ALMOST-PERIODIC PERTURBATIONS OF NON-HYPERBOLIC EQUILIBRIUM POINTS VIA P¨OSCHEL-R ¨USSMANN KAM METHOD

. This paper focuses on almost-periodic time-dependent perturbations of a class of almost-periodically forced systems near non-hyperbolic equi- librium points in two cases: (a) elliptic case, (b) degenerate case (including completely degenerate). In elliptic case, it is shown that, under suitable hy- pothesis of analyticity, nonresonance and nondegeneracy with respect to perturbation parameter (cid:15), there exists a Cantor set E ⊂ (0 ,(cid:15) 0 ) of positive Lebesgue measure with suﬃciently small (cid:15) 0 such that for each (cid:15) ∈ E the system has an almost-periodic response solution. In degenerate case, we prove that, ﬁrstly, the almost-periodically perturbed degenerate system in one-dimensional case admits an almost-periodic response solution under nonzero average condition on perturbation and some weak non-resonant condition; Secondly, imposing further restriction on smallness of the perturbation besides nonzero average, we prove the almost-periodically forced degenerate system in n -dimensional case has an almost-periodic response solution under small perturbation with- out any non-resonant condition; Finally, almost-periodic response solution can still be obtained with weakened nonzero average condition by used Herman method but non-resonant condition should be strengthened. Some proofs of main results are based on a modiﬁed P¨oschel-R¨ussmann KAM method, our results show that P¨oschel-R¨ussmann KAM method can be applied to study the existence of almost-periodic solutions for almost-periodically forced non- conservative systems. Our results generalize the works in [14, 13, 23, 20] from quasi-periodic case to almost-periodic case and also give rise to the reducibility of almost-periodic perturbed linear diﬀerential systems.


1.
Introduction. It is well known that the question of the existence of a.p. (almost periodic) solutions of ordinary differential equations is a hard problem. Much literature exists [6] on the a.p. solutions of small perturbed almost-periodically forced systems with hyperbolic linear part, but the real difficulty is non-hyperbolic case, especially the case where all eigenvalues of linear part have zero real part. In this case, there is no quality of exponential dichotomies and center manifold to take for | 1 | and | 2 | near 0, where δ 2 > δ 1 > 0 are constant, they proved that for 0 < 0 1 there exists a Cantor set E ⊂ (0, 0 ) of almost full Lebesgue measure such that for each ∈ E the system (2) has a quasi-periodic solution with the same frequency ω ∈ R d as forcing, called response solution, near equilibrium point O. But as far as we knows, it seems very few results to study existence of response solution of almost-periodically forced systems in elliptic case because it is not easily to deal with 'small divisor' problem for infinite-dimensional frequencies. Thus, the first goal of this paper is to consider the following question.
• Question 1: For almost-periodically forced system (2), can we get the existence of a.p. response solutions with the same frequency ω as forcing ?
In this paper, we first consider n-dimensional systeṁ where E has n distinct pure imaginary eigenvalues λ i 's and λ i = 0 for all 1 ≤ i ≤ n, h = O(x 2 )(x → 0), and f is a small perturbation for small and C 1 -smooth with respect to . Here, f and h are both almost-periodic in t with same frequency ω ∈ R Z and admit the same spatial series expansion, the conception of which was first presented by Pöschel in [15]. In section 3, by Pöschel-Rüssmann KAM method we will prove that system (3) can be reduced to a suitable normal form with zero as equilibrium point by an almost-periodic transformation. Hence, the almostperiodically forced system (3) has an a.p. response solution near the equilibrium point.
