PLANAR DIFFERENCE EQUATIONS: ASYMPTOTIC BEHAVIOR OF SOLUTIONS AND 1-1 RESONANT POINTS

In order to study the global behavior of difference equations, it is necessary to understand the local behavior in a neighborhood of a equilibrium point of the difference equation. This thesis focuses on two aspects of the local behavior of planar difference equations: the asymptotic behavior of a solution converging to a hyperbolic fixed point, and the local qualitative behavior of a non isolated fixed point whose jacobian matrix has a particular structure. Manuscript 2 describes how closely a convergent solution {xn} of (real or complex) difference equations xn+1 = J xn + fn(xn) can be approximated by its linearization zn+1 = J zn in a neighborhood of a fixed point; where xn is a mvector, J is a constant m×m matrix and fn(y) is a vector valued function which is continuous in y for fixed n, and where fn(y) is small in a sense. Manuscript 3 describes completely the local qualitative behavior of a real planar map in a neighborhood of a non-isolated fixed point whose jacobian matrix is similar to ( 1 1 0 1 ), also called a non-isolated 1-1 resonant fixed point. Theorem 3 gives conditions for four non-conjugate dynamical scenarios to occur.

The phase portrait of system (103). The dynamical behavior of (103) is described by Theorem 5 part (c). Every point on the x-axis is a fixed point (blue line). The curves depict the path and direction followed by orbits under forward iteration of (103). The dashed line is the sector boundary curve C. . . . . . 59 12 The phase portrait of system (106) with a = 1, b = 1, and B = 2. The dynamical behavior of (106) is described by Theorem

Introduction
The main goal of the study of difference equations is to establish the global dynamical behavior of the orbits of the difference equation. A common strategy used to help understand difference equations in general is to restrict the analysis to classes of difference equations [7], [15]. In particular, since the early 1990s there has been considerable interest in the study of rational difference equations and monotone difference equations. Much work has been done in these areas by Elaydi, Smith, Agarwal, Ladas, Grove, Kulenović, Merino [7], [8], [14], [15], [16] and others cited therein. In order to understand the global dynamical behavior of a difference equation, one must first understand the local dynamical behavior near the fixed points of the difference equation. In difference equations, fixed points are classified by the modulus of the eigenvalues of the jacobian matrix associated with the fixed point. In the plane, there are three scenarios for the modulus of the eigenvalues of a jacobian matrix of the map of a difference equation. I will consider each scenario below: In the first case when the eigenvalues |λ 1 | = 1 and |λ 2 | = 1, the local dynamics of this case are well understood. Phillip Hartman (see [7] for a secondary source) proved for a C 2 -diffeomorphism that in a neighborhood of a hyperbolic fixed point, that the map is C 1 conjugate to its linearization at the fixed point. This is the discrete analog to the Hartman-Grobman Theorem [6].
In the second case when |λ 1 | = 1 and |λ 2 | = 1, the dynamics are also understood in some circumstances. If λ 1 and λ 2 are complex conjugates with λ 1,2 = e ±iqθ = 1 for q = 0, 1, 2, 3, or 4 (these exceptions are called strong resonances), then techniques from KAM theory can be used to establish the local dynamical behavior, see [16] for an overview or [2] for an application of the theory.
A fixed point is called 1-1 resonant if the jacobian of the map at the fixed point is similar to ( 1 1 0 1 ). The local dynamical behavior near a 1-1 resonant fixed point is still unresolved because this type of fixed point has a strong resonance.
In the third case when |λ 1 | = |λ 2 | = 1, the local dynamical behavior has not been completely established. However, the existence of invariant center-stable and center-unstable manifolds has been investigated completely. In 1977 Hirsch, Pugh, and Shub [11] proved that in the neighborhood of a pseudo hyperbolic fixed point (this covers the first and third cases mentioned above), a C k -system possesses a C k -unstable manifold, establishing the existence of a center-unstable manifold associated with the fixed point. In the case when the map is invertible, their theorem also establishes a center-stable manifold associated with the fixed point. The local behavior in this case has been established for some subclasses; for example in 2010 Kulenović and Merino [16] described the local dynamical behavior in the case when λ 1 < λ 2 = 1 for monotone systems of difference equations. There is no general theory that can be applied to the remaining configurations of λ 1 , λ 2 or for more general classes of functions. Manuscript 3 classifies the local dynamical behavior of near a non-isolated fixed point whose jacobian has both eigenvalues equal to 1.
Even when the local dynamics of a system are understood, as they are in the case when the eigenvalues |λ 1 | = 1 and |λ 2 | = 1, there is still more analysis to be done. In this case, if one of the eigenvalues lies within the unit circle in the complex plane, by the discrete version of the Hartman-Grobman Theorem [6] we know that the behavior of the map is conjugate to its linearization. In particular, there exists a stable manifold on which orbits under forward iteration of the map will converge to the fixed point. If we consider a solution that converges to the fixed point, the question still remains, How close can orbits of the linearization be to orbits from the original map? This question is important in practice because if you approximate a nonlinear map with its linearization in a neighborhood of the fixed point, it is imperative to understand how much error you are introducing by making the approximation. There have been several papers [17] - [20] studying error estimates between a map and its linearization that have been published in the last 15 years. Manuscript 2 gives a improved bound on the error.
To conclude the introduction, we review some basic notation and definitions [5] Y. A. Kuznetsov

Abstract
We give asymptotic results for convergent solutions {x n } of (real or complex) difference equations x n+1 = J x n + f n (x n ), where x n is a m-vector, J is a constant m × m matrix and f n (y) is a vector valued function which is continuous in y for fixed n, and where f n (y) is small in a sense. In addition, we obtain asymptotic results for solutions {x n } of the Poincaré difference equation

1). An application and examples
illustrate the results.

