Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in (mathbb{R}^4) with (mathbb{Z}_2)-symmetry and integral of motion

We consider a (mathbb{Z}_2)-equivariant flow in (mathbb{R}^{4}) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit (Gamma). We provide criteria for the existence of stable and unstable invariant manifolds of (Gamma). We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schr\"odinger equations is considered.


Background
Consider a Hamiltonian system (or more generally, a system with a smooth first integral) defined for x ∈ R 2n , n ≥ 2 with an integral H, and a hyperbolic equilibrium O at the origin. An orbit Γ = {x (t) : t ∈ R} of this system is said to be 'homoclinic to O' or 'a homoclinic loop' if it belongs to both stable and unstable invariant manifolds of O, or equivalently, x (t) → O as t → ±∞. Existence of homoclinic orbits for systems with a smooth first integral is known to be a robust phenomenon. This is due to the fact that the n-dimensional stable and unstable invariant manifolds of O lie in the same (2n − 1)-dimensional level H = constant, and they may intersect transversely in that level along the homoclinic orbits. A natural question which arises here is the possible dynamics near homoclinic orbits in the level H (x) = H (O).
When x(t) oscillates as it converges to O (this happens when the leading, i.e. the nearest to the imaginary axis, eigenvalues of the linear part of the system at O are complex), the dynamics in H −1 (O) is highly non-trivial [Dev76, BS90, Ler91, Ler00, Ler97, BS96,BIR16]. On the other hand, when the leading eigenvalues are real and x (t) decays to zero monotonically as t → ±∞, the generic dynamics in H −1 (O) are very simple. Thus, the only orbits staying in a small neighborhood of a finite bunch of generic homoclinic loops to a saddle with real leading eigenvalues are only the homoclinic loops themselves and the point O [TS89,Tur14].
In this case, to have interesting behavior, we need degeneracies or symmetries. It was shown in [ST97] (based on an earlier work [EKTS89]) that symmetries can lead to the emergence of the so-called super-homoclinic orbits which, in turn, serve as limits of infinite series of multi-pulse homoclinic loops. Namely, as shown in [ST97,Tur01], in certain situations, the homoclinic loops or bunches of homoclinic loops can have stable or unstable invariant manifolds; the super-homoclinics correspond to the intersection of these manifolds.
In the non-conservative setting, super-homoclinic orbits and the non-trivial dynamics associated with them were discovered and studied by Homburg [Hom96]. Eleonsky et al. [EKTS89] spotted super-homoclinic orbits in the numerical investigation of an electromagnetic field in a nonlinear medium. Barrientos et al. [BRR19] found super-homoclinics near a homoclinic loop to a saddle-focus in the context of reversible systems, similar to the structure described for the Hamiltonian case by Belyakov and Shilnikov in [BS90]. Chawanya and Ashwin [CA10] built an example of a heteroclinic network that possesses a super-homoclinic in the sense of an orbit which connects sub-networks. In general, super-homoclinic orbits may potentially appear in heteroclinic networks, especially if the network undergoes a chaotic behavior, see e.g. [NADP20].
In this paper, we consider the simplest case of Z 2 -symmetry which results in the emergence of stable and unstable invariant manifolds for homoclinic loops, enabling the creation of superhomoclinic orbits, and describe the multi-pulse homoclinics associated to them.

Problem setting and results
Consider a C ∞ -smooth 4-dimensional system of differential equations (1.1)ẋ = X(x), x ∈ R 4 , with a C ∞ -smooth first integral H : R 4 → R, i.e.
Assumption 1. X has a hyperbolic equilibrium state O at the origin.
Assumption 2. The quadratic part of H at O is a nondegenerate quadratic form.
It is easy to see (see e.g. [Bak20]) that when Assumptions 1 and 2 are satisfied, system (1.1) near O can be brought to the following form by a linear transformation: where u = (u 1 , u 2 ) ∈ R 2 , v = (v 1 , v 2 ) ∈ R 2 and A is a matrix whose eigenvalues have positive real parts. Moreover, the first integral takes the form: where ·, · is the standard inner product on R 2 . Assumption 3 implies that the plane {u 1 = v 1 = 0} is invariant with respect to the flow of system (1.3). Note that we can always assume that H is invariant with respect to symmetry (1.5), i.e.
The equilibrium state O is a saddle with 2-dimensional stable and unstable invariant manifolds W s (O) and W u (O) which are tangent at O to the u-plane and v-plane respectively. Both the invariant manifolds lie in the 3-dimensional level {H = 0} and may intersect transversely in that level, producing a number of homoclinic loops. We consider the following specific case: Assumption 4. In the invariant plane {u 1 = v 1 = 0}, there exists a homoclinic loop Γ of the transverse intersection of W s (O) and W u (O) (see Figure 1).
Since the action of this symmetry commutes with the linear part of system (1.3), the matrix A is diagonal and takes the form for some positive real numbers λ 1 and λ 2 . Without loss of generality, let λ 1 ≤ λ 2 .
This is not a technical assumption. Indeed, we will see that the cases λ 2 < 2λ 1 and 2λ 1 < λ 2 are dynamically different. Let U be a sufficiently small neighborhood of Γ ∪ {O} in the zero level-set {H = 0}. The main issue which is addressed in this paper is giving a complete description of dynamics in U.
Definition 1.1. Let A = Γ 1 ∪ Γ 2 ∪ · · · ∪ Γ m , where Γ i are homoclinic to an equilibrium O, and m ≥ 1. Consider a sufficiently small open neighborhood U of A ∪ {O} in the energy level of O. The local stable (resp. unstable) set of A, denoted by W s loc (A, U) (resp. W u loc (A, U)), is the union of A itself and the set of the points in U whose forward (resp. backward) orbits lie in U and their ω-limit sets (resp. α-limit sets) coincide with A ∪ {O}. We may use the notations W s loc (A) and W u loc (A) for the stable and unstable sets of A when the neighborhood U is clear from the context. By this definition, the local stable and unstable sets of Γ always contain Γ. Note that these sets lie in the zero-level set {H = 0}. Denote by W s U (O) (resp. W u U (O)) the set of the points in W s glo (O) (resp. W u glo (O)) whose forward (resp. backward) orbits lie entirely in U. Obviously,

Dynamics near a single homoclinic orbit
Our first result is the following:  By this theorem, to understand the dynamics near the homoclinic orbit Γ, we need to study the local stable and unstable sets of this orbit. This is done in the following two theorems.
Theorem A2. If λ 2 < 2λ 1 and Assumptions 1-4 hold, then W s loc (Γ) = W u loc (Γ) = Γ. Thus, by Theorem A1, the forward and backward orbits of any point in U either leave U or converge to O.
The next theorem describes the local stable and local unstable sets of Γ when 2λ 1 < λ 2 . The formulation uses a specific choice of coordinates near the equilibrium O. We introduce this coordinate system in Section 2 (see normal form (2.12)). For this choice of coordinates, system (1.3) keeps its form and its invariance with respect to symmetry (1.5). Moreover, the first integral takes the form  Figure 2). On each of the cross-sections Π s and Π u , the variables u 2 and v 2 are uniquely determined by (u 1 , v 1 ) (see Lemma 2.14). This allows us to parametrize Π s and Π u by the (u 1 , v 1 )-coordinates.
Orbits which lie in U define a Poincaré map T that takes a subset of Π s to Π s . This map can be written as a composition of a local map T loc from a subset of Π s to Π u which is defined by the flow inside the δ-neighborhood of O, and a global map T glo from Π u to Π s which is defined by the flow near the global piece of Γ outside the δ-neighborhood of O (see Figure  2). Since the flight time from Π u to Π s is bounded, the global map T glo is a diffeomorphism. Consider the points M s = Γ ∩ Π s and M u = Γ ∩ Π u (note that both points correspond to (0, 0) in Π s,u ). The Taylor expansion of T glo at M u has the form (1.8) T glo (u 1 , v 1 ) = (au 1 + bv 1 + o (u 1 , v 1 ) , cu 1 + dv 1 + o (u 1 , v 1 )) , for some a, b, c, d ∈ R. Since the local unstable manifold of O corresponds to {u 1 = 0}, and the local stable manifold corresponds to {v 1 = 0}, the transversality assumption (see Assumption 4) is equivalent to d = 0. Theorem A3. Let 2λ 1 < λ 2 and Assumptions 1-4 hold. Suppose that system (1.3) near the equilibrium O is brought to the form (2.12) and let b, c and d in (1.8) be non-zero.
(i) If cd > 0, then W s loc (Γ) = Γ. If cd < 0, then W s loc (Γ) is a C 1 -smooth 2-dimensional invariant manifold which is tangent to W s glo (O) at every point of Γ.
(ii) If bd < 0, then W u loc (Γ) = Γ. If bd > 0, then To stress the smoothness of W s loc (Γ) and W u loc (Γ), we further call them local stable and unstable invariant manifolds.
We remark a parallel to the case of a general homoclinic loop, considered in detail by Homburg [Hom96], see also [Tur84,San00,ST99], for which the existence of stable/unstable manifolds depends on the sign of a certain coefficient in the Poincaré map (the so-called separatrix value [SSTC98]).

Dynamics near a homoclinic figure-eight
Next, we consider the existence of a pair of homoclinic loops in the invariant plane {u 1 = v 1 = 0}: Assumption 6. There exist two homoclinic loops Γ 1 and Γ 2 of transverse intersection of W s (O) and W u (O) in the invariant plane {u 1 = v 1 = 0} such that they leave and enter O along opposite directions (see Figure 3).
Such scenario happens generically, when the level-set {H = 0} is compact. Let V be a small neighborhood of Γ 1 ∪ {O} ∪ Γ 2 in the level-set {H = 0} and denote by W s ) whose forward (resp. backward) orbits lie entirely in V. Then Theorem B1. Under Assumptions 1-3, 5 and 6, the forward (resp. backward) orbit of a point in V lies entirely in V if and only if it belongs to W s The next two theorems give analogues of Theorems A2 and A3 for the case of homoclinic figure-eight.
Theorem B3. Assume 2λ 1 < λ 2 and Assumptions 1-3 and 6. Suppose that system (1.3) near the equilibrium O is brought to the form (2.12) and let b i , c i and d i (i = 1, 2) be non-zero.
Let 0 < γ = λ 1 λ 2 ≤ 1, and consider a one-parameter family {X γ } of the vector fields of the form (1.3) that satisfy the assumptions stated above. In particular, suppose that X γ possesses a homoclinic orbit Γ γ which persists as γ varies. Then, according to our results, when γ > 0.5, there is no dynamics near the homoclinic orbit Γ γ in its energy level, while when γ < 0.5, depending on how the global map behaves (i.e. what the coefficients a, b, c and d are), there may exist stable and unstable invariant manifolds to the homoclinic loop Γ γ . This lets us to conjecture that saddle periodic orbits can be born in the level H = 0 as γ crosses 1 2 . This question requires a further investigation.