Another direction of non-hyperbolic case is to study perturbations of degenerate equilibrium points. In degenerate case, this problem becomes more difficult to handle because the linear term cannot control the shift of equilibrium point very well, and so the standard KAM theory is not directly applicable to this perturbation problem. However, the degenerate systems are also relevant in view of applications to physical real word nonlinear models, such as system (1) with b = 0. In 1998, You ([27]) first considered the hyperbolic-type degenerate Hamiltonian systems where (x, y, u, v) ∈ T n ×R n ×R×R. He proved there is an ω * close to ω such that an n-dimensional invariant torus exits provided that the perturbation P is analytic and small enough. In above work, the Hamiltonian system (4) is partially degenerate. Subsequently, Han, Li and Yi ( [9]) investigated the degenerate Hamiltonian system H = ω, y + 1 2 z, M (ω)z + P (x, y, z, ω), where (x, y, z) ∈ T n × R n × R 2m , ω is an independent parameter varying over a positive set O ⊂ R n and M (ω) can be singular on O. They proved the persistence of lower-dimensional tori in Hamiltonian systems of the form (5). In which, they imposed some restrictions on the perturbation P but their result can be applied to the both hyperbolic and elliptic cases and is suitable for partially or completely degenerate case. In recent years, many authors are devoted to investigate quasi-periodic perturbations of degenerate systems. In 2010 and 2013, Xu ([24,25]) considered the the perturbation of 2-dimensional nonlinear quasi-periodic system with partially degenerate equilibrium point, that is ẋ = y + h 1 (x, y, t) + f 1 (x, y, t), y = x 2m+1 + h 2 (x, y, t) + f 2 (x, y, t), for m = 1 or m > 1, respectively. It was proved that, by a nonlinear quasi-periodic transformation, system (6) can be reduced to a normal form with an equilibrium point at the origin under the Diophantine condition. Hence, a quasi-periodic response solution of the system (6) is obtained. But its method seems not valid to completely degenerate systems, i.e., systems with vanished linear part. The existence of quasi-periodic response solutions for quasi-periodically forced systems with completely degenerate equilibrium point, under small quasi-periodic perturbations, was also investigated recently. In 2016, Zhang, Jorba and Si ( [28]) considered some specific completely degenerate differential system's Poincaré map   x = x + y m + f 1 (θ, x, y, ) + h 1 (θ, x, y, ), y = y + x n + f 2 (θ, x, y, ) + h 2 (θ, x, y, ), θ = θ + ω, where mn > 1, n ≥ m, h 1 and h 2 are higher order terms, f 1 and f 2 are lower order perturbations. It is shown that if f = (f 1 , f 2 ) T satisfies some non-zero average hypotheses and the frequency vector ω is Diophantine, then the above map exists weakly hyperbolic invariant torus. In 2017 and 2018, Si and Si ([18,19]) considered the perturbations of 4-dimensional quasi-periodically forcing systems with completely hyperbolic type degenerate equilibrium point and completely elliptic type degenerate equilibrium point respectively, in which we proved that these two kinds of completely degenerate systems have a quasi-periodic response solution by means of the Pöschel-Rüssmann KAM method.
Moreover, system (7) has strong physical background which can describe oscillations of n weakly connected oscillators (Please see Remark 3.13 in the paper [20]). In [20], we proved that system (7) admits a quasi-periodic response solution if the frequency vector ω satisfies Brjuno-Rüssmann's non-resonant condition. And specially, system (7) in one-dimensional case admits a quasi-periodic response solution under some non-resonant condition weaker than Brjuno-Rüssmann's and even without any non-resonant condition in some special perturbations. However, In almost-periodic case, the existence of a.p. response solutions of system (7) becomes more difficult because it is not only necessary to control the shift of equilibrium point with vanished linear part but also necessary to deal with the 'small divisor' produced by the integer combination of infinite many frequencies. Thus, the second goal of this paper is to consider the following question.
• Question 2: For almost-periodically forced degenerate system (7), can we get the existence of a.p. solutions with the same frequency ω as forcing ?
In order to deal with the 'small divisor' in almost-periodic case, as shown in paper [15] the infinite-dimensional frequencies ω should satisfy the following non-resonant condition where the concrete definition of (8) can be seen later in next section. It can be seen that non-resonant condition (8) plays very important role in the process of KAM iteration. Thus, the third goal of this paper is to consider the following question.
• Question 3: Can almost-periodically forced degenerate system (7) admit a.p. solutions with the same frequency ω as forcing under some non-resonant condition weaker than (8) or without any non-resonant condition ?
