Melnikov method for non-conservative perturbations of the restricted three-body problem

Abstract

We consider the planar circular restricted three-body problem, as a model for the motion of a spacecraft relative to the Earth–Moon system. We focus on the collinear equilibrium points $L_1$ and $L_2$. There are families of Lyapunov periodic orbits around either $L_1$ or $L_2$, forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either $L_1$ or $L_2$, as well as heteroclinic orbits between the Lyapunov manifold around $L_1$ and the one around $L_2$. The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, which gives the future asymptotic of a homoclinic orbit as a function of the past asymptotic. In contrast to the more customary Melnikov theory, we do not need to assume that the asymptotic orbits have a special nature (periodic, quasi-periodic, etc.). We add a non-conservative, time-dependent perturbation, as a model for a thrust applied to the spacecraft for some duration of time, or for some other effect, such as solar radiation pressure. We compute the first-order approximation of the perturbed scattering map, in terms of fast convergent integrals of the perturbation along homoclinic/heteroclinic orbits of the unperturbed system. As a possible application, this result can be used to determine the trajectory of the spacecraft upon using the thrust.

Appendices

Appendix A: Normally hyperbolic invariant manifolds

We briefly recall the notion of a normally hyperbolic invariant manifold.

Definition A.1

Let M be a ${\mathcal {C}}^r$-smooth manifold, $\Phi ^t$ a ${\mathcal {C}}^r$-flow on M. A submanifold (with or without boundary) $\Lambda$ of M is a normally hyperbolic invariant manifold (NHIM) for $\Phi ^t$ if it is invariant under $\Phi ^t$, and there exists a splitting of the tangent bundle of TM into sub-bundles over $\Lambda$

$$\begin{aligned} T_z M=E^{\mathrm{u}}_z \oplus E^{\mathrm{s}}_z \oplus T_z \Lambda , \quad \forall z \in \Lambda \end{aligned}$$

(A.1)

that are invariant under $D\Phi ^t$ for all $t\in {\mathbb {R}}$, and there exist rates

$$\begin{aligned} \lambda _-\le \lambda _+<\lambda _c<0<\mu _c<\mu _-\le \mu _+ \end{aligned}$$

and a constant ${C}>0$, such that for all $x\in \Lambda$ we have

$$\begin{aligned} \begin{aligned} {C}e^{t\lambda _- }\Vert v\Vert&\le \Vert D\Phi ^t(z)(v)\Vert \le {C}e^{t\lambda _+}\Vert v\Vert \text { for all } t\ge 0, \text { if and only if } v\in E^{\mathrm{s}}_z,\\ {C}e^{t\mu _+ }\Vert v\Vert&\le \Vert D\Phi ^t(z)(v)\Vert \le {C}e^{t\mu _- }\Vert v\Vert \text { for all } t\le 0, \text { if and only if }v\in E^{\mathrm{u}}_z,\\ {C}e^{|t|\lambda _c }\Vert v\Vert&\le \Vert D\Phi ^t(z)(v)\Vert \le {C}e^{|t|\mu _c}\Vert v\Vert \text { for all } t\in {\mathbb {R}}, \text { if and only if }v\in T_z\Lambda . \end{aligned} \end{aligned}$$

(A.2)

In the case when $\Phi ^t$ is a Hamiltonian flow, the rates can be chosen so that

$$\begin{aligned}\lambda _-=-\mu _+,\, \lambda _+=-\mu _-,\,\text { and }\lambda _c=-\mu _c.\end{aligned}$$

The regularity of the manifold $\Lambda$ depends on the rates $\lambda ^-$, $\lambda ^+$, $\mu ^-$, $\mu ^+$, $\lambda _c$, and $\mu _c$. More precisely, $\Lambda$ is ${\mathcal {C}}^{\ell }$-differentiable, with $\ell \le r-1$, provided that

$$\begin{aligned} \begin{aligned}&\ell {\mu }_c + {\lambda }_+ < 0, \\&\quad \ell {\lambda }_c + {\mu }_- > 0. \end{aligned} \end{aligned}$$

(A.3)

The manifold $\Lambda$ has associated unstable and stable manifolds of $\Lambda$, denoted $W^{\mathrm{u}}(\Lambda )$ and $W^{\mathrm{s}}(\Lambda )$, respectively, which are ${\mathcal {C}}^{\ell -1}$-differentiable. They are foliated by 1-dimensional unstable and stable manifolds (fibers) of points, $W^{\mathrm{u}}(z)$, $W^{\mathrm{s}}(z)$, $z\in \Lambda$, respectively, which are as smooth as the flow.