Dynamics near a super-homoclinic orbit
Coming back to the case of the single homoclinic loop Γ, we consider the case in which both W s loc (Γ) and W u loc (Γ) are non-trivial. Notice that, according to Theorem A3, in order for these two manifolds to coexist, we require cd < 0 and bd > 0. Continuing these two local manifolds by the flow of the system gives the global stable and unstable invariant manifolds of Γ, denoted by W s glo (Γ) and W u glo (Γ), respectively. These manifolds lie in the 3-dimensional level {H = 0} which means that it would be reasonable if we assume that they intersect transversely in that level. Any orbit at this intersection is bi-asymptotic, or in other words, homoclinic to the union of Γ and the equilibrium O, i.e. converges to Γ ∪ {O} as t → ±∞. We refer to such an orbit as 'homoclinic to homoclinic' or 'super-homoclinic' orbit.
Assumption 7. There exists a super-homoclinic orbit S of the transverse intersection of W s glo (Γ) and W u glo (Γ).
Theorem C1. Under Assumptions 1-4 and 7, there exist infinitely many multi-pulse homoclinic loops in a small neighborhood of S ∪ Γ ∪ {O} (i.e. the closure of S).
A similar result holds for a homoclinic to homoclinic figure-eight: Assumption 8. There exists a super-homoclinic orbit S of the transverse intersection of W s glo (Γ 1 ∪ Γ 2 ) and W u glo (Γ 1 ∪ Γ 2 ).
Theorem C2. Under Assumptions 1-3, 5, 6 and 8, there exist infinitely many multi-pulse homoclinic loops in a small neighborhood of S ∪ {O} ∪ Γ 1 ∪ Γ 2 (i.e. the closure of S). The multi-pulse homoclinic orbits in Theorem C1 (and Theorem C2) refer to homoclinic loops that intersect the cross-section Π s (resp. Π s 1 ∪ Π s 2 ) in n points. We call such orbits npulse homoclinic. We prove that the existence of super-homoclinic orbits implies the existence of n-pulse homoclinic orbits for arbitrarily large n.

Coupled nonlinear Schrödinger equations
The coupled nonlinear Schrödinger equation (CNLSE) is one of the basic models for light propagation. This equation also has various applications in different branches of physics since it appears as a universal model of behavior near a threshold of instability (see e.g. [KSM92]). In this section, we discuss this equation as an application of our theory.
The CNLSE is written as where Ψ and Φ are complex-valued functions of (t, x). We consider the case where α and β are positive real constants. We consider the steady-state solutions of (1.9) which are of the form for some real valued functions ψ and φ. By a rescaling, we can assume ω 1 = 1 and ω 2 = ω (ω > 0). Thus, the stationary solutions of CNLSE satisfy (1.10)ψ where H = 1 2 ψ 2 2 + φ 2 2 − ψ 2 1 − ω 2 φ 2 1 + αψ 4 1 + 2ψ 2 1 φ 2 1 + βφ 4 1 . This system is Hamiltonian with two degrees of freedom. Diagonalizing the linear part reduces this system to (1.11)u where E 1 and E 2 are cubic functions of (u, v), and transforms the Hamiltonian H to the form This system is invariant with respect to symmetry (1.5) and the symmetry (u 2 , v 2 ) ↔ (−u 2 , −v 2 ). Therefore, assuming 1 ≤ ω = 2, system (1.11) meets all Assumptions 1-3 and 5. In addition, it possesses a pair of homoclinic solutions (homoclinic figure-eight): (1.12) These solutions correspond to the following solutions of (1.9): (1.13) We consider the case where the homoclinic figure-eight (1.12) is transverse, i.e. Assumption 6 is met. Therefore, the dynamics near this homoclinic figure-eight in the level {H = 0} can be analyzed by Theorems B2 and B3. For ω < 2, Theorem B2 implies that both forward and backward orbits of any point close to the homoclinic figure-eight leave a small neighborhood of it (in the level {H = 0}) unless it lies on the stable or unstable invariant manifolds of O. For the case of ω > 2, in order to apply Theorem B3, one needs to reduce system (1.11) to normal form (2.12) and compute the coefficients a i , b i , c i and d i (i = 1, 2).
System (1.11) is reversible with respect to the linear involution u 1 ↔ v 1 , u 2 ↔ v 2 . In general, a systemẋ = f (x) on R n is said to be reversible with respect to an involution R, i.e. a diffeomorphism on R n with the property R 2 = id, if dR • f = −f • R. It is easily seen that when x (t) is a solution, so does R • x (−t). The reversibility of system (1.11) implies Reducing system (1.11) to normal form (2.12) preserves the invariance of the system with respect to the symmetry (u 2 , v 2 ) ↔ (−u 2 , −v 2 ). This implies that the loops Γ 1 and Γ 2 are symmetric, and a 1 = a 2 = a, b 1 = b 2 = b, c 1 = c 2 = c and d 1 = d 2 = d. Because of the symmetry, Proposition 1.2 implies Proposition 1.3. Simultaneously, all the manifolds W u loc (Γ 1 ), W s loc (Γ 1 ), W u loc (Γ 2 ) and W s loc (Γ 2 ) are either trivial or non-trivial.
Concerning the coefficients a, b, c and d, notice that the Hamiltonian structure of the equations implies that the map T glo is area-and orientation-preserving; hence ad − bc = 1. Therefore, a b c d −c a , so the reversibility implies b = −c. Recall also that the transversality condition implies d = 0.
It follows from Theorem B3 that if bd > 0, then the local unstable invariant manifold of each of the loops Γ 1 and Γ 2 is non-trivial, while the local unstable invariant manifold of the homoclinic figure-eight Γ 1 ∪ Γ 2 is trivial (i.e. coincides with Γ 1 ∪ {O} ∪ Γ 2 ). In contrast, when bd < 0, the local unstable invariant manifold of the homoclinic figure-eight is non-trivial, while the local unstable invariant manifold of each of the loops Γ 1 and Γ 2 is trivial. The same conclusion holds for the corresponding stable manifolds. This analysis together with Propositions 1.2 and 1.3 yields Proposition 1.4. Let ω > 2 and suppose that the coefficients b, c and d are non-zero. Then, one (and only one) of the following two scenarios holds: To figure out which of the scenarios above happens for CNLSE, one needs to find the corresponding coefficients a, b, c and d. For any particular values of α, β and ω, this can be done numerically by solving the linearization of system (1.10) along the solution (1.13). It is also easy to show that as ω → ∞, the coefficient d changes sign infinitely many times. When d = 0, we have b 2 = 1, so crossing d = 0 leads to a change of sign of bd. Thus, both cases of the above proposition are realized in CNLSE. In either case, there are non-trivial local stable and unstable invariant manifolds of the homoclinic orbits in the CNLSE. Globalizing these manifolds, we conjecture that they intersect transversely along some super-homoclinic orbits: Conjecture 1.5. The coupled nonlinear Schrödinger equations given by (1.9) possess transverse super-homoclinic orbits.
Our conjecture is based on the fact that the stable and unstable manifolds of the homoclinic loops are 2-dimensional manifolds lying in the same compact 3-dimensional energy level, andby Poincaré recurrence -come infinitely close to each other. Hence, they are likely to intersect transversely along super-homoclinic orbits. Moreover, numerical evidence [EKKS93,EK96,Yan97,Yan98] points to the existence of infinitely many multi-pulse homoclinic orbits in the CNLSE. This supports our conjecture since, by Theorems C1 and C2, the existence of these multi-pulse homoclinics might be a bi-product of the existence of super-homoclinic orbits.

The methods and organization of the paper
The standard approach for investigating the dynamics near homoclinic orbits is to study the Poincaré maps along these orbits. The main difficulty in dealing with these maps is that the Poincaré map along a homoclinic orbit is a singular map defined on a non-trivial domain. Let Γ be a homoclinic orbit, Σ be a small cross-section to it, T be the Poincaré map Due to these properties, the smooth theory of Hadamard-Perron cannot be applied directly to study the invariant manifolds of the Poincaré maps along homoclinics. Our approach for investigating the invariant manifolds is applying the method of Shilnikov cross-maps [SSTC98,GL10]. To describe this method, assume that the Poincaré map T is written by g (x, y) = (0, 0) , and the point M corresponds to (x, y) = (0, 0). Suppose that y in the second equation can be solved in terms of (x, y), i.e. y = G (x, y) for some function G. This leads us to introduce the map T × defined by x = F (x, y) , y)). In other words, the Poincaré map T takes (x, y) to (x, y) if and only if the cross-map T × maps (x, y) to (x, y). Denote the domain of T × by D × . Note that the point M = (x, y) = (0, 0) is not in the domain D × , however, it lies in the closure of D × . The advantage of dealing with the cross-map T × over the Poincaré map T is that despite dT (x, y) → ∞ as (x, y) → M , the limit lim (x,y)→M T × (x, y) may exist. This property, if the (x, y) coordinates system on Σ is chosen appropriately, enables us to extend the cross-map T × to an open neighborhood of M smoothly. Then, according to Theorem D.3 (see Appendix D), if extended T × satisfies certain properties, the Poincaré map T possesses an invariant manifold that contains the ω-limit points of every forward orbit of the domain. This is exactly the procedure that we follow in this paper to prove the existence of invariant manifolds of the Poincaré maps along the homoclinic orbits.
In order to obtain necassary estimates for the Poincaré map, we first bring our system near the equilibrium state O to a normal form. Notice that our system is not necessarily linearizable. Indeed, since the spectrum of the linear part of the system is {−λ 2 , −λ 1 , λ 1 , λ 2 }, for some 0 < λ 1 ≤ λ 2 , resonances prevent the linearization. In this paper, we follow the approach of Ovsyaninkov and Shilnikov [OS86] (see also [Tur01] and [SSTC98]) for the normal form reduction. In contrast to the standard normal form approach in which the system near a hyperbolic equilibrium reduces to a polynomial vector field that consists of only resonant terms up to some order (see e.g. [IL99] and [BK92]), in the approach of [OS86], some non-resonant terms remain in the normal form, while some resonant terms of low-orders are eliminated.
Once the system is brought to a normal form, we need to investigate the behavior of the orbits near the equilibrium state O. This enables us to compute the Poincaré maps. To do this, we apply the method of successive approximations (more specifically, Shilnikov's method of solving boundary value problems, see [SSTC98]) to estimate the flow near the equilibrium O.
This paper is organized as follows: Section 2 is dedicated to the study of the local map T loc . In Section 2.1, we define this map and its domain precisely. Then, in Section 2.2, we bring our system near the equilibrium state O to a normal form. In Section 2.3, we investigate the behavior of the orbits near the equilibrium state O. Finally, in Section 2.4, we analyze the domain and the behavior of the local map.
In Section 3, we use the results of Section 2 to study the dynamics near the homoclinic orbits. In Section 3.1, we introduce some notations. In Section 3.2, we study the dynamics near the homoclinic orbit Γ when λ 2 < 2λ 1 . Theorems A2 is proved in this section. The dynamics near Γ when 2λ 1 < λ 2 is studied in Section 3.3. We prove Theorem A3 in this section. Theorem A1 is also proved in these two sections. The case of homoclinic figure-eight is studied in Section 3.4. The proofs of Theorems B1, B2 and B3 are provided in this section. Finally, we discuss the case of superhomoclinics in Section 3.5. We prove Theorems C1 and C2 in this section.
Most of the technical lemmas and calculations are postponed to appendices. We also give a brief introduction to the method of cross-maps in Appendix D.

