} =-\kappa _0\int _{-i\infty }^{i\rho }{\mathcal {K}}_{11}(t)dt\nonumber \\{} & {} = -i\kappa _0\int _{-\infty }^{\rho }{\mathcal {K}}_{11}(i\, r)dr, \end{aligned}$$

(71)

which implies

$$\begin{aligned} \Theta = \kappa _0 -i\kappa _0\int _{-\infty }^{\rho }{\mathcal {K}}_{11}(i\, r)dr +{\mathcal {E}}_{\Theta }. \end{aligned}$$

(72)

To obtain this integral, we need an approximation of the coefficient ${\mathcal {K}}_{11}$.

Lemma 4.1

The function ${\mathcal {K}}_{11}$ introduced in (26) associated to Eq. (61) satisfies

$$\begin{aligned} \begin{aligned} {\mathcal {K}}_{11}(s)&= \frac{i\beta _1 }{s^4} +\frac{\beta _2}{ s^5}+ \frac{i\, \beta _3 }{ s^6}+\frac{\beta _4}{ s^7} + \mathcal {EKT}_{11},\\ \beta _1&=3, \ \beta _2= -6, \ \beta _3=-68, \beta _4=48, \end{aligned} \end{aligned}$$

(73)

and $\mathcal {EKT}_{11}$ satisfies

$$\begin{aligned} |\mathcal {EKT}_{11}| \le \frac{B{\mathcal {R}}}{|s|^7}+\frac{B_{11}+B_{12}+B_{13}+B_{14}}{|s|^8}, \end{aligned}$$

(74)

where

$$\begin{aligned} \begin{aligned} B&=\frac{5\pi }{32} \left( M^4_{11}(\rho )+ M_{12}^4(\rho )\right) M_0+ \frac{225 \pi }{2 },\\ {\mathcal {R}}&=\frac{ 1+ 4 M_0(1+\frac{1}{\rho })+ 3\frac{ M_0^2}{\rho ^4}}{\left( 1-\frac{M_0^2}{\rho ^4}\right) ^3},\\ B_{11}&= \frac{M_*^4 (1+\frac{1}{\rho })}{\left( 1-\frac{M_*^2}{\rho ^4}\right) ^2},\\ B_{12}&=M_*^5\frac{ (2M_*(1+\frac{1}{\rho })+1)(3+\frac{2M_*^2}{\rho ^4})}{\rho ^4\left( 1-\frac{M_*^2}{\rho ^4}\right) ^2},\\ B_{13}&=2 M_*^3 \left( 2M_* \left( 1+\frac{1}{\rho }\right) +1 \right) ,\\ B_{14}&=800\left( 1-\frac{1}{\rho }\right) , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} M_*(\rho )=1+\frac{4}{\rho }+\frac{20}{\rho ^2}+\frac{120}{\rho ^3}. \end{aligned}$$

This lemma is proven in Sect. 4.1.

Now take $\rho _{\diamond }$ be such that $M_{*}(\rho )\le M_0(\rho )$ for $\rho \ge \rho _{\diamond }$. Using the expression of ${\mathcal {K}}_{11}$ given in (73) to compute (71) and using the remainder estimates in (74) we obtain:

$$\begin{aligned} \begin{aligned} -i\kappa _0\int _{-\infty }^{\rho }{\mathcal {K}}_{11}(i\, r)dr&= \kappa _0\int _{-\infty }^{\rho }\frac{\beta _1}{r^4}-\frac{\beta _2}{r^5}- \frac{\beta _3 }{r^6}+ \frac{\beta _4 }{r^7} dr +\mathcal{E}\mathcal{S}\\&=\kappa _0\left( -\frac{\beta _1}{3 \rho ^3}+\frac{\beta _2}{4\rho ^4}- \frac{\beta _3 }{5 \rho ^5}- \frac{\beta _4 }{6\rho ^6}\right) +\mathcal{E}\mathcal{S}, \end{aligned} \end{aligned}$$

(75)

and

$$\begin{aligned} \begin{aligned} |\mathcal{E}\mathcal{S}|&=|\mathcal{E}\mathcal{S}(\rho )| \le \kappa _0 \int _{\rho }^{\infty } \frac{B{\mathcal {R}}}{r^7}+\frac{B_{11}+B_{12}+B_{13}+B_{14}}{r^8} \\&= \frac{B{\mathcal {R}}}{6\rho ^6}+\frac{B_{11}+B_{12}+B_{13}+B_{14}}{7\rho ^7}. \end{aligned} \end{aligned}$$

Finally, using the expression (72) for $\Theta$ we obtain

$$\begin{aligned} \begin{aligned} \Theta&= \kappa _0\left( 1-\frac{\beta _1}{3 \rho ^3}+\frac{\beta _2}{4\rho ^4}- \frac{\beta _3 }{5 \rho ^5} -\frac{\beta _4 }{6\rho ^6} \right) +\mathcal{E}\mathcal{T},\\ \mathcal{E}\mathcal{T}&= \mathcal{E}\mathcal{T}(\rho )= \mathcal{E}\mathcal{S} +{\mathcal {E}}_{\Theta }. \end{aligned} \end{aligned}$$

As we know the constants $\beta _i$, one can use this formula to improve the computation of $\Theta$.

Indeed, using the above approach one obtains

$$\begin{aligned} \Theta \in [1.047906, 1.049289]+[-0.00070294, 0.00070294]i. \end{aligned}$$

(76)

We can see that the accuracy of the computation is roughly $10^{-3}$, which is an improvement when compared to the accuracy $2\cdot 10^{-1}$ from (66). (For (76) we have used the same $\rho _*$ and the computed value of $\Delta \psi (-i\rho _*)$ as for (66).)

