Non-floquet invariant tori in reversible systems

In this paper we obtain a theorem about the persistence of non-floquet invariant tori of analytic reversible systems by an improved KAM iteration. This theorem can be applied to solve the persistence problem of completely hyperbolic-type degenerate invariant tori for a class of reversible system.

1. Introduction and main results. Consider the existence of n-dimensional invariant tori of the following dynamical system: where (x, y, u, v) ∈ T n × R m × R p × R p (n ≤ m), ω(y) = (ω 1 (y), · · · , ω n (y)) ∈ R n is called frequency vector, and y ∈ M ⊂ R m , where M is a bounded open domain. A and B are p × p matrices, f 1 , f 2 , f 3 and f 4 are small perturbations.
In the special case of p = 0 and m ≥ n, that is, where there is no normal frequency, Arnold [1] and Sevryuk [21] proved the existence of invariant tori under the non-degeneracy condition that the unperturbed frequency map y → ω(y) is submersive in M, i.e., Rank ∂ω ∂y = n, for all y ∈ M. (1.5) In the case of p > 0, the invariant n-tori of reversible system (1.2) are called lower dimensional. Many authors studied the persistence of lower dimensional invariant tori under the non-degeneracy condition (1.5) and obtained many kinds of KAM theorems [2,3,4,5,22,23,24,35].
The above mentioned results about the persistence of invariant tori in reversible systems deal only with the Diophantine condition: where α > 0 and τ > n − 1 are some constants. The natural question to ask is whether the Diophantine (1.6) can be further weakened and in what case the frequencies of the persisting invariant tori for reversible systems can persist. Recently, Zhang, Xu and Wang obtained a result for the above problem under Brjuno-Rüssmann's non-resonant condition [42]. The persistence of invariant tori for reversible systems of the form (1.1) requires the condition that the matrix A(y) is non-singular. When the matrix A is nonsingular, we can use the linear term Av to control the shift of lower-order terms from small perturbation in KAM steps and so we can completely control the shift of equilibrium point. When A(y) is singular, the invariant tori are said to be degenerate.
There are already some results on degenerate lower-dimensional invariant tori for Hamiltonian systems [9,12,39]. Li and Yi [12] studied the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms and proved a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori. Han, Li and Yi [9] considered the persistence problem of lower-dimensional, possibly degenerate, invariant tori for Hamiltonian of the form: where (x, y, z) ∈ T n × R m × R 2m , is a small parameter, and M (ω) can be singular. The result [9] replaced the condition DetM = 0 by the following restrictive condition on the perturbation: with a real analytic family z . Here [P ] is the mean value of the function P (x, 0, z (ω), ω, 0) over T n (see (1.12) below). Recent years, the existence of degenerate lower dimensional invariant tori for reversible systems has been studied by many authors. Liu [15] considered the reversible systems of the following form: where (x, y, u, v) ∈ T n × R n × R p × R q , ω is an independent parameter varying over a positive measure set M ⊂ R n . Liu [15] replaced the condition DetA = 0 by the condition Rank(A, C) = p in the reversible system (1.7). A natural question is what happens when Rank(C, A) < p? Wang, Xu and Zhang [30,31] obtained some results about the persistence of degenerate lower-dimensional tori in reversible systems of the following form: where (x, y, u, v) ∈ T n × R m × R × R (m ≥ n + 1), y = (y 1 , y 2 , · · · , y m ) ∈ R m , n 1 and n 2 is a positive integer, P 1 , P 2 , P 3 and P 4 are small perturbations. The system (1.8) is actually partially degenerate case. If B = 0, the equilibrium point of the unperturbed system (1.8) is completely degenerate case.
In this paper, we are mainly interested in the persistence of non-floquet invariant tori. There are already some results on non-floquet invariant tori for Hamiltonian systems [8,41]. Zehnder [41] considered the Hamiltonian function of the following form: where (x, z + ) ∈ T n ×R m and (y, z − ) ∈ R n ×R m are canonically conjugate variables, P is a small perturbation, Q is an n × n-matrix valued function on T n , Ω is an m × m-matrix-valued function satisfying Re Ω(x)ξ, ξ ≥ σ|ξ| 2 , (1.10) for all ξ ∈ C m and some σ > 0. The author proved that all the hyperbolic lower dimensional tori with Diophantine frequency of Hamiltonian systems survive small perturbations. For a similar result also see [8].
Motivated by [8,41], we want to obtain some information on the persistence of non-floquet invariant tori in reversible system. To be more precise, we consider the reversible system of the following form: andΩ(x) are real analytic in x on a complex domain D(s) = x ∈ C n /2πZ n |Im x| ≤ s .
Moreover, the perturbation terms f 1 , f 2 and g are all real analytic with respect to (x, y, u, v) on a complex domain D(s, r), where D(s, r) = (x, y, u, v) |Im x| ≤ s, |y| ≤ r, |u| ≤ r, |v| ≤ r Obviously, the frequencies of the invariant tori for the unperturbed system admit the different dimensions of angle variables, then the invariant tori for the unperturbed system are said to be non-floquet invariant tori. The purpose of this paper is to study the persistence of the non-floquet invariant torus with given frequencies ω. Before formulating our theorem, we first give some notations and assumptions. Let f (x 1 , · · · , x n ) be a continuous function with period 2π in every x i , i = 1, 2, · · · , n, denote the average of f by Denote the norm of f on D s by f s = k∈Z n |f k |e s|k| . Let Define a norm of f by where Mf k = l,i,j |f klij |y l u i v j and |Mf k | r denotes the sup-norm of Mf k over the domains D(s, r) with respect to y, u, v.
Then it is easy to see that B s is a Banach space. We assume that Ω is hyperbolic on M. Define a linear operator L on B s by with h(x) ∈ B s . Denote the norm of the linear operator L on B s by