Set-up and notations
Our approach for studying the dynamics near the homoclinic loop Γ (and homoclinic figureeight Γ 1 ∪ Γ 2 ) is based on the study of the behavior of the corresponding Poincaré map(s). As was mentioned earlier, the Poincaré map T along the homoclinic loop Γ can be written as the composition of a global and a local map. This section is dedicated to the study of the behavior of the local map T loc . To this end, we first need to choose appropriate coordinates near the equilibrium state O of system (1.3). This is done in Section 2.2 below. We consider three different cases of λ 1 = λ 2 , λ 1 < λ 2 < 2λ 1 and 2λ 1 < λ 2 , and introduce a specific normal form for each case. In Section 2.3, we employ the Shilnikov technique for solving boundary value problems to compute the flow near the equilibrium O. This allows us to find an approximation for the local map. Finally, in Section 2.4, we study the behavior of this map and investigate some of its properties.
In comparison to the global map, the local map has more complicated behavior. Indeed, T glo is a diffeomorphism and can be approximated by its Taylor polynomial while the local map T loc is a singular map with a non-trivial domain.
Let us now give a more precise meaning to the above terminologies. Recall the cross-sections Π s and Π u . In all of the normal forms considered in Section 2.2, the local stable and local unstable as well as the local strong stable and local strong unstable invariant manifolds of O are straightened. Therefore, the homoclinic loop Γ intersects Π s and Π u at M s = (0, δ, 0, 0) and M u = (0, 0, 0, δ), respectively. As it is proved later (see Section 2.4.1), we can choose a (u 1 , v 1 ) coordinate-system on each of these cross-sections. Both M s and M u correspond to (0, 0) in this coordinate-system.
Consider a point (u 10 , v 10 ) on Π s close to M s (e.g. the green point in Figure 2) whose forward orbit goes along the homoclinic loop Γ, after a certain time τ it crosses Π u at a point (u 1τ , v 1τ ) (e.g. the blue point in Figure 2), and after a finite time it comes back to Π s at a point (u 10 , v 10 ) (e.g. the red point in Figure 2). Obviously, τ → ∞ as (u 10 , v 10 ) → M s . Let D ⊂ Π s be the set of all such points (u 10 , v 10 ) that satisfy for some sufficiently small constants 0 < ≤ u < δ (see Figure 5). It is trivial that M s / ∈ D. When D = ∅, we define the Poincaré map T : D → Π s by (u 10 , v 10 ) −→ (u 10 , v 10 ). The local map T loc : D → Π u is defined by The global map is defined on the u -ball B u in Π u centered at M u , i.e. T glo : B u → Π s , and its restriction to T loc (D) ⊂ B u is Obviously, T = T glo • T loc . Not every orbit starting from Π s goes along Γ and intersects the cross-section Π u . Trivial examples are the orbits that start at W s loc (O)∩Π s . Other examples are the orbits that go along  Definition 2.1. We denote by D the set of the points (u 10 , v 10 ) on Π s close to M s whose forward orbits go along the negative branch of W u loc (O), and after a certain time τ they cross Σ at (u 1τ , v 1τ ) such that (2.1) holds (see Figure 6). An orbit starting from D 1 ⊂ Π s 1 (resp. D 2 ⊂ Π s 2 ) goes along Γ 1 (resp. Γ 2 ) and intersects Π u 1 (resp. Π u 2 ), while an orbit which starts from D 1 ⊂ Π s 1 (resp. D 2 ⊂ Π s 2 ) goes along the negative (resp. positive) side of v 2 -axis and intersects Π u 2 (resp. Π u 1 ). We introduced the Poincaré, local and global maps along a single homoclinic loop above. For the case of homoclinic figure eight, we also define these maps for each loop: Notation 2.3. We denote by T i , T loc i and T glo i the Poincaré, local and global maps along Γ i (i = 1, 2), respectively (see Figure 7).
To study the case of a homoclinic figure-eight, we consider two extra local maps: and T loc 2 , respectively. The Poincaré maps T 1 (along Γ 1 ) and T 2 (along Γ 2 ) are defined by T 1 = T glo 1 • T loc 1 and T 2 = T glo 2 • T loc 2 , respectively.

Choice of coordinates near the equilibrium state O
This section is dedicated to finding suitable coordinate systems near the equilibrium state O. As it was mentioned above, we consider three different cases of λ 1 = λ 2 , λ 1 < λ 2 < 2λ 1 and 2λ 1 < λ 2 , and for each case we bring system (1.3) into a particular normal form. The proofs of the results stated below are postponed to Appendix A. We start with the following: Lemma 2.5. Consider system (1.3) and first integral (1.4). There exists a C ∞ -smooth change of coordinates which brings system (1.3) to the form where the functions f ij , g ij are C ∞ -smooth and vanish at the origin, i.e.
and transforms first integral (1.4) to Moreover, system (2.4) remains invariant with respect to symmetry (1.5). In particular, The statement of Lemma 2.5 holds for arbitrary 0 < λ 1 ≤ λ 2 . However, we will particularly use this normal form for analyzing the case λ 1 = λ 2 .
Lemma 2.6. Consider system (1.3) and first integral (1.4), and assume λ 1 < λ 2 . There exists a C ∞ -smooth change of coordinates which brings system (1.3) to the form where the functions f ij , g ij are C ∞ -smooth and satisfy the identities This change of coordinates transforms first integral (1.4) to where H 1 and H 2 are C ∞ functions that vanish at O. We can write (2.10) as Moreover, normal form (2.8) and first integral (2.10) remain invariant with respect to symmetry (1.5). In particular, (2.7) holds.
The statement of Lemma 2.6 holds for arbitrary λ 1 < λ 2 . However, we will particularly use this normal form to analyze the local dynamics near O when λ 1 < λ 2 < 2λ 1 . The normal form that is used for analyzing the case 2λ 1 < λ 2 is given by the following: Lemma 2.7. Consider system (1.3) and first integral (1.4) and assume 2λ 1 < λ 2 . Let q be the largest integer such that qλ 1 < λ 2 . There exists a C q -smooth change of coordinates which brings system (1.3) to the form where f ij and g ij are C q−1 -smooth and satisfy identities (2.9). This change of coordinates transforms first integral (1.4) to where H is C q , and H 1 , H 2 , H 3 and H 4 are some C q−1 , C q , C q−2 and C q−2 functions, respectively, such that H 1 (O) = H 2 (O) = 0. Moreover, system (2.12) and first integral (2.13) remain invariant with respect to symmetry (1.5). In particular, (2.7) holds.
Remark 2.8. For simplicity, we can write (2.13) as . A common structure of all of normal forms (2.4), (2.8) and (2.12) is that the local stable and unstable as well as the local strong stable and strong unstable invariant manifolds of the equilibrium O are straightened, i.e. W s where F 1 , F 2 , G 1 and G 2 and their first derivatives vanish at the origin. By [SSTC98] (Theorem 2.9), for given τ ≥ 0 and sufficiently small u 10 , u 20 , v 1τ and v 2τ there exists a unique solution , v 2 (t)) of system (2.14) such that The dependence of this solution on each of the variables τ , u 10 , u 20 , v 1τ and v 2τ is as smooth as the original system (2.14).
The following lemmas estimate the solutions of systems (2.4), (2.8) and (2.12) that satisfy boundary condition (2.15). We prove these lemmas in Appendix B.

Local maps and their properties
In this section, we use the results of the previous two sections to study the local maps for each of systems (2.4), (2.8) and (2.12). Recall (2.2) and write where η 1 and η 2 are some functions. In the previous section, for each of systems (2.4), (2.8) and (2.12), we have approximated the unique solution (u * , v * ) which satisfies boundary conditions (2.15) (see Lemmas 2.9, 2.10 and 2.11). We write this solution as to emphasize that it explicitly depends on t, τ , u 10 , u 20 , v 1τ and v 2τ . This solution represents an orbit which at t = 0 is at the point (u 10 , u 20 , v 10 , v 20 ) and at t = τ is at the point To study the map T loc , we consider the case in which u 20 = v 2τ = δ, i.e. the points (u 10 , u 20 , v 10 , v 20 ) and (u 1τ , u 2τ , v 1τ , v 2τ ) belong to Π s and Π u , respectively. Evaluating the first equation of (2.22) at t = τ and the last two equations of (2.22) at t = 0 gives (2.23) which is an implicit relation between u 10 , v 10 , v 20 , u 1τ = η 1 (u 10 , v 10 ), v 1τ = η 2 (u 10 , v 10 ) and τ . On the other hand, τ and v 20 can be expressed as functions of (u 10 , v 10 ). This allows us to approximate the functions η 1 and η 2 .

Choice of coordinates on the cross-sections
We point out here that we can always choose (u 1 , v 1 )-coordinate system on each of our cross-sections: Lemma 2.14. In each of the cases λ 1 = λ 2 , λ 2 < 2λ 1 and 2λ 1 < λ 2 , for any arbitrary point (u 1 , u 2 , v 1 , v 2 ) of each of the cross-sections Π s , Π u , Π s 1 , Π u 1 , Π s 2 , Π u 2 and Σ, the variables u 2 and v 2 are uniquely determined by (u 1 , v 1 ).
Proof. We only prove the statement for Π s and Π u . The proof for the other cross-sections is the same.