We give the answers of Questions 2 and 3 in section 4. We will apply the non-zero average hypothesis on perturbation to prove the almost-periodically forced degenerate system (7) in one-dimensional case admits an a.p. response solution under non-resonant condition weaker than (8). We also prove that the almost-periodically forced degenerate system (7) can admit an a.p. response solution without any nonresonant condition but smallness of perturbation must be restricted beside non-zero average condition. Finally, we prove that non-zero average hypothesis on perturbation can also be weakened under non-resonant condition (8).
In this paper, our main results are Theorem 3.1 and Theorems 4.1-4.3 below. Let us make some comments on the results: 1. A novelty of KAM skill in this paper is using Pöschel-Rüssmann KAM method in almost-periodic case. The Pöschel-Rüssmann KAM method is developed by Rüssmann [17] and Pöschel [16] who considered a class of dynamical systems of polynomial character and a constant vector field on n-torus, respectively. Compared with the traditional KAM iteration, the advantage of Pöschel-Rüssmann KAM method is that this KAM iteration containing an artificial parameter q, 0 < q < 1, makes the steps of KAM iteration infinitely small in the speed of the exponential function q n rather than super exponential function like classical KAM theory, which greatly simplifies the calculation process and as stated in [16], such result may be the shortest complete KAM proof for perturbations of integrable vector fields available.
Our results show that the Pöschel-Rüssmann KAM method can also be applied to study the perturbation problem of linear vector field. And even for the Pöschel-Rüssmann KAM method itself, we also have new improvements. It is worth to mention that we first use Pöschel-Rüssmann KAM method to almost-periodic case. The most important idea of Pöschel-Rüssmann KAM method in quasi-periodic case is sufficiently taking advantage of the polynomial structure in the truncation skill. Different from that, in almost-periodic case, we should find a new polynomial structure in the truncation (see (14) and (15) in the later). Furthermore, non-resonance condition in this paper is Brjuno-Rüssmann type. As far as we know, Xu and You had considered, in an old paper [22], almost periodic reducibility problem via the traditional KAM method and also used similar Brjuno-Rüssmann non-resonance conditions.
2. Our Theorem 3.1 and Theorems 4.1-4.3 are the direct generalization of works in [14,23,20] from quasi-periodic case to almost-periodic case. Actually, using the same method, we believe that the results in literatures [18,19,24,25,28] can also be generalized from quasi-periodic case to almost-periodic case. It is worth pointing out that the proof of Theorem 4.1 in this paper requires only that the frequency vector satisfies some weaker non-resonant conditions in which the approximate function does not necessary satisfies integral condition (see (11)). And the proof of Theorem 4.2 does not require any non-resonant condition on the frequency vector.
3. An important part of this paper is to consider the almost-periodic perturbations of degenerate systems. Since the systems we consider include completely degenerate case, new skill should adopted. A technical breakthrough to overcome completely degeneracy is that we make a translation transformation and suitable scalings for completely degenerate systems on the basis of equilibrium point of the averaged system to reduce the system into a normal forṁ where η is small perturbation parameter and G(θ, x, η) = O(x 2 )(x → 0). Only when l 2 > l 1 and l 3 > 2l 1 hold, can the KAM method be valid. Generally speaking, it is hard to reach the order with l 2 > l 1 and l 3 > 2l 1 unless we impose more restrictions on the perturbation like non-zero average condition and smallness condition (see Theorems 4.1 and 4.2). Herman method is also used in this paper to weaken nonzero average condition (see Theorem 4.3). Herman method is a well-known KAM technique that introduces an artificial external parameter to make the unperturbed system highly non-degenerate. 4. In the last years, many authors have been devoted to studies of quasi-periodic bifurcation for conservative and dissipative dynamical systems and obtained many important results, see [1,2,3,4,5,7,8,20,21] and references therein. We think it is also important and interesting to study almost-periodic degenerate bifurcations for almost-periodically forced systems. We believe that the method used in the present paper can also be used to consider almost-periodic degenerate bifurcations for almost-periodically forced conservative and dissipative systems. This is one of the subjects of future work.