4.1 Proof of Lemma 4.1

Lets call ${\mathcal {K}}_{ij}(s)$ the 4 elements of the matrix ${\mathcal {K}}$. To obtain expansions for these coefficients we compute first an expansion for $\psi ^\pm$ associated to Example 1 in (60).

Lemma 4.2

The functions $\psi ^\pm$ can be written as

$$\begin{aligned} \psi ^\pm =\psi _*+{\mathcal {E}}^\pm , \end{aligned}$$

where $\psi _*=(\phi _*,\varphi _*)$ with

$$\begin{aligned} \begin{aligned} \phi _*&= -\frac{i}{s^3} -\frac{4}{ s^4}+\frac{20i}{s^5}+\frac{120}{s^6}, \\ \varphi _*&= \frac{i}{ s^3} -\frac{4}{s^4}-\frac{20i}{s^5} +\frac{120}{s^6}, \end{aligned} \end{aligned}$$

which satisfy

$$\begin{aligned} |\phi _*(s)|,|\varphi _*(s)|\le \frac{M_*(\rho )}{|s|^3}, \end{aligned}$$

(77)

for

$$\begin{aligned} M_*(\rho )=1+\frac{4}{\rho }+\frac{20}{\rho ^2}+\frac{120}{\rho ^3}, \end{aligned}$$

(78)

and the remainders ${\mathcal {E}}^\pm =({\mathcal {E}}^\pm _\phi ,{\mathcal {E}}^\pm _\varphi )$ satisfy

$$\begin{aligned} |{\mathcal {E}}^\pm _\phi |,|{\mathcal {E}}^\pm _\varphi |\le \frac{B}{|s|^6}. \end{aligned}$$

(79)

with

$$\begin{aligned} B=\frac{5\pi }{32} \left( M^4_{11}(\rho )+ M_{12}^4(\rho )\right) M_0+ \frac{225 \pi }{2 }. \end{aligned}$$

Proof

By (62), the first iteration ${\mathcal {F}}^-(0)$ analyzed in Lemma 2.2 for Example 1 is given by

$$\begin{aligned} \begin{aligned} {\mathcal {F}}_1^-(0)&=s\int _{-\infty }^0\frac{1}{(s+t)^4} e^{i t}dt,\\ {\mathcal {F}}_2^-(0)&=s\int _{-\infty }^0\frac{1}{(s+t)^4} e^{-i t}dt. \end{aligned} \end{aligned}$$

Integrating by parts, we obtain

$$\begin{aligned} \begin{aligned} {\mathcal {F}}_1^-(0)=&\, -\frac{i}{ s^3}+\frac{4s}{i}\int _{-\infty }^0\frac{1}{(s+t)^5} e^{i t}dt\\ =&\, -\frac{i}{ s^3} -\frac{4}{ s^4}+\frac{20i}{ s^5} +\frac{120}{s^6}+ E_1^-(s) =\phi _*(s)+E_1^-(s), \\ \end{aligned} \end{aligned}$$

where

$$\begin{aligned} |E_1^-(s)|=\left| 720 s\int _{-\infty }^0\frac{1}{(s+t)^8}e^{i t}dt\right| \le \frac{720}{|s|^6}\int _0^\infty \frac{1}{(t^2+1)^4}dr = \frac{720}{|s|^6} \frac{5 \pi }{32}=\frac{225 \pi }{2 |s|^6}. \end{aligned}$$

Analogously, for the second component

$$\begin{aligned} \begin{aligned} {\mathcal {F}}_2^-(0)=\, \frac{i}{s^3} -\frac{4}{ s^4}+\frac{20i}{s^5} +\frac{120}{s^6} +E_2^-(s) = \varphi _*(s)+E_2^-(s), \end{aligned} \end{aligned}$$

where $E_2^-$ has the same bounds as $E_1^-$:

$$\begin{aligned} |E_2^-(s)|\le \frac{225 \pi }{2|s|^6}. \end{aligned}$$

Let us call $\psi _*=(\phi _*,\varphi _*)$ and $E^-=\left( E_1^-,E_2^-\right)$. Observe that by Lemma 2.1 and Corollary 2.4 and recalling that, for Example 1, one has $M_{11}^j=M_{12}^j=0$ for $j=1,2,3$,

$$\begin{aligned} \begin{aligned} \left\| {\mathcal {F}}^-(\psi )-{\mathcal {F}}^-(0)\right\| _{6}&\le \frac{5\pi }{32}\left\| {\mathcal {S}}(\psi ,s)-{\mathcal {S}}(0,s)\right\| _{7}\\&\le \frac{5\pi }{32} \left( M^4_{11}(\rho )+ M_{12}^4\right) \left\| \psi \right\| _{3}. \end{aligned} \end{aligned}$$

Now we use that $\psi ^-$ is a fixed point of operator ${\mathcal {F}}^-$ and therefore

$$\begin{aligned} \begin{aligned} \left\| \psi ^- -\psi _*\right\| _{6}&\le \left\| \psi ^- -{\mathcal {F}}^-(0)\right\| _{6}+ \left\| E^-(0)\right\| _{6}\\&\le \frac{5\pi }{32} \left( M^4_{11}(\rho )+ M_{12}^4(\rho )\right) M_0+ \frac{225 \pi }{2 } =B. \end{aligned} \end{aligned}$$