Introduction
Consider the difference equation where x n is an m-vector, J is a constant m × m matrix and f n (y) is a vector valued function which is continuous in y for fixed n, and where f n (y) is small in some sense as (n, y ) → (∞, 0). Equation (2) (2) is very general and it includes as a particular case the (matrix) Poincaré equation where x n is an m-vector, and A, B n are an m × m matrices for n = 1, 2, . . . such that B n → 0.
Equation (2) has been studied by O. Perron, C. V. Coffman, and others [1] [2], who obtained asymptotic behavior results for solutions {x n } of (2). Coffman's Theorem 5.1 in [2] is a refinement of results of Perron [1], and it states that if f n (y) / y → 0 as (n, y) → (∞, 0), solutions x = x n to (2) that converge to zero either have x n = 0 for large n or satisfy where λ is an eigenvalue of J. Although not stated explicitly, Coffman further established (Theorems 5.1 and 8.1 of [2] ) that 0 < |λ| < 1 implies the existence of y ∈ C d and κ ∈ N for which the following asymptotic relation holds: The origin of the study of relation (2) can be traced to Poincaré [3], who investigated the non-autonomous scalar linear difference equation for which the following limits are assumed to exist: Equation (5) has a limiting equation Poincaré proved under the hypothesis that the roots of the characteristic equation of (7) have distinct moduli, that for every solution ζ n of (5) for which ζ n = 0 for all large n, the ratios ζ n+1 /ζ n approach one of the characteristic roots of (7) [3]. Poincaré assumed the coefficients p ,n in (5) to be rational functions of n, but this condition may be dropped if p 0,n is required to be nonzero for all n (see the statement and proof of Theorem 2.13.1 in [6]). Poincaré's result was generalized by Perron [1], and by Gelfond et al [4] to systems (2) and (3) (see [2]).
In [17] R. P. Agarwal and M. Pituk studied the scalar equation (5) if at least one such λ j exists, and q i = 0 otherwise. Define q := max{q i : 1 ≤ i ≤ d}. Then (3) has for n sufficiently large a fundamental matrix satisfying where p can be explicitly estimated.
The formula p = 4t − 3 for the parameter p in (8)  give an asymptotic estimate more precise than (8): the parameter q is removed from the asymptotic formula, and an estimate for the exponent p of the polynomial term is obtained which, in many cases, improves upon the one given in [18]. This estimate is obtained with a method of proof that is different than that in [18]. In addition, the matrix A is not required to be invertible. More precisely, Theorem 2 presented in Section 2.4 gives the following asymptotic expansion for a solution x n of (3) such that lim sup x n 1/n = ρ andỹ is a suitable vector: According to Theorem 2 the parameter β may be chosen in (9) so that β ≤ 3 t − 2, and often β ≤ 3 (t − 1) suffices. Our proof of Theorem 1 gives a procedure for approximating the vectorỹ in (16), see Corollary 2 in Section 2.4.
In order to compare formulas (8) and (9), apply formula (8) to an initial vector x with corresponding solution x n satisfying lim sup x n (1/n) = ρ. Then formula (8) becomes When η < q it can be easily seen by comparing equations (9) and (10) that the asymptotic formula (9) is more accurate than (8). When η ≥ q, the exponential terms in both (9) and (8) are the same. However β ≤ p, and thus (9) is a better asymptotic estimate than (8) or in the worst case, equivalent.
To illustrate the situation outlined in the previous paragraph, consider the Poincaré difference system The matrix A has eigenvalues λ 1 = 1 2 and λ 2 = 1 4 . Suppose that a solution {x n } satisfies lim sup x n (1/n) = 1 2 = ρ (this is the generic case). In the notation of (8) we have η = 1 3 , q = 1 2 , and since q = η, we have p = 0. Theorem BL1 implies that whereas under the same assumptions Theorem 2 states that there exists aỹ such that Note that in relation (11) the parameter β of the asymptotic formula (9) is zero since the condition of Theorem 2 that λ = η ρ for every eigenvalue λ of A is satisfied.
In [18] Bodine and Lutz also obtain an asymptotic relation for convergent sequences {x n } →x generated by iteration of a map T whose jacobian atx is Lipschitz continuous.
T (x) =x. Assume the following: (i) there exists an open convex set Ω containinḡ x such that the jacobian matrix J x exists for x ∈ Ω, (ii) the eigenvalues λ 1 , . . . , λ d of J x (x) do not have modulus 0 or 1, and (iii) J x is Lipschitz continuous in a neighborhood ofx. Then, for any solution {x n } of x n+1 = T (x n ) for n ≥ n 0 such that x n =x for large n and satisfying x n →x, there exists an eigenvalue of Jx with modulus ρ ∈ (0, 1) and corresponding linear combinations of characteristic solutions of the limiting system y n+1 = Jx y n such that where δ is an arbitrary number satisfying δ > max ρ, max Our Corollary 5 presented in Section 2.4 gives the following asymptotic relation whereỹ is a suitable vector and β is a suitable nonnegative integer that can be estimated. Clearly relation (13) is more precise than Bodine-Lutz's relation (12).
The hypotheses of our Corollary 5 require T to be of class C 2 , which is a small strengthening of the requirement in Theorem BL2 that J x be Lipschitz continuous.
The hypothesis on J x in Corollary 5 may be easily modified to match that of Theorem BL2 (see the proof of Theorem 4 in [18] up to (38)), however this is not done here. We also point out that the y n in (13) is not restricted to be a linear combination of characteristic solutions of the limiting system, as it is the case in (12). Finally, Corollary 5 does not require that Jx be invertible.
The rest of this paper is structured as follows. In Section 2.2 we introduce the main hypothesis, and two results of C.V. Coffman are given for easy reference.  [20].

Main hypothesis and results of C. V. Coffman
In this paper we obtain asymptotic results valid for large classes of problems, including the following: (i) Autonomous, nonlinear: The sequence {x n } satisfies x n+1 = T (x n ) for n ≥ 0 and converges to zero. Here T is a map on a subset of C m that is two-times continuously differentiable in a neighborhood of zero, or at least T satisfies where J is a given matrix, and r(x) ≤ c x α for some c > 0, α > 1, and x in some neighborhood of zero.
(ii) Linear, non-autonomous: where B n is a m × m matrix valued function of n ∈ N such that B n converges to the zero matrix at a geometric rate η < 1.
The asymptotic relation (4) was established in [2] under highly technical conditions on the f n (see Theorem B in Section 2), which in particular apply in cases (i) and (ii) above. We shall use a strengthened version of those hypotheses, which while still covering cases (i) and (ii), will be shown in Theorem 1 to be sufficient to obtain a strengthened version of relation (4). The key hypothesis in the main result is the following.
(H) There exist η ∈ (0, 1], α ∈ [1, ∞), c > 0, and a neighborhood V of 0 ∈ C m such that α and η are not both equal to 1 and is a solution to (2) for which x n = 0 for all n large and ρ = lim x n 1/n satisfies 0 < ρ < 1, then there existsỹ ∈ C m and a positive integer β such that Relation (16) is a significant improvement over Bodine and Lutz's formula (8).
Condition (H) is less general than the hypotheses of Coffman's Theorem 8.1 of [2] (see Theorem B in Section 2.2 below), but it is easier to verify, and it is general enough to be applicable to many classes of problems. Furthermore, our proof of Theorem 1 gives a procedure for approximating the vectorỹ in (16); see Corollary 2 in Section 2.4.
For convenience we review some of C. V. Coffman's results from [2]. For the most part, we follow the notation and setup from [2]. Assume that the matrix J in (2) is in block-diagonal form where for 1 ≤ i ≤ R, J i is an m i × m i elementary Jordan block with associated eigenvalue λ i . Assume that the eigenvalues are given in nondecreasing order of magnitude, and let s 1 < s 2 < · · · < s f be the f distinct numbers among the |λ i |. Let some integer t, 1 ≤ t ≤ f be chosen.
An integer j, 1 ≤ j ≤ g will be designated by p = p(t), q = q(t), or r = r(t) For any x ∈ R d denote with x 1 the 1-norm of x, i.e., Thus The following result is Theorem 5.1 in [2] with hypotheses stated explicitly. While the original result is given in terms of · 1 , clearly it can be stated in terms of the Euclidean norm · in R d without any other changes and without affecting the validity of the result.
Theorem A [Coffman] Let f n (y) satisfy Let y = y n be a solution of (2) defined for all sufficiently large n and such that y n = 0 for large n, and y n → 0 as n → ∞. Then there exists an integer t 0 , and The following result is Theorem 8.1 in [2], with hypotheses stated explicitly, and restated here in terms of the Euclidean norm · .
Suppose that φ(n, r) has the form φ(n, r) = ψ(n) r, where or suppose that φ(n, r) satisfies conditions (i), (ii), and (iii): (i) φ(n, r) is nondecreasing in r for each n, (ii) φ(n, r) is nondecreasing in n for each fixed r, Let y = y n be a solution of (2) satisfying (18) and (19). Then for some j 0 , 0 ≤ j 0 < h * and some constants c qk not all 0, y satisfies

A Preliminary Lemma
Denote with σ 1 (L), . . . , σ m (L) the singular values of L ∈ C m×m , in nonincreasing order. Thus σ 1 (L) = L = the operator norm of matrix L. The following lemma is probably known, but the authors of this note did not find a reference for it.

Lemma 1.
Let L ∈ C m×m be similar to an elementary Jordan block with main diagonal entries equal to µ = 0. Then there exist positive constants C 1 and C 2 such that Proof. Suppose L = P −1 J P , where J is an elementary m × m Jordan block with µ = 0 on the diagonal. Then The matrix J n is an upper triangular Toeplitz matrix whose j-th super-diagonal has entries equal to n j µ n−j , j = 1, 2, . . . m − 1. Thus the entries a j on the j-th super-diagonal of the m × m matrix 1 n m−1 | µ | n J n satisfy, for n ≥ j, Relation (25) implies that σ 1 1 n m−1 | µ | n J n is bounded as a function of n ∈ N\{0}. This observation together with (24) guarantee the existence of a constant C 2 > 0 such that The matrix L −1 is similar to an elementary Jordan block with main diagonal entries equal to 1/µ. Apply relation (26) to L −n to obtain that for some positive constant C, By setting C 1 := 1/C in (27) and from the relation 1/σ m (L n ) = σ 1 (L −n ) we obtain which completes the proof of the lemma.