5. It worth to mention that almost-periodic solution in asymmetric oscillation and invariant curves of smooth quasi-periodic mappings have already been obtained in work [10,11] by the authors P. Huang, X. Li and B. Liu.
The rest of this paper is organized as follows. In Section 2, we introduce some notations and definitions applied in the sequent. In Section 3, we give the existence of a.p. response solutions in elliptic case. In Section 4, we give the existence of a.p. response solutions in degenerate case.

2.
Preliminaries. In this section, we first introduce some notations and definitions. The first thing is to define analytic functions on some infinite dimensional space where U is an open subset of X, is called analytic if f is continuous on U , and f | U ∩X1 is analytic in the classical sense as a function of several complex variables for each finite dimensional subspace X 1 of X.
Definition 2.2. A function f : R → R is called almost periodic with the frequency ω = (· · · , ω λ , · · · ) λ∈Z ∈ R Z , if there exists a continuous function which admits a rapidly converging Fourier series expansion and S is a family of finite subsets A of Z with Z ⊆ A∈S A, k, θ = λ∈Z k λ θ λ , such that f (t) = F (ωt) for all t ∈ R, where F is 2π-periodic in each variable. If F (θ) is analytic on T Z s := {θ = (· · · , θ λ , · · · ) ∈ C Z /(2πZ) Z , sup λ∈Z |Imθ λ | ≤ s} for some s > 0, then we call f (t) is analytic on T Z s Here F (θ) is the shell function of f (t). We remark that this family S is not totally arbitrary. It is because for an almost-periodic function f (t) which has the Fourier exponents {Λ λ : λ ∈ Z}, its basis is {ω λ : λ ∈ Z}. For any λ ∈ Z, Λ λ can be expressed into Λ λ = r λ1 ω λ1 + · · · + r λ j(λ) ω λ j(λ) , where r λ1 , · · · , r λ j(λ) are rational numbers. Therefore, S = {(λ 1 , · · · , λ j(λ) ) : λ ∈ Z}. Rather, S has to be a spatial structure on Z characterized by the property that the union of any two sets in S is again in S, if they intersect: Further, we define Z Z S = {k = (· · · , k λ , · · · ) ∈ Z Z : suppk ⊂ A, A ∈ S}. Thus, f (t) can be represented as a Fourier series Define a nonnegative weight function where > 2 is a constant.
Let M be a compact set in C without zero and B r := {z = (z 1 , . . . , z n ) ∈ C n | |z| ≤ r} be the n-dimensional sphere with radius r in C n . Define ∆ s,r = T Z s × B r for each given r > 0 and s > 0. For an analytic function F (θ, x, ξ) : ∆ s,r × M → C, which admits the following spatial series expansion we define the norm of F (θ, x, ξ) by where |f A,k (x, ξ)| r,M = sup (x,ξ)∈Br×M |f A,k (x, ξ)|. Thus, for all above bounded analytic functions, we can define a Banach space C ω m (∆ s,r × M, C) = {F : F s,r,m,M < ∞}. Similarly, we can denote C ω m (∆ s,r , C), C ω m (T Z s × M, C) and C ω m (T Z s , C) the space of analytic function F : ∆ s,r → C, the space of analytic function F : T Z s × M → C and the space of analytic function F : T Z s → C, respectively. Their norms are represented by · s,r,m , · s,m,M and · s,m respectively. Let C ω m (∆ r,s × M, gl(n, C)) and C ω m (∆ s,r × M, C n ) be the set of all analytic n × n matrix functions and ndimensional vector functions mapping from ∆ s,r ×M to gl(n, C) and C n respectively. For each F = (F 1 , . . . , F n ) ∈ C ω m (∆ s,r × M, C n ), we define its norm F s,r,m,M = 1≤i≤n F i s,r,m,M . For each F = (F i,j ) 1≤i,j≤n ∈ C ω m (∆ s,r × M, gl(n, C)), we define its norm F s,r,m,M = 1≤i≤n sup 1≤j≤n F i,j s,r,m,M . Similarly, we can consider C ω m (∆ s,r , gl(n, C)), C ω m (∆ s,r , C n ), C ω m (T Z s × M, gl(n, C)), C ω m (T Z s × M, C n ), C ω m (T Z s , gl(n, C)) and C ω m (T Z s , C n ) in the same way. A function f ∈ C ω m (∆ s,r × M, C) is called a real analytic function if it gives real values to real arguments. Denote by C ω m (∆ s,r × M, R) the set of all such real analytic functions in C ω m (∆ s,r × M, C). Then C ω m (∆ s,r × M, R) is a subspace of C ω m (∆ s,r ×M, C) under · s,r,m,M . Similarly, we can consider C ω m (∆ s,r , R), C ω m (T Z s × M, R), C ω m (T Z s , R), C ω m (∆ r,s × M, gl(n, R)), C ω m (∆ s,r × M, R n ), C ω m (∆ s,r , gl(n, R)), C ω m (∆ s,r , R n ), C ω m (T Z s × M, gl(n, R)), C ω m (T Z s × M, R n ), C ω m (T Z s , gl(n, R)) and C ω m ( T Z s , R n ).

