Asymptotic Dynamics of a Difference Equation with a Parabolic Equilibrium

The aim of this work is the study of the asymptotic dynamical behaviour, of solutions that approach parabolic fixed points in difference equations. In one dimensional difference equations, we present the asymptotic development for positive solutions tending to the fixed point. For higher dimensions, through the study of two families of difference equations in the two and three dimensional case, we take a look at the asymptotic dynamic behaviour. To show the existence of solutions we rely on the parametrization method.


Introduction and Main Results
In studying difference equations, one interesting problem is to know the asymptotic dynamical behaviour, of the positive solutions that approach equilibrium points. This question naturally arises, for instance, when describing applications in biological or economical systems. See [10,18,22,23,32,35], for instance. In particular, how popula-  1 tions evolve to equilibrium states and, more concretely, if they tend to rest points in the same or different asymptotically way is of great interest in mathematical modeling.
This problem appears not only in modeling real-life processes but also in Celestial Mechanics [16,24,25,28,30], in complex analytical dynamics [26] and in many more others fields where the subject is concerned with the study of invariant submanifolds of fixed parabolic points of maps.
Related to the determination of the complete asymptotic expansion of solutions tending to fixed points in difference equations are the next three examples appearing in the literature, in which this issue is unsolved.
The first one was proposed by Berg and Stević in 2002, see [6,33] respectively, concerning the difference equation for the case k = 1. In [6] the asymptotic expansion of x n is bounded as 2 n + 2 n 2 ln n + a n 3 ln 2 n ≤ x n ≤ 2 n + 2 n 2 ln n + b n 3 ln 2 n, where a < 2 < b. As a consequence, a proof of the existence of a particular solution of this equation such that {x n } n → 0 is also given. The second example concerns the difference equation (1) in the case k = 2. In general, i.e. for each k ∈ N, Stević in 2006, see [36], proved the existence of a positive solution, {x n } n , converging to zero, by assuming that the first five terms in the asymptotic expansion of x n have the following form: x n ∼ a n + b ln n + c n 2 + d ln 2 n + e ln n n 3 .
for some a, b, c, d, e ∈ R. Finally, the third example concerns the difference equation when k = 3. This equation was introduced by Berg in 2008 and by Berg-Stević in 2011, see [8,9] respectively. In these works the existence of a solution such that {x n } n → 0 is proved for all k, by assuming that the asymptotic expansion of x n is given by where the coefficients are fixed in such a way that x n is proved to be a solution of Eq. (2). In our work, we answer to the problem of determining the complete asymptotic expansion of x n , for one dimensional difference equations having a parabolic equilibrium point. As a consequence, we solve the problem of giving the asymptotic development of the solution, x n , mentioned in the above first example. This problem is still open in the other two previous examples.
It is worth mentioning some previous works that address the problem of obtaining the first terms of asymptotic developments and motivated our work. See, for instance, [5,7,19,31,34,37,38]. These articles, using various tools, manage to obtain the first terms of the development of some families related to those presented in our work. Among all these references, due to their proximity to some of the techniques that we use in our work, we want to highlight the article of Stević, [31], where the first three terms of the asymptotic development are found.
The study of the asymptotic development in terms of n, for positive solutions, {x n } n∈N , approaching parabolic fixed points, of difference equations is the general subject of our work. Since the topic is classical, some of the results in this paper appear scattered in this or that form, but we have not managed to find concrete references for all of them. So we have decided to make our paper as self-contained as possible.
By assuming that the fixed point is at the origin, this problem can be stated as follows.
Let us take the difference equation where F : is an analytic function defined on an open subset of the origin, U ⊂ R k 1 × R k 2 , with a parabolic fixed point at x = (0, 0). Additionally, from now on, we will consider positive solutions of Eq. (3), i.e. only solutions of the form {x n } n∈N , x n ∈ R + k 1 +k 2 will be considered.
At this point, we would recall that the origin of R k 1 × R k 2 is a parabolic fixed point of F if F(0, 0) = (0, 0) and D F(0, 0) = Id; i.e. that map is tangent to the identity. Also, the set of points whose positive iterates converge to the fixed point is invariant by the map and it is called stable invariant set or stable invariant manifold.
To decide whether a parabolic fixed point has associated a stable or unstable manifold is still, in general, an open problem. However, it is worth mentioning the existence and uniqueness results in some cases. See [1][2][3]15,17,29], for instance. To develop algorithms for the computation of local approximations of invariant manifolds of parabolic fixed points is also of interest. See [4], for instance. Conditions implying a kind of weak hyperbolicity for the fixed point will be specified later.
In this work we will prove that there are families of difference equations (3) such that, on the stable manifold of the origin, they present positive solutions, tending to the fix point, with different asymptotic behaviours.