Local maps: case λ 1 = λ 2
We prove D = ∅ by showing that (2.1) never holds. This implies that the Poincaré map along Γ cannot be defined when λ 1 = λ 2 . This also proves Theorem A2 for the particular case of λ 1 = λ 2 .
Let λ = λ 1 = λ 2 and consider the case u 20 = v 2τ = δ. Evaluating the first two equations of (2.17) at t = τ and the last two equations at t = 0 gives Substituting (2.24) into this relation gives e −λτ = u 10 v 10 For a given sufficiently small δ, we have This means that (2.1) does not hold when and u are chosen sufficiently small. On the other hand, it is easily seen that the same happens for the points (u 10 , v 10 ) in D. The same also holds for the case of homoclinic figure-eight. Therefore, 2.4.3 Local maps: case λ 1 < λ 2 < 2λ 1 Let λ < λ 2 < 2λ 1 and consider the case u 20 = v 2τ = δ. Evaluating the first two equations of (2.19) at t = τ and the last two equations at t = 0 gives This, in particular, implies and therefore By these relations, we rewrite (2.27) as Substituting this into the equation of u 1τ in (2.26) gives Let us now explore the domain D of the map T loc . By choosing δ sufficiently small such that |O (δ)| ≤ 1, we have Therefore, for any given (fixed) sufficiently small δ, we have u 1τ = v 1τ O 2 . Thus, By virtue of (2.28), we see that if (u 10 , v 10 ) ∈ D, then u 10 v 10 > 0. It is also easy to see that analogous statements hold for the points in D. This gives: Proposition 2.16. Let λ < λ 2 < 2λ 1 . For a given sufficiently small δ, we can choose and u so that the domain D (resp. D) becomes the set of all points (u 10 , v 10 ) in Π s such that 0 < u 10 v 10 (resp. u 10 v 10 < 0), (u 10 , v 10 ) < and (2.32) holds (see Figure 8).
Remark 2.17. The case of homoclinic figure-eight is the same. Relation u 1τ = v 1τ O 2 holds for any (u 10 , v 10 ) on Π s i , and the domains D i and D i are given by Proposition 2.16 (i = 1, 2). Let 2λ 1 < λ 2 . Evaluating the first two equations of (2.20) at t = τ and the last two equations at t = 0 gives For the particular case of u 20 = v 2τ = δ, we have This, in particular, implies Substituting this into the equation of u 1τ in (2.34) gives Then, local map (2.21) maps (u 10 , v 10 ) to (u 1τ , v 1τ ), where u 1τ and v 1τ are as in (2.36) and (2.35), respectively, and τ is a function of (u 10 , v 10 ). It is not as straightforward as the previous two cases to express τ as a function of (u 10 , v 10 ). This is not straightforward either to find the domain D of T loc . Below, we divide D into three regions (it is shown that D = ∅) and study each region separately. Let B be the -ball in Π s centered at M s . For a given m > 1 define : m|u 10 | < |v 10 |} (see Figure 9). Obviously, Analogously, we define We may drop the subscript and m, when no confusion arises. For (u 10 , v 10 ) ∈ Y m 1 ∪ Y m 2 , we have |v 10 | ≤ m|u 10 | and therefore v 10 = O (u 10 ). By virtue of this relation and taking into account that first integral (2.13) vanishes at (u 10 , δ, v 10 , v 20 ) ∈ Π s , we derive This relation together with (2.34) implies that any point ( Like the preceding two cases, relation (2.39) yields (2.28) and (2.29). Let δ be sufficiently small. With (2.35), (2.36) and some straightforward calculation, we derive This gives the following: Proposition 2.18. For given m, sufficiently small δ and sufficiently small u , we can choose sufficiently small such that for i = 1, 2 we have We have |u 10 | ≤ m|v 10 | and hence u 10 = O (v 10 ). By virtue of this relation and relation (2.33), we obtain This relation together with (2.33) gives . Note that it was relation (2.39) that enabled us to, first, identify the points in Y m i for which (u 1τ , v 1τ ) < u holds, and second, distinguish D i from D i for i = 1, 2. For the case of i = 3, we cannot deduce such a relation from first integral (2.13). However, as we see later, the dynamics on Y m 3 is quite simple and can be analyzed without knowing D 3 and D 3 precisely. Meanwhile, we have shown the following

Analysis near homoclinics and super-homoclinics
The purpose of this section is to study the dynamics near (single and figure-eight) homoclinic and super-homoclinic orbits. In particular, we prove in this section, all the theorems stated in the Introduction. In the first section below, we introduce some concepts and notations. The second and the third sections are dedicated to study the dynamics near a single homoclinic orbit. We prove Theorems A1, A2 and A3 in these two sections. The ideas and techniques which are used to prove these theorems are also used in the later sections. In the fourth section, we extend the results obtained for a single homoclinic to the case of the homoclinic figure-eight. The proofs of Theorems B1, B2 and B3 are provided in this section. Finally, we study the case of a super-homoclinic and prove Theorems C1 and C2 in the fifth (and the last) section.

Set-up and notations
Choose a sufficiently small δ > 0 such that all the statements of the previous sections hold. Fix this δ. According to (2.3) and (1.8), for (u 10 , v 10 ) ∈ D ⊂ Π s , we have where a, b, c and d are real constants (in fact, these coefficients are functions of δ but since δ is assumed to be fixed, we treat these coefficients as constants). Our job is to analyze this map for different values of a, b, c and d, and for each of the cases λ 2 < 2λ 1 and 2λ 1 < λ 2 . In this strand, we first introduce some notations: Notation 3.1. Let N ⊂ M be two arbitrary sets and f : N → M be an injective map. We denote the set of the points in N whose forward orbits lie entirely in N by Λ s N ,f or Λ s N , when no confusion arises. Indeed, We denote the set of the points in N whose backward orbits lie entirely in N by Λ u N ,f or Λ u N , when no confusion arises. Indeed, we can reformulate Theorem A1 as follows: The forward (resp. backward) orbit of any point on Λ s D,T (resp. Λ u D,T ) converges to the homoclinic orbit Γ.
Notation 3.3. Given a point (u 10 , v 10 ) on a given cross-section, we denote the quantity v 10 u 10 (when u 10 = 0) by w(u 10 , v 10 ) or w. Consider the case (u 10 , v 10 ) ∈ D and let (u 10 , v 10 ) ∈ Π s be its image under the Poincaré map T . We denote the quantity v 10 u 10 (when u 10 = 0) by w (u 10 , v 10 ) or w.  3.2. Dynamics near the homoclinic orbit Γ: case λ 2 < 2λ 1 Here, we show that when λ 2 < 2λ 1 , any point in the domain D of the Poincaré map T leaves D by both forward and backward iterations of the Poincaré map. The proof of the case λ 1 = λ 2 directly follows from Proposition 2.15 in which we have shown that the domain D of the Poincaré map is empty. For the case of λ 1 < λ 2 < 2λ 1 , we prove that the image of the domain D under the Poincaré map T has no intersection with D (see Figure 10). We formalize this discussion in the following lemma: Proof. When λ 1 = λ 2 , the statement follows from Proposition 2.15. Suppose λ 1 < λ 2 < 2λ 1 . By Proposition 2.16, the domain D of the Poincaré map is where 1 2 < γ = λ 1 λ 2 < 1 and K u is some constant (see (2.32)). Since u 1τ = v 1τ O 2 , Poincaré map (3.1) can be written as This means that the images of the points in the domain D under the Poincaré map T accumulate near * . However, for a fixed δ and a sufficiently small , this line has no intersection with the domain D (see Figure 10). This implies Λ s D,T = Λ u D,T = ∅, as desired.
We can now prove Theorems A1 (case λ 2 < 2λ 1 ) and A2. The proof of Theorem A1 for the case 2λ 1 < λ 2 is provided in the next section.
Proof of Theorem A1: case λ 2 < 2λ 1 . The proof is an immediate consequence of Remark 3.2 and Lemma 3.5.
Proof of Theorem A2. Any orbit in W s loc (Γ) other than Γ must intersect Π s at Λ s D,T . However, by Lemma 3.5, we have Λ s D,T = ∅. This implies W s loc (Γ) = Γ. The proof of W u loc (Γ) = Γ is the same. In this section, we study the dynamics near the homoclinic orbit Γ for the case 2λ 1 < λ 2 , and prove Theorems A1 (case 2λ 1 < λ 2 ) and A3.