WEN SI, FENFEN WANG AND JIANGUO SI
represents the average of quasi-periodic function F A (θ).
Next, we will define the strongly non-resonant condition on the frequency vector ω = (· · · , ω λ , · · · ) λ∈Z . For k ∈ Z Z S , we define the weight of its support Then the non-resonant condition reads where γ > 0, |k| = λ |k λ | and ∆ is some fixed approximation function. A function ∆ : [1, ∞) → [1, ∞) is called an approximation function, if ∆ is non-decreasing, ∆(1) = 1 and In the following we will give a criterion for the existence of strongly nonresonant frequencies. It is based on the growth conditions on the distribution function 3. There exist a constant N 0 and an approximation function Φ such that The proof of this Lemma can be found in [15]. We omit it here. Throughout this paper, we denote by c, C the universal positive constants if we do not care their values, denote the absolute value (or norm of vector, or norm of matrix) by | · |. In the sequel, we still denote the shell of an almost-periodic function h(t) by h(ωt), for the sake of simplicity.
3. Elliptic case. In this section, we will be devoted to the construction of a.p. response solutions of system (3) by adapting Pöschel-Rüssmann KAM method. To this end, we introduce the extended phase space T Z × R n , and so system (3) can be written as θ = ω, where Q(θ, ) = ∂f (θ,0, ) ∂x and F (θ, ) = f (θ, 0, ) and G(θ, x, x, which are analytic on ∆ s,r for some positive numbers s, r and admit spatial series expansions Since there exists an invertible matrix B such that B −1 EB is diagonal in complex field, we can assume that, without loss of generality, E = diag(λ 1 , λ 2 , . . . , λ n ). Moreover, we writeλ i ( ), i = 1, · · · , n, as the eigenvalues of wherex(t, ) is the a.p.solution of the equationẋ = Ex + F (t, ), existence ofx(t, ) can be obtained by solving the first homological equation in the later (see (21)), the definition of P (·) can be seen in (14) and the definition of P (·) can be seen in (13).
We have the following theorem.
Theorem 3.1. Consider analytic system (12) defined in ∆ s,r . Assume that the following hypotheses hold: where δ > 0. Then, for sufficiently small constant¯ > 0 there exist a Cantor set E ⊂ (0,¯ ) of almost full Lebesgue measure and s * < s, r * < r such that for each ∈ E there exists a analytic transformation and C 1 -smooth with respect to , which transforms system (12) intȯ Remark 1. Linear non-autonomous differential systemṡ is called reducible, if there exists a non-singular change of variables x = Φ(t)y, for which Φ(t) and Φ(t) −1 are bounded, such that system is transformed toẏ = By, where B is a constant matrix. The reducibility of linear differential systems has been studied widely by many scholars. The earliest result in this field is the wellknown Floquet theory ( [6]), which states that any linear periodic system is always reducible. Reducibility for quasi-periodic differential systems can also be obtained not for all A(t) (see [12,13]). In 1996, reducibility for almost-periodic differential systems was obtained by Xu and You in [22]. Actually, if G and F in system (12) are both zero, then the system (12) can be written as follows, Then, our Theorem 3.1 can reduce to Theorem A shown in the paper [22].