We want to comment that, for system (3), we are not only interested in to get the asymptotic development of positive solutions tending to the fixed point located at the origin but also to prove that these are all the positive sequences that tend to the origin. These results are achieved by using the parameterization method [11][12][13]20,21], in the simultaneous search for an invariant stable manifold, W s , of the origin as an immersion K : V ⊂ R k 1 → R k 1 × R k 2 , and a map R : V → V , where V is a domain which contains 0 on its boundary. This is a novel approach for this type of problems.
We note that K is a parametrization of the manifold W s , and R is a representation of the dynamics of F on W s , satisfying the invariance equation We use the parametrization method in the proof of Theorem 1.3. We also use the invariance equation in finding numerical evidences of how W s can be, assuming that this invariant stable manifold of the origin exists. The corresponding results are stated in Proposition 3.1.
Concerning the one dimensional case of the difference equation (3), i.e. when k 1 +k 2 = 1, the asymptotic behaviour study of {x n } n∈N satisfying that lim n→∞ x n = 0, can be performed by taking for some k ≥ 1 with a 1 = 0, and 0 ≤ |λ| ≤ 1. As in the complex case, by using conformally conjugate functions to F, it is possible reduce this study to more simpler cases according λ. More specifically, the use of the linearization theorem applies when 0 < |λ| < 1, meanwhile the conjugation theorem can be used when λ = 0. In Sect. 2 we prove an adapted theorem to the dynamics of the real case for these values of λ.
In the particular case |λ| = 1, the asymptotic behaviour is known in some cases. In complex dynamics, Resman, in [27,Prop. 3], considers the difference equation defined by the parabolic diffeomorphism for some k ∈ N. In this setting, the asymptotic development of z n is given by where the coefficients g i = g i (k, A, a 2 , . . . , a i ), i = 1, . . . , k +1, are complex-valued functions, and A = (−ka 1 ) − 1 k . When λ = 1, in the one dimensional real case, we present in next theorem a key result giving the asymptotic development of the solution of the difference equation (5). Although we have not found explicit references, their results seem to be of common knowledge. In any case, in next section we include a detailed proof.
As we will see, Theorem 1.1, together with the parametrization method, give the clue for understanding the appearance of logarithms in the asymptotic developments in some solutions of difference equations of order bigger than one, like for instance equations (1) or (2). Theorem 1.1 (Asymptotic development of x n ) Let us consider the one dimensional difference equation (5) being k a positive integer number and x,a i ∈ R, for all i ≥ 1. Let x 0 be an initial condition belonging to an attracting domain of the origin, i.e. such that lim n→∞ x n = 0. Then, for each positive integer m, the asymptotic development of x n is given by where coefficients g i, j Remark 1. 2 We note that, since we are just involved into the control how the functions ln i n/n j/k i = 0, 1, 2, . . . , j = 1, 2, . . . , emerge in the asymptotic development of x n , we are not interested into fix the coefficients g i, j p in the expression (7).
We also note that a related result to Theorem 1.1 can be found, for example, in [27] for the complex case and in [31] for the real one.
To illustrate Theorem 1.1, its consequences, and the role of the parameter x 0 in the asymptotic developments of its statement we consider two simple examples: (a) The sequence satisfying the bilinear difference equation with initial condition x 0 near the origin, can be explicitly obtained by introducing the new sequence y n = 1/x n . We get where c = 1/x 0 . Hence the sequences tending to zero are always parameterized by one parameter c that in this example can be easily related with x 0 . Moreover, notice that for this particular simple case, all coefficients associated to logarithmic terms are zero.
(b) It is clear that the difference equation associated to the celebrated logistic map has a parabolic fixed point at the origin when μ = 1. For this special case the sequences with initial conditions satisfying 0 < x 0 < 1 tend to the origin. From the above theorem we can compute their asymptotic development at 0. For instance, we obtain that In this case the dependence between c and x 0 is not made explicit by the theorem. For the case λ = −1, as a consequence of previous theorem, in Proposition 3.1 we obtain a similar result.
At this point, we wonder if the previous asymptotic behaviour can be extrapolated to higher dimensions. To give an insight on what happens in dimension higher than one, we present two results concerning difference equations in two and three dimensions, see Theorem 1.3 and Proposition 1.4, respectively.
In the two dimensional case we study a given family based on the difference equation (1) when k = 1. In this case, Eq. (1) is the first component of the two-dimensional scheme iteration F, where F is defined through the shift function G as In next theorem we study the difference equation given by F, and we prove that in a open region of the first quadrant, the stable invariant manifold of the origin is given by the graph of an analytic function K .
Even more, we also prove that on K the dynamics of the first component of F is given by expression (7) and, in this sense, the dynamical asymptotic behaviour on K is unique.
Before proving next theorem, let us introduce some notation. Consider the two dimensional difference equation (3) when k 1 = k 2 = 1, defined in a neighbourhood, U , of the origin. In this case, by way of notation, we introduce the projectors π 1 (x, y) = x, and π 2 (x, y) = y and, for each r > 0, we define where V = (0, r ), as the stable invariant manifold of the origin, for the map F, restricted to the first quadrant. In next theorem we summarize previous results.