Invariant Manifolds of Homoclinic Orbits
Recall from Section 2.4.4 that when 2λ 1 < λ 2 , we divide the domain D of the Poincaré map T into three subsets D 1 , D 2 and D 3 , i.e. D = D 1 ∪ D 2 ∪ D 3 (see Figure 11). In order to understand the dynamics near the homoclinic loop Γ, we need to investigate the set of the points on the domain D whose forward or backward orbits (under the iterations of the Poincaré map T ) lie in D, i.e. the sets Λ s D,T and Λ u D,T (see Notation 3.1). To this end, we take the following three steps: • Step 1: Investigating the set of the points in D 2 ∪ D 3 whose forward or backward orbits lie entirely in D 2 ∪ D 3 , i.e. the sets Λ s D 2 ∪D 3 ,T and Λ u D 2 ∪D 3 ,T . • Step 2: Investigating the set of the points in D 1 whose forward or backward orbits lie entirely in D 1 , i.e. the sets Λ s D 1 ,T and Λ u D 1 ,T . Obviously, Λ s D 1 ,T and Λ s D 2 ∪D 3 ,T are subsets of Λ s D,T . In addition, Λ u D 1 ,T and Λ u D 2 ∪D 3 ,T are subsets of Λ u D,T . In the third step, we show that the reverse directions also hold: Notice that the statement of Step 3 is not trivial. In fact, at the first stage, one can consider the possibility of the existence of a point x ∈ D such that its forward orbit lies entirely in D, i.e. x ∈ Λ s D,T , but it does not lie entirely in only one of the sets D 1 or D 2 ∪ D 3 , i.e. x / ∈ Λ s D 1 ,T and x / ∈ Λ s D 2 ∪D 3 ,T . In other words, the forward orbit of x stays in D but switches between D 1 and D 2 ∪ D 3 . In Step 3, we indeed show that this scenario does not happen.
We take Step 1 in the following lemma. This lemma helps us to understand the dynamics of the Poincaré map T on the set D 2 ∪ D 3 . We explore in this lemma how T behaves on this set, with which rate the orbits of this set grow, and how Λ s D 2 ∪D 3 ,T and Λ u D 2 ∪D 3 ,T look like. From a technical point of view, part (vii) of this lemma which shows the existence of the unstable manifold of the Poincaré map T is the main result of this section. The techniques which are used in the proof of this part are also used in Section 3.4 for the proof of the existence of the unstable manifold of the homoclinic figure-eight. We prove Lemma 3.6 in Section 3.3.1.
(vii) when bd > 0, the set {M s } ∪ Λ u D 2 ∪D 3 ,T is a one-dimensional C 1 -manifold which is tangent to * at M s .
It follows from this lemma that the image of D 2 ∪ D 3 under the Poincaré map T lies near * , and the Poincaré map increases the norm of any point of this set. Informally speaking, for the particular case of bd > 0, this means that the Poincaré map T preserves and expands the region D 2 ∪ D 3 . A geometrical picture of this behavior is illustrated in Figure 12.
We now take the second step in the next lemma. In this lemma, we study the dynamics of T −1 on the set D 1 . Most of the statements of the following lemma are analogous to the statements of the preceding lemma. This is not a coincidence. In fact, we see later in the proof of Lemma 3.7 that the dynamics of T −1 on D 1 can be obtained from the dynamics of T on D 2 ∪ D 3 by a permutation and reversion of time. The proof of this lemma is postponed to Section 3.3.2.
(iv) if cd < 0, then T −1 (u 10 , v 10 ) remains in D 1 unless it leaves B . ( is a onedimensional C 1 -manifold which is tangent to the horizontal axis at M s .
In the preceding two lemmas, we have shown that the sets Λ s D 2 ∪D 3 ,T and Λ u D 1 ,T are always empty. It was also shown that Λ u D 2 ∪D 3 ,T = Λ u D 2 ,T . This allows us to reformulate Step 3 as in the following lemma: Proof. Let x ∈ D 2 ∪ D 3 . It follows from parts (i), (ii) and (iii) of Lemma 3.6 that if bd < 0, then T (x) / ∈ D, and if bd > 0, then for some k, T k (x) / ∈ B . Thus, any point in Λ s D,T must belong to D 1 . This proves part (i).
To prove part (ii), notice that if Λ u D,T = ∅, then Λ u D 2 ,T = ∅ and therefore Λ u D,T = Λ u D 2 ,T . So we assume that Λ u D,T is non-empty. Let x ∈ Λ u D,T . We need to show x ∈ D 2 . To do this, we first prove x / ∈ D 1 . Assume the contrary, i.e. x ∈ D 1 . It follows from parts (i) and (iii) of Lemma 3.6 that if T −1 (x) ∈ D 2 ∪ D 3 , then x = T T −1 (x) either belongs to D 2 or lies outside the domain D which contradicts the assumption x ∈ D 1 . Therefore, T −1 (x) / ∈ D 2 ∪ D 3 , and so T −1 (x) ∈ D 1 . By virtue of part (i) of Lemma 3.7, this relation implies cd < 0. On the other hand, when cd < 0, it follows from parts (iii) and (iv) of Lemma 3.7 that there exists a k > 0 such that T −k (x) / ∈ B and hence T −k (x) / ∈ D. This contradicts the preliminary assumption x ∈ Λ u D,T . Therefore, if x ∈ Λ u D,T , then x / ∈ D 1 , or equivalently, T −n (x) / ∈ D 1 for all n ≥ 0. To finish the proof, it is sufficient to show that x / ∈ D 3 . Assume the contrary, i.e. x ∈ D 3 . Since x ∈ Λ u D,T implies T −n (x) / ∈ D 1 for all n ≥ 0, we have T −1 (x) / ∈ D 1 . On the other hand, parts (i) and (iii) of Lemma 3.6 imply that if T −1 (x) ∈ D 2 ∪ D 3 , then x = T T −1 (x) either belongs to D 2 or lies outside the domain D which contradicts the assumption x ∈ D 3 . Therefore, x / ∈ D 3 , as desired.
Recall that the local stable (unstable) set of the homoclinic loop Γ, denoted by W s loc (Γ) (W u loc (Γ)), is the union of Γ itself and the set of the points in a sufficiently small neighborhood U of Γ whose forward (backward) orbits lie in U and their ω-limit sets (α-limit sets) coincide with Γ ∪ {O}. By this definition, the intersection of W s loc (Γ) and Π s must belong to {M s } ∪ Λ s D , and the intersection of W u loc (Γ) and Π s must belong to {M s } ∪ Λ u D . On the other hand, we have shown in the above lemmas that when Λ s D (Λ u D ) is non-empty, any point on this set converges to M s by the forward (backward) iterations of the Poincaré map T . This leads to the following: Proposition 3.9. Let φ t be the flow of system (2.12). Then for t ≤ 0.
In system (2.12), the local unstable invariant manifold of the equilibrium O is straightened, i.e. W u loc (O) = {u = 0}. Thus, the intersection of this manifold and the cross-section Π u = {v 2 = δ} ∩ {H = 0} is the straight line {u 1 = 0}, i.e. v 1 -axis. Consider the restriction of this line to a small neighborhood of M u (in Figure 13, it is shown by blue color on Π u ). The global map T glo maps this restricted piece to a curve, denote it by γ u , on Π s (shown by blue color on Π s in Figure 13). This curve is in fact at the intersection of the global unstable invariant Thus, the blue curve (v 1 -axis restricted to a small neighborhood of M u in Π u ) lies at the intersection of the local unstable invariant manifold of O and the cross-section Π u . This curve is mapped to the blue curve on Π s by T glo which means that the blue curve on Π s lies in W u glo (O) ∩ Π s . Since v 1 -axis on Π u is mapped to * on Π s by dT glo , the straight line * is tangent to the blue curve on Π s at M s . manifold of O and the cross-section Π s . Since T glo is a diffeomorphism and the vector ( 0 1 ) is tangent to v 1 -axis at M u , the vector dT glo ( 0 1 ) = b d is tangent to γ u at M s in Π s , i.e. γ u is tangent to * at M s (recall that * is the line in Π s whose slope is d b ). Therefore, it follows from  By virtue of the above results, we can prove Theorems A1 (case 2λ 1 < λ 2 ) and A3: Proof of Theorem A1: case 2λ 1 < λ 2 . The proof is an immediate consequence of Remark 3.2 and Lemmas 3.6, 3.7 and 3.8.
Proof of Theorem A3. By Proposition 3.9 and the preceding Lemmas we have that W u loc (Γ) = Γ when bd < 0, and W s loc (Γ) = Γ when cd > 0. The rest of the theorem is already proved (see Proposition 3.10).
The following remark suggests an alternative formulation of Theorem A3: Remark 3.11. Consider the global stable and unstable invariant manifolds of the equilibrium O of system (2.12). Let γ u (resp. γ s ) be a curve at the intersection of the global unstable (resp. stable) invariant manifold of O and the cross-section Π s (resp. Π u ) which passes through M s (resp. M u ). Following the discussion above, the slope of the tangent line to the curve γ u at M s is d b . Moreover, the slope of the tangent line to the curve γ s at M u is −c d . This suggests an alternative way to detect sgn (bd) and sgn (cd) which are required in the statement of Theorem A3. Indeed, instead of computing the coefficients a, b, c and d in Theorem A3, one can look at the slopes of the intersection curves of the global stable and unstable invariant manifolds of the equilibrium O with the cross-sections Π u and Π s at the points M u and M s .
Proof of part (iii). By part (i) of Lemma 3.6, T (u 10 , v 10 ) is somewhere close to the line * and since, for bd > 0, the restriction of * \ {M s } to B lies in D 2 we have that if T (u 10 , v 10 ) lies in B , then it must belong to D 2 .

Proofs of parts (iv), (v) and (vi). All are easy consequences of the previous parts.
Proof of part (vii). The proof is based on a theory of invariant manifolds for cross-maps (see [SSTC98] and [GL10]). An introduction to this theory is provided in Appendix D.
According to this lemma and the implicit function theorem, the variable z is a C q -smooth (q is as in Lemma 2.7) function of (w, z) for w ∈ m −1 , m and z ∈ g (Y). Denote this function by G. Regarding the domain of this function, note that not every (w, z) necessarily belongs to the domain of G. In other words, for an arbitrary (w, z), there might not exist z ∈ [− 2 α , 2 α ] \ {0} such that z = G (w, z). However, by (3.5), this relation holds if z is chosen sufficiently small, i.e. for a sufficiently small θ > 0 we have Denote this set by R, i.e. R = m −1 , m × ([−θ, θ] \ {0}). Without loss of generality, assume θ < 2 α . Having the function G in hand means that we can write the Poincaré map T in cross-form: we define the cross-map T × : (w, z) → (w, z) by and (w, z) ∈ domain (G). It follows from part (i) of Lemma 3.6 (proved earlier), relation (3.6) and the fact that z = 0 if and only if z = 0 (follows from (3.5)) that T × (R) ⊂ R. Hereafter, we focus on the restriction of T × on R. Our approach to prove the existence of the desired invariant manifold for the Poincaré map T is to apply Theorem D.3 (see Appendix D) on the cross-map T × . However, to do this, there are two issues that we need to take care of. The first is that the domain R does not satisfy the assumption of Theorem D.3 (in that proposition, the domain must be written as a Cartesian product of two convex closed sets but R is not of this form since it does not contain the line z = 0). Second, we need to compute the partial derivatives of the cross-map T × . The second issue is resolved by the following lemma: This lemma is proved in Appendix C. We now extend the domain R to R, where R = m −1 , m × [−θ, θ]. We also extend the map T × to the map T × defined on R by Lemma 3.13 implies that for a fixed sufficiently small α, the map T × : R → R is a C 1 -smooth extension of T × to R. Now, let us come back to the Poincaré map T defined on Y. We extend this map to It is clear that the map T × is in fact the cross-map of T on R. Note that since θ < 2 α , we have R ⊂ Y. Thus, both of the maps T and T × are defined on R. Therefore, for a sufficiently small θ, the map T × satisfies the assumptions of Theorem D.3 and Proposition D.4. This implies that the map T possesses a C 1 -smooth invariant manifold Choosing a sufficiently small also guarantees that M * is a connected piece of M * and hence is a C 1 -manifold.
The manifold M * is our desired manifold if we show Λ u D 2 ,T = M * \ {M s }. So far, we have shown that Λ u D 2 ,T ⊂ M * \ {M s } and so it is sufficient to show M * \ {M s } ⊂ Λ u D 2 ,T . However, this is just a direct consequence of part (ii) of Lemma 3.6 (proved earlier). The fact that M * is tangent to * at M s is also a direct consequence of part (i) of this lemma. This ends the proof of part (vii).
Remark 3.14. Lemma 3.6 states that when bd > 0, the set {M s } ∪ Λ u D 2 ∪D 3 ,T is a C 1 -smooth curve which is tangent to * at M s , and any point on this curve converges to M s by the backward iterations of the Poincaré map T . It follows from part (iii) of Theorem D.3 and the proof of Lemma 3.6 that, when bd > 0, if we take a curve ζ in D 2 , then {T n (ζ) | D 2 } ∞ n=1 converges uniformly to the curve {M s } ∪ Λ u D 2 ∪D 3 ,T .

Proof of Lemma 3.7
Reverse the time direction in system (2.12) (i.e. t → −t) and exchange the stable and unstable components, i.e. apply the linear change of coordinates This gives a system which is of the form of system (2.12), where all the assumptions of Lemma 2.7 are satisfied. The global map along Γ for this system is J T glo −1 J −1 , where J = 0 1 1 0 and T glo is the global map of system (2.12). Thus, the differential of this map at M s is This implies that if we replace conditions bd > 0 and bd < 0 in Lemma 3.6 by cd < 0 and cd > 0, respectively, and the line * by the straight line whose slope is −d c , then all the statements of Lemma 3.6 also hold for this system and the region D 2 ∪ D 3 ⊂ Π s . Consequently, by applying the inverse of change of coordinates (3.8), all the statements of Lemma 3.6 also hold for the system which is derived from system (2.12) by a reversion of time and the region {(u 1 , v 1 ) ∈ B u ⊂ Π u : 0 < v 1 u 1 ≤ m, u 1 = 0} ⊂ Π u . In this case, the line * is replaced by the straight line in Π u whose slope is −c d . The homoclinic loop Γ in this system leaves and enters O along the positive sides of u 2 and v 2 , respectively, and the corresponding Poincaré map, call it T , is defined on Π u . Therefore, the statements of Lemma 3.6 also hold for the map (3.9) T glo • T • T glo −1 and the set (3.10) where the line * is replaced by the horizontal axis in Π s . The later one is simply because Notice that map (3.9) is conjugate to the inverse of the Poincaré map T −1 .
The map (3.9) coincides with T −1 on T (D). Note that, for sufficiently large m, the set T glo {(u 1 , v 1 ) ∈ B u ⊂ Π u : u 1 = 0, m < v 1 u 1 } has no intersection with D 1 . Therefore, Lemma 3.7 will be proved once we show that D 1 ⊂ K if cd < 0, and K ∩ D 1 = ∅ if cd > 0. However, this is an immediate consequence of the discussion above. In fact, it follows from the above discussion that the line * passes through D 2 ∪ D 3 if and only if the horizontal axis passes through K. The later case, for sufficiently large m, is equivalent to the condition D 1 ⊂ K and happens if and only if cd < 0. This ends the proof of Lemma 3.7.
Remark 3.15. Lemma 3.7 states that if cd < 0, then the set {M s } ∪ Λ u D 1 ,T −1 is a C 1smooth curve which is tangent to the horizontal axis at M s . Moreover, any point on this curve converges to M s by the forward iterations of the Poincaré map T . It follows from Remark 3.14 and the proof of Lemma 3.7 that if we take a curve ζ in K ∩ B , then {T −n (ζ) | K∩B } ∞ n=1 converges uniformly to the curve {M s } ∪ Λ u D 1 ,T −1 .