We conclude

$$\begin{aligned} \begin{aligned} \left| \phi ^- -\left( - \frac{i}{s^3} -\frac{4}{s^4}-\frac{20i}{s^5} +\frac{120}{s^6}\right) \right|&\le \frac{B}{|s|^6}, \\ \left| \varphi ^- -\left( \frac{i}{s^3} -\frac{4}{s^4}+\frac{20i}{ s^5} +\frac{120}{ s^6}\right) \right|&\le \frac{B}{|s|^6}. \end{aligned} \end{aligned}$$

$\square$

Using the previous result, we can compute a better asymptotic expansion of ${\mathcal {K}}_{11}$. We rely on the expression

$$\begin{aligned} {\mathcal {K}}_{11}(s)=\int _0^1 \partial _\phi {\mathcal {S}}_{1}(\psi ^-(s)+t(\psi ^+(s) -\psi ^-(s)) dt. \end{aligned}$$

Note that, for Example 1,

$$\begin{aligned} \partial _\phi {\mathcal {S}}_1=\frac{2\mathtt A\varphi \phi +\mathtt B\varphi ^2\phi ^2+ \mathtt C\varphi }{\left( 1+\mathtt D\varphi \phi \right) ^2}, \end{aligned}$$

where

$$\begin{aligned} \mathtt A=\left( i-\frac{1}{s}\right) s^2, \quad \mathtt B=s^4 \left( i-\frac{1}{s}\right) , \quad \mathtt C=\frac{1}{s},\quad \mathtt D=s^2. \end{aligned}$$

We define the function

$$\begin{aligned} g(r)=\partial _\phi {\mathcal {S}}_{1}\left( \psi _*(s)+r\left( {\mathcal {E}}^-(s)+t\left( {\mathcal {E}}^+(s) -{\mathcal {E}}^-(s)\right) \right) \right) , \end{aligned}$$

and using the fundamental theorem of calculus, $g(1)=g(0)+\int _0^1 g'(r)dr$, we have that

$$\begin{aligned} \begin{aligned}&\partial _\phi {\mathcal {S}}_1(\psi ^-(s)+t(\psi ^+(s) -\psi ^-(s)) =\partial _\phi {\mathcal {S}}_1(\psi _*(s))\\&+ \left( {\mathcal {E}}_\phi ^-(s)+t({\mathcal {E}}_\phi ^+(s) -{\mathcal {E}}_\phi ^-(s)\right) \int _0^1\partial _{\phi \phi } {\mathcal {S}}_1 \left( \psi _*(s)+r\left( {\mathcal {E}}^-(s)+t({\mathcal {E}}^+(s) -{\mathcal {E}}^-(s)\right) \right) \ dr \\&+ \left( {\mathcal {E}}_\varphi ^-(s)+t({\mathcal {E}}_\varphi ^+(s) -{\mathcal {E}}_\varphi ^-(s)\right) \int _0^1\partial _{\phi \varphi } {\mathcal {S}}_1 \left( \psi _*(s)+r\left( {\mathcal {E}}^-(s)+t({\mathcal {E}}^+(s) -{\mathcal {E}}^-(s)\right) \right) \ dr. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} {\mathcal {K}}_{11}&=\partial _\phi {\mathcal {S}}_1(\psi _*(s))\\&\quad +\int _0^1 \left[ \left( {\mathcal {E}}_\phi ^-(s)+t({\mathcal {E}}_\phi ^+(s) -{\mathcal {E}}_\phi ^-(s)\right) \int _0^1\partial _{\phi \phi } {\mathcal {S}}_{1}\left( \psi _*(s)\right. \right. \\&\quad \left. \left. +r\left( {\mathcal {E}}^-(s)+t({\mathcal {E}}^+(s) -{\mathcal {E}}^-(s)\right) \right) \, dr \right] \, dt\\&\quad + \int _0^1\left[ \left( {\mathcal {E}}_\varphi ^-(s)+t({\mathcal {E}}_\varphi ^+(s) -{\mathcal {E}}_\varphi ^-(s)\right) \int _0^1\partial _{\phi \varphi } {\mathcal {S}}_{1}\left( \psi _*(s)\right. \right. \\&\quad \left. \left. +r\left( {\mathcal {E}}^-(s)+t({\mathcal {E}}^+(s) -{\mathcal {E}}^-(s)\right) \right) \, dr \right] \, dt. \end{aligned} \end{aligned}$$

(80)

One can easily check that

$$\begin{aligned} \partial _{\phi \phi } {\mathcal {S}}_{1}(\psi )= \frac{2\mathtt A\varphi - 2\mathtt D\mathtt C\varphi ^2}{\left( 1+\mathtt D\varphi \phi \right) ^3},\qquad \partial _{\phi \varphi } {\mathcal {S}}_{1}(\psi )= \frac{\mathtt C+ 2\mathtt A\phi - \mathtt D\mathtt C\varphi \phi }{\left( 1+\mathtt D\varphi \phi \right) ^3}. \end{aligned}$$

Moreover, by (52) and (77), we know that

$$\begin{aligned} |\psi _*(s) +r ({\mathcal {E}}^-(s) + t ({\mathcal {E}}^+(s) - {\mathcal {E}}^-(s)))|\le \frac{M_0}{|s|^3}. \end{aligned}$$