These manifolds are defined by:

$$\begin{aligned} \begin{aligned} W^\mathrm{s}(\Lambda )&= \{ y \,|\, d( {\Phi }^t_\varepsilon (y), \Lambda ) \rightarrow 0 \text { as }{t \rightarrow +\infty } \} \\&= \{ y \,|\, d( {\Phi }^t_\varepsilon (y), \Lambda ) \le C_y e^{t\lambda _+} ,t \ge 0\}, \\ W^\mathrm{u}(\Lambda )&= \{ y \,|\, d( {\Phi }^t_\varepsilon (y), \Lambda ) \rightarrow 0 \text { as }{t \rightarrow -\infty }\} \\&= \{ y \,|\, d( {\Phi }^t_\varepsilon (y), \Lambda ) \le C_y e^{t\mu _- } ,t \ge 0\},\\ W^{\mathrm{s}}(x)&= \{y \,|\, d( \Phi ^t (y),\Phi ^t (x) )< C_y e^{t\lambda _+ },\, t\ge 0 \}, \\ W^{\mathrm{u}}(x)&= \{y \,|\, d( \Phi ^t (y), \Phi ^t (x) ) < C_y e^{t\mu _- },\, t\le 0 \}. \end{aligned} \end{aligned}$$

(A.4)

The fibers $W^{\mathrm{u}}(x)$, $W^{\mathrm{s}}(x)$ are not invariant by the flow, but equivariant in the sense that

$$\begin{aligned} \begin{aligned} \Phi ^t(W^\mathrm{u}(z))&=W^\mathrm{u}(\Phi ^t(z)),\\ \Phi ^t(W^\mathrm{s}(z))&=W^\mathrm{s}(\Phi ^t(z)). \end{aligned}\end{aligned}$$

(A.5)

Since $W^{\mathrm{s},\mathrm{u}}(\Lambda )=\bigcup _{z\in \Lambda } W^{s,u}(z)$, we can define the projections along the fibers

$$\begin{aligned} \begin{aligned} \Omega ^{+}:W^{\mathrm{s}}(\Lambda )\rightarrow \Lambda ,\quad \Omega ^+(z)&=z^+\text { iff } z \in W^{\mathrm{s}}(z^{+}),\\ \Omega ^{-}:W^{\mathrm{u}}(\Lambda )\rightarrow \Lambda ,\quad \Omega ^-(z)&=z^-\text { iff }z \in W^{\mathrm{u}}(z^{-}).\end{aligned}\end{aligned}$$

(A.6)

The point $z^+\in \Lambda$ is characterized by:

$$\begin{aligned} d( {\Phi }^t(z), {\Phi }^t(z^+) ) \le C_z e^{t \lambda _+}, \quad \text { for all } t \ge 0. \end{aligned}$$

(A.7)

and the point $z^- \in \Lambda$ by

$$\begin{aligned} d( {\Phi }^t(z), {\Phi }^t(z^-) ) \le C_z e^{t \mu _-}, \quad \text { for all } t \le 0, \end{aligned}$$

(A.8)

for some $C_z>0$.

For our applications, the most important result about NHIMs is that they persist when we perturb the flow. This is the fundamental result of Fenichel (1971), Hirsch et al. (1977) and Pesin (2004).

The standard assumption for persistence is that the unperturbed NHIM is a compact manifold without a boundary. The persistence of the NHIM also holds when the compactness assumption is replaced with the assumption that the perturbation has uniformly bounded derivatives in all variables (Hirsch et al. 1977, Section 6). There are also proofs of persistence in the infinite-dimensional case which do not require compactness, such as in Bates et al. (1999) and Bates et al. (2008).