Theorem 1.3 Let us consider the two dimensional difference equation (3) given by
(2) the dynamics on W s V is given by the analytic function R, (3) on W s V , the first terms of the asymptotic development of the solutions corresponding to the difference equation are given by where being c a constant parameter that is fixed by initial condition of the orbit.
For three dimensional dynamics we present positive orbits, {x n } n∈N , of Eq. (3), exhibiting different asymptotic behaviours when approaching the origin. We proceed by using the invariance equation, to obtain the first terms of their asymptotic developments and so their numerical approaches.
We remark that due to the facilities to implement the parametrization method, almost any computer algebra system (CAS) can be used to obtain the rational coefficients appearing in the asymptotic expansions given in the above theorem and in forthcoming Propositions 1.4 and 1.5. In our work we have used CAS Maple ©.
More concretely, we provide two families of difference equations (3). For the former one, we give two positive orbits, {x n } n∈N , with different asymptotic speeds developments in n when approaching the origin. For the latter one, we obtain another asymptotic speed development expression on n, for the dynamics of an orbit tending to the origin.
Both examples agree with the following scheme. Consider the family of "shift" functions where g is an analytic function defined in a neighborhood of the origin, with g(0, 0, 0) = 0. For each fixed function g, function G defines a difference equation. We note that, even though the origin is a fixed point of this difference equation, it is not a parabolic one. One way to have a parabolic fixed point at the origin is to consider the difference equation (3) given by for some suitable function g.
In the three dimensional case we study two families of difference equations based on the difference equations (1) and (2), where g is given either by or We remark that the first component of the three-dimensional scheme iteration defined by g as in (14), agrees with the difference equation (1) when k = 2, proposed in [36], while the family defined by g as in (15) is related to the difference equation (2) proposed in [8].
Next proposition shows the results obtained on the asymptotic behaviour for both families. In Sect. 3 we present, besides its proof, some details on their dynamics. (3), defined in a neighbourhood of the origin, where F and G are given by (13) and (12) with g as in (14). Then, there exist numerical evidences on the following facts.

Proposition 1.4 Let us consider the three dimensional difference equation
(1) there exist three invariant manifolds of dimension two, K i (t, s), i = 1, 2, 3, whose first analytic development terms are given by: such that the curve K i (t, 3/2), i = 1, 2, 3, is a non-negative solution of Eq. (3), the asymptotic behaviour of it agrees with expression (11).

