A KAM THEOREM FOR THE ELLIPTIC LOWER DIMENSIONAL TORI WITH ONE NORMAL FREQUENCY IN REVERSIBLE SYSTEMS

. In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, and prove that if the frequency mapping ω ( y ) ∈ R n and normal frequency mapping λ ( y ) ∈ R satisfy that where ω 0 = ω ( y 0 ) and λ 0 = λ ( y 0 ) satisfy Melnikov’s non-resonance conditions for some y 0 ∈ O , then the direction of this frequency for the invariant torus persists under small perturbations. Our result is a generalization of X. Wang et al[Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems, Discrete and Continuous Dynamical Systems series B, 14 (2010), 1237-1249].

A mapping Φ : (x, y, u, v) → (x + , y + , u + , v + ) is called compatible with the involution G if Φ and G commute. The compatible transformations transform reversible systems into systems reversible with respect to the same involution. If P j = 0 (j = 1, 2, 3, 4), then reversible system (1.1) becomeṡ x = ω(y),ẏ = 0,u = A(y)v,v = −A(y)u. (1. 3) The reversible system (1.3) has an invariant subspace {u = 0, v = 0}, which foliated by a family of invariant tori T n × {y 0 } × {0} × {0} with the frequency ω(y 0 ) for all y 0 ∈ M. Some of the invariant tori can be destroyed by an arbitrarily small perturbation. Whether some invariant tori can persist under small perturbation is an important problem in the perturbation theory of reversible system and is studied by many authors.
In the special case of p = 0, that is, where there is no normal frequency, the classical KAM theorem says that if m ≥ n and the frequency mapping y → ω(y) is submersive in M, i.e., Rank ∂ω ∂y = n, for all y ∈ M, (1.4) then the majority of invariant tori survive of small perturbations [1,24]. In the case of p > 0, the invariant n-tori of reversible system (1.3) are called lower dimensional. Many authors studied the persistence of lower dimensional invariant tori for reversible systems of the form (1.1) under the non-degeneracy condition (1.4) and obtained many kinds of KAM theorems [5,6,7,8,25,26,27,38]. There are also some similar results in the Hamiltonian framework [10,11,12,23] and in the context of celestial mechanics [3,4,13].
Later, the non-degeneracy condition (1.4) has been weakened to Rüssmann's non-degeneracy condition [5,30]: a 1 ω 1 (y) + a 2 ω 2 (y) + · · · + a n ω n (y) ≡ 0 on O, (1.5) for all a = (a 1 , a 2 , · · · , a n ) ∈ R n \ {0}. However, under Rüssmann's non-degeneracy condition, the range of the frequency mapping ω may be on a sub-manifold and the frequencies of persisting invariant tori may not come from unperturbed ones. Therefore, it is difficult to provide accurate information about the frequencies of KAM tori under Rüssmann's non-degeneracy condition.
In this paper, we are mainly interested in the persistence of invariant tori with prescribed frequency. There are already some results about on the above problem for reversible systems [31,32,35].
Recently, some authors turn to study the persistence of invariant tori with prescribed frequency by the theory of topological degree [31,39]. We first recall the definition for Brouwer degree. If f (y) is a continuous mapping from O ⊂ R m into R m satisfying the condition p 0 / ∈ f (∂O) (∂O is the boundary of O), then one can define Brouwer degree deg(f, O, p 0 ) by using an approximation scheme introduced by Nagumo [21]. The idea consists in defining it first for f smooth and a regular value of p 0 through the formula with Df (y) the Jacobian of f , and then to approximate the continuous function f and the point p 0 above by a sequence of such functions and points for which this definition holds. This is possible by the Weierstrass approximation theorem and the Sard theorem. The degrees of the approximations stabilize to a common value, denoted by deg(f, O, p 0 ) and being an algebraic count of the number of counterimages p 0 of f under in O, which is stable for small perturbations of p 0 and f . For the details, see [2,15,18].
In the paper [31], the authors considered the reversible system (1.1) with m = n + p. Let λ(y) = (λ 1 (y), · · · , λ p (y)), Ω(y) = (ω(y), λ(y)) and Ω 0 = (ω 0 , λ 0 ) = (ω(y 0 ), λ(y 0 )) with a certain y 0 ∈ M satisfying the following non-resonance condition: Noting that m = n + p and y ∈ R m , the system (1.1) has enough variables and parameters to control, simultaneously, the values of the the tangential frequency and the normal frequency of the torus. A natural question is what happens when m = n? By observation we find that a similar result also holds under the condition m = n if p = 1. Moreover, the non-degeneracy condition (1.6) can be weakened to The aim of this paper is to prove that a similar result also holds under the above non-degeneracy condition for the case of m = n and p = 1.
Although ω(y) satisfies Rüssmann's non-degeneracy condition, the previous KAM theorems cannot provide any information on the persistence of the lower dimensional invariant tori T n × {0} × {0} × {0} for the reversible system (1.7).