We remark that the particular case when the perturbation is periodic or quasi-periodic in time can be reduced to the compact case. More precisely, we can rewrite system (2.2) as

$$\begin{aligned}\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}z&= {\mathcal {X}}^0(z) + \varepsilon {\mathcal {X}}^1(z, \theta ),\\ \frac{\mathrm{d}}{\mathrm{d}t} \theta&=\omega , \end{aligned}\end{aligned}$$

where $\theta$ ranges over a torus ${\mathbb {T}}^d$, and $\omega \in {\mathbb {R}}^d$ is a rationally independent vector when $d>1$. If the flow of $\frac{\mathrm{d}}{\mathrm{d}t}z = {\mathcal {X}}^0(z)$ admits a compact NHIM $\Lambda _0$, the extended system for $\varepsilon = 0$ admits a compact NHIM $\Lambda _0\times {\mathbb {T}}^d$ which persists for small enough $\varepsilon$, using the standard theory. The torus ${\mathbb {T}}^d$ is sometimes called ‘the clock manifold’.

In the case when the manifold has a boundary, the persistence result requires a step of extending the flow. This makes that the persistent manifold is not invariant but only locally invariant and not unique (it depends on the extension).

When we are given a family of flows, it is possible to choose the extensions depending smoothly on parameters and obtain that the manifolds depend smoothly on parameters.

The precise meaning of the smooth dependence is that the we can find parametrizations $k_\varepsilon : \Lambda _0 \rightarrow \Lambda _\varepsilon$. The parametrizations $k_\varepsilon$ can be chosen so that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d} \varepsilon } k_\varepsilon (z) \in E^\mathrm{u}_{z} \oplus E^\mathrm{s}_{z}, \end{aligned}$$

where the splitting $E^\mathrm{u}_{z} \oplus E^\mathrm{s}_{z}$ corresponds to the invariant manifold of the perturbed system. In this case, we obtain $\Lambda _\varepsilon$ as a graph over the central variables on $\Lambda _0$; see Delshams et al. (2008).

The maps $k_\varepsilon (x)$ are jointly $C^r$ as functions of $x, \varepsilon$. The proof of this well-known result is not very difficult. It suffices to consider an extended flow ${{\tilde{\Phi }}}^t(x, \varepsilon ) = ( \Phi _\varepsilon ^t(x), \varepsilon )$, which is a small perturbation of ${{\tilde{\Phi }}}^t_0(x, \varepsilon ) = ( \Phi _0^t(x), \varepsilon )$. The regularity of the NHIM of ${{\tilde{\Phi }}}^t$ gives the claimed regularity of the NHIM of $\Phi ^t$ with respect to parameters.

From the same proof (using the invariant objects of the extended flow), it easily follows the regularity with respect to parameters of the stable and unstable bundles and the stable and unstable manifolds.

Appendix B: Scattering map

Assume that $W^\mathrm{u}(\Lambda )$, $W^\mathrm{s}(\Lambda )$ have a transverse intersection along a manifold $\Gamma$ satisfying:

$$\begin{aligned} \begin{aligned} T_z\Gamma&= T_zW^\mathrm{s}(\Lambda )\cap T_zW^\mathrm{u}(\Lambda ), \text { for all } z\in \Gamma ,\\ T_zM&=T_z\Gamma \oplus T_zW^\mathrm{u}(z^-)\oplus T_zW^\mathrm{s}(z^+), \text { for all } z\in \Gamma . \end{aligned} \end{aligned}$$

(B.1)

Under these conditions, the projection mappings $\Omega ^\pm$ restricted to $\Gamma$ are local diffeomorphisms. We can restrict $\Gamma$ if necessary so that $\Omega ^\pm$ are diffeomorphisms from $\Gamma$ onto open subsets $U^\pm$ in $\Lambda$.

Definition B.1

A homoclinic channel is a homoclinic manifold $\Gamma$ satisfying the strong transversality condition B.1, and such that

$$\begin{aligned}\Omega ^\pm _{\mid \Gamma }:\Gamma \rightarrow U^\pm :=\Omega ^\pm (\Gamma )\end{aligned}$$

are ${\mathcal {C}}^{\ell -1}$-diffeomorphisms.