Then, taking into account the definitions of $\mathtt A,\mathtt C,\mathtt D$, one can obtain the following bounds for $|s|\ge \rho$,

$$\begin{aligned} \begin{aligned} |\partial _{\phi \phi } {\mathcal {S}}_{1}(\psi )|&\le \frac{1 }{s} \frac{ 2 M_0(1+\frac{1}{\rho })+ 2\frac{ M_0^2}{\rho ^4}}{\left( 1-\frac{M_0^2}{\rho ^4}\right) ^3}, \\ |\partial _{\phi \varphi } {\mathcal {S}}_{1}(\psi )|&\le \frac{1}{s} \frac{ 1+ 2 M_0(1+\frac{1}{\rho })+ \frac{ M_0^2}{\rho ^4}}{\left( 1-\frac{M_0^2}{\rho ^4}\right) ^3}. \end{aligned} \end{aligned}$$

Using (80) and the bounds (79), we obtain that ${\mathcal {K}}_{11}$ satisfies

$$\begin{aligned} {\mathcal {K}}_{11}(s)= \partial _{\phi } {\mathcal {S}}_{1}(\psi _*(s))+ \mathcal{E}\mathcal{K}_{11}, \end{aligned}$$

(81)

with

$$\begin{aligned} |\mathcal{E}\mathcal{K}_{11}|\le \frac{B}{|s|^7}{\mathcal {R}},\qquad \text {where}\qquad {\mathcal {R}}=\frac{ 1+ 4 M_0(1+\frac{1}{\rho })+ 3\frac{ M_0^2}{\rho ^4}}{\left( 1-\frac{M_0^2}{\rho ^4}\right) ^3}. \end{aligned}$$

(82)

Last step is to compute $\partial _{\phi } {\mathcal {S}}_{1}(\psi _*)$ using the formula of $\psi _*$ in Lemma 4.2. We recall that

$$\begin{aligned} \begin{aligned} \partial _{\phi } {\mathcal {S}}_{1}(\psi _*)&=\frac{\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) \left( 2+\varphi _*\phi _* s^2\right) +\frac{1}{s}\varphi _*}{\left( 1+\varphi _*\phi _* s^2\right) ^2}, \end{aligned} \end{aligned}$$

and we write

$$\begin{aligned} \begin{aligned} \partial _{\phi } {\mathcal {S}}_{1}(\psi _*)&= \frac{2\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) +\frac{1}{s}\varphi _*}{\left( 1+\varphi _*\phi _* s^2\right) ^2}+ \mathcal{E}\mathcal{R}_1, \end{aligned} \end{aligned}$$

with

$$\begin{aligned} | \mathcal{E}\mathcal{R}_1 |= \left| \frac{\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) \varphi _*\phi _* s^2}{\left( 1+\varphi _*\phi _* s^2\right) ^2} \right| \le \frac{M_*^4 (1+\frac{1}{\rho })}{\left( 1-\frac{M_*^2}{\rho ^4}\right) ^2}\frac{1}{|s|^8} =\frac{B_{11}}{|s|^8}. \end{aligned}$$

(83)

Now, using that

$$\begin{aligned} \frac{1}{(1+x)^2}=1-2x+\frac{x^2(3+2x)}{1+x^2}, \end{aligned}$$

we write

$$\begin{aligned} \begin{aligned} \partial _{\phi } {\mathcal {S}}_{1}(\psi _*)&= \left( 2\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) +\frac{1}{s}\varphi _*\right) \left( 1-2\varphi _*\phi _* s^2\right) + \mathcal{E}\mathcal{R}_1 +\mathcal{E}\mathcal{R}_2\\&= 2\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) +\frac{1}{s}\varphi _* + \mathcal{E}\mathcal{R}_1 +\mathcal{E}\mathcal{R}_2+\mathcal{E}\mathcal{R}_3, \end{aligned} \end{aligned}$$

with

$$\begin{aligned} | \mathcal{E}\mathcal{R}_2 |= & {} \left| 2\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) +\frac{1}{s}\varphi _*\right| \left| \frac{\varphi _*^2\phi _*^2 s^4\left( 3+ 2\varphi _*\phi _* s^2\right) }{\left( 1+\varphi _*\phi _* s^2\right) ^2}\right| \nonumber \\\le & {} \frac{M_*^5}{|s|^{12}}\frac{ (2M_*(1+\frac{1}{\rho })+1)(3+\frac{2M_*^2}{\rho ^4})}{\left( 1-\frac{M_*^2}{\rho ^4}\right) ^2} \nonumber \\\le & {} \frac{M_*^5}{|s|^{8}}\frac{ (2M_*(1+\frac{1}{\rho })+1)(3+\frac{2M_*^2}{\rho ^4})}{\rho ^4\left( 1-\frac{M_*^2}{\rho ^4}\right) ^2} =\frac{B_{12}}{|s|^8}, \nonumber \\ |\mathcal{E}\mathcal{R}_3|= & {} \left| \left( 2\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) +\frac{1}{s}\varphi _*\right) \left( -2\varphi _*\phi _* s^2\right) \right| \nonumber \\\le & {} \frac{2 M_*^3}{|s|^{8}} \left( 2M_* \left( 1+\frac{1}{\rho }\right) +1 \right) =\frac{B_{13}}{|s|^8}. \end{aligned}$$

(84)