Results
We begin by giving a corollary to Theorem A.
Corollary 1. Let f n : C m → C m be continuous for each n ∈ N be such that (H) is satisfied, and let J ∈ C m×m . If {x n } is a sequence that satisfies x n+1 = J x n + f n (x n ) such that x n = 0 for all large n, then ρ := lim x n 1/n exists and equals the modulus of an eigenvalue of J.
Proof. Let c, V , α and η be as in condition (H). If either α > 1 or η < 1 are such that (15) holds, it follows that hypothesis (17) of Theorem A is satisfied. Thus the conclusion of Theorem A holds.
The following is the central result of this paper.
Theorem 1. Let f n : C m → C m be continuous for each n ∈ N be such that (H) is satisfied, and let J ∈ C m×m . Let {x n } be a sequence that satisfies x n+1 = J x n + f n (x n ) and x n = 0 for all large n, and let ρ be the number guaranteed by Corollary 1.
Let k denote one less than the largest geometric multiplicity of the eigenvalues of J, and let k * denote one less than the largest geometric multiplicity of the eigenvalues of J whose modulus equal ρ. If 0 < ρ < 1, then there existsỹ ∈ R m such that with β = α k * + 2 k + 1, Furthermore, if |λ| = η ρ α for every eigenvalue λ of J, then β = α k * + 2 k is valid in (28).
From (15) it follows that (17) of Theorem A holds, and with φ(n, r) = c ρ n r inequality (20) is valid. Also, the second statement in (21) is equivalent to n h 0 −1 ρ n < ∞ which is a true statement. Thus condition (21) of Theorem B is satisfied. It follows that in all cases considered, the conclusion of Theorem B is true. Let h 0 , κ and j 0 be as in Theorem B. Note that the exponent Thus we have from (23) and (30), By (15) and (31) there exists C > 0 such that f n (x n ) < C n αk * η n ρ αn , n = 1, 2, . . . .
Henceforth we shall assume, without loss of generality, that J is in block-diagonal Jordan block with associated eigenvalue λ i , arranged in order of nonincreasing magnitude. Define i := |λ i | for each i = 1, 2, . . . , R, and define p to be the number of non-zero terms i if one such value exists, else set p to be zero. Then Thus J i is invertible for 1 ≤ i ≤ p and nilpotent for p < i ≤ R. Vectors in x ∈ C m will be partitioned as follows where x (i) consists of the entries of x, corresponding to J i . For each n ≥ 0 let y n ∈ C m be given by To begin we assume i ∈ {1, . . . , p}, so in particular i > 0 and J i is invertible.
From Lemma 1 and the fact that the set of indices i is finite, there exists a constant C > 0 independent of i such that for n ≥ 0, which together with the inequalities m i − 1 ≤ k for i = 1, . . . , p, imply that there exists a constant C 2 > 0 independent of i such that for n ≥ 0, From (32), (34), and (35), for n ≥ 0, we have LetC be an arbitrary positive constant, and β = α k * + k + 1. For n ≥ 0, define n to be the set in C m i whose image under J n i is a ball with center x (i) n and radius C n β η n ρ αn respectively. Then, x (i) n + B(0;C n β η n ρ αn ) = y (i) n + J −n i B(0;C n β η n ρ αn ) .
The smallest singular value of J −n i is σ m i (J −n i ). If follows from (38) that From (39) and from Lemma 1, there exists C 1 > 0 (which due to the fact that the set of indices i is finite may be assumed to be independent of i) such that for n ≥ 0, We shall need the following constants: To prove the claim, fix i ∈ {1, . . . , p} and consider the cases (a) i > η ρ α , n } is a Cauchy sequence and thus converges, say to z i . Furthermore, by (36) we have for n ≥ 1 By (42) and (43) ForC > D, (44) implies Then by (40), (47), and (48), which implies that Then (37) and (49) give (45), which completes the proof of part (a).
To continue with the proof of the theorem, we assume i ∈ {p+1, . . . , R}, so i = 0.
To finish the proof of the theorem, chooseC > D and set where z i for each i is given in Claim 1. Then Claims 1 and 2 imply that J nỹ ∈ B(x n ;C n β (η ρ α ) n ) for n ≥ 1, which in turn implies (28).
If |λ| = η ρ α for every eigenvalue λ of J, then case (c) in the proof of Theorem 1 does not occur, and β in the conclusion of Theorem 1 may be taken to be As a final remark, note that from proof of Claim 1 one can see that the components of the vectorỹ obtained in case (a) are determined uniquely (i.e., From the remark at the end of the proof of Theorem 1 we have the following two results.

Corollary 2. Suppose the matrix J is in Jordan canonical form
Then the entriesỹ (i) ofỹ corresponding to i such that i > η ρ α are uniquely determined, while the entries ofỹ corresponding to i such that i ≤ η ρ α may be set to be any predetermined number. A particular vectorỹ for which the asymptotic formula (28) holds isỹ = (ỹ (1) , · · · ,ỹ (r) ), wherẽ Corollary 3. If every eigenvalue λ of J satisfies |λ| > η ρ α , then the vectorỹ in (28) is uniquely determined by the relatioñ For real difference equations, we have the following result. Proof. The result follows from taking real part of both members of equation (28).
In the case of autonomous difference systems with a solution that converges to a fixed point we have the following corollary.
Corollary 5. Let T be a map on a set R ⊂ R m with a fixed pointx in the interior of R, such that T is of class C 2 on a neighborhood ofx. Let {x n } be such that x n+1 = T (x n ) for n = 0, 1, . . . with x 0 ∈ R. If x n →x, then either x n =x for all n large, or the following statements are true.
(a) There exists ρ with 0 ≤ ρ ≤ 1 such that ρ is the modulus of an eigenvalue of the jacobian matrix J of T atx and such that lim x n −x 1/n = ρ.
(b) Let k denote one less than the largest geometric multiplicity of the eigenvalues of J, and let k * denote one less than the largest geometric multiplicity of λ such that |λ| = ρ. If 0 < ρ < 1, then there existsỹ ∈ R m such that with β = 2 k * + 2 k + 1, If the Taylor expansion of a map at a fixed point contains no nonlinear terms of degree − 1 or smaller, then K(x) ≤ c x −x for some c > 0 and x in a neighborhood ofx. In this case, the asymptotic formula (62) can be made more accurate: For scalar autonomous difference equations an equilibrium pointζ satisfies g(ζ, . . . ,ζ) =ζ. With g , = 1, . . . , m denoting first partial derivatives of g and withz := (ζ, . . . ,ζ), the linearization of (63) about the equilibriumζ is the equation The companion matrix associated with (64) is the matrix We have the following result for solutions to (63) that converge to an equilibrium point.
whereζ is an equilibrium point of (63), and let {ζ n } be a solution of equation (63). If ζ n →ζ, then either ζ n =ζ for all n large, or the following statements are true.
(b) Let J denote the companion matrix associated with the linearization (64) of equation (63). Let k denote one less than the largest geometric multiplicity of the eigenvalues of J, and let k * denote one less than the largest geometric multiplicity of λ such that |λ| = ρ. If 0 < ρ < 1, then there exists a solution {φ n } to the linearized equation (64) such that with β = 2 k * + 2 k + 1, The lim sup rather than just lim in item (a) of Corollary 6 is necessary, as it was observed by M. Pituk in [12], p. 205. A modification of Pituk's example illustrates the point: consider the equation ζ n+1 = 1 2 ζ n−1 , n = 1, 2, . . .. A particular solution that converges to the equilibriumζ = 0 is {ζ n }, where ζ n = ( 1 2 ) n for n even and ζ n = 0 for n odd. Thus lim sup |ζ n | 1/n = 1 2 , while lim |ζ n | 1/n does not exist.
The next result gives an asymptotic expression valid for any solution of a Poincaré difference system. Part (a) of Theorem 2 below is Theorem 1 in [12], and it is included here to have a more comprehensive statement.
Theorem 2. Let A ∈ C m×m , and let B n ∈ C m×m for n ∈ N, η ∈ (0, 1) and c > 0 be such that Let {x n } ⊂ C m be a solution to the Poincaré difference system such that x n = 0 for n sufficiently large. Let k denote one less than the largest geometric multiplicity of the eigenvalues of A. Then the following statements are true.
(a) The limit ρ = lim x n 1/n exists and equals the modulus of an eigenvalue of A.
Proof. Following Perron (as mentioned in [12], page 206), choose a number µ > A and set J := µ −1 A. Then J has eigenvalues µ −1 λ , where λ is an eigenvalue of A. In particular, J is a contraction. Set y n = µ −n x n , and for y ∈ C m and n ≥ 0 set f n (y) := µ −1 B n y. Thus y n = 0 for n sufficiently large. Equation (67) becomes y n+1 = J y n + f n (y n ), n = 0, 1, . . . .
Thus (15) holds with α = 1. By part (a) of Theorem 1, the limit ρ := lim y n 1/n = lim µ −n x n 1/n exists, it is equal to the modulus of an eigenvalue of J, andρ < 1 since J is a contraction. Thus ρ := lim x n 1/n = µρ is equal to the modulus of an eigenvalue of A. Also, y n+1 ≤ ( J + c µ −1 η n ) y n for n = 1, 2, . . ., and since J < 1 we have y n → 0 as n → ∞. If ρ = 0 is assumed, then 0 <ρ < 1, and by part (b) of Theorem 1 there exists ofỹ ∈ C m such that Relation (68) follows from substituting y n = µ −n x n and J n = µ −n A n in (69).
Since any m-th order scalar difference equation may be formulated as a mvector first order equation (p. 117 in [5]), Theorem 2 has the following corollary about the scalar Poincaré equation.
Let k denote one less than the largest geometric multiplicity of the eigenvalues of the companion matrix J associated with the linearized equation about the origin.
Proof. The condition q 1 = 0 implies that zero is not a characteristic root of the linearized equation. Thus ρ = 0 in Theorem 2. The result follows.