Proposition 1.5 Let us consider the three dimensional difference equation
(1) there exist a positive invariant curve, K (t), whose first analytic development terms are given by: (2) The dynamics on K is approached by the analytic function R (3) the first terms of the asymptotic development of R are given by

Asymptotic Dynamical Properties in One Dimension
In this section we present some properties about asymptotic dynamic behaviour of the solutions, near the fix point at the origin, of the one dimensional difference equation (5).
First, in Lemma 2.1, we introduce a result recalling the asymptotic behaviour, except in the rationally neutral cases, i.e. except when |λ| = 1.
In Theorem 1.1, the rationally neutral case λ = 1 is studied. Additionally, in Lema 2.3 we present some results on the asymptotic dynamic behaviour for the λ = −1 case. For this value of λ, it is worthwhile to mention that the asymptotic behaviour of the solutions tending to the origin of the difference equation (5) are studied by considering the iterates of the difference equation taking either x 0 or F(x 0 ) as the initial condition, as we detail later. (5). For each x 0 close to zero, the solution x n with initial condition x 0 satisfies that:

Lemma 2.1 Consider the difference equation
where α 1 = 0 and ϕ is an analytic diffeomorphism at 0 such that ϕ(0) = 0, and y 0 = ϕ −1 (x 0 ). In the λ = 0 case, the origin of Eq. (5) is a superattracting fixed point and, following Boettcher result, see [14,Th. 4.1] for instance, we get that there is a complex analytic map y = φ(x), with ϕ = φ −1 , of a neighbourhood of the origin onto itself which conjugates F(x) to y p . So, the proof follows because the sequences generated by y n+1 = y p n are y n = y p n 0 .
Next lemma is an easy technical result which is useful to prove Theorem 1.1. Its proof is included for the sake of completeness.

Lemma 2.2 Given n 0 ∈ N, let us consider the recurrence relation
where f is a real and continuous non-negative function, defined on R. Suppose, additionally, that u n 0 > 0 and that f is a monotonous function on [n 0 , +∞). Consider a function F, F ∈ C 1 (R), such that and suppose that one of the following hypotheses holds: (1) lim x→+∞ F(x) = ∞ and lim n→+∞ f (n)/ f (n + 1) = 1,

Then there exists a function G(x), x ≥ n 0 , such that G (x) = f (x) and satisfying u n = G(n) + o (G(n)) . (26)
Proof To prove equality (26), let us prove the equivalent condition lim n→+∞ u n G(n) = 1.

Straightforward calculations show that
where ξ n ∈ (n, n + 1). By assuming that f is a monotonous decreasing function on [n 0 , +∞), we have that for all n ∈ N. By considering the limiting case of former inequalities sequence, when n tends to infinity, and by using the second hypothesis of case (1), we conclude that the limit of expression (28) exists and, hence, that desired conclusion follows. In the case when f is a monotonous increasing function on [n 0 , +∞), by using analogous arguments, the same conclusion holds. Assume now hypothesis (2). Since lim x→+∞ F(x) = a ∈ R then, we have that f is a monotonous decreasing to zero function, by one hand, and that it is not restrictive to assume that a > 0, on the other.
Since, for all n ∈ N and for all p ∈ N, p ≥ n + 1, we have that then, from the first inequality, then we also have that ∞ k=n+1 f (k) <∞. Concerning the sequence {u m } m∈N , since o ( f (n + 1)) , (30) we get that {u m } m∈N is a Cauchy sequence and, then there exists l ∈ R, such that lim p→∞ u p = l.
By considering the limiting case of the inequalities sequence given by (29), when p tends to infinity, using equality (30), we get ( f (n)) .
From the first and second inequalities we get respectively. As a consequence, where From the recurrence relation (25) and using the hypothesis, we have that l > 0. Using this fact, it can be proved that As a consequence, by considering the limiting case when n tends to infinity in the expression (31), equality (27) holds.
By using the previous lemma, we prove our main result on the asymptotic development of the solution x n of the difference equation (5), in the rationally neutral case λ = 1.