Definition B.2

Given a homoclinic channel $\Gamma$, the scattering map associated with $\Gamma$ is defined as

$$\begin{aligned}\begin{aligned} \sigma&:=\sigma ^\Gamma : U^- \subseteq \Lambda \rightarrow U^+ \subseteq \Lambda ,\\ \sigma&= \Omega ^+ \circ (\Omega ^{-})^{-1}. \end{aligned} \end{aligned}$$

Equivalently, $\sigma (z^-) = z^+$, provided that $W^\mathrm{u}(z^-)$ intersects $W^\mathrm{s}(z^+)$ at a unique point $z\in \Gamma$.

The meaning of the scattering map is that, given a homoclinic excursion, it has two orbits in the manifold is asymptotic to. It is asymptotic to an orbit in the past and to another orbit in the future. The scattering map considers the future asymptotic orbit as a function of the asymptotic in the past. When we consider all the homoclinic orbits in a homoclinic channel, we obtain a scattering map from an open domain. The intuition of the scattering map is that if we observe the orbit for long times, we just measure the effect of the homoclinic excursion on the asymptotic behavior. The scattering map is a very economical way of studying these excursions since it is a map only on the NHIM. Furthermore, as we will see now, it satisfies remarkable geometric properties.

Due to (A.5), the scattering map satisfies the following property

$$\begin{aligned} \Phi ^T \circ \sigma ^{\Gamma }=\sigma ^{\Phi ^T(\Gamma )}\circ \Phi ^T \end{aligned}$$

(B.2)

for any $T\in {\mathbb {R}}$.

If M is a symplectic manifold, $\Phi ^t$ is a Hamiltonian flow on M, and $\Lambda \subseteq M$ is symplectic, then the scattering map is symplectic. If the flow is exact Hamiltonian, the scattering map is exact symplectic. For details, see Delshams et al. (2008).

In a similar fashion, we can define heteroclinic channels and associated scattering maps.

Given two NHIM’s $\Lambda ^1$ and $\Lambda ^2$, we can define the projection mappings $\Omega ^{\pm ,i}:W^{\mathrm{s},\mathrm{u}}(\Lambda ^i)\rightarrow \Lambda ^i$ for $i=1,2$. Assume that $W^\mathrm{u}(\Lambda ^1)$ intersects transversally $W^\mathrm{s}(\Lambda ^2)$ along a heteroclinic manifold $\Gamma$ so that:

$$\begin{aligned} \begin{aligned} T_z\Gamma&= T_zW^\mathrm{u}(\Lambda ^1)\cap T_zW^\mathrm{s}(\Lambda ^2), \text { for all } z\in \Gamma ,\\ T_zM&=T_z\Gamma \oplus T_zW^\mathrm{u}(z^-)\oplus T_zW^\mathrm{s}(z^+), \text { for all } z\in \Gamma , \end{aligned} \end{aligned}$$

(B.3)

where $z^-=\Omega ^{-,1}(z) \in \Lambda ^1$ and $z^+=\Omega ^{+,2}(z)\in \Lambda ^2$.

We can restrict $\Gamma$ so that $\Omega ^{-,1}:\Gamma \rightarrow \Lambda ^1$ and $\Omega ^{+,2}:\Gamma \rightarrow \Lambda ^2$ are diffeomorphisms onto their corresponding images.

Definition B.3

A heteroclinic channel is a heteroclinic manifold $\Gamma$ satisfying the strong transversality condition B.3, and such that

$$\begin{aligned}\begin{aligned} \Omega ^{-,1}_{\mid \Gamma }:\Gamma \rightarrow U^-:=\Omega ^{-,1}(\Gamma )\subseteq \Lambda ^1,\\ \Omega ^{+,2}_{\mid \Gamma }:\Gamma \rightarrow U^+:=\Omega ^{+,2}(\Gamma )\subseteq \Lambda ^2, \end{aligned}\end{aligned}$$

are ${\mathcal {C}}^{l-1}$-diffeomorphisms.

Definition B.4

Given a heteroclinic channel $\Gamma$, the scattering map associated with $\Gamma$ is defined as

$$\begin{aligned}\begin{aligned} \sigma&:=\sigma ^\Gamma : U^- \subseteq \Lambda ^1 \rightarrow U^+ \subseteq \Lambda ^2,\\ \sigma&= \Omega ^{+,2} \circ (\Omega ^{-,1})^{-1}. \end{aligned} \end{aligned}$$

From the result of the regularity with respect to parameters of the stable and unstable manifolds and the fact that the scattering map is expressed in terms of transverse intersections, we obtain that the scattering map depends smoothly on parameters. Thus, the goal of this paper is not to prove the derivative of the scattering map with respect to the perturbation parameter exists, only to give explicit formulas knowing that the derivative exists.