An Application
In this section we present results for smooth difference systems in the plane or second order scalar difference equations and an example. If the characteristic roots at an equilibrium of a system in the plane are a pair of complex conjugate numbers, then there is only one rate at which solutions approach such equilibrium, namely the modulus ρ of the roots. Furthermore, since the eigenvalues of the jacobian matrix J at the equilibrium are complex and have geometric multiplicity one, we have k * = k = 0 in Corollaries 5 (b) and 6 (b). Also, under the given setup there does not exist an eigenvalue whose modulus is ρ 2 , thus β = 0 is valid in the asymptotic relation. These considerations provide a justification for the following two corollaries.
Corollary 8. Let T be a map on a set R ⊂ R 2 with a fixed pointx in the interior of R such that T is of class C 2 on a neighborhood ofx. Let {x n } be such that Suppose the eigenvalues of the jacobian matrix J of T atx are a complex conjugate pair with modulus ρ. If x n →x, then either x n =x for all n large, or the following statements are true.
Letζ be an equilibrium point, and assume that g is of class C 2 in a neighborhood of (ζ,ζ). Suppose the roots of the characteristic equation of the associated linearized equation atζ are complex conjugate numbers with modulus ρ. If ζ n →ζ, then either ζ n =ζ for all n large, or 0 ≤ ρ ≤ 1 and lim sup |ζ n −ζ| 1/n = ρ .
In the latter case, if 0 < ρ < 1 then the limit exists, and the solution {ψ n } to the linearized equation with initial values given by Example 1: The difference equation has been studied in [27], where it was shown that for p > q > 0, the unique positive equilibrium is globally asymptotically stable on the positive quadrant.
To simplify calculations, we follow [27] and introduce a transformation in terms of new parameters u > 1 and α > 0. Substitute The equilibrium of equation (73) is preciselyφ = u.
We now verify that the only solution {φ n } for which φ n = u for all large n is the equilibrium solution, i.e., that for which φ 0 = φ 1 = u. Let N be the first The linearized equation of (73) at the equilibrium u is The characteristic roots of the linearized equation at the equilibrium are the complex conjugate pair Let J be the jacobian matrix of the map associated with equation (73), at the equilibrium. Then J and its inverse are given by By Corollaries 3 and 9, relation (75), and the Main Theorem in [27] we have the following result.

The equation is
where ψ n is a precisely the solution to the linearized equation (74) for which See Fig. 1.

Numerical Examples
In In the examples presented in this section, all calculations were performed with extended precision arithmetic using 200 decimal places of precision.  Figure 2 (b) is a log-plot of the relative error J nỹ − x n / x n between the linear and non-linear iterations as a function of n. From this plot it can be seen that as n increases, the error is diminishing quickly relative to the norm of x n .  shows that x n 1/n → 0.25 = ρ. Figure 4 (b) shows that the relative error between the linear and non-linear iterates becomes small quickly.
As in Example 2, the structure of J implies that k * = 1 and k = 1, and f (x) is again such that α = 2. In contrast to Example 2, here 2 = ρ α = (0.25) 2 = 0.0625.   As a numerical example, we set B n = (0.1) n U n , where the matrices U n were chosen with entries of the form i 100 with i a randomly generated integer chosen uniformly between −100 and 100. If the initial vector is x 0 = (−0.62, 0.14, −0.84), then lim n→∞ x n = 0, andỹ ≈ (0.3307208470, −0.5738212397, −0.6766558855) T . Figure   6 (a) illustrates the relation lim x n 1/n = ρ = 0.25. One can see from the structure where β = 3k + 1 = 4. Figure 7 shows plots of J nỹ − x n /(n i (.025) n ) for various values of i as a function of the number of iterations n. As predicted, the plot corresponding to i = 4 appears to be bounded.
Here η = 0.1, ρ = 0.25, α = 1, and since η ρ α = 0.1·0.25 = 0.025, by Corollary 2 a vectorỹ exists and is unique so that Acknowledgment. The authors are grateful to an anonymous referee, who offered valuable suggestions for the improvement of this paper.

Introduction
Planar maps with a curve of fixed points may have fixed points exhibiting 1-1 resonance. The main purpose of this paper is to give a complete classification of all possible dynamical behavior scenarios valid in a neighborhood of such points for maps that are real analytic.
Since the early 1990s there has been a large amount of activity in the study of difference equations in general, especially in rational difference equations and monotone difference equations. A considerable amount of work has been done in these areas by many authors, see Elaydi [12], Smith [8], Agarwal [13], Ladas [15], Kulenović [16] and references cited therein.
The study of the dynamical behavior near an equilibrium point of a planar difference equation or system (i.e. a fixed point of the associated map) is separated into cases depending on whether the fixed point is hyperbolic or non-hyperbolic.
It is assumed from now on that the map in question is at least continuously differentiable on a planar domain. In the hyperbolic case, the fixed point is either a sink, a source, or a saddle, and local dynamics are well understood [14]. In from Kolmogorov-Arnold-Moser (KAM) theory can be used to establish the local dynamical behavior; see [2] for an overview or [3] for an application of the theory.
These cases where KAM theory cannot be applied are called strong resonances (see [5] p.396.) In this paper, we are interested in a special case of a strong resonance, called 1-1 resonance. A fixed point of a planar map is said to be 1-1 resonant if the jacobian of the map at the fixed point is similar to ( 1 1 0 1 ). The local dynamical behavior near a 1-1 resonant fixed point is not yet resolved.
In the case when one characteristic value is not on the unit circle and the second one is ±1, the local dynamical behavior has not been completely established.
However, the existence of invariant center-stable and center-unstable manifolds has been established. If one is able to find center manifolds associated with a In 1977 Hirsch, Pugh, and Shub [8] proved that in the neighborhood of an isolated fixed point whose jacobian matrix has at least one eigenvalue outside the unit circle, a C k -system possesses a C k -unstable manifold, establishing the existence of a center-unstable manifold associated with the fixed point. In the case when the map is invertible, their theorem also establishes a center-stable manifold associated with the fixed point. The local behavior in this case has been established for some subclasses; for example Kulenović and Merino [4] described the local and global dynamical behavior in the case when |λ 1 | < λ 2 = 1 for competitive and cooperative systems of difference equations. Unfortunately, any general procedure to identify a center manifold, for example the Lyapunov-Perron method or the Hadamard Graph Transform method (see [9] for an overview), require that the fixed point has at least one hyperbolic eigenvalue. There is not a general theory that can be applied to the remaining configurations of λ 1 , λ 2 or for more general classes of functions. A 1-1 resonant fixed point has only non-hyperbolic eigenvalues, so finding center manifolds in this case is problematic.
We shall be concerned with the case of non-isolated fixed points, which has not been considered so far in the literature. Then it is clear that (1, 0) T or (0, 1) T is an eigenvector with eigenvalue 1.
To illustrate the situation when a system has a non-isolated 1-1 resonant fixed point, consider the following example: Global behavior of solutions was not completely determined in [10], [11]. With tools developed later [4], it can be shown that each of these fixed points other than (0, 1 − a) has, respectively, an associated local stable, respectively unstable, paper describes the local qualitative behavior for the system studied in [10], [15], and any other system containing a non-isolated 1-1 resonant fixed point.
A continuous analog to 1-1 resonance for ordinary differential equations has been studied in the 1950s and 1960s by N. A. Gubar [1] (see also [16] for an English translation). In [1], a classification of the local dynamical behavior of a system of planar differential equations near an isolated fixed point whose jacobian was similar to ( 0 1 0 0 ) was made under the assumption that the planar system is real analytic and that the fixed point is isolated (this case corresponds to 1-1 resonance in the difference equations context). This suggests that a classification would be possible for a 1-1 resonant fixed point of an analytic system of planar difference equations.
Computer simulations suggest that the criteria for the classification made in the differential equations setting are similar to the those needed for the classification in the difference equations setting. In the continuous case, the classification for isolated 1-1 resonant fixed points made by Gubar [1] was obtained by putting the system into a normal form that is particularly helpful. We shall use a normal form analogous to Gubar's to produce a classification of the possible dynamical scenarios near a 1-1 resonant non-isolated fixed point of a map. We shall show in Section 3.2 that every 1-1 resonant fixed point of a real analytic map is conjugate to, and hence can be reduced to, the adaptation of the normal form used by Gubar.
Thus results for the normalized form are valid for any real analytic map with a 1-1 resonant non-isolated fixed point. In order to describe the dynamical behavior, we will introduce in Section 3.3 the notion of a sector, which is also used in [16] in a different form.
Gubar [1] defines systems of differential equations with a 1-1 resonance at the origin to be in normal form to be those of the form for all (x, y) in a neighborhood of the origin, and g(x, y) is a real valued analytic function that satisfies ∂ ∂y g(0, 0) = 0 and g(x, 0) = 0 for (x, 0) in the neighborhood of the origin. Any such real analytic function g(x, y) can be written as where P, Q, and R are real analytic functions such that P (0), Q(0) = 0. The main theorem in [3] makes the classification in terms of the sign of P (x) and parity of k.
Our main result, Theorem 5 given in Section 3.4, is a discrete version of Gubar's classification for differential equations systems. However, the method of proof is completely different for the one used by Gubar.
We now give an overview of the manuscript. In Section 3.2 we introduce normal forms. Conditions are given for the normal form to exist and it is shown how to obtain information relevant to the normal form without obtaining the normal form explicitly. Section 3.3 introduces sectors and sector boundary curves, which we will use to describe dynamical behavior in the statement of the main theorem, Theorem 5. The proof of the main theorem, which relies on Theorems 6 and 7, is given in Section 3.4. The proofs of Theorems 6 and 7 are presented separately from the proof of the main theorem in Sections 3.5 and 3.6.