Remark 1.
If To state the main result, we need the following assumptions: We suppose that the rank of the matrix [Q] is n.
Theorem 1.1. Consider the reversible system (1.11). Suppose that the above assumptions (1)-(3) hold. Then, there exists a sufficiently small γ > 0, which is independent of α and usually depends on δ, τ, σ, there is a compatible transformation which transforms the reversible system (1.11) intȯ is an invariant torus of reversible system (1.11) with frequencies ω.

Remark 3. An example to which the above theorem can be applied is
. Assume that Rank[Q] = n and ω satisfies the nonresonant conditions (1.14) . If Ω s ≤ 7, then it follows that By Theorem 1.1, the non-floquet invariant torus with given frequencies ω survives small perturbations. In contrast to the conditions of Zehnder and Graff [8,41], our condition (1.13) does not require Ω(x) to satisfy the positivity condition (1.10). Moreover, if m > n, we have analytic (m − n)-parameter families of invariant n-tori with frequency vector ω, since we only need n components of y to control the shift of frequency and the other n − m components can introduce some parameters.
Remark 4. The key features of the system (1.11) are that the tangential frequencies and the normal frequencies of the invariant tori for the unperturbed system admit angle variables, that is, the presence of the terms Q(x)y andΩ(x)z and the dependence of Q andΩ on x.
for small ε > 0. Then the system (1.11) becomes where the terms εC 1 (x), εC 2 (x)v can be treated as perturbations. So Theorem 1.1 can be regarded as a particular case of the previous results [2,3,15,22,23,24,29]. However, ifQ(x) = 0 andΩ(x) = 0 , the previous results cannot tell whether the reversible system (1.11) has a torus with prescribed frequencies ω. Our result shows that the reversible system (1.11) has a torus with the frequencies ω.
As an application of Theorem 1.1, we give a result about the persistence problem of lower-dimensional, possibly degenerate (i,e.the matrix A may be singular), invariant tori for reversible system of the form: is real analytic in x on a complex domain D(s). Moreover, the perturbation terms P 1 , P 2 , P 3 and P 4 are all real analytic with respect to (x, y, u, v, ) on a complex domain D(s, r) × (0, 1).
The rest of the paper is organized as follows. In Section 2 the detailed proof of Theorem 1.1 is described, which consists of KAM step, setting the parameters and iteration, and convergence of iteration. In Section 3, we prove that Theorem 1.2 can be reduced to Theorem 1.1.

2.
Proof of Theorem 1.1. In this section we use an improved KAM iteration to prove Theorem 1.1. In the proof of this theorem, we can remove the shifts of frequencies ω by a small translation of coordinates of y in KAM steps. The existence of such translation of coordinates can be guaranteed by the condition that Rank[Q] = n. The condition (1.13) is used to ensure that Ω +Ω(x) is hyperbolic on D(s).