We now substitute the expressions $\psi _*$ in Lemma 4.2 which give

$$\begin{aligned} \begin{aligned} \phi _*\varphi _*&=\frac{1}{ s^6}-\frac{24}{s^8}+\frac{400}{s^{10}},\\ 2\varphi _*\phi _* s^2\left( i-\frac{1}{s}\right) +\frac{1}{s}\varphi _*&= \frac{3i}{ s^4}-\frac{6}{s^5}-\frac{68i}{s^6}+\frac{48}{ s^7}+\frac{800i}{s^8}-\frac{800}{ s^9}, \end{aligned} \end{aligned}$$

which gives

$$\begin{aligned} \begin{aligned} \partial _\phi {\mathcal {S}}_{1}(\psi _*)&=\frac{3i}{s^4}-\frac{6}{ s^5}-\frac{68i}{ s^6}+\frac{48}{ s^7}\\&\quad +\mathcal{E}\mathcal{R}_1 +\mathcal{E}\mathcal{R}_2+\mathcal{E}\mathcal{R}_3+\mathcal{E}\mathcal{R}_4, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} |\mathcal{E}\mathcal{R}_4|\le \frac{800}{|s|^8} \left( 1-\frac{1}{\rho }\right) = \frac{B_{14}}{|s|^8}. \end{aligned}$$

(85)

Using these approximations in (81), we obtain the statement of the lemma taking

$$\begin{aligned} \mathcal {EKT}_{11}= \mathcal{E}\mathcal{K}_{11} + \mathcal{E}\mathcal{R}_1 +\mathcal{E}\mathcal{R}_2+\mathcal{E}\mathcal{R}_3+\mathcal{E}\mathcal{R}_4. \end{aligned}$$

Using the bounds (82), (83), (84), (85), we get

$$\begin{aligned} |\mathcal {EKT}_{11}| \le \frac{ B{\mathcal {R}}}{|s|^7}+\frac{B_{11}+B_{12}+B_{13}+B_{14}}{|s|^8}. \end{aligned}$$

(86)

.

Appendices

Proof of Lemma 2.1

To prove Lemma 2.1, we need first the following lemma.

Lemma A.1

If m is even

$$\begin{aligned} \int _{-\infty }^0 \frac{1}{(t^2 + 1)^{m/2}} \, dt = \frac{\pi }{2} \frac{(m-3))!!}{(m-2)!!} =:B_{m}, \end{aligned}$$

and, for m odd

$$\begin{aligned} \int _{-\infty }^0 \frac{1}{(t^2 + 1)^{m/2}} \, dt = \frac{(m-3)!!}{(m-2)!!} =:B_m. \end{aligned}$$

Proof

Integrating by parts,

$$\begin{aligned} \begin{aligned} I_{k}:&= \int _{-\infty }^0 \frac{1}{(t^2 + 1)^{k/2}} \, dt = k\int _{-\infty }^0 \frac{t^2}{(t^2 + 1)^{\frac{k}{2}+1}} \\&= kI_k - kI_{k+2}. \end{aligned} \end{aligned}$$

Therefore $I_{k+2}= \frac{k-1}{k} I_{k}$ which implies $I_{k}=\frac{k-3}{k-2}I_{k-2}$ and therefore

$$\begin{aligned} I_{k}=\frac{(k-3)!!}{(k-2)!!} J, \end{aligned}$$

with $J=I_3$ if k is odd and $J=I_2$ if k is even. Since $I_3=1$ and $I_2=\frac{\pi }{2}$ we are done. $\square$

We use this lemma to prove Lemma 2.1.

Proof of Lemma 2.1

Let $\psi =(\phi ,\varphi )\in {\mathcal {X}}_{\nu }$ The first component of ${\mathcal {B}}^-(\psi )$ is

$$\begin{aligned} {\mathcal {B}}^-_1(\psi ) = s \int _{-\infty }^0 \frac{e^{i\alpha t}}{s+t}\phi (s+t)\, dt. \end{aligned}$$

(87)

We first prove the first item. Indeed, using that $|s+t|^2 \ge |s|^2 + t^2$ for $s\in {\mathcal {D}}^-_\rho$, one can prove that

$$\begin{aligned} \left| {\mathcal {B}}^-_1(\psi )\right| \le |s| \Vert \phi \Vert _\nu \int _{-\infty }^0 \frac{1}{|s+t|^{\nu +1}}\, dt \le \frac{\Vert \phi \Vert _{\nu }}{|s|^{\nu -1}} \int _{-\infty }^0 \frac{1}{\big (t^2 + 1\big )^{\frac{\nu +1}{2}}} = B_{\nu +1} \frac{\Vert \phi \Vert _{\nu }}{|s|^{\nu -1}}. \end{aligned}$$

Analogously we deal with ${\mathcal {B}}^-_2(\psi )$ and we obtain the result in the first item, taking into account that the product norm is the supremum norm.

Now we deal with the second item. By the geometry of ${\mathcal {D}}^-_\rho$, and using the Cauchy’s theorem, we can change the path of integration in the integral  (87) defining ${\mathcal {B}}_1^-(\psi )$ to $t e^{i\gamma }$, $t\in (-\infty ,0]$, with $0\le \gamma \le \beta$. We obtain then

$$\begin{aligned} {\mathcal {B}}^-_1(\psi )(s) = s \int _{-\infty }^0 \frac{e^{i\alpha te^{i\gamma }}}{s+te^{i\gamma }} \phi (s+te^{i\gamma })e^{i\gamma } \, dt. \end{aligned}$$

Notice that $s+te^{i\gamma } \in {\mathcal {D}}^-_\rho$ and

$$\begin{aligned} |s+te^i\gamma | \ge |s|\sin \left( \frac{\pi }{2}-\gamma \right) = |s|\cos \gamma . \end{aligned}$$

Therefore

$$\begin{aligned} \left| {\mathcal {B}}^-_1(\psi )\right| \le \frac{\Vert \phi \Vert _\nu }{|s|^\nu \big (\cos \gamma \big )^{\nu +1}}\int _{-\infty }^0 e^{\alpha \sin \gamma t} \, dt = \frac{\Vert \phi \Vert _\nu }{|s|^\nu \alpha \big (\cos \gamma \big )^{\nu +1} \sin \gamma }. \end{aligned}$$

The function $\big (\cos \gamma \big )^{\nu +1} \sin \gamma$ has only a maximum in $\left( 0,\frac{\pi }{2}\right)$ in $\gamma _*$ such that $(\nu +1)\sin ^2 \gamma _* =1$.