Curves of Fixed Points and Normal Forms
Near any 1-1 resonant non-isolated fixed point, the set of zeros of a real analytic planar map is a real analytic curve. This is stated formally in the next result. Without loss of generality it may be assumed that (x,ȳ) = (0, 0) and that the jacobian matrix of S at (0, 0) is ( 1 1 0 1 ). Therefore there exist real analytic functions f and g such that with In particular, 1 + f (0,1) (x, y) = 0 for (x, y) close enough to (0, 0). Set Thus every fixed point of S in W must be of the form (x, ξ(x)) for some x near 0. We now show that points of the form (x, ξ(x)) satisfy g(x, ξ(x)) = 0, and consequently they are fixed points of S. Since (0, 0) is a non-isolated fixed point, there exists a sequence (x n , y n ) of fixed points that converges to (0, 0). By the previous discussion, y n = ξ(x n ) for n larger than some N ∈ N, and thus we have g(x n , ξ(x n )) = 0, n ≥ N . Then g(·, ξ(·)) is a real analytic function on I that has a sequence of zeros that accumulates in the interior of I. By Corollary 1.2.6 in [6], g(x, ξ(x)) = 0 for every x ∈ I. Finally, relations (84) and (85) give 1+f (0,1) (0,0) = 0, which is the last statement in the theorem. The definition below is an adaptation of the normal form used by Gubar in [1]. Definition 1. A real analytic map T defined on a neighborhood V ⊂ R 2 of the origin is said to be in normal form if there exist real analytic functions Q and R where either Q(·) is the zero function, or Q(0) = 0 and there exists an integer For (x, y) ∈ V , T (x, y) = (x, y) if and only if y = 0. It should be noted that the normal form T is not unique-for example, if T is a normal form, then T conjugated with a map reflecting over the origin also will be a normal form for a possibly different choice of W . However, the local dynamical behavior of a map S and any of its normal forms will be the same. Theorem 4 shows that every real analytic planar mapping with a 1-1 resonant fixed point has a normal form. Proof. Without loss of generality we assume (x,ȳ) = (0, 0) and that the jacobian matrix of S at (0, 0) is ( 1 1 0 1 ). By Theorem 3 the set of fixed points C 0 near the origin has a real analytic parametrization of the form y = ξ(x) such that ξ(0) = 0 and ξ (0) = 0. The map Θ(x, y) := (x, y − ξ(x)), defined on a neighborhood U of the origin chosen so that Θ(U ) ⊂ W , is real analytic, 1-1, and satisfies Θ(0, 0) = (0, 0) and Θ −1 (x, y) = (x, y + ξ(x)) for (x, y) near the origin.
The conjugate map of S through Θ given by T = Θ −1 SΘ satisfies Since (0, 0) is a 1-1 resonant fixed point, there exist real analytic functions f and g such that with Write (x,ỹ) = T (x, y). By the real analytic character of T and by (87), there exist real analytic functions φ and ψ such that x = x + y + y φ(x, y) y = y + y ψ(x, y) , Set z = y + φ(x, y) y andz =ỹ + φ(x,ỹ)ỹ.
Combine (90) and (91) to obtain x = x + z z = (y + y ψ(x, y)) φ(x, y + y ψ(x, y)) + y + y ψ(x, y) , The function φ satisfies φ(0, 0) = 0, so there exists a neighborhood U ⊂ U of the origin such that The latter relation, the first equation of (91), and the Real Analytic Implicit Function Theorem (Theorem 1.8.3 in [6]) imply that there exist a neighborhood V ⊂ U of the origin and a real analytic function h on V such that y = h(x, z) for (x, z) ∈ V . From substituting y = h(x, z) in (92), the latter may be rewritten as where H(x, z) is a real analytic function on V . Now H(x, 0) = 0 for (x, 0) ∈ V , by the choice of U ⊃ V and by the first equation of (91). Thus there exists H 1 real analytic on V such that Next we verify that H 1 (0, 0) = 1. By (91) and (93), z = 0 if and only if y = 0, so (91), (94) and (95) imply that for (x, z) ∈ V with z = 0, Since the last term in (96)  T (x, y) = ( x + y + y φ(x, y) , y + y ψ(x, y) ) , where φ and ψ are real analytic on U . Then T has a normal form (86) where Q(x) x = ψ(x, 0) for all x close enough to 0.

Sector Boundary Curves and Sectors
We now introduce the concepts of sector boundary curves and sectors. Sector boundary curves are curves that serve as boundaries of regions (sectors) with different dynamical behavior.  a. For every (x, y) ∈ C, S(x, y) ∈ C, S n (x, y) → (x,ȳ), and either for every > 0 there exists (z, w) ∈ B((x, y), ) ∩ V such that S n (z, w) ∈ V for some n > 0, or S n (z, w) → (x,ȳ).
b. For every (x, y) ∈ C, S −1 (x, y) ∈ C, S −n (x, y) → (x,ȳ), and either for every c. For every (x, y) ∈ C, S(x, y) = (x, y), and for every > 0 there exists Thus under iteration of the map, all points on a sector boundary curve march towards the fixed point, or away from it, or are fixed points. In addition, under iteration of the map or its inverse, some points near a point on a sector boundary curve will behave in a different manner than points on the curve. c. a repelling parabolic sector of S relative to (x,ȳ) and V if for every (x, y) ∈ R, there exists n ∈ N such that S (x, y) ∈ R for −∞ < < n, S n+1 (x, y) ∈ See Figure 9 for an illustration. ii. iii.
iv. Figure 9. A neighborhood (shaded region) with two sector boundary curves C 1 and C 2 . The resulting sectors are i. hyperbolic, ii. repelling parabolic, iii. elliptic, and iv. attracting parabolic.