Remark 2.
Noting that y ∈ R n+p in the paper [31], the reversible system in [31] has enough parameters to remove the shifts of the tangential frequency and the normal frequency and make the frequency ω 0 and λ 0 fixed in KAM steps. The principal difficulty of this paper is that the system (1.7) does not have enough parameters to control, simultaneously, the values of the tangential frequency and the normal frequency of the torus in the KAM steps. However, by observation we find that it is not necessary to remove the shift of the normal frequency by adjusting parameters in the KAM steps for the case of p = 1. In the proof of this theorem, we first reduce the system (1.7) to a parameterized system by the transformation y = ξ + y + . Then we introduce an artificial external parameter γ and rewrite the normal frequency λ j as in the KAM steps, whereλ j is the shift of the normal frequency. By adjusting the parameters and the implicit function theorem, we can also rewrite the tangential frequency ω j as in the KAM steps. Noting that ω 0 and λ 0 satisfy Melnikov's non-resonance conditions (1.8), then ω j and λ j also meet the Melnikov's non-resonance conditions in the KAM steps. The above factorization method plays an important role in weakening the Brouwers degree condition (1.6).

Remark 3.
In the proof of this paper, we use the topological degree theory to obtain a solution corresponding to an invariant torus of the reversible system (1.7). Actually, the reversible system (1.7) may admit many invariant tori because the solution obtained by the topological degree theory is not unique.
2. Proof of Theorem 1.1. Motivated by [23,31,39], we first reduce the system (1.7) to a parameterized system. By a transformation y = ξ + y + , we introduce a parameter ξ and then the reversible system (1.7) is equivalent to a parameterized system: where and ξ ∈ O is regarded as parameter. Note that for simplicity we have used y instead of the new variables y + in the transformed equations. Below we consider the persistence of invariant tori for the reversible systems (2.1) instead of the original We first introduce some notations. If f (x, y, u, v; ξ) is analytic on D(s, r) × Π d , we expand f in Taylor-Fourier series as: where |Mf k | r;d denotes the sup-norm of Mf k over the domain D(s, r) × Π d . Let f = (f 1 , f 2 , f 3 , f 4 ) be a vector field depending on x, y, u, v and ξ. Define a weighted norm by Suppose that ω 0 and λ 0 satisfy the non-resonant condition (1.8) and deg(ω/λ, Π, ω 0 /λ 0 ) = 0. Then there exists a sufficiently small positive constant > 0, such that if then there exists ξ * ∈ Π such that the reversible system (2.1) at ξ * has an invariant torus with the tangential frequency ω * = λ(ξ * )+λ * (ξ * ) λ0 ω 0 and the normal frequency Remark 4. By the above discussions, Theorem 1.1 can be reduced to Theorem 2.1. Actually, the invariant torus for (2.1) at the parameter ξ * corresponds to an invariant tours in Theorem 1.1. So below we mainly prove Theorem 2.1.
We use the Herman method to prove Theorem 2.1. The Herman method is a well-known KAM technique that introduces an artificial external parameter to make the unperturbed system highly non-degenerate. This method has been used in [5,27,28,31,39]. Now we introduce an artificial external parameter γ and consider the following reversible system where γ = (γ 1 , γ 2 , · · · , γ n ) ∈ R n is an external parameter. Obviously the reversible system (2.1) corresponds to the reversible system (2.2) with γ = 0. We will first give a KAM theorem for the reversible system (2.2) with parameters (ξ, γ) and delay the proof of Theorem 2.1 later.
In order to prove Theorem 2.1, we first give the following theorem.

KAM-
Step. In this section, we outline the formal process of one cycle of the KAM iteration. To simplify notations, in what follows, the quantities without subscripts refer to those at the j-th step, while the quantities with subscripts " +" denote the corresponding ones at the (j + 1)-th step. We will use the same notation c to indicate different constants, which are independent of the iteration process. Suppose at the j-th step,, we have arrived at the following reversible system: where ω(ξ, γ) = γ + ω 0 (ξ) +ω(ξ, γ), with λ(ξ, γ) = λ 0 (ξ) +λ(ξ, γ). We describe one step of KAM iteration in more details in the following lemma. Assume that Λ(ξ, γ) =ω(ξ, γ) − ω0 λ0λ (ξ, γ) satisfies
We divide the proof of Lemma 3.1 into the following several parts.
A. Truncation. If f (x; ξ, γ) is analytic in x on T n , expanding f as Fourier series with respect to x, we have Define the truncation of the Fourier series by Denote by B. Constructing compatible transformation. Define a transformation where ω + = ω + ∆ω and A + = A + ∆Â with ∆ω and ∆Â being decided later, ∂ ω a and ∂ ω b are defined similarly. Note that the parameters (ξ, γ) are implied in the above equations. Moreover, we have Here we have used (x, z) instead of the new variables (x + , z + ) in the transformed equations for simplicity.