As in the first item, ${\mathcal {B}}_2^-(\psi )$ can be treated in the same way, changing here the integration path to $te^{-i\gamma }$. $\square$

Computing the bound on $\Delta \psi \left( -i\rho ^{*}\right)$

Here we provide an explicit rigorous estimate, using interval arithmetic bounds, for the distance between $\psi ^{+}\left( -i\rho \right)$ and $\psi ^{-}\left( -i\rho \right)$ for a given $\rho >0$ which we have used for our Examples 1 and 2 (discussed in Sects. 3.1, 3.2 and 4).

We start with Example 1. We work with the system (11) with $F_{1}=-s^{-3}$, $F_{2}=0,$ $H=0$.

By writing

$$\begin{aligned} \phi =x_{1}+iy_{1},\qquad \varphi =x_{2}+iy_{2},\qquad s=s_{1}+is_{2}, \end{aligned}$$

we can rewrite (11) as

$$\begin{aligned} x_{1}^{\prime }&=\left( 1+\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) y_{1}+\frac{s_{1}x_{1}}{s_{1}^{2}+s_{2}^{2}}-\frac{s_{1}^{3}-3s_{1}s_{2}^{2} }{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}},\nonumber \\ y_{1}^{\prime }&=-\left( 1+\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) x_{1}+\frac{s_{1}y_{1}}{s_{1}^{2}+s_{2}^{2}}-\frac{s_{2}^{3}-3s_{1}^{2}s_{2} }{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}},\nonumber \\ x_{2}^{\prime }&=-\left( 1-\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) y_{2}+\frac{s_{1}x_{2}}{s_{1}^{2}+s_{2}^{2}}-\frac{s_{1}^{3}-3s_{1}s_{2}^{2} }{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}},\nonumber \\ y_{2}^{\prime }&=\left( 1-\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) x_{2}+\frac{s_{1}y_{2}}{s_{1}^{2}+s_{2}^{2}}-\frac{s_{2}^{3}-3s_{1}^{2}s_{2} }{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}},\nonumber \\ s_{1}^{\prime }&=-2s_{1}s_{2}\left( x_{1}y_{2}+x_{2}y_{1}\right) +\left( x_{1}x_{2}-y_{1}y_{2}\right) \left( s_{1}^{2}-s_{2}^{2}\right) +1,\nonumber \\ s_{2}^{\prime }&=2s_{1}s_{2}\left( x_{1}x_{2}-y_{1}y_{2}\right) +\left( x_{1}y_{2}+x_{2}y_{1}\right) \left( s_{1}^{2}-s_{2}^{2}\right) . \end{aligned}$$

(88)

This is an ODE in ${\mathbb {R}}^{6}$, with real time $\tau$. We shall write $\Phi _{\tau }$ for the flow induced by (88).

Note that $\left( s_{1},s_{2}\right) =\left( 0,-\rho \right)$ corresponds to the complex $s=-i\rho$.

For $x\in {\mathbb {R}}^{6}$, we shall write

$$\begin{aligned} \tau ^{+}\left( x\right)&=\sup \left\{ \tau <0:\pi _{s_{1}}\Phi _{\tau } (x)=0\right\} ,\\ \tau ^{-}\left( x\right)&=\inf \left\{ \tau >0:\pi _{s_{1}}\Phi _{\tau } (x)=0\right\} , \end{aligned}$$

and define

$$\begin{aligned} {\mathcal {P}}^{+},{\mathcal {P}}^{-}&:{\mathbb {R}}^{6}\rightarrow \left\{ s_{1}=0\right\} ,\\ {\mathcal {P}}^{+}\left( x\right)&:=\Phi _{\tau ^{+}\left( x\right) }(x),\\ {\mathcal {P}}^{-}\left( x\right)&:=\Phi _{\tau ^{-}\left( x\right) }(x). \end{aligned}$$

We do not assume that ${\mathcal {P}}^{+}$ and ${\mathcal {P}}^{-}$ are globally defined. Whenever we write ${\mathcal {P}}^{+}\left( x\right)$ or ${\mathcal {P}}^{-}\left( x\right)$, we will always validate that the considered point x lies in the domain of the map.