Dynamics near the homoclinic figure-eight
In this section, we study the dynamics near the homoclinic figure-eight Γ 1 ∪Γ 2 . In particular, we prove Theorems B1, B2 and B3 in this section. We start with recalling some definitions and notations from Section 2.1.
For i = 1, 2, we denote by D i the set of the points (u 10 , v 10 ) on Π s i whose forward orbits go along the homoclinic orbit Γ i and intersect Π u i at (u 1τ , v 1τ ) such that (3.11) (u 10 , v 10 ) < and (u 1τ , v 1τ ) < u , for some sufficiently small constants 0 < ≤ u < δ. We denote by D 1 (D 2 ) the set of the points (u 10 , v 10 ) on Π s 1 (Π s 2 ) whose forward orbits go along the negative (positive) side of v 2axis and intersect Π u 2 (Π u 1 ) at (u 1τ , v 1τ ) such that (3.11) holds (see Figure 16). We also denote by T i , T loc = domain (T 2 ) = D 2 . We also define the map T loc 12 : D 1 ⊂ Π s 1 → Π u 2 (resp. T loc 21 : D 2 ⊂ Π s 2 → Π u 1 ) by (u 10 , v 10 ) → (u 1τ , v 1τ ), where (u 10 , v 10 ) ∈ D 1 (resp. ∈ D 2 ) and (u 1τ , v 1τ ) ∈ Π u 2 (resp. ∈ Π u 1 ) (see Figure 7). Let V be a sufficiently small neighborhood of Γ 1 ∪ Γ 2 and define Ξ = D 1 ∪ D 1 ∪ D 2 ∪ D 2 . For any x ∈ Ξ, we correspond a (finite or infinite) sequence {x k } to x in the following way: (i) x 0 = x, (ii) if x k ∈ Ξ (k ≥ 0), we define x k+1 to be the first intersection point of the forward orbit of x k and Π s 1 ∪ Π s 2 . Similarly, if x k ∈ Ξ (k ≤ 0), we define x k−1 to be the first intersection point of the backward orbit of x k and Π s 1 ∪ Π s 2 . In order to understand the dynamics in V, we need to find the set of the points whose forward or backward orbits lie entirely in V, i.e. the set of the points x ∈ Ξ for which the sequence {x k } is well-defined for all k ≥ 0 or k ≤ 0.
When λ 2 < 2λ 1 , the dynamics near the homoclinic figure-eight is quite similar to the case of a single homoclinic loop: the forward and backward orbit of any arbitrary point in V leaves V. When λ 1 = λ 2 , it follows from Proposition 2.15 that Ξ = ∅ and so there is no dynamics near the homoclinic figure-eight. For the case of λ 1 < λ 2 < 2λ 1 , we show in the next proof that for any x ∈ Ξ whose corresponding x 1 is defined, the point x 1 lies close to the straight lines with , and hence, it lies outside of the set Ξ (see Figure 17).
Proof of Theorem B2. The proof for the case λ 1 = λ 2 is an immediate consequence of Proposition 2.15. Figure 16: The homoclinic figure-eight Γ 1 ∪Γ 2 and the the cross-sections Π s 1 , Π u 1 , Π s 2 and Π u 2 are shown. We consider -neighborhoods of M s 1 and M s 2 (green dashed circles) in Π s 1 and Π s 2 , respectively, as well as u -neighborhoods of M u 1 and M u 2 (red dashed circles) in Π u 1 and Π u 2 , respectively. The set D 1 (resp. D 2 ) is the set of the points in the -neighborhood in Π s 1 (resp. Π s 2 ) whose forward orbits go along Γ 1 (resp. Γ 2 ) and intersect the u -neighborhood in Π u 1 (resp. Π u 2 ). The blue point on Π s 1 and the brown point on Π s 2 belong to D 1 and D 2 , respectively. We denote by D 1 (resp. D 2 ) the set of the points in the -neighborhood in Π s 1 (resp. Π s 2 ) whose forward orbits go along the negative (resp. positive) side of v 2 -axis and intersect the u -neighborhood in Π u 2 (resp. Π u 1 ).
Suppose λ 1 < λ 2 < 2λ 1 . By Proposition 2.16, we have Ξ = {(u 10 , v 10 ) : (u 10 , v 10 ) < , and 0 < |v 10 | < K u |u 10 | where K u > 0 is some constant and γ = λ 1 λ −1 2 > 0.5 (see Figure 17). Consider (u 10 , v 10 ) ∈ Ξ. Since u 1τ = v 1τ O 2 (see Remark 2.17), the forward orbit of this point intersects one of the cross-sections Π u 1 or Π u 2 at a point close to the vertical axis and then it ends up either in the cross-section Π s 1 close to the straight line with the slope d 1 b 1 or in the cross-section Π s 2 close to the straight line with the slope d 2 b 2 . In both cases, this point is outside of the set Ξ (see Figure  17). This proves Theorem B2.
A point in Π s 1 ∪ Π s 2 whose forward orbit lies entirely in V and does not lie on the stable manifold of O must belong to Ξ. We denote the set of these points by Λ s . The same holds for backward orbits. We also define the set Λ u analogously. In order to understand the dynamics in V, we need to investigate these two sets. For the case of λ 2 < 2λ 1 , Theorem B2 states that both of these sets are empty. Our approach to investigate Λ s and Λ u for the case 2λ 1 < λ 2 is similar to what we have done in the previous section for the case of a single homoclinic loop.
Recall from Section 2.4.4 that when 2λ 1 < λ 2 , we divide each of the sets D 1 , D 1 , D 2 and D 2 into three regions, i.e. for i = 1, 2, we write Figure 18). Write Ξ = I ∪ J ⊂ Π s 1 ∪ Π s 2 , where Definition 3.16. We define Λ s I (Λ u I ) as the set of the points in I whose forward (backward) orbits intersect Ξ infinitely many times and all the intersection points belong to I. More precisely, The sets Λ s J and Λ u J are defined analogously. Similar to the case of a single homoclinic, we take three steps to investigate the sets Λ s and Λ u . In the first step, we investigate the sets Λ s J and Λ u J . This is done in Lemma 3.17. From technical point of view, part (viii) of this lemma which proves the existence of an unstable invariant manifold of the homoclinic figure-eight is the main result of this section. The techniques which are used in the proof of this part rely on the proof of part (vii) of Lemma 3.6. In the second step, we investigate the sets Λ s I and Λ u I . This is also done in Lemma 3.18. Finally, in Lemma 3.19, we clarify the relations between the sets Λ s J , Λ u J , Λ s I and Λ u I , and the sets Λ s and Λ u . This enables us to prove Theorem B2. We start with the following: Lemma 3.17. Assume 2λ 1 < λ 2 and let w be as in Notation 3.3. For x ∈ J , we have Here, o (1) stands for a function of x that converges to zero as x → M s 1,2 .
More precisely, for each i = 1, 2, the union of M s i and Λ u J ∩ D i 2 is a one-dimensional C 1 -manifold in Π s i which at M s i is tangent to the straight line with slope d i b i . Moreover, the backward orbit of any point in Λ u J intersects these two manifolds alternately, i.e. for any x ∈ Λ u J , all the points x k for even and negative ks belong to only one of the manifolds and all the other x k (odd and negative ks) belong to the other manifold.
Proof. The same techniques that were used in the proof of Lemma 3.6 also prove parts (i), (ii) and (iii).
Part (iv) is an immediate consequence of (ii) and (iii).
In the rest of the proof, we assume x ∈ Λ u J . Notice that x = x 0 ∈ Λ u J implies that x k is defined for all k ≤ 0 and x k ∈ Λ u J . Since Λ u J ⊂ J , we have two possibilities for x k : Our strategy for proving the rest of this lemma is to consider both of these possibilities and keep track of the sequence x k , x k+1 , · · · , x −1 , x 0 . We analyze the behaviors and patterns of this sequence for arbitrary x ∈ Λ u J . To prove part (v), suppose b 1 d 1 > 0 and b 2 d 2 < 0. By part (i), for x −2 , we observe According to this observation, x ∈ Λ u J implies x ∈ D 1 2 . In other words, Λ u J is in fact the set of all x ∈ D 1 2 whose backward orbits only intersect Π s 1 (and not Π s 2 ), and all the intersection points belong to D 1 2 . It follows from Theorem A3 that this set is nothing but W u loc (Γ 1 ) ∩ D 1 2 . This proves part (v).
The proof of part (vi) is analogous to the proof of part (v). To prove part (vii), let b 1 d 1 > 0 and b 2 d 2 > 0. By (i), for x k−2 (k ≤ 0) we observe This observation holds for any arbitrary k ≤ 0 which means that the set Λ u J consists of two disjoint sets: the first is the set of all x ∈ D 1 2 whose backward orbits intersect Ξ infinitely many times and every time at D 1 2 , and the second is the set of all x ∈ D 2 2 whose backward orbits intersect Ξ infinitely many times and every time at D 2 2 . According to Theorem A3, the first set is in fact W u loc (Γ 1 ) ∩ D 1 2 and the second one is W u loc (Γ 2 ) ∩ D 2 2 . This proves part (vii). To prove part (viii), let b 1 d 1 < 0 and b 2 d 2 < 0. By (i), for x k−1 (k ≤ −4), we observe (viii) if c 1 d 1 > 0 and c 2 d 2 > 0, then Λ s I ⊂ D 1 1 ∪ D 2 1 . More precisely, for each i = 1, 2, the union of M s i and Λ s I ∩ D i 1 is a one-dimensional C 1 -manifold in Π s i which at M s i is tangent to the horizontal axis. Moreover, the forward orbit of any point in Λ s I intersects these two manifolds alternately, i.e. for any x ∈ Λ s I , all the points x k for even and negative ks belong to only one of the manifolds and all the other x k (odd and negative ks) belong to the other manifold.
The following Lemma states that the forward (resp. backward) orbit of a point in V lies in V if and only if it intersects the cross-sections Π s 1 and Π s 2 only at I (resp. J ).
Lemma 3.19. We have Λ u = Λ u J and Λ s = Λ s I . Proof. It follows from parts (ii) and (iii) of Lemma 3.18 that if x ∈ I, then the sequence {x k } is not defined for all k ≤ 0. Indeed, For some k 0 ≤ 0, we have {x k 0 , · · · , x −1 } ⊂ I such that x k 0 −1 lies outside the -balls around M s 1 or M s 2 . This means that if x belongs to Λ u , then it must belong to J . Therefore, x ∈ Λ u implies x ∈ Λ u J . On the other hand, we know Λ u J ⊂ Λ u . This proves the first part of the lemma. The proof of the other part is the same.
By virtue of the preceding lemmas, we are now in a position to prove Theorem B3.
Proof of Theorem B3. The local stable (resp. unstable) set of the homoclinic figure-eight Γ 1 ∪ Γ 2 , denoted by W s loc (Γ 1 ∪ Γ 2 ) (resp. W u loc (Γ 1 ∪ Γ 2 )), is the union of Γ 1 ∪ Γ 2 itself and the set of the points in a sufficiently small neighborhood V of Γ 1 ∪ Γ 2 whose forward (resp. backward) orbits lie in V and their ω-limit sets (resp. α-limit sets) coincide with Γ 1 ∪ Γ 2 ∪ {O}. By this definition, the intersection of W s loc (Γ 1 ∪ Γ 2 ) and any of the cross-sections Π s 1 and Π s 2 must belong to {M s 1 , M s 2 } ∪ Λ s . Similarly, the intersection of W u loc (Γ 1 ∪ Γ 2 ) and the cross-sections Π s 1 and Π s 2 must belong to {M s 1 , M s 2 } ∪ Λ u . It follows from Lemma 3.18 that in any cases except the case c 1 d 1 > 0 and c 2 d 2 > 0, the ω-limit set of any orbit in Λ s coincides with either Γ 1 ∪ {O} or Γ 2 ∪ {O}. Therefore, in all of these cases, we have W s loc (Γ 1 ∪ Γ 2 ) = Γ 1 ∪ Γ 2 . Denote the flow of system (2.12) by φ t . When c 1 d 1 > 0 and c 2 d 2 > 0, it follows from parts (ii) and (viii) of Lemma 3.18 that the set Γ 1 ∪ Γ 2 ∪ φ t (Λ s ) for t ≥ 0 is a 2-dimensional C 1 manifold, and the forward orbit of any point on this manifold converges to Γ 1 ∪ {O} ∪ Γ 2 as t → ∞. This means that this manifold is in fact the local stable set of the homoclinic figureeight Γ 1 ∪ Γ 2 . The fact that this manifold is tangent to W s glo (O) at every point of Γ 1 ∪ Γ 2 is an straightforward consequence of the discussion before Proposition 3.10.
The proof for the case of W u loc (Γ 1 ∪ Γ 2 ) is the same. This ends the proof.
Corollary 3.20. Let φ t be the flow of system (2.12). Then Finally, we prove Proof of Theorem B1. Denote the set W s loc (Γ 1 ) ∪ W s loc (Γ 2 ) ∪ W s loc (Γ 1 ∪ Γ 2 ) by W s . By definition, the forward orbit of any point on W s V (O) ∪ W s lies in V. Consider a point in V \ W s V (O) whose forward orbit lies entirely in V. The forward orbit of this point must intersect Π s 1 ∪ Π s 2 at Λ s . Therefore, it follows from the proof of Theorem B2 (for the case λ 2 < 2λ 1 ) and Corollary 3.20 (for the case 2λ 1 < λ 2 ) that this point lies on W s . This finishes the proof for the case of forward orbits.
The proof of the case of backward orbits is the same. This finishes the proof.