Theorem 1.1 Let us consider {x n } n∈N , a solution of
where a 1 = 0, and satisfying that lim n→∞ x n = 0. Let us consider the change of variables We remark that, since lim n→∞ x n = 0, the case k even and a 1 > 0 can not be taken into account. Hence, the change of variables is well defined. Even more, the convergence to zero of x n means that we choose the initial condition x 0 ≈ 0 such that either, k is odd and x 0 a 1 < 0, or k is even and a 1 < 0. As a consequence, we remark that the convergence to zero only depends on a 1 , k and x 0 ; that is, as n goes to infinity, for all a i , i ≥ 2.
Using previous change of variables applied to the difference equation (32), we obtain the following recurrence relation for ω n for some real values c i , depending on the initial coefficients a j . Recurrence (35) can be re-written as as n goes to infinity. By applying Lemma 2.2.1 to the recurrence on ω n , we get that Let us define ω n = n. In this way, we introduce the recurrence In accordance with previous definition, we have for some d i real value, and where q is an arbitrary, but fixed, natural number q k. By applying Lemma 2.2.2 to the previous recurrence on p n , we get that From equality (36), we obtain If we undo the change of variables (33) on previous expression of ω n , we obtain where Now, we are going to prove that if we use Taylor development for expression (37), on the variable x, in a neighbourhood of the origin, up to order p on x, then x n writes as p+1 ln p+1 n + R p (n), (38) for some coefficients g i, j x 0 , a 1 , a 2 , . . . ) which are real valued functions; and where R p (n) is a function including the Taylor remainder term. Let us prove formula (38) by using mathematical induction on p.
For p = 0, performing a Taylor expansion of first order on x in expression (37), we get where R 0 (n) is the Taylor remainder term. This expression agrees with the one given by formula (38). We observe that, in this case, Hence, we proved the base case of the induction process.
Let us prove the induction step, i.e. that if formula (38) holds for p, then it holds for p + 1.
If we define the functions γ i, j (n) = ln i n/n j/k , i = 0, 1, 2, . . . j = 1, 2, . . . , we note that, x and each x n, p is a linear combination of a subset of them. For x n, p+1 , let us check that we obtain the functions γ i, j that appear in expression (38).
To get x n, p+1 we need a Taylor expansion, of order p + 2, of expression (37), including the x p+2 term. Hence, concerning the γ i, j functions, those appearing in x n, p+1 are the ones we had up to order x p+1 plus the new ones coming from the x p+2 term. So, we need to check that expression (38) for x n, p+1 includes these new functions, that are From the induction hypothesis, we can obtain x n, p+1 by multiplying x n, p times x. We observe that the functions γ i, j (n) involved in the expression of x n, p , i.e. in the terms of the Taylor development, up to order x p+1 , are i = 0, 1, . . . , p, j = 1, 2, . . . , k( at least, and the ones involved in the expression of x are at least. We note that the functions γ i, j (n) given in expression (39) can be obtained from the product of the ones given in expression (40) by the ones given in expression (41), what finishes the proof.
In the next proposition, which is a sequel of Theorem 1.1, we present some results on the asymptotic development of the solution x n of the difference equation (5), in the rationally neutral case λ = −1.
We note that, we are only interested in those initial conditions belonging to the stable manifold of the fixed point at the origin, i.e. we only consider x 0 such that lim n→∞ F n (x 0 ) = 0. As a consequence, in next result, the case k even and a 1 < 0 is excluded. (5) with λ = −1, that is