We know that the two solutions $\psi ^{\pm }$ of (12) (with $F_{1}=-s^{-3}$, $F_{2}=0,$ $H=0$) satisfy

$$\begin{aligned} \left\| \psi ^{-}(s)\right\|&\le \left| s\right| ^{-3} M_{0}\le \left| \Re s\right| ^{-3}M_{0}\qquad \text {for } \Re s<0, \end{aligned}$$

(89)

$$\begin{aligned} \left\| \psi ^{+}(s)\right\|&\le \left| s\right| ^{-3} M_{0}\le \left| \Re s\right| ^{-3}M_{0}\qquad \text {for } \Re s>0. \end{aligned}$$

(90)

Fig. 1

Illustration for Lemma B.1. Here we depict the projection of the flow onto the $s=(s_1,s_2)$ coordinates. The sets ${\mathcal {P}}^+({\textbf{p}}^+)$ and ${\mathcal {P}}^-({\textbf{p}}^-)$ which lead to the bound (92) for $\Delta \psi$ lie on the section $\{s_1=0\}$

Lemma B.1

Consider $s^{-},s_{l}^{+},s_{u}^{+}\in {\mathbb {R}}^{2}$ of the form $s^{-}=( s_{1}^{-},s_{2}^{-}) ,$ $s_{l}^{+}=( s_{1}^{+},s_{2,l}^{+}) ,$ $s_{u}^{+}=( s_{1}^{+},s_{2,u} ^{+})$, where $s_{1}^{-}<0<s_{1}^{+}$ and $s_{2,l}^{+}<s_{2,u}^{+}$. (The subscripts l and u stand for ‘lower’ and ‘upper’; see Fig. 1.) Let ${\textbf{s}}^{+}\subset {\mathbb {R}}^{2}$ be the vertical interval joining $s_{l}^{+}$ and $s_{u}^{+}$ and let

$$\begin{aligned} {\textbf{p}}^{-}&:=\left\{ p\in {\mathbb {R}}^{6}:\pi _{s}p=s^{-},\left\| \pi _{x,y}p\right\| \le \left| s_{1}^{-}\right| ^{-3}M_{0}\right\} ,\\ {\textbf{p}}^{+}&:=\left\{ p\in {\mathbb {R}}^{6}:\pi _{s}p\in {\textbf{s}} ^{+},\left\| \pi _{x,y}p\right\| \le \left| s_{1}^{+}\right| ^{-3}M_{0}\right\} ,\\ {\textbf{p}}_{u}^{+}&:=\left\{ p\in {\mathbb {R}}^{6}:\pi _{s}p=s_{u} ^{+},\left\| \pi _{x,y}p\right\| \le \left| s_{1}^{+}\right| ^{-3}M_{0}\right\} ,\\ {\textbf{p}}_{l}^{+}&:=\left\{ p\in {\mathbb {R}}^{6}:\pi _{s}p=s_{l} ^{+},\left\| \pi _{x,y}p\right\| \le \left| s_{1}^{+}\right| ^{-3}M_{0}\right\} . \end{aligned}$$

If

$$\begin{aligned} \pi _{s_{2}}{\mathcal {P}}^{+}\left( {\textbf{p}}_{l}^{+}\right)<\pi _{s_{2} }{\mathcal {P}}^{-}\left( {\textbf{p}}^{-}\right)<\pi _{s_{2}}{\mathcal {P}} ^{+}\left( {\textbf{p}}_{u}^{+}\right) <0, \end{aligned}$$

(91)

then there exists a $\rho ^{*}\in -\pi _{s_{2}}{\mathcal {P}}^{-}\left( {\textbf{p}}^{-}\right)$ such that

$$\begin{aligned} \Delta \psi \left( -i\rho ^{*}\right) =\psi ^{+}\left( -i\rho ^{*}\right) -\psi ^{-}\left( -i\rho ^{*}\right) \in \pi _{x,y}\left( {\mathcal {P}} ^{+}\left( {\textbf{p}}^{+}\right) -{\mathcal {P}}^{-}\left( {\textbf{p}} ^{-}\right) \right) . \end{aligned}$$

(92)

Proof

From (89–90) we see that

$$\begin{aligned} \psi ^{-}\left( s^{-}\right)&\in \pi _{x,y}{\textbf{p}}^{-},\\ \psi ^{+}\left( {\textbf{s}}^{+}\right)&\subset \pi _{x,y}{\textbf{p}}^{+}. \end{aligned}$$

By the Bolzano theorem applied to

$$\begin{aligned} {\textbf{s}}^{+}\ni s^{+}\mapsto \pi _{s_{2}}\left( {\mathcal {P}}^{+}\left( \psi ^{-}\left( s^{+}\right) ,s^{+}\right) -{\mathcal {P}}^{-}\left( \psi ^{-}\left( s^{-}\right) ,s^{-}\right) \right) , \end{aligned}$$

from (91), we see that there there exists a $s_{*}^{+} \in {\textbf{s}}^{+}$ such that

$$\begin{aligned} {\mathcal {P}}^{+}\left( \psi ^{-}\left( s_{*}^{+}\right) ,s_{*} ^{+}\right) ={\mathcal {P}}^{-}\left( \psi ^{-}\left( s^{-}\right) ,s^{-}\right) . \end{aligned}$$

By definition of ${\mathcal {P}}^{\pm }$ we know that $\pi _{s}{\mathcal {P}}^{\pm }\left( q\right) \in \left\{ 0\right\} \times {\mathbb {R}}$, so

$$\begin{aligned} \pi _{s}{\mathcal {P}}^{+}\left( \psi ^{-}\left( s_{*}^{+}\right) ,s_{*}^{+}\right) =\pi _{s}{\mathcal {P}}^{-}\left( \psi ^{-}\left( s^{-}\right) ,s^{-}\right) =\left( 0,-\rho ^{*}\right) , \end{aligned}$$