Dynamics near super-homoclinic orbits
In this section, we prove Theorem C1. The idea of the proof is to show that there exist sequences of curves {l u k } ∞ k=1 ⊂ W u glo (O) ∩ Π s and {l s k } ∞ k=1 ⊂ W s glo (O) ∩ Π s that accumulate to W u loc (Γ) ∩ Π s and W s loc (Γ) ∩ Π s , respectively (see Figure 19). Then, the flow near the superhomoclinic orbit defines a map which maps the first sequence to a sequence of curves, denoted by {m u k } in Figure 19, such that each of the curves {m u k } intersects each of the curves {l s k } at a single point. Each of these intersection points correspond to a homoclinic orbit. The proof of Theorem C2 is exactly the same. Figure 19: The blue and green curves belong to the intersection of Π s and the global unstable and stable invariant manifolds of the equilibrium O, respectively. The blue curves accumulate to W u and the green curves accumulate to W s , where W u = W u loc (Γ) ∩ Π s and W s = W s loc (Γ) ∩ Π s . Let q u ∈ W u and q s ∈ W s be at the intersection of the super-homoclinic orbit and the cross-section Π s . The flow near the super-homoclinic orbit defines a map on a small neighborhood B u of q u onto a small neighborhood B s of q s . This map maps the blue curves restricted to B u to the blue curves in B s . The blue and green curves in B s intersect transversely. Any point of these intersections belongs to both stable and unstable invariant manifolds of O. Thus, the orbits passing through these points are homoclinic to O.
Proof of Theorem C1. Let W s = W s loc (Γ) ∩ D 1 and W u = W u loc (Γ) ∩ D 2 . We have shown in Section 3.3 (after Proposition 3.9) that T glo (W u loc (O) ∩ Π u ) intersects Π s at a curve which is tangent to * at M s . For a sufficiently small , the restriction of this curve to B \ {M s } lies in D 2 . Denote this restricted curve by L u 0 , and let L u k (k ≥ 1) be the restriction of T L u k−1 to B \ {M s }. By Remark 3.14, the sequence {L u k } ∞ k=1 converges to W u uniformly. Now, consider the restriction of W s loc (O) ∩ Π s to B \ {M s } and denote it by L s 0 . We have L s 0 ⊂ K, where K is as in (3.10). Let L s k (k ≥ 1) be the restriction of T −1 L s k−1 to B \ {M s }. By Remark 3.15, the sequence {L s k } ∞ k=1 converges to W s uniformly.
The super-homoclinic orbit S intersects Π s at W u and W s infinitely many times. Denote the furthest points of S ∩ W u and S ∩ W s from M s by q u and q s , respectively. Let B u be a sufficiently small open ball in D 2 centered at q u . The orbits starting from B u leave the small neighborhood U of Γ and go along the super-homoclinic orbit S, and after a finite time, they come back and intersect Π s at some points close to q s . These orbits induce a global map along the super-homoclinic orbit S, where B s = T S (B u ) and T S (q u ) = q s . Since B u is sufficiently small and the map T S is a diffeomorphism, the neighborhood B s is small, connected and convex.
Define l u = W u ∩ B u and l s = W s ∩ B s . Since the sequence {L s k } ∞ k=1 converges to W s uniformly, there exists a sufficiently large k s such that for all k ≥ k s , the curve L s k intersects B s . Let l s k = L s k ∩ B s for k ≥ k s . This implies that l s k unif − − → l s . Similarly, for some sufficiently large k u , all the curves L u The map T S maps B u to B s . Thus, the curves l u and l u k in B u are mapped to some curves in B s by T S . Let m u = T S (l u ) and m u k = T S (l u k ) for k ≥ k u . Since the super-homoclinic orbit S is at the transverse intersection of the stable and unstable invariant manifolds of the homoclinic orbit Γ, the curves m u and l s intersect each other transversely. On the other hand, the sequences of the curves m u k and l s k converge to m u and l s , respectively. This implies that the curves m u k intersect the curves l s k transversely. Moreover, without loss of generality, we can assume that the integers k u and k s are large enough such that the curves m u i and l s j intersect each other at a unique point p i,j for any i ≥ k u and j ≥ k s . The orbits passing through the points p i,j are the desired multi-pulse homoclinic orbits. This proves Theorem C1.
A. Proofs of Lemmas 2.5, 2.6 and 2.7 In this appendix, we prove normal form lemmas. We start with a brief discussion on some materials needed for the proofs of the lemmas, and then proceed to the proofs.
Definition A.1. By straightening the invariant manifold M, we mean applying a change of coordinates of the form ( x, y) = (x, y − ϕ (x)).
Since H vanishes at every point of the local unstable invariant manifold {u 1 = u 2 = 0}, it can be written as for some C ∞ -smooth H 1 , H 2 : R 4 → R such that H 1 and H 2 and their first derivatives vanish at O. On the other hand, H vanishes at every point of the local stable invariant manifold {v 1 = v 2 = 0}. This implies 0 = H (u 1 , u 2 , 0, 0) = λ 1 u 1 H 1 (u 1 , u 2 , 0, 0) − λ 2 u 2 H 2 (u 1 , u 2 , 0, 0) . Therefore This suggests that, without loss of generality, we can assume that H 1 and H 2 vanish at {v 1 = v 2 = 0}. Now, consider the change of coordinates Since H 1 (u 1 , u 2 , 0, 0) = H 2 (u 1 , u 2 , 0, 0) = 0, applying this change of coordinates on system (2.4) keeps the local stable and local unstable invariant manifolds straightened and therefore keeps the form (2.4) of the system such that (2.5) still holds. However, this change of coordinates reduces the first integral H to the form (2.6).
It is a direct consequence of Remark A.4 that change of coordinates (A.4) preserves the symmetric structure of the system and the first integral. Concerning the change of coordinates (A.6), note that since H in (A.5) satisfies (1.6), we have This implies that the change of coordinates (A.6) commutes with symmetry (1.5), and therefore, preserves the invariance of the system with respect to symmetry (1.5). This ends the proof of Lemma 2.5.
Our proof of Lemma 2.6 is based on a theorem in [SSTC98] (Theorem A.1). A special case of this theorem that we need for the proof of that lemma is stated below: , Theorem A.1) Consider system (2.4) and assume λ 1 < λ 2 . There exists a C ∞ -smooth change of coordinates which brings system (2.4) to the form where the functions f ij , g ij are C ∞ -smooth and Proof. See [SSTC98].
As a matter of comparison between this lemma and Lemma 2.6, the functions f i1 and g i1 (i = 1, 2) in (2.8) do not depend on u 2 and v 2 , respectively, and (2.9) includes all conditions (A.8) as well as two extra constraints Remark A.6. The desired change of coordinates in Lemma A.5, denote it by Φ, is in fact a composition of several changes of coordinates, each describing some invariant manifolds. One can observe that each of these changes of coordinates commutes with symmetry (1.5) (see [Bak20]). Moreover, due to this symmetric property, each of these changes of coordinates can be written in the form where o (1) and O (1) stand for C ∞ -smooth functions of (u, v) which converge to zero and are bounded above by a constant, respectively, as (u, v) → O. A straightforward calculation shows that making changes of coordinates of this form preserves the form (2.6) of the first integral H. On the other hand, first integral (2.6) is already of the form (2.10). This implies that the change of coordinates Φ transforms first integral (2.6) to the form (2.10).
Proof of Lemma 2.6. According to Lemmas 2.5 and A.5, there exists a change of coordinates which brings system (1.3) to system (A.7) where the functions f ij , g ij are C ∞ -smooth and satisfy (A.8). We show that there exists a change of coordinates which brings system (A.7) into the form (2.8), where f ij , g ij are C ∞ -smooth and satisfy (2.9).
Consider system (A.7) and, for i = 1, 2, let It is easy to see that Q 1 vanishes at u = 0 and also the updated f ij and g ij in system (A.15) satisfy all the conditions (2.9) except for (A.9) and (A.10). In order to get (A.9), it is sufficient to find q 1 (v) such that Q 2 (v) ≡ 0, i.e. q 1 (v) satisfies the relation Consider the C ∞ -smooth system where (U, v 1 , v 2 ) ∈ R 3 . The linear part of this system at the origin is with the spectrum {0, λ 1 , λ 2 }. Therefore, this system has a C ∞ -smooth 2-dimensional local unstable invariant manifold defined by the equation {U = q 1 (v 1 , v 2 )} for some function q 1 . Moreover, this function satisfies (A.16) because this relation is nothing but the condition of the invariance of the local unstable invariant manifold with respect to the flow of system (A.17) (see Definition A.3). Thereby, as we required, a C ∞ -smooth function q 1 (v 1 , v 2 ) that fulfills (A.16) exists.
We have now shown our first claim: the underlined term J 2 (v)u 2 in system (A.12) can be eliminated by performing a change of coordinates of the form (A.13). The proof for the second claim that the term J 4 (u)v 2 can be eliminated by applying a change of coordinates of the form (A.14) can be accomplished analogously (see [Bak20]).
To finish the proof, note that changes of coordinates (A.13) and (A.14) are of the form (A.11). Therefore, by Remark A.6, applying changes of coordinates (A.13) and (A.14) together with the change of coordinates used in Lemma A.5 transforms first integral (2.6) to the form (2.10). This ends the proof of Lemma 2.6.
Proof of Lemma 2.7. By Lemma 2.6, there exists a change of coordinates which commutes with symmetry (1.5) and brings system (1.3) and first integral (1.4) to (2.8) and (2.10), respectively. System (2.8) possesses a C q -smooth three dimensional extended unstable invariant manifold W uE defined by {(u, v) : u 2 = φ uE (u 1 , v 1 , v 2 )}, and a C q -smooth extended stable invariant manifold W sE defined by {(u, v) : v 2 = φ sE (u 1 , u 2 , v 1 )} (see [SSTC98]). We claim that straightening W uE , i.e. applying the C q -smooth change of coordinates and straightening W sE , i.e. applying the C q -smooth change of coordinates reduce system (2.8) to system (2.12), where (2.9) is satisfied, and transforms first integral (2.10) to (2.13). On the other hand, by Remark A.4, straightening these manifolds keeps the invariance of system (2.8) and first integral (2.10) with respect to symmetry (1.5). Thus, we are done once we prove this claim. To this end, we use the following lemma Lemma A.7. The following hold for the C q -smooth functions φ uE and φ sE : The following are immediate consequences of this lemma: Corollary A.8. We can write φ uE and φ sE as u 2 , v 1 ), where p sE 1 and p uE 1 are some C q−1 -smooth functions and p sE 2 and p uE 2 are some C q−2 -smooth functions such that p uE 1 = u 1 p uE 2 and p sE 1 = v 1 p sE 2 .
Corollary A.9. We have We prove Lemma A.7 later. Taking into account that as a result of applying change of coordinates (A.18), the equation ofu 2 vanishes at {u 2 = 0}, one can easily see that change of coordinates (A.18) reduces system (2.8) to Here, x := φ uE (u 1 , v), the functions f ij and g ij are as in (2.8), and P j (u, v) (j = 1, .., 7) are some functions such that P 1 vanishes at {v = 0} (see [Bak20] for more details). Moreover,f ij are C q−1 -smooth andg ij are C q -smooth. Using Lemma A.7 and Corollaries A.8 and A.9 and taking into account that the expression u 2 + φ uE (u 1 , v) vanish at u = 0, and also the functions f ij and g ij satisfy (2.9), one can easily show thatf ij andg ij satisfy (2.9) as well. System (A.20) is of the form (2.8) where f 21 (u 1 , v) ≡ 0. Similar to the case of straightening the extended unstable manifold, one can use Lemma A.7 and Corollaries A.8 and A.9 and show that making change of coordinates (A.19) reduces system (A.20) to system (2.12) where the corresponding f ij and g ij are C q−1 -smooth and satisfy (2.9). This ends the proof of the first part of Lemma 2.7.
Denote the H 1 and H 2 in (2.10) by H • 1 and H • 2 , respectively, and let x := (u 1 , u 2 , v 1 ), y := u 1 , v 1 , v 2 + φ sE (x) and z := u 1 , u 2 + φ uE (y) , v 1 , v 2 + φ sE (x) . Applying changes of coordinates (A.18) and (A.19) brings (2.10) to which by Corollary A.8, can be written in the form (2.13), for This proves the second part of Lemma 2.7. All that remains to finish the proof of Lemma 2.7 is proving Lemma A.7. We only prove part (i) of this lemma; the proof of part (ii) is the same.
To prove the identity we consider the condition of the invariance of the manifold W uE with respect to the flow of system (2.8) (see Definition A.3), i.e.
where x := φ uE (u 1 , v). Both sides of this relation are C q−1 -smooth (q ≥ 2 because 2λ 1 < λ 2 ) functions of u 1 , v 1 and v 2 . Taking (A.21) as well as conditions (2.9) and Corollary A.9 into account, we can differentiate this relation with respect to u 1 at u 1 = 0 and obtain Define z = z(v) = φ uE u 1 (0, v). Then, (A.23) can be written as (A.24) where z(0) = 0 (note that φ uE u 1 (0, 0, 0) = 0). be invariant with respect to the flow of the C q−1 -smooth system which is defined on a small neighborhood of the origin in R 3 . (Indeed, relation (A.24) is the condition of the invariance of (A.25) with respect to the flow of system (A.26).) Therefore, the uniqueness of the solution of (A.24) can be proved by showing that system (A.26) has a unique invariant manifold of the form (A.25). To do this, first, notice that this system possesses a unique two dimensional stable invariant manifold of form (A.25). Second, we observe that any orbit on manifold (A.25) converges to the origin of system (A.26): the first two equations in (A.26) are independent of z and have (v 1 , v 2 ) = (0, 0) as an asymptotically stable equilibrium. Therefore, as t → ∞, an orbit (v(t), z(v(t))) of system (A.26) which belongs to invariant manifold (A.25) converges to (0, z (0)). Since z (0) = 0, this means that any invariant manifold of the form (A.26) must be a subset of the stable manifold of system (A.26). However, since both manifolds are 2-dimensional, they must be the same. Therefore, system (A.26) has a unique invariant manifold of the form (A.25) which is in fact its stable invariant manifold. This ends the proof of Lemma A.7 and hence the proof of Lemma 2.7.
B. Proofs of Lemmas 2.9, 2.10 and 2.11 We only prove Lemma 2.9. The proofs of Lemmas 2.10 and 2.11 are similar. We refer the reader to [Bak20] for the proofs of these lemmas. We start with a discussion on the method of the proof and then proceed to the proof of Lemma 2.9.