3.1.1. Outline of our proof. We will adapt the Pöschel-Rüssmann KAM method (see [16] and [17]), which was used in the perturbation problem of a constant vector field on n-torus, to prove Theorem 3.1. Our the result shows that the Pöschel-Rüssmann KAM method is also feasible for the perturbation problem of linear vector field with elliptic equilibrium point.
Before stating and proving the step lemma accurately, we first present a quick overview describing how Pöschel-Rüssmann KAM method works in the proof.
For any matrix function The most important idea of Pöschel-Rüssmann KAM method in quasi-periodic case is sufficiently taking advantage of the polynomial structure in the truncation skill. Different from that, in almost-periodic case, we should find a new polynomial structure in the truncation. Let P (θ) ∈ C ω m (T Z s , C) admits a spatial series expansion For e −τ σ = 1 − a, we can truncate P (θ) = P (θ) + P (θ) with and We have the following estimate because e −τ σ = 1 − a and function (1 − (1 − a)e tσ ) under the sup is monotonically decreasing for 0 ≤ t ≤ τ and equals a at t = 0. On the other hand, the polynomial rest P (θ) is bounded on a larger domain. Indeed, as the function (1 − (1 − a)e tσ ) under the sup is monotonically decreasing for 0 ≤ t ≤ τ and equals a at t = 0.
In this outline we hide in the system for simplicity. In order to get the reducibility of the systeṁ where , we first make the splitting Q = Q + Q and F = F + F respectively, and we will try to eliminate the term F and Q by means of almost-periodic change of variables and put the term F and Q into the new perturbation term. To this end, we must consider the following linear equations where H(θ) = Q(θ) + ∂G ∂z (θ, V (θ)) and ∂ ω := λ∈Z ω λ ∂ ∂θ λ . The solvability of this equations needs the non-resonant conditions where λ i (i = 1, 2, . . . , n) are the eigenvalues of E andλ i (i = 1, 2, . . . , n) are the eigenvalues of E+ Q(θ)+ ∂G ∂z (θ, V (θ)). In the next subsubsection, we will know above equations are solvable and . Now we can perform the change of variables z = V (θ) + (I + U (θ))z + to (18) to obtainż

WEN SI, FENFEN WANG AND JIANGUO SI
One can check that the term Q(θ)V (θ) + G(θ, V (θ)) in F + (θ) is much smaller than F (θ), whose order is O( 2 ). And the term F (θ) can also be smaller than F (θ), whose order is O(2(1 − a) ) (see (16)). Finally, Here a, b, q are defined as in next subsection. One can also check . And the term Q(θ) can also be smaller than Q(θ), whose order is O(2(1 − a) ) (see (16)). Finally, In this way, after l steps, system will look likė . The scheme will be convergent to an equation likė In order to guarantee that the non-degeneracy conditions can be persist at each KAM step, we estimate the derivative of each functions. Actually by 2 ). Thus, the choice of q should be small suitably (see (19) below). Different from this proof of Theorem 3.1, the choice of parameter q in the proof of Theorem 4.3 can be arbitrary value in (0, 1) because in Theorem 4.3 we do not need the persistence of non-degeneracy conditions.