Proposition 2.3 Consider the difference equation
Let x 0 be an initial condition belonging to an attracting domain of the origin, i.e. such that lim n→∞ x n = 0. Then, for each positive integer m, the asymptotic development of x n is given by (1) if k is even, then (2) if k is odd, then for some integer number k > k, where g i, j x 0 , a 1 , a 2 , . . . ) are real valued functions.
Proof We get the asymptotic behaviour of an orbit, {x n } n∈N , from the study of the two sub-orbits {x 2n } n∈N and {x 2n+1 } n∈N , i.e. from the study of the behavior of the iterations through F 2 , starting with initial conditions x 0 and F(x 0 ), respectively.
We note that In the case k even, since a 1 = 0, we apply Theorem 1.1 to the iterations of F 2 on the corresponding initial conditions. Hence, the asymptotic behaviours of {x 2n } n∈N and {x 2n+1 } n∈N are, both, given by expression (42).
In the case k odd, we observe that the coefficient of x k+1 in F 2 vanishes. Hence, either there exist a positive integer number j > 1 and a j = 0, such that Since x 0 belongs to an attracting domain of the origin, the latter case can not be. Whence, we can also apply Theorem 1.1 to the iterations of F 2 . Hence, the asymptotic behaviours of {x 2n } n∈N and {x 2n+1 } n∈N are, both, given by expression (43). We remark that, since j > 1, the non-linear terms in F 2 are of order o(x k+1 ) and, hence, expression (43) is taken with a certain k > k.

Some Results in Higher Dimensions
In this section, to give an insight to the asymptotic behaviour in higher dimensions, we prove Theorem 1.3, Propositions 1.4 and, 1.5, concerned to two particular families of difference equations, in two and three dimensions, respectively.
Next technical lemma give us some properties of the difference equation (3) by taking F as in (8). (3), defined in a neighbourhood of the origin U , given by (8)

Lemma 3.1 For k = 2, let us consider the two dimensional difference equation
Then, the following properties follow.
(2) Points on the coordinate axes are fixed points.
Proof of Theorem 1.3 Let us consider the two dimensional difference equation (3), defined in a neighbourhood of the origin U , given by (8).
Let us prove statement (1) of this theorem. First, we remark that from Lemma 3.1, the origin, (0, 0), is a parabolic fixed point. Now, let us prove that the hypotheses of Theorem 4.1 of [1] are fulfilled and, as a consequence, the invariant manifold of the origin, W s V , is the graph of an analytic function. Following the notation in [1], for each r > 0, we write V (r ) = (0, r ) ⊂ R + . By way of notation, for each ρ > 0, we define V 1 (ρ) = {ρ}.
In a neighbourhood of the origin, we observe that F is an analytic function given by We apply the above mentioned theorem to the map F, after performing the analytic change of variables ϕ, given by If we take ρ > 0 such that then the hypotheses, H1-H4 of [1,Theorem 4.1], are fulfilled for F(x 1 , y 1 ). Consequently, there exists r > 0 such that W s V is the graph of an analytic function in V .
Undoing previous change of variables, we have that the first quadrant of the (x 1 , y 1 )plane corresponds to an open region of the (x, y)-plane whose boundary is given by two disjoint curves. One of these curves is concave and tangent to the y = x straight line at the origin. The other one, reaches the origin with slope −a 10 /a 01 > 1. Locally, both curves bound an open region on which the invariant manifold of the origin, W s ϕ −1 (V ) , is the graph of an analytic function, h.
Since h is an analytic function, an approximation of its Taylor expansion is obtained as in expression (9).
To prove statement (2), i.e. to obtain a representation, R, of the dynamics on the curve K , we use the parametrization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points introduced in [2]. We check that the hypotheses of Theorem 2.1 of [2] are satisfied and, hence, there exist a C ∞ map K which is a parametrization of the one-dimensional invariant manifold of map F, and a polynomial R(t) such that As in statement (1), we apply the above mentioned theorem to the map In this setting, we have F(0, 0) = 0, D F(0, 0) = Id, and as we would prove. The computation of R is done by matching powers of t in (44). Statement (3) of this theorem follows from Theorem 1.1, applied to the difference equation where R is given by expression (10).
To give an insight on the three dimensional asymptotic behaviour, next we present two examples of difference equations (3). By assuming analytical behaviour of their solutions, we numerically approach them. These examples exhibit different dynamical asymptotic development in terms of n, for positive solutions, {x n } n∈N , tending to the origin.
Next Lemmas 3.2 and 3.3 are technical results useful to prove Proposition 1.4 and Proposition 1.5, respectively. (3) where F and G are given by (13) and (12) with g as in (14). Then the following properties follow.

