Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced mapsin a degenerate case

In this work we consider a class of degenerate 
analytic maps of the form 
\begin{eqnarray*} 
 \left\{ 
\begin{array}{l} 
\bar{x} =x+y^{m}+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\ 
\bar{y}=y+x^{n}+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\ 
\bar{\theta}=\theta+\omega, 
\end{array} 
\right. 
\end{eqnarray*} 
where $mn>1,n\geq m,$ $h_1 \ \mbox{and} \ h_2$ are of order $n+1$ in $z,$ and $\omega=(\omega_1,\omega_2,\ldots,\omega_{d})\in \Bbb{R}^{d}$ is a 
vector of rationally independent frequencies. It is shown that, under 
a generic non-degeneracy condition on $f$, if 
$\omega$ is Diophantine and $\epsilon>0$ is small enough, the map has 
at least one weakly hyperbolic invariant torus.

1. Introduction and main results. The existence and persistence of invariant manifolds are fundamental topics in nonlinear dynamical systems. Geometrically, invariant tori describe the quasiperiodic motions for dynamical systems. Indeed, quasiperiodic forcing is not only a natural extension of periodic forcing, it also occurs in many physically relevant situations as there are many systems subject to external forcing depending on several frequencies. For instance, Harper map with quasiperiodic potential and the quasiperiodically forced Arnold circle map serve as models of quasiperiodic crystals respectively in [2] and [5].

TINGTING ZHANG,ÀNGEL JORBA AND JIANGUO SI
In this paper, we consider a map of the form where T d = R d /Z d , F is analytic with respect to z, θ and , and ω is a vector of rationally independent frequencies. In this case, R 2 is called the fiber space, and F the fiber map. This kind of skew product has been studied in many works [2,3,5,6,9,11] (see also references therein). Many of these works focus on breakdown of invariant tori into strange nonchaotic attractors. In this paper, we study the persistence of invariant tori under perturbation in a degenerate situation: we assume that F (0, θ, 0) = 0 (∀θ ∈ T d ), that D z F (0, θ, 0) does not depend on θ, and that Spec (D z F (0, θ, 0)) = {1}. Degenerate systems appear commonly in Celestial Mechanics [12,13,4,1], and degenerate volume preserving maps are also studied in [14,19]. The setting considered in this paper is the following:   x = x + Ωy m + f 1 (x, y, θ, ) + h 1 (x, y, θ, ), y = y + x n + f 2 (x, y, θ, ) + h 2 (x, y, θ, ), where Ω > 0, mn > 1, n ≥ m and ω = (ω 1 , ω 2 , . . . , ω d ) ∈ R d . Moreover, f i and h i are of the form f 2 (x, y, θ, ) = 0≤i+j≤n f 2ij (θ, )x i y j , h 2 (x, y, θ, ) = i+j≥n+1 h 2ij (θ, )x i y j , (2) with f (0, 0, θ, ) = 0 and h(0, 0, θ, ) = 0, where f = (f 1 , f 2 ) T and h = (h 1 , h 2 ) T . Therefore, if = 0, u(θ) = 0 is a parabolic invariant torus. We say that f are lower order terms, and h higher order terms. Here the minimum order of h 1 is n+1 rather than m + 1, since x in the solution has larger size than y has if n > m and we expect an uniform size of the high order terms to make the results correct. Throughout this paper, and without loss of generality, we assume that n ≥ m. If n < m, an analogous result can be obtained.
We focus on invariant tori with a prescribed vector of fixed frequencies ω. Hence they can be seen as the response of the autonomous system (when = 0) to the effect of the quasiperiodic forcing (when > 0). Analogous situations are discussed in [7,8,10,15,16,17,18] for the elliptic or weakly hyperbolic cases.
Moreover, the results of this paper can be applied to some specific class of degenerate differential equations by means of a suitable Poincaré section. A construction of a skew product transformation derived from a quasiperiodic vector field will be found in Appendix A. In particular, we consider the differential equation ẋ = y m + l 1 (x, y, t) + q 1 (x, y, t), y = x n + l 2 (x, y, t) + q 2 (x, y, t), as an example (see details in Appendix A), where the time dependence is quasiperiodic. As a result, when m = 1, n > 1, the Poincaré map of (4) defined by a complete revolution of one of the angles of the quasiperiodic time-dependence has the following form and when n ≥ m > 1, it has the form where f contains lower order terms and h contains higher order terms. These degenerate maps are the topic of this paper.
The existence of a weakly hyperbolic quasiperiodic solution for the case m = 1 and m, n both odd is mentioned as an open problem in Remark 4 of [16]. In this paper, we provide an answer for this problem. What we do is even more general because m and n can also be even. Hence, it is not just a generalization of results in [15,16,17]. Moreover, we stress that the ideas of the proof are totally different from the proofs in [15,16,17].
Under suitable hypothesis, the results in this paper show that this map has also at least one hyperbolic invariant torus. Therefore, the cases studied in [15] and [16] by Xu are included here. Moreover, m and n are not necessarily odd as they are in [15] and [16].
As it is usual, we need a Diophantine condition to control the small divisors appearing during the KAM iterations.
The following theorems are the main results of this paper. In order to state the theorems simply and clearly, we denote by [f ] the average of f (θ) with respect to θ. if n is odd.
Then there exists a sufficiently small 0 > 0, such that if < 0 then the map (3) has at least one weakly hyperbolic and analytic invariant torus.  (1) and (2) respectively, where i = 1, 2. Moreover, we assume if n is odd.
Then there exists a sufficiently small 0 > 0, such that if < 0 then the map (6) has at least one weakly hyperbolic and analytic invariant torus.