The KAM
Step. We will give an iterative lemma to prove Theorem 3.1. Let ∆ be an approximation function as defined in (11). Denote Λ(t) = t 2 ∆ 12 (t). we can find two constants c 1 and C 1 such that We choose δ, a, b and q such that δ = q One can check that Finally, we choose 0 =¯ small sufficiently, where¯ only relies on the constants mentioned above. Next, we define the sequences and (γ ν ) ν≥0 in the following manner: We will see that m ν , r ν and s ν have a positive limit. Indeed, for N > τ 0 It follows that Hence, we can achieve that ν≥0 We suppose that after ν steps, the transformed system defined in the domain ∆ sν ,rν is of the forṁ Suppose that there is a γ > 0 for which the frequency vector ω satisfies non-resonant condition (10). Let us consider the analytic almost-periodic system and G l (z, θ, ) are C 1 -smooth with respect to and satisfy the following hypotheses: Then there exists a change of variables where z are 'old' variable and z + are 'new' variable, U ν (θ) ∈ C ω mν (T Z sν , gl(n, C)) V ν (θ) ∈ C ω mν (T Z sν , C n ), such that the system (Eq) ν is transformed into the system (Eq) ν+1 and conditions (l.1) − (l.3) are fulfilled by replacing l by ν + 1 and replacing z by z + , respectively. Proof of Lemma 3.2. In this proof, we hide if there is no confusion. We divide the proof of Lemma 3.2 into the following several parts.
One can also check that B. Construction of a transformation.
First of all, we solve the first equation of (21). For sufficiently small * > 0, we define two sets by We solve the first equation of (21).

WEN SI, FENFEN WANG AND JIANGUO SI
for each i, if q < min{ }. In the same way, we can also get c 1 < This proves assumption (l.1) holds true with l = ν + 1.
Next, we solve the second equation of (21). Define set by By the second equation of (21), we can obtain where [H ii ν (θ)] = 0. Thus, (23) is solvable. Similar to first equation of (21), if ∈ D 2 ν , we can get , for 1 ≤ i, j ≤ n. This implies that U ν (θ) ∈ C ω (T d sν , gl(2n, C)) is the solution of second equation of (21) and C 1 -smooth with respect to .
Finally, choosing C 2 > max{ 2 c1γ , 4C1 D. Estimates of perturbation terms. By the third and fourth inequalities of (24) we have as δ ≤ 1/32. Thus, according to (24) and (25) we have the following several estimates: provided that 0 is small enough. This proves assumption (l.2) holds true with l = ν + 1.
Similarity, in view of (22) we also show that E ν converges to a matrix E * . Then, (Eq) 0 is changed to where G * = O(z 2 )(z → 0). It is obvious that (29) possesses an invariant torus It follows from (24), that 3.1.4. Measure estimate. In this section, we will estimate the measure of E which will defined later. We first give two Lemmas which can be found in [15].
where N 0 , t i are given in Lemma 2.3.
Note that This means that E has relative positive measure. The proof of Theorem 3.1 is complete.
4. Degenerate case. In this section, we will consider the existence of a.p. response solutions for the perturbation system of a n-dimensional almost-periodically forced system with degenerate equilibrium point, the n-dimensional system (7). If we introduce the extended phase space T Z × R n , where writing θ = (. . . , θ λ , . . .) = (. . . , ω λ t, . . .) for the toral variable yields the desired time-dependence, then the following system is equivalent to system (7) where φ(x) = (a 1 x l1 1 , · · · , a n−1 x ln−1 n−1 , a n x ln n ) T is a vector field that is comprised of n power functions with 0 = a i ∈ R, l i ∈ N(i = 1, 2, . . . , n), l n ≥l := max{l 1 , · · · , l n−1 } and h = (h 1 ,· · ·, h n ) T , with h = O(x ln+1 ), is a high order term and f = (f 1 , · · · , f n ) T , with f (θ, x, 0) = 0, is a low order perturbation term. If = 0, then the unperturbed system (30) has the origin as a degenerate equilibrium and has the invariant torus here 0 n denotes n-dimensional zero vector, carrying a quasi-periodic flow θ(t) = ωt + θ 0 with same frequencies ω as the forcing. Its normal space is described by the x = (x 1 , · · · , x n )-coordinates. Obviously, the existence of response solutions of system (7) is equivalent to the persistence of such a degenerate response torus T 0 under sufficiently small perturbation f.
For system (30) with n = 1, we have the following theorem: where Ω(t) is a positive function satisfying ln Ω(t) t → 0, as t → ∞; Then for sufficiently small positive constant , system (30) has a degenerate response torus, i.e. the torus persists under small perturbations.