The Classification Theorem
The next theorem is the main result of this paper. It gives a classification of possible dynamic scenarios near (x,ȳ) in terms of the curve of fixed points and of the normal form.
Theorem 5. Let S be a real analytic planar map on an open set W ⊂ R 2 containing a type 1-1 resonant, non-isolated fixed point (x,ȳ). Suppose W is so small that the set of fixed points of S in W is a real analytic curve C 0 , and that there exists a normal form T of S relative to (x,ȳ) and W . Let Q and be associated to T and W as in equation (86).
If Q = 0, then there exists a closed neighborhood V ⊂ W of (x,ȳ) such that V \ C 0 consists of two hyperbolic sectors relative to (x,ȳ) and V . If Q = 0, there exists a closed neighborhood V of (x,ȳ) and a smooth curve C 1 in V with endpoints in ∂V and tangential to C 0 at (x,ȳ), such that V \ (C 0 ∪ C 1 ) consists of four sectors relative to (x,ȳ) and V . Furthermore, (a) If is odd, the four sectors, in either clockwise or counterclockwise orientation, are of elliptic, attracting parabolic, hyperbolic, and repelling parabolic type, respectively. Also, C 1 \ {(x,ȳ)} has two connected components C − 1 and C + 1 that are sector boundary curves such that S −n (x, y) → (x,ȳ) for every (x, y) ∈ C − 1 and S n (x, y) → (x,ȳ) for every (x, y) ∈ C + 1 .
(c) If Q(0) < 0 and is even, all four sectors are of attracting parabolic type, and S n (x, y) → (x,ȳ) for every (x, y) ∈ C 1 .
The four cases in Theorem 5 are illustrated in Figure 10. The map, for a such that 0 < a < 1, is In this example, the y-axis is a curve of fixed points, and the jacobian matrix T (x, y) = x + y − xy + y 2 (a − 1) + y , y − xy 1 + x .

Example 3.
Consider the following map, studied by Brett et al. [15]: 1 0 x y Figure 11. The phase portrait of system (103). The dynamical behavior of (103) is described by Theorem 5 part (c). Every point on the x-axis is a fixed point (blue line). The curves depict the path and direction followed by orbits under forward iteration of (103). The dashed line is the sector boundary curve C.
where a, b, B > 0. In equation (106), the x-axis is a curve of fixed points, and the jacobian at the fixed point (b/B, 0) is Comparing equation (107) with equation (97), we have − By(bx−ay) −b 2 −bBx+aBy = y ψ(x, y) so that Thus Q(0) < 0 and = 1, so by Theorem 5 the local dynamical behavior of (106) is conjugate to the dynamics pictures in Figure 10 (a), see Figure 12.
iii. The set B δ ∩ Q 1 \ C − 1 has two connected components, henceforth denoted by S 1 and S 2 , such that where S 1 is a repelling parabolic sector of T relative to (0, 0) and B δ , and for (x, y) in S 2 , both T n (x, y) and T −n (x, y) eventually leave B δ ∩ Q 1 .
iv. Every nonzero point (x, y) in B δ ∩ Q 4 belongs to the unstable manifold of a fixed point of T .
iii. The set B δ \ C + 1 has two connected components, henceforth denoted by S 1 and S 2 , where S 1 is a repelling parabolic sector of T relative to (0, 0) and B δ , and for (x, y) in S 2 , both T n (x, y) and T −n (x, y) eventually leave B δ ∩ Q 2 .
iv. Every nonzero point (x, y) in B δ ∩ Q 3 belongs to the stable manifold of a fixed point of T on the negative x semi-axis.

Proof of the Classification Theorem
For . . , 4 denote with Q the usual closed quadrants with respect to the origin.
Also, let Let T be a map in normal form, defined on a neighborhood V of the origin.
There are some properties of T that follow immediately from (86). In particular, every point in V of the form (x, 0) is a fixed point of T , and any sufficiently small neighborhood V of the origin has the property (x, y) ∈ V and (x , y ) = T (x, y) =⇒ x < x and y > 0 whenever y > 0, and x > x and y < 0 whenever y < 0.
For small enough δ > 0, T is injective on B δ .
Establishing additional dynamical behavior characteristics of T near the origin takes considerably more work. Next, we will combine Theorems 6 and 7 to see the complete dynamical picture. There are four non-conjugate dynamic scenarios; we will consider each one separately. Finally, we claim that if S 2 and S 2 are defined as in Theorems 6 and 7, then S 2 ∪ S 2 is a hyperbolic sector. By Theorem 7, if (x, y) ∈ S 2 , then there exist indices m, k ∈ N such that T k (x, y) ∈ B δ ∩ Q 1 and T −m (x, y) ∈ B δ ∩ Q 1 . By (110), thus S 1 ∪ S 1 is a hyperbolic sector with respect to the origin and B δ , completing Case 1.  We wish to show the existence of two more repelling parabolic sectors. By Theorem 7, if (x, y) ∈ B δ ∩ Q 2 , then (x, y) lies in the unstable manifold of a fixed point (x, 0) lying on the negative x semi-axis. We claim that these unstable manifolds can be extended to intersect the set B δ ∩ Q 1 . Consider an unstable manifold through the point (0, δ ) for δ > 0. Since (0, δ ) ∈ B δ , then by Theorem  Let T n (x, y) := (x n , y n ). We will reach a contradiction by showing that an orbit remaining in B δ must converge to a point on the x-axis, and then show that no orbit can converge to the x-axis. Choose δ > 0 such that Suppose that (x n , y n ) ∈ B δ for all n ≥ 1. Then by the definition of T , |x n+1 − x n | = |y n | for all n ≥ 1.
and since (x n , y n ) → (x, 0) then by (110) we have additionally that That is, In light of inequalities (116) and (117), the slopes of the line segment between (x n , y n ) and (x, 0) form a decreasing sequence of negative terms, and thus converge to a fixed negative constant m in the extended real line. Further we have Set m := lim y n /(x n −x). Taking the limit of both sides of equation (118) as n → ∞, since R(x n , y n )y n → 0 as (x n , y n ) → (x, 0) we have Applying Proposition 11 together with Corollary 12 and repeating the argument above withT gives that {T −n (x, y)} must eventually leave B δ , completing the case.

Proof of Theorem 6
The proof of the Theorem 6 will be broken into two parts. In the first part, we will show the existence of an invariant curve using the Hadamard graph transform; see [8] for an introduction. Informally, the Hadamard graph transform T # of a transformation T : R 2 → R 2 can be pictured as follows-consider the image of the graph of a function under T . If the image of the function can also be parametrized as a function of x, then let T # (x) be the resulting function. If there exists a function that is a fixed point for T # , then the function must be an invariant function of T .
The normal course of action is to find a Banach space of functions over which T # is a contraction and use the Banach contraction principle to conclude that the space has a unique fixed point under iteration of T # ; see [8] Section 5 for an example.
Proofs like those in [8] rely on one eigenvalue of the jacobian at the fixed point being off the unit circle in order to find a metric for which T # is a contraction; we were not able to use the same method. We shall find a space R δ that has the following monotonicity property : If graph(ψ), graph(φ) ⊂ R δ , then ψ(x) < φ(x), ). This monotonicity property can be used to find a sequence {τ n (x)} generated by iteration of T # which is monotone and bounded, and thus converges pointwise to a function τ (x) whose graph we later show is invariant under T # .
In order to show that τ (x) is continuously differentiable, we find bounds on To begin the proof of Theorem 4, assume Q(0) > 0, and choose arbitrary real numbers α, β that satisfy We will need the following definitions: Definition 5. For a function φ : A → R where A is a set of real numbers, the graph of φ is the set graph(φ) = {(x, φ(x)) : x ∈ A}, and the graph function of φ is the function Γ(φ) given by Γ(φ)(x) = (x, φ(x)) for x ∈ A.