Appendix C: Gronwall’s inequality

In this section, we apply Gronwall’s Inequality to estimate the error between the solution of an unperturbed system and the solution of the perturbed system, over a time of logarithmic order with respect to the size of the perturbation.

Lemma C.1

Consider the following differential equations:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{z}(t)= & {} {\mathcal {X}}^0(z,t) \end{aligned}$$

(C.1)

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{z}(t)= & {} {\mathcal {X}}^0(z,t)+\varepsilon {\mathcal {X}}^1(z,t,\varepsilon ) \end{aligned}$$

(C.2)

Assume that ${\mathcal {X}}^0,{\mathcal {X}}^1$ are uniformly Lipschitz continuous in the variable z, $C_0$ is the Lipschitz constant of ${\mathcal {X}}^0$, and ${\mathcal {X}}^1$ is bounded with $\Vert {\mathcal {X}}^1\Vert \le C_1$, for some $C_0,C_1>0$. Let $z_0$ be a solution of Eq. (C.1) and $z_\varepsilon$ be a solution of Eq. (C.2) such that

$$\begin{aligned} \Vert z_0(t_0)-z_\varepsilon (t_0)\Vert <c\varepsilon . \end{aligned}$$

(C.3)

Then, for $0<\varrho _0<1$, $k\le \frac{1-{\varrho _0}}{C_0}$, and $K=c+\frac{C_1}{C_0}$, we have

$$\begin{aligned} \Vert z_0(t)-z_\varepsilon (t)\Vert < K\varepsilon ^{\varrho _0}, \text { for } 0\le t-t_0\le k\ln (1/\varepsilon ). \end{aligned}$$

(C.4)

For a proof, see Gidea et al. (2021).

Appendix D: Master lemmas

In this section, we recall some abstract Melnikov-type integral operators and some of their properties from Gidea et al. (2021).

Consider a system as in (2.2) and the extended systems as in (2.6).

Assume that, for some $\varepsilon _1>0$, and for each $\varepsilon \in (-\varepsilon _1,\varepsilon _1)$, there exists a normally hyperbolic invariant manifold ${{{\tilde{\Lambda }}}}_\varepsilon$ for ${\tilde{\Phi }}^\tau _\varepsilon$, as well as a homoclinic channel ${\tilde{\Gamma }}_\varepsilon$, which depend ${\mathcal {C}}^{\ell }$-smoothly on $\varepsilon$. Associated with ${\tilde{\Gamma }}_\varepsilon$, we have projections $\Omega ^\pm :{\tilde{\Gamma }}_\varepsilon \rightarrow \Omega ^\pm ({\tilde{\Gamma }}_\varepsilon )\subseteq {{{\tilde{\Lambda }}}}_\varepsilon$, which are local diffeomorphisms. We are thinking of ${\tilde{\Phi }}^\tau _\varepsilon$, ${{{\tilde{\Lambda }}}}_\varepsilon$, ${\tilde{\Gamma }}_\varepsilon$ as perturbations of ${\tilde{\Phi }}^\tau _0$, ${{{\tilde{\Lambda }}}}_0$, ${\tilde{\Gamma }}_0$.

For ${\tilde{z}}_0\in {\tilde{\Gamma }}_0$ let ${\tilde{z}}_\varepsilon \in {\tilde{\Gamma }}_\varepsilon$ be the corresponding homoclinic point satisfying (5.4). Because of the smooth dependence of the normally hyperbolic manifold and of its stable and unstable manifolds on the perturbation, ${\tilde{z}}_\varepsilon$ is $O(\varepsilon )$-close to ${\tilde{z}}_0$ in the ${\mathcal {C}}^\ell$-topology, that is

$$\begin{aligned} {\tilde{z}}_\varepsilon = {\tilde{z}}_0+O(\varepsilon ). \end{aligned}$$

(D.1)

Let $( {\tilde{z}}_\varepsilon ,\varepsilon )\in {\widetilde{M}}\mapsto \mathbf{F}( {\tilde{z}}_\varepsilon ,\varepsilon )\in {\mathbb {R}}^k$ be a uniformly ${\mathcal {C}}^{1}$-smooth mapping on ${\widetilde{M}}\times {\mathbb {R}}$.