Remark 2. Non-resonant condition (31) is weaker than the classical non-resonant condition (10). That is because if Ω(t) is an approximation function in non-resonant condition (10), then for each x > 1, we have     [20] from quasiperiodic case to almost-periodic case but also from 1-dimensional system to ndimensional system, and Theorem 4.3 is the direct generalization of Theorem 3.12 in [20]. Moreover, if we apply Theorem 4.3 to 1-dimensional system, then conditions (ii), (iii) and (v) vanish naturally. This shows that perturbation in Theorem 4.3, for 1-dimensional case, does not need any restriction which implies Theorem 4.3 also generalizes the work in [23] from quasi-periodic case to almost-periodic case.

The proofs of Theorems 4.1 and 4.2.
In this subsection, we intend to give the proofs of theorems 4.1 and 4.2 which will divide the following two parts. 4.1.1. Normal form. We first reduce system (30) to a normal form for which KAM method can be applied. According to the different conditions of the two theorems, we represent the procedure of the normal forms of two theorems in two lemmas respectively. Actually, the main idea in these two lemmas is that the desired normal form transformation is achieved by first moving the equilibrium solution of the average system to the origin and then removing some lower order terms by means of a suitable normalizing transformation (only Lemma 4.4 use the nonresonance conditions on the frequency vector ω of the forcing). See the proofs of the following two Lemmas for more details. of the form with U s,r,m ≤ 1 + Cη l1 and V s,r,m ≤ Cη, such that system (30) is transformed into system θ = ω, z = (e(η) + d(θ, η))z + p(θ, η) + q(θ, z, η), , q(θ, z, η) ∈ C ω m (∆s ,r , R) and the following estimates hold: Proof of Lemma 4.4. First of all, we consider the averaged system of system (30): We wish to investigate the solutions of system (30) near the equilibrium of the averaged system (35). To do this, we consider the implicit equation In view of f 1 (θ, x, ) = O( ), using a Taylor expansion, we have Putting x = Then the implicit function theorem asserts that the equation (38) (39) System (39) can write as the following system θ = ω, y = (ẽ(η) +d(θ, η))y +p(θ, η) +q(θ, y, η), q(θ, y, η) =O(y 2 )(y → 0).
In addition, it is easy to see the estimate (34).
Proof of Lemma 4.7. In this proof, we hide parameter κ 0 for simplicity if there is no confusion. One can apply the transformation z = z + + V ν (θ) to systeṁ Once the homological equation is solved, then system (47) becomeṡ

Now, we solve equation (48). For each
we define an operator T : where I is identity matrix, and an operator W : In fact, by (49) and E is a real matrix, we can get (here · denotes the complex conjugate of ·), and (50) implies T (V (θ)) is a real analytic function. Then the homological equation (48) can be written as By the definition of the operator T , we can get Thus T −1 W ≤ T −1 W ≤ cκ 0 , which implies the operator T + W is invertible, . Therefore, the equation (48) is solvable and V ν (θ) = (T + W ) −1 (F ν (θ)). We have the estimation One can check that r ν+1 + V ν (θ) sν+1,mν+1 ≤ r ν , which implies By the definition of G ν+1 (θ, z + ), one can also check which implies that for each 0 <r ≤ r ν+1 , for each ν.

Iteration
Let us take E 0 = E, Q 0 = Q, Q 0 = 0, F 0 = F, G 0 = G. So, it is easy to check that system (46) satisfies all hypotheses of Lemma 4.7 with l = 0. By induction, we can prove that for any ν ≥ 0 there is a sequence Φ ν of transformations such that Then, after the transformation Φ ν , system (46) is changed tȯ By the inductive assumptions of KAM iteration, for sufficiently small κ 0 , we have
Proof of Lemma 4.9. We divide the proof of Lemma 4.9 into the following several parts, and in the following discussions we do not write parameters (ξ, λ) and 0 explicitly if there is no confusion for simplicity.
This completes the proof of Theorem 4.8.