Lemma 3.2 Let us consider the difference equation
(1) The origin, (0, 0, 0), is a parabolic fixed point and V (x, y, z) = x 2 + y 2 + z 2 is a Lyapunov function. (2) The planes x = 0, y = 0 and z = 0 are invariants by the iteration of the difference equation. Furthermore, all points of the coordinate axes are equilibrium points. (3) On each coordinate plane, the dynamical behaviour of F coincides with the one given in Theorem 1.3. (3) where F and G are given by (13) and (12) with g as in (15). Then the following properties follow.

Proof of Proposition 1.4
To prove statements (1) and (2) of this proposition, let us take g as given in expression (14), and let us apply the parameterization method for looking an invariant stable manifold of the origin.
By imposing the invariant equation (4), we get the parameterization of the first analytic development terms, of invariant stable curves, in terms of an extra parameter, s, such that s ≤ 3/2. Let's call K i (t, s), i = 1, 2, 3, to such non-negative curves as they are established in expressions (16), (17) and (18) in the statement (1). We point out that, since this method provides us a representation of the dynamics of F on the invariant stable curves, the asymptotic behaviour of K i (t, 3/2), i = 1, 2, 3, agrees with expression (11). Furthermore, from Lemma 3.2. (2) and (3), by using Theorem 1.3, we get on each coordinate plane the existence of an analytic solution given by expression (11).
Additionally, the parameterization method also provides us the first analytic development terms of a positive invariant stable curve, K (t), on the invariant stable manifold of the origin, as it is given in expression (19) in statement (2). Furthermore, and also as a consequence of the application of the previous parametrization method, the first analytic terms of the dynamics on K , R, are approached by expression (20) in the statement (3).
Finally, by applying Theorem 1.1 to the first analytic terms of R, we get expression (21) in the statement (4). (1) and (2) of this proposition, we proceed as in the proof of Proposition 1.4. More concretely, let us take g as it is given in expression (15) and, looking for an invariant stable manifold of the origin, we use the parameterization method. As a consequence, we get expression (22) giving the first analytical terms of the invariant curve K (t) and, additionally, we also get the approach of the dynamics on K , given by R(t), as in expression (23). Then, by applying Theorem 1.1 to the first analytic terms of R, we get expression (24).

Proof of Proposition 1.5 To prove statements
In Fig. 1, we locally depict the invariant surface (three leaves) on which the solutions of the difference equation (3) where F and G are given by (13) and (12) with g as in (14) present two different behaviours, according to their initial value. More concretely, solutions tending to the origin and solutions going to the fixed points on Fig. 1 According to Proposition 1.4, local depict of the stable invariant manifolds, K i (t, s) and K (t) (see developments (16), (17), (18) and (19), respectively), of the origin, of the difference equation (3), where F and G are given by (13) and (12) with g as in (14). Different perspectives. Two different speeds, 1/(2n) and 1/n, are observed in the asymptotic developments Fig. 2 According to Proposition 1.5, local depict of the stable invariant manifold curve, K (t), of the origin, as it is given in expression (22). This is the case of the difference equation (3), where F and G are given by (13) and (12) with g as in (15). Speed 1/ √ 2n is observed in the asymptotic development the axes. On the intersection of the three leafs given by K (t), given by expression (19), there is a solution, {x n } n , having asymptotic expansion given by expression (21); that is, this solution goes to the origin as 1/(2n). On each coordinate plane, as it proved in Lemma 3.1, there is a solution {x n } n , whose asymptotic expansion is given by expression (11), that is a solution tending to the origin as 1/n. Finally, solutions in between the two aforementioned ones, and on the invariant surface, belong to the stable invariant manifold of the origin, W s V . In Fig. 2, we locally depict the stable invariant manifold of the origin, W s V , of the difference equation (3), where F and G are given by (13) and (12) with g as in (15). The manifold W s V is depicted as the invariant curve K (t) given by expression (22). On this curve the solutions {x n } n tend to the origin as 1/ √ 2n. Coordinate planes, as it proved in Lemma 3.3, are fulfilled with equilibrium points.