B.1. The method of the proof
Here, we present the main procedure which is used in the proofs of Lemmas 2.9, 2.10, 2.11 and also Lemma C.1 (see Appendix C). Consider system (2.14) and denote its unique solution that satisfies boundary condition (2.15) by (u * , v * ), where u * = (u * 1 , u * 2 ) and v * = (v * 1 , v * 2 ). We may also write this as to emphasise that in addition to time variable t, this solution explicitly depends on τ and the boundary conditions u 10 , u 20 , v 1τ and v 2τ as well. It is easy to see that (u * (t), v * (t)) is a solution of this system with boundary conditions (2.15) if and only if For a given τ , denote by I the set of all vector valued functions (u 1 (t) , u 2 (t) , v 1 (t) , v 2 (t)) defined for t ∈ [0, τ ] on some small neighborhood of the origin in R 4 . Then, the right-hand side of (B.2) defines an integral operator on I, denote it by T, as follows: The solution (u * (t), v * (t)) is in fact the fixed point of this integral operator. According to [SSTC98] (Theorems 2.9 and 5.11), this integral operator is a contraction and its fixed point is the limit of the sequence of successive approximations where u (0) , v (0) ≡ (0, 0) and Let A be a closed subset of I such that (u (t) , v (t)) ≡ (0, 0) ∈ A and T (A) ⊂ A. Since u (0) , v (0) ≡ (0, 0) ∈ A, the invariance of A implies that u (n) (t), v (n) (t) belongs to A for all n > 0, and so does the solution (u * (t), v * (t)).
Our approach for proving Lemmas 2.9, 2.10 and 2.11 (and Lemma C.1) is based on this remark. We construct the integral operator, introduce the invariant set A and find an estimate for the image of the elements of this set under T. Then, this estimate holds for the solution (u * , v * ) too.
Meanwhile, we have shown that the image of any element of A under T can be written in the form (2.16) such that the corresponding ξ 1 , ξ 2 , ζ 1 and ζ 2 satisfy the estimates given in the statement of the lemma. However, since u (0) , v (0) = (0, 0) ∈ A, it follows from Remark B.1 that the same holds for the solution (u (t) , v (t)) that satisfies boundary condition (2.15). This ends the proof of Lemma 2.9.
C. Proofs of Lemmas 3.12 and 3.13 The main part of this appendix is the proof of the following lemma: Lemma C.1. Let (2.21) be the local map of system (2.12) and suppose (u 10 , v 10 ) ∈ D 2 . Write x := (u 10 , v 10 ). We have It is straightforward to derive Lemmas 3.12 and 3.13 from Lemma C.1 (see [Bak20]). Indeed, by this lemma, an estimate for ∂(u 10 ,v 10 ) ∂(u 10 ,v 10 ) can be obtained. One can use this estimate and the implicit relations between (u 10 , v 10 ), (w, z), (u 10 , v 10 ) and (w, z) to prove Lemmas 3.12 and 3.13. We refer the reader to [Bak20] for further details.
The next theorem provides a setting in which if a map T possesses a cross-map T × which satisfies certain properties, then it has an invariant manifold that contains ω-limit points of every forward orbit of the domain. This theorem becomes powerful when one is looking for the invariant manifolds of a non-smooth map whose cross-map is smooth. This result was first obtained by Afraimovich and Shilnikov [AS77] for maps defined on an annulus. The following formulation of this result which holds for arbitrary Banach spaces is stated in [SSTC98].
Theorem D.3. ( [SSTC98], Theorem 4.3) With the setting in Definition D.1, let X and Y be two convex closed subsets of X * and Y * , respectively, such that R = X × Y ⊂ U, T × is defined on R and T × (R) ⊂ R. Let F and G in (D.1) be C 1 -smooth and satisfy sup (x,y)∈X×Y where ϕ (x, y) • = sup (x,y)∈X×Y ϕ (x, y) for any vector-valued or matrix-valued function ϕ. Then (ii) for any x = (x, y) ∈ R and any arbitrary > 0, there exists an integer N x ∈ N such that for any n > N x if {T i (x)} i=n i=0 ⊂ R, then dist (T n (x) , M * ) < . In particular, M * contains the ω-limit set of any point of R whose forward orbit lies entirely in R.
(iii) if R is bounded, then the integer N x given above can be chosen independent of x, i.e. for any arbitrary > 0, there exists an integer N ∈ N such that for any n > N and any x ∈ R if {T i (x)} i=n i=0 ⊂ R, then dist (T n (x) , M * ) < . Proof. The proof is by contradiction. Assume x / ∈ M * . This implies dist (x, M * ) > 0. Choose an 0 < < dist (x, M * ) and consider N given by Theorem D.3. We have dist M * , T N T −N (x) < and thereby dist (M * , x) < , which is a contradiction.