for some $\rho ^{*}>0$ (the sign follows from (91)) and hence

$$\begin{aligned} \psi ^{-}\left( -i\rho ^{*}\right)&=\pi _{x,y}{\mathcal {P}}^{-}\left( \psi ^{-}\left( s^{-}\right) ,s^{-}\right) \in \pi _{x,y}{\mathcal {P}} ^{-}\left( {\textbf{p}}^{-}\right) ,\\ \psi ^{+}\left( -i\rho ^{*}\right)&=\pi _{x,y}{\mathcal {P}}^{-}\left( \psi ^{-}\left( s_{*}^{-}\right) ,s_{*}^{-}\right) \in \pi _{x,y}{\mathcal {P}}^{+}\left( {\textbf{p}}^{+}\right) , \end{aligned}$$

which implies (92), as required. $\square$

In our computer-assisted proof, we have taken ${\bar{\rho }}:=$ 16.00008679 and

$$\begin{aligned} s^{-}&=\left( -10^{3},-{\bar{\rho }}\right) , \end{aligned}$$

(93)

$$\begin{aligned} s_{u}^{+}&=\left( 10^{3},-{\bar{\rho }}+10^{-6}\right) , \end{aligned}$$

(94)

$$\begin{aligned} s_{l}^{+}&=\left( 10^{3},-{\bar{\rho }}-10^{-6}\right) . \end{aligned}$$

(95)

(the choice of ${\bar{\rho }}$ is dictated by the fact that then $\rho ^{*}\approx 16$; see (63)). Then, we have validated that, with such choice of $s^{-},s_{u}^{+},s_{l}^{+}$, Lemma B.1 leads to the bound (64). The computation of ${\mathcal {P}}^{+}\left( {\textbf{p}}^{+}\right) ,{\mathcal {P}}^{-}\left( {\textbf{p}}^{-}\right)$ required a long integration time, due to the number $10^{3}$ in our choice of $s^{-}_1,s_1^{+}$. The benefit of such large value in $\Re s$ is that then $\left| \Re s\right| ^{-3}$ is a very small number, leading to small sets ${\textbf{p}}^{\pm },{\textbf{p}}_{l}^{+},{\textbf{p}}_{u}^{+}$. This results in good bounds on $\Delta \psi$. Such choice of $\Re s$ was reached by trial and error.

The computer-assisted validation of (64) took under 20 seconds, running on a single thread of a standard laptop.

The computation of $\Delta \psi$ in the second example also follows from Lemma B.1. The only difference is the formula for the vector field, which is

$$\begin{aligned} x_{1}^{\prime }&=\frac{s_{1}}{s_{1}^{2}+s_{2}^{2}}x_{1}+\left( \alpha +\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) y_{1}-\frac{s_{1}^{3}-3s_{1} s_{2}^{2}}{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}}\\&\quad +\frac{1}{s_{1}^{2}+s_{2}^{2}}\left( s_{2}\left( x_{1}^{2}-y_{1} ^{2}-x_{2}^{2}+y_{2}^{2}\right) +2s_{1}\left( x_{2}y_{2}-x_{1}y_{1}\right) \right) ,\\ y_{1}^{\prime }&=-\left( \alpha +\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) x_{1}+\frac{s_{1}}{s_{1}^{2}+s_{2}^{2}}y_{1}-\frac{s_{2}^{3}-3s_{1}^{2}s_{2} }{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}}\\&\quad +\frac{1}{s_{1}^{2}+s_{2}^{2}}\left( s_{1}\left( x_{1}^{2}-x_{2} ^{2}-y_{1}^{2}+y_{2}^{2}\right) +2s_{2}\left( x_{1}y_{1}-x_{2}y_{2}\right) \right) ,\\ x_{2}^{\prime }&=\frac{s_{1}}{s_{1}^{2}+s_{2}^{2}}x_{2}-\left( \alpha -\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) y_{2}-\frac{s_{1}^{3}-3s_{1} s_{2}^{2}}{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}}\\&\quad +\frac{1}{s_{1}^{2}+s_{2}^{2}}\left( s_{2}\left( x_{1}^{2}-y_{1} ^{2}-x_{2}^{2}+y_{2}^{2}\right) +2s_{1}\left( x_{2}y_{2}-x_{1}y_{1}\right) \right) ,\\ y_{2}^{\prime }&=\left( \alpha -\frac{s_{2}}{s_{1}^{2}+s_{2}^{2}}\right) x_{2}+\frac{s_{1}}{s_{1}^{2}+s_{2}^{2}}y_{2}-\frac{s_{2}^{3}-3s_{1}^{2}s_{2} }{\left( s_{1}^{2}+s_{2}^{2}\right) ^{3}}\\&\quad +\frac{1}{s_{1}^{2}+s_{2}^{2}}\left( s_{1}\left( x_{1}^{2}-x_{2} ^{2}-y_{1}^{2}+y_{2}^{2}\right) +2s_{2}\left( x_{1}y_{1}-x_{2}y_{2}\right) \right) ,\\ s_{1}^{\prime }&=1-2bs_{1}s_{2}\left( x_{1}y_{2}+x_{2}y_{1}\right) +b\left( s_{1}^{2}-s_{2}^{2}\right) \left( x_{1}x_{2}-y_{1}y_{2}\right) ,\\ s_{2}^{\prime }&=2bs_{1}s_{2}\left( x_{1}x_{2}-y_{1}y_{2}\right) +b\left( s_{1}^{2}-s_{2}^{2}\right) \left( x_{1}y_{2}+x_{2}y_{1}\right) . \end{aligned}$$

We take the same $s^{-},s_{l}^{+},s_{u}^{+}$ as in (93 –95) which, with the aid of Lemma B.1 and interval arithmetic integration, leads to the bounds (69 –70).

The computer-assisted validation of (70) took under 25 seconds, running on a single thread of a standard laptop.