Properties of R δ
In this subsection, we will show that R δ possesses useful properties with respect to T # and the parametrically defined derivative of T # .
Define the north-east partial ordering ne on R 2 by (x, y) ne (z, w) if and only if x ≤ z and w ≤ y, and the south-east partial ordering se on R 2 by (x, y) se (z, w) if and only if x ≤ z and y ≤ w.
Proposition 3 (Properties of the restriction of T to R δ ). For all δ > 0 sufficiently small, the following statements are true. Figure 14. The graph transform T # of T . The lower curve is the image of φ(x) under T , and T # is the parametrization of the lower curve as a function of x. Here, Then g (1,0) (x, y) > 0 and 1 + g (0,1) (x, y) > 0, provided δ > 0 is sufficiently small.
Also, we have for δ > 0 sufficiently small that For such δ, the jacobian matrix of T at (x, y), has positive entries for all (x, y) in R δ \ {(0, 0)}. Then the components of T are increasing in each variable.
Proposition 4 (Properties of T # in R δ ). For all δ > 0 sufficiently small, the following statements are true.
1. T # exists and is well-defined for any φ ∈ W δ .
By the Intermediate Value Theorem and equation (130), there must exist s ∈ [0, t] such that I(s) = s + φ(s) = t. Since I(s) is increasing on [0, δ], the point s is unique in this interval, concluding the proof.
The former case implies that x ≤ z and w ≤ y. By the monotonicity of φ, we have contradicting w ≤ y. Thus it must be the case that (z, w) se (x, y) .
The proof of (3) is similar to the proof of (2) so we skip it.
In order to establish the differentiability of C − 1 , it will be useful to have an expression for the derivative of an image of a function under the graph transformation of T . Define T # (φ)(t) and T # (φ)(t) to be respectively the first and second parametric derivatives of T evaluated at (t, φ(t)). That is, for s + φ(s) = t, and We will later use T # and T # to show that a subsequence of the sequence {τ n (t)} of derivatives of {τ n (t)} converges uniformly as n → ∞. Let us first establish some monotonicity properties for T # : 1. For all σ , τ ∈ W δ and σ , τ ∈ W δ , 2. For all σ , τ ∈ W δ and σ , τ ∈ W δ , 3. If σ ∈ W δ and σ ∈ W δ , then T # (σ)(t) ∈ W δ .
By (151) and the fact that s 1 ≥ s 2 , we have for δ sufficiently small The proof of (2) is similar to the proof of (1) and we skip it.
If σ ∈ W δ and σ ∈ W δ , then By part 2) of this proposition, we have A proof analogous to that of the continuity of T # (σ)(t) shows that T # (σ)(t) is continuous, thus completing the proposition.
The remainder of this subsection will be devoted to showing that there exist sequences of functions which converge to the invariant curve C − 1 , and that they converge in such a way that their limit C − 1 is continuously differentiable. These sequences will be the iterates of the boundaries of W δ under T # , which are defined at each t ∈ [0, δ] as follows: σ 0 (t) := α t +1 and σ n (t) := T # (σ n−1 )(t), n = 1, 2, . . .

Existence of an Invariant Curve
We are now ready to prove the existence of the invariant center-unstable man- Proposition 7. There exists δ > 0 and a unique τ ∈ W δ such that T # (τ ) = τ and Proof: Consider the sequences {σ n } and {τ n }. An induction argument and statement (2) of Proposition 4 imply that σ n (t) < σ n+1 (t) and τ n (t) > τ n+1 (t) for t ∈ (0, δ], n ∈ N. We claim that there exist σ, τ such that (a) σ n → σ and τ n → τ pointwise, and By taking the limit as n → ∞ in (175) one gets T # (σ)(t) = σ(t). Since t was arbitrary, it follows that σ is invariant under T # . One can prove in similar fashion that τ is invariant as well. This concludes the claim.
Since τ n (t) is continuous for all n ∈ N, it follows that ζ(t) is continuous as well.
Therefore σ = τ , concluding the proof of Theorem 6 parts i) and ii).

Behavior off the Invariant Curve
In this subsection, we will establish the behavior of solutions lying in the region {(x, y) ∈ Q 1 : max{x, y} < δ} off of C − 1 . To facilitate the proof, we must break this region into the following three pieces, defined for each δ > 0 fixed: It is clear that A 1 , A 2 , A 3 depend upon δ > 0, but to ease the notation we will suppress this dependence. Proposition 8 below will show that A 3 is a repelling hyperbolic sector with respect to the origin and Proposition 9 will show that A 1 ∪A 2 is a subset of a hyperbolic sector with respect to the origin. Proposition 8. If (x, y) ∈ A 3 , then T −n (x, y) → (x, 0) for somex ∈ (0, δ).
We begin the proof by showing that the preimage of a vertical line segment , so that the endpoints of this line segment are (u 0 , v 0 ) and (x, 0). For Equation (184) implies that T 2 (u, v) = v + g(u, v) is monotone decreasing as u increases along the line v = −u + x. By the invariance of C − 1 and the x-axis under T , we have that T (u 0 , v 0 ) ∈ C − 1 and T (x, 0) = (x, 0) respectively. Thus By the continuity of T 2 and the Intermediate Value Theorem, for all (x, y) ∈ A 3 By (183) we have that T (u, v) = (x, y).
Now that we have established the preimage of the set {(u, v) ∈ A 3 : u = x}, we can use its structure to show that the x-coordinates of a backwards orbit of a point (x, y) ∈ A 3 must converge. Let (x −n , y −n ) := T −n (x, y). Proposition 2 tells us that (x −n , y −n ) ∈ B δ has a unique preimage under T in B δ , thus Equation (186) shows that x −n is a decreasing sequence which is bounded below by 0, so {x −n } converges tox for somex > 0. Turning our attention to the ycoordinate, y −n = y −(n+1) +g(x −(n+1) , y −(n+1) ), and g(x, y) = Q(0)x +x +1 O(1) > 0 for (x, y) ∈ A 3 , so {y n } is also a decreasing sequence bounded below by 0, and thus converges toȳ. The only fixed points in the closure of A 3 lie on the x-axis, so T −n (x, y) → (x, 0), completing the proposition.
Proposition 9. For every (x, y) ∈ A 1 ∪ A 2 , there exists indices m, k such that T n (x, y) ∈ A 1 ∪ A 2 for −m ≤ n ≤ k, and T n (x, y) ∈ A 1 ∪ A 2 for n = −m − 1 or n = k + 1.
Define (x n , y n ) := T n (x, y) for all n ∈ Z. Suppose that (x n , y n ) ∈ A 1 ∪ A 2 .
First, we narrow down the locations where (x n+1 , y n+1 ) may lie. Notice that x n+1 = x n + y n and x n , y n > 0, so (x n+1 , y n+1 ) must lie in either Q 1 or Q 4 . By (186), the unique preimages of points in A 3 that lie in B δ must also lie in A 3 , so (x n+1 , y n+1 )) ∈ A 3 . To see that y n+1 > 0, choose δ sufficiently small so that for x ∈ (0, δ), Equation (187) shows that (x n+1 , y n+1 ) ∈ Q 4 . Now consider the sequence {(x n , y n )}, n ≥ 0. The sequence {x n } is increasing, and since C(x) is increasing on the interval [0, δ], we have that so there must exist an index k such that (x k , y k ) ∈ A 1 ∪ A 2 and x k+1 > δ, that is that (x k+1 , y k+1 ) ∈ A 1 ∪ A 2 , concluding the first part of the proof.
We will now show that there exists an index m such that T −m (x, y) ∈ Q 2 .
Suppose that there exists (x, y) ∈ A 1 ∪ A 2 whose backwards orbit remains in In order to remain in A 1 , (and thus Q 1 ), the sequence {x −n } must converge to somex since {x −n } is decreasing. This can happen only if {y −n } → 0.
The only fixed point in the closure of A 1 ∪ A 2 is the origin, so our problem can be reduced to showing that the backwards orbit of any point in A 1 ∪ A 2 does not converge to the origin.
We will consider two cases. First, suppose that {(x −n , y −n )} ⊂ A 2 for all n ∈ N. If (x, y) ∈ A 2 , then for δ > 0 sufficiently small we have so T is orientation preserving in this region. Choose (z, w) ∈ C such that (z, w) se (x, y). Repeating the same argument as for the uniqueness of τ in Theorem 1, we have 0 < z − x < z −n − x −n , and since z −n → 0, it must be that x −n 0 < 0 for some n 0 ∈ N, completing the first case.
For the second case, suppose there exists an index n 0 such that (x −n 0 , y −n 0 ) ∈ A 1 . If (x, y) ∈ A 1 and δ sufficiently small, Equation (189) implies that It can be easily seen geometrically by considering the concavity of y = x and equation (190) that (x −(n+1) , y −(n+1) ) ∈ A 1 . Thus the position vector angle y −n /x −n is increasing with each n ≥ n 0 . Suppose that {(x −n , y −n )} remains in A 1 . Then it must be that {y −n /x −n } converges, since {y −n /x −n } is a monotone and bounded sequence, necessarily to an m that is strictly greater than 0. Further we have Set m := lim y −n /x −n . Assuming that (x −n , y −n ) → (0, 0) and taking the limit as n → ∞ to both sides, we have to which only m = 0 is a solution, leading to a contradiction. Setting S 1 := A 1 ∩Q δ and S 2 := (A 2 ∪ A 3 ) ∩ Q δ completes the proof of part iii) of Theorem 6.