Remark 1.
For concreteness, if m and n are both even, we get two weakly hyperbolic and analytic invariant tori, otherwise, we just get one.

Remark 2.
In the nondegenerate case m = n = 1, h i and f i do not need to be analytic. It is enough if they are of class C k with k ≥ 1. Then, the corresponding invariant torus is also of class C k . The proof can be found in Appendix B. Hence, in this work we mainly discuss the proof of Theorem 1.2.
2. Sketch of the proof. To simplify the notation, we will not write the dependence of f and g on . As we will see, this dependence does not have any impact on the proofs. Hence, we consider the map   x = x + y m + f 1 (x, y, θ) + h 1 (x, y, θ), y = y + x n + f 2 (x, y, θ) + h 2 (x, y, θ), θ = θ + ω, where nm > 1 and n ≥ m. Moreover, f i and h i are as in (1) and (2) skipping , with f (0, 0, θ) = 0 and h(0, 0, θ) = 0. If = 0, the fiber map of (7) has a fixed point at the origin.
It is natural to consider the average map of the fiber map of (7) where [f ] denotes the average of f with respect to θ, which is If = 0, the map (8) has a fixed point at the origin. However, if > 0, the fixed point may split into several fixed points and we want to show that at least one of them is real. These fixed points are roots of the combined equations By rescaling the variables as follows equation (9) becomes where τ = 1 n . Assume thatF (z, τ ) = 0 denotes the combined equations (11) wherẽ z = (x,ỹ) T . It is easy to see thatF is real analytic onz and C 1 on τ. In order to use the Implicit Function Theorem to show the existence of solutions for τ = 0, we assume and Let us considerF (z, 0) = 0, which is Equations (14) have at least a nonzero real root by conditions (12) and (13). Here if n is even we denote by (−[f 200 ])   is regular, the Implicit Function Theorem ensures that, for each τ close enough to 0, there exist a valuez 0 (τ ) such thatF (z 0 (τ ), τ ) = 0 and, moreover, z 0 (τ ) = (x 0 (τ ),ỹ 0 (τ )) = (x 00 ,ỹ 00 ) + O(τ ).