We define the integral operators

$$\begin{aligned} \begin{aligned} {\mathfrak {I}}^+({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , {\tilde{z}}_\varepsilon )&=\int _{0}^{+\infty } \left( {\mathbf {F}}({\tilde{\Phi }}^\tau _\varepsilon ( {\tilde{z}}_\varepsilon ^+))-{\mathbf {F}}({\tilde{\Phi }}^\tau _\varepsilon ( {\tilde{z}}_\varepsilon ))\right) \mathrm{d}\tau ,\\ {\mathfrak {I}}^-({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , {\tilde{z}}_\varepsilon )&=\int _{-\infty }^{0} \left( {\mathbf {F}}({\tilde{\Phi }}^\tau _\varepsilon ( {\tilde{z}}^-_\varepsilon ))-{\mathbf {F}}({\tilde{\Phi }}^\tau _\varepsilon ( {\tilde{z}}_\varepsilon )\right) \mathrm{d}\tau . \end{aligned}\end{aligned}$$

(D.2)

Lemma D.1

(Master Lemma 1) The improper integrals (D.2) are convergent. The operators ${\mathfrak {I}}^+({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , z_\varepsilon )$ and ${\mathfrak {I}}^-({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , z_\varepsilon )$ are linear in ${\mathbf {F}}$.

Lemma D.2

(Master Lemma 2)

$$\begin{aligned} \begin{aligned} {\mathbf {F}}( {\tilde{z}}^+_\varepsilon )-{\mathbf {F}}( {\tilde{z}}_\varepsilon )&=-{\mathfrak {I}}^+(({\mathcal {X}}^0+\varepsilon {\mathcal {X}}^1){\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , {\tilde{z}}_\varepsilon ),\\ {\mathbf {F}}( {\tilde{z}}^-_\varepsilon )-{\mathbf {F}}( {\tilde{z}}_\varepsilon )&={\mathfrak {I}}^-(({\mathcal {X}}^0+\varepsilon {\mathcal {X}}^1){\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , {\tilde{z}}_\varepsilon ). \end{aligned} \end{aligned}$$

(D.3)

Lemma D.3

(Master Lemma 3)

$$\begin{aligned} \begin{aligned} {\mathfrak {I}}^+({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , {\tilde{z}}_\varepsilon )&={\mathfrak {I}}^+({\mathbf {F}},{\tilde{\Phi }}^\tau _0, {\tilde{z}}_0)+O(\varepsilon ^{\varrho }),\\ {\mathfrak {I}}^-({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , {\tilde{z}}_\varepsilon )&={\mathfrak {I}}^-({\mathbf {F}},{\tilde{\Phi }}^\tau _0, {\tilde{z}}_0)+O(\varepsilon ^{\varrho }), \end{aligned} \end{aligned}$$

(D.4)

for $0<\varrho <1$. The integrals on the right-hand side are evaluated with ${\mathcal {X}}^1={\mathcal {X}}^1(\cdot ;0)$.

Lemma D.4

(Master Lemma 4) If $\mathbf{F} =O_{{\mathcal {C}}^1}(\varepsilon )$, then

$$\begin{aligned} \begin{aligned} {\mathfrak {I}}^+({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , z_\varepsilon )&={\mathfrak {I}}^+({\mathbf {F}},{\tilde{\Phi }}^\tau _0, z_0)+O(\varepsilon ^{1+\varrho }),\\ {\mathfrak {I}}^-({\mathbf {F}},{\tilde{\Phi }}^\tau _\varepsilon , z_\varepsilon )&={\mathfrak {I}}^-({\mathbf {F}},{\tilde{\Phi }}^\tau _0, z_0)+O(\varepsilon ^{1+\varrho }), \end{aligned} \end{aligned}$$

(D.5)

for $0<\varrho <1$. The integrals on the right-hand side are evaluated with ${\mathcal {X}}^1={\mathcal {X}}^1(\cdot ;0)$.

The proofs of the above lemmas can be found in Gidea et al. (2021), and similar arguments can be found in Gidea and de la Llave (2018).