Behavior in the Fourth Quadrant
In this section, we will establish the dynamical behavior of solutions lying in B δ ∩ Q 4 . To begin, for each δ > 0 define We will construct a region satisfying part iv) of Theorem 6 that lies inside ∇ δ . We wish to first show that orbits lying in ∇ δ will eventually enter Q 3 . Choose δ > 0 so that for all (x, y) ∈ ∇ δ , Q(x)x > 0 and |R(x, y)| < 1, and so that the angle between the two eigenvectors of the jacobian of each of the fixed points (x, 0) ∈ ∇ δ is less than π 4 . This is possible because the eigenvectors of the jacobian of T at (x, 0) are given by 1 0 and and thus the angle between the eigenvectors in (196) θ(x) := arccos satisfies θ → 0 as x → 0. By the continuity θ(x) for x close to 0, we can choose δ so that |θ(x)| < π 4 . Notice that if (x, y) ∈ ∇ δ , by (194) and (195) respectively, we have T 2 (x, y) = y + g(x, y) = y 1 + g(x, y) y < 0, and 0 > y(Q(x)x +1 + (xR(x, y) − 1)y) Next, we must find the regions in which the image of a point (x, y) ∈ ∇ δ may lie. By (198), T (x, y) cannot lie in Q 1 ∪ Q 2 . If T (x, y) ∈ Q 4 \ ∇ δ , then since (x, y) ∈ ∇ δ , so that equation (200), (x, y) ∈ ∇ δ and T (x, y) ∈ Q 4 imply that Equation (201) implies that |y| < |g(x, y)|.
Thus it must be that T (x, y) ∈ ∇ δ ∪ Q 3 . Now, we must rule out the possibility that an orbit remains in ∇ δ . Since T 1 (x, y) > 0 and (x, y) ∈ ∇ δ , then so if (x, y) ∈ ∇ δ , then the angle formed between the origin and (x n+1 , y n+1 ) is less than the angle formed between the origin and (x, y). Suppose that {x n , y n } ⊂ ∇ δ for all n ∈ N, where {x n , y n } := T n (x 0 , y 0 ) for an arbitrary (x 0 , y 0 ) ∈ ∇ δ . Now by (199) and (203), we have that {y n /x n } is a decreasing sequence for all n ∈ N, and thus is either less than −1 for some index or is bounded below by a constant greater than −1. If y k /x k ≤ −1 for some index k, then T 1 (x k , y k ) = x k + y k ≤ 0, which together with equation (198) implies that (x k+1 , y k+1 ) ∈ Q 3 .
If {y n /x n } is bounded below, then {y n /x n } → m for some −1 < m < 0. Since (x n , y n ) ∈ ∇ δ ⊂ Q • 4 , we have x n+1 = x n + y n < x n for all n ∈ N.
If (x n , y n ) ∈ ∇ δ for all n, clearly {x n } is bounded below by 0, and thus converges, say tox. Let > 0 and choose N ∈ N such that for all n ≥ N , we have x n −x < /2. Then Since > 0 was arbitrary, it must be that (x n , y n ) → (x, 0). However, ifx = 0, the sequence {y n /x n } → 0, which contradicts equation (203). Thus {(x n , y n )} → (0, 0). Further we have y n+1 x n+1 = y n + g(x n , y n ) x n + y n = y n x n + g(x n , y n ) x n 1 + y n x n = y n x n + y n x n (Q(x n )x n + R(x n , y n )y n ) Set m := lim y n /x n . Taking the limit of both sides of equation (206) as n → ∞, since Q(x n )x n + R(x n , y n )y n → 0 as (x n , y n ) → (0, 0) we have which has no solutions in (−1, 0), leading to a contradiction. Thus there exists an index k such that (x k , y k ) ∈ Q 3 .
By Theorem A in the appendix, the fixed point (δ, 0) has a local unstable manifold; define U δ to be the subset of the manifold that lies below the x-axis.
The manifold U δ must be tangent to the eigenvector of the jacobian of T at (δ, 0) which has vector angle less than π/4, which implies that U δ ⊂ ∇ δ . By the claim, T (U δ ) ⊂ ∇ δ ∪ Q 3 and T n (U δ ) ∩ Q 3 = ∅ for n sufficiently large. Thus U δ can be extended until it intersects the negative y-axis. Define R U δ to be the region bounded by the x-axis, the y-axis, and U δ .
We claim that if (x, y) ∈ R U δ , then T −1 (x, y) ∈ R U δ . Define (x −n , y −n ) := T −n (x, y) for n ∈ N. Then we have and If (x −n , y −n ) ∈ R U δ ⊂ ∇ δ , then by (194) and (208), we have y −(n+1) < 0. In turn, Consider any path from the origin to (x −n , y −n ) that is a subset of R U δ . Suppose that T −1 (x −n , y −n ) ∈ R U δ . By the discussion in the previous paragraph, the preimage of the path under T must lie in Q 4 , so it must intersect U δ at some point.
This leads to a contradiction, since U δ is invariant under T .
Finally, we have that {(x −n , y −n )} n∈N ⊂ R U δ , so by (209), {x −n } is an increasing sequence which is bounded above by δ. Using arguments analogous to those used in A 3 in the proof of part iii) of Theorem 6 shows that T −n (x, y) → (x, 0) for some 0 <x < δ. Choosing δ > 0 such that B δ ⊂ R U δ completes the proof of part iv) of Theorem 6.
The following corollary describes the behavior of the invariant curve near the origin.

Proof of Theorem 7
To study the left half plane, we will focus on the scenario when Q(0) > 0 and odd or when Q(0) < 0 and even. The other combinations of signs for Q(0) and parity of can be reduced to these cases through reflections about the origin. We will show in subsections 3.6.1 and 3.6.2 that the behavior of T −1 in the left half plane for Q(0) > 0 and odd or Q(0) < 0 and even is conjugate to the behavior of T in the right half plane for Q(0) > 0, which satisfies the hypothesis of Theorem 6.

(213)
SinceT is invertible near the origin, both x and y are functions ofx andỹ, so for a suitable function Φ we have Φ(x,ỹ) = −g(−x, y).
By invertibility ofT and the fact that fixed points ofT are precisely the points on the x-axis, we have y = 0 if and only ifỹ = 0, and in either case x =x.
The second equality in (221) follows from (215) and (218). We now proceed to prove the first equality in (221).
We now claim that there exist˜ an integer greater than or equal to one and a functionQ that is real analytic near 0 ∈ R withQ(0) > 0, such that Φ(x +w,w) =Q(x)x˜ w +R(x,w)w 2 .
Forx near zero andỹ = 0, it must be the case that x is also near zero, and g (0,1) (−x, 0) ≈ 0 follows. Hence from the latter relation and (229)  Corollary 12. Let T andT be as in Proposition 11. Then a conjugacy map Θ for which T −1 = Θ −1T Θ on a neighborhood V of the origin can be chosen to satisfy Proof. The conjugacy map θ is the composition of two mappings-the first mapping is a reflection over the y-axis. Clearly the upper-and lower-half planes and the x-axis are invariant under this reflection. The second mapping is the change of coordinates w = y + Φ(x, y), see equation (217). Since Φ(x, 0) = 0 for all (x, 0) ∈ V and Φ is real analytic, Φ(x, y) = yΦ(x, y) for some real analyticΦ in V , and V can be chosen small enough so that |Φ(x, y)/y| < 1. Thus, if y > 0, w = y + Φ(x, y) = y 1 + Φ(x, y) y > 0 and similarly for y < 0, completing the proof of the corollary.
d. Every nonzero point (x, y) in B δ ∩ Q 4 belongs to the unstable manifold of a fixed point ofT .
Finally, to prove (iv), choose δ > 0 such that θ(B δ ∩ Q 3 ) ⊂ B δ ∩ Q 4 . If (x, y) ∈ B δ ∩Q 3 , then θ(x, y) ∈ B δ ∩Q 4 , so θ(x, y) belongs to the unstable manifold of a fixed point ofT on the positive x semi-axis. By an argument nearly identical to equation (233) we have that every nonzero point (x, y) ∈ B δ ∩ Q 3 belongs to the stable manifold of a fixed point of T on the negative x semi-axis.

Appendix
The following theorem is an adaptation of Theorem 5.1 in [8].
Theorem A. Let S be a real analytic map on a set R ⊂ R 2 with nonempty interior, and let (x,ȳ) be an interior fixed point of S. Let the characteristic values of S at (x,ȳ) be α and β, and let E α be the eigenspace associated to α. If α and β are real numbers and satisfy |α| < 1 ≤ β, then there exists a neighborhood N δ of (x,ȳ) and a C 1 invariant curve C ⊂ N δ that is tangential to E α at (x,ȳ) such that S n (x, y) ∈ N δ and S n (x, y) → (x,ȳ) whenever (x, y) ∈ C. If α and β