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: , y = f 2 (x, y, z), z = Ω +Ω(x) z + g(x, y, z), Q andΩ are n × m and 2p × 2p matrices, respectively. Let f = (f 1 , f 2 , f 3 , f 4 ) and [Q] be the average of Q(x). We summarize one KAM step in the following lemma.
The above lemma is actually one KAM step. We divide the proof of Lemma 2.1 into several parts.
A. Constructing compatible transformation. In the following, we will construct a compatible transformation Φ which changes the reversible system (2.1) into (2.3). Let Φ be defined by Let a(x) = (a 1 (x), a 2 (x)), b(x) = (b 1 (x), b 2 (x)), S 1 = diag(I m , −I p , I p ) and S 2 = diag(−I p , I p ). It is easy to see that Φ is compatible with the involution G if and only if (2.7) Let h(x) ∈ B s , define ∂ ∆ h by Let Ω(x) = Ω +Ω(x) and set where f 1 y (x, 0) = ∂f 1 ∂y | (y,z)=0 , f 1 z (x, 0) = ∂f 1 ∂z | (y,z)=0 , and g z (x, 0) = ∂g ∂z | (y,z)=0 . Under the transformation Φ the system (2.1) is changed into . Note that we have used (x, y, z) instead of the new variables (x + , y + , z + ) in the transformed equations for simplicity. We hope to find h(x), a 1 (x), a 2 (x), b 1 (x), b 2 (x) and d(x) such that We first solve the equation (2.17). Note that f 2 s,r ≤ r and f 2 is odd function with respect to x. Therefore [f 2 ] = 0. Let Then we have It follows that with a 1 0 being determined later. Since the system (2.1) is reversible, we have

By (2.19) it follows that
Define a mapping T on Banach space B s by It is easy to see that T : B s → B s . Then the problem is reduced to finding the fixed point of T on B s . Obviously, In view of 1+δ 2 < 1, by the contraction mapping theorem, there exists a unique fixed point a of the mapping T on B s . Moreover, we have a 2 s = Lg + L Ω(x)a 2 s ≤ Lg s + 1 + δ 2 β 0 a 2 s ≤ L · g s + 1 + δ 2 a 2 s .

2.2.
Convergence of iteration. Now we choose some suitable parameters so that the above iteration can go on infinitely. At the initial step, we set Q 0 ( Then, it is easy to see that s j , r j , ρ j , E j , j are all well defined for j ≥ 0. In the following we are going to check all assumptions in the iteration lemma 2.1 to ensure that KAM steps are valid for all j ≥ 0.
With the aid of (2.4), we can prove that the transformation Φ j is convergent to Φ * on D(s/2, r/2). The proof is the same as in the case of Hamiltonian systems(in fact simpler), so we omit the details and refer the reader to [19,20]. By (2.5) and in view of E j → 0 as j → ∞, Then it is easy to see that Φ * transforms the reversible system (2.1) into the following form: Noting that j → 0 as j → ∞, it is easy to see that f j * (x, 0, 0, 0) = 0, (j = 1, 2, 3, 4). This completes the proof of Theorem 1.1.
3. Proof of Theorem 1.2. By compatible transformations, we prove that the reversible system (1.15) can be reduced to a suitable normal form which Theorem 1.1 can be applied.
Define a compatible transformation Φ 1 by the system 1.1 is transformed into where f j = P j (x, y, u, v + v , )(j = 1, 2, 4) and f 3 = P 3 (x, y, u, v + v , ) + Av . Let Then system (3.1) is written aṡ Denote w + = (y + , u + , v + ) T . Φ 2 is written in a more compact form: Let S = diag(I m , −I p , I p ). It is easy to see that Φ 2 is compatible with the involution G if and only if Under the transformation Φ 2 the system (3.2) is changed into where A + = A +Â withÂ being decided later, f 1 w (x, 0) = ∂f 1 ∂w | w=0 , G w (x, 0) = ∂G ∂w | w=0 and . Note that we have used (x, w) instead of the new variables (x + , w + ) in the transformed equations for simplicity. Let We hope to find h(x), a(x) and b(x) such that
By (1.18) and in the same way as in [15,27,35], it follows that the equation If |k| = 0, in the same way as in [15,27,35], it follows that the equation (3.11) has a solution b(x) ∈ B 0 s−2ρ with b(x) = O( ). In the same way as in [15,27,30], we can prove that Φ 2 is a compatible transformation. Then the reversible system (3.12) is written as Obviously, the shifts of normal matrixΩ(x) satisfies Ω (x) s = O( 2 ) . Thus, the assumption L · Ω s ≤ c < 1 holds in KAM steps. In the same way as Theorem 1.1, we can prove Theorem 1.2.