6604
TINGTING ZHANG,ÀNGEL JORBA AND JIANGUO SI the map (7) becomes As τ is a small parameter, if m < n the first part of the map (15) is more important than the second one. Hence, we rescale the variables as follows and where D(δ) = D 0 + B −1 0 C(δ)B 0 and h is of second order in z, and of order 0 in δ. We are going to apply a sequence of transformations such that the final fiber map has a fixed point at the origin. The main idea is based on a KAM iteration. Before starting a KAM iteration, we will simplify the map (17) so that this iteration can be carried out in an easier way. The main idea is to make the size of the independent term g smaller. What we exactly do is to translate the variables by an approximation of the invariant torus. This approximation u(θ, δ) is obtained from the linearization z = (I + δ n−1 D(δ))z + δ n g(θ, δ), θ = θ + ω.
If we look for the quasiperiodic solution u of this system, we face the small divisors |e 2π . As we will see in Lemma 4.3, u is of order δ n when [g] = 0. After applying the transformation we obtain where To have a quadratically convergent scheme, we need to find a new change of variables making δ n Q 1 smaller. To this end, we rewrite the map (19) as Making the change of variables where P is a quasiperiodic matrix determined by the condition the map (19) becomes Now, for the linear part of (21), we have that, as [g] = 0, the invariant torus is not of order δ 2n (for details see Lemma 4.3). Then after two transformations (the first for making the independent term smaller, and the second for making the new term δ 2nQ smaller), we obtain Here we use Lemma 4.2 to show that all the corresponding eigenvalues are real numbers and this allows to control the divisors |e 2π Replacing δ n−1 by η, the map (22) can be written as Now, we split η 3 δ 4 g * 1 (θ, δ) into the two factors η 2 and ηδ 4 g * 1 (θ, δ). Here we use the part η 2 to deal with the η appearing in ηA * 1 (δ), and let the part ηδ 4 g * 1 (θ, δ) be a KAM iteration term which means that in each KAM step we make the size of this term smaller than it is in the above step. We also rewrite η 3 δ 4 Q * 1 (θ, δ) in the same way. Then the map (23) becomes where . Moreover, it is easy to see that Q 1 and g 1 are of order η. We are going to apply a sequence of transformations such that the fiber map of the final map has a fixed point at the origin. We take the map (24) as the initial map in KAM iteration.
In the j-th KAM step, we have a map of the form where Q j ≤ ν j , g j ≤ ν j (ν 1 = η). By transformations of the type (18) and (20), the map (25) becomes where Q j+1 ≤ ν j+1 , g j+1 ≤ ν j+1 . Roughly speaking, we will have ν j+1 ν 2 j (the exact iterative formula of ν j can be found in section 6). In view of ν 1 = η, Q j+1 and g j+1 are of order η 2 j . Then, the scheme will be convergent to a map which has a hyperbolic fixed point at the origin. Therefore, the original map has a hyperbolic invariant torus near the origin, more precisely, near the point 1 m ) which is the main part of a root of the averaged map (8). Moreover, the eigenvalues of I + ηA ∞ are of the form of 1 ± O(η), so we say that the invariant torus is weakly hyperbolic. Furthermore, when m and n are both even, we can get one more weakly hyperbolic invariant torus near the which is also an appropriate root ofF (z, 0) = 0.
3. Notations. For z ∈ R 2 , z denotes the sup norm of z and, if A is a matrix, A is the corresponding sup norm. We denote the complex torus by T d Let us define and we define C ω r,ρ (R 2 × T d , R 2 ) similarly to C ω r,ρ (R 2 × T d , R). If f (z, θ, δ) is analytic with respect to z and θ on D(r, ρ) and continuous with respect to δ on ∆ δ0 , we define f (z, θ, δ) r,ρ,δ0 = sup Given k ∈ N, we denote by C k (T d ρ , R 2 ) the set of C k functions u : T d ρ → C 2 such that u(T d ) ⊂ R 2 , endowed with the norm where · denotes sup norm of D i u(θ). C ω (T d ρ , R 2 ) is defined in the same way, with the sup norm 4. Some technical lemmas. In order to prove the main theorem, we will first give some lemmas. The proof of this Lemma can be found in [8]. Letλ 1 ,λ 2 be the eigenvalues of D + A, and let U 1 , U 2 be the following disks U 1 = {u ∈ C : |u − (λ 1 + a 11 )| < |a 12 |}, By Gerschgorin theorem, if U 1 U 2 = ∅,λ 1 lies in one of the disks, andλ 2 lies in the other disk. This implies thatλ 1 , andλ 2 must be real numbers.
First of all, let us prove that the following bound holds: In view of A ∈ B α (D 0 ), let B be the matrix found in Lemma 4.1. Then the change of variables (u, θ) = (Bv, θ) conjugates the map (28) to v = (I + δ q D)v + δ l f (θ), θ = θ + ω.

4.
D z+z+ h + r+,ρ+ ≤ K (I+δ l P ρ + ) 2 2(1−δ l P ρ + ) , where 0 < (I + δ l P )r + ≤ r * and δ is small enough such that δ l L 2 Q * ρ−σ 1 Proof. Under the transformation H 2 , the fiber map of (30) becomes where we denoteP = P (θ + ω), and it satisfies the homological equation Therefore, where when k = 0. When k = 0, s k ij = 0 (recall that Q * has zero average). Then we have . Now, we are going to estimate the functions in map (32). According to the bound of P ρ+ and if δ is small enough it is easy to see that δ l P ρ+ < 1. Hence we have Therefore, . Then the initial map becomes (31), where A + = A * + δ l−q [Q 2 ], g + = g 2 , h + = h 2 and, moreover, and 5. KAM theorems. To prove Theorem 1.2, we reduce the original map (7) to a map of the form (see the details in Section 6) We are not going to start the KAM iteration from map (33). In the following theorem, we simplify (33) such that these iterations are easier to perform.
As lim j→∞ a j < α/2, the sequence A j converges to A ∞ and A ∞ < D 0 + α.