Quasi-effective stability for nearly integrable Hamiltonian systems

This paper concerns with the stability of the orbits for nearly 
integrable Hamiltonian systems. Based on Nekehoroshev's original 
works in [14], we present the definition of quasi-effective 
stability and prove a theorem on quasi-effective stability under the 
Russmann's non-degeneracy. Our result gives a relation 
between KAM theorem and effective stability. A rapidly converging 
iteration procedure with two parameters is designed.

1. Introduction and main result. KAM theory and effective stability are two important contexts in the area of Hamiltonian dynamical systems. The former is established by Kolmogorov, Arnold andMoser, in 1954-1963s [7, 1, 13]. The latter is developed by Nekhoroshev in 1977 [14]. On the one hand, the classical KAM theory shows that under appropriate non-degeneracy such as the classical non-degeneracy or Rüssmann's non-degeneracy of the integrable Hamiltonian, the nearly integrable systems persist or keep the majority of invariant tori of integrable systems. Hence, the majority of orbits, which is in the invariant tori, is perpetual stable. On the other hand, Nekhoroshev's theorem points out that under the steepness of the integrable systems the action variables slowly evolve over exponentially long time interval under sufficiently small Hamiltonian perturbations.
A question is whether there are any relations between the KAM theory and the effective stability. As is known to all, their similarities are that they can be used to describe the stability of orbits in the phase space for Hamiltonian systems. A common condition of them is convexity of the integrable Hamiltonian [9,17]. In 68 FUZHONG CONG, JIALIN HONG AND HONGTIAN LI 1995 Morbidelli and Giorgilli considered a kind of nearly integrable Hamiltonian systems, and found a connection between KAM theorem and effective stability in the sense of the diffusion speed [12]. Later on Delshams and Gutiérrez discussed the similar problem [4]. They investigated the quasiconvex systems, and gave a common approach to the proofs of KAM and Nekhoroshev's theorems by applying Nekhoroshev's iteration with some modifications.
An interesting topic is that under the conditions of KAM theorem, such as Rüssmann's non-degeneracy, one is wondering if there is a Nekhoroshev type result. In this paper we investigate stability of the orbits in nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy and obtain a result about quasi-effective stability.
Consider a nearly integrable Hamiltonian system in the forṁ with the Hamiltonian for nonnegative small parameter . Here p ∈ D are the action variables, D is some bounded domain in R n , while q ∈ T n are the conjugate angle variables, T n = R n /2πZ n is a usual torus. Moreover, all our Hamiltonian functions are assumed to be real analytic in all arguments. The phase space of system (1) is D ×T n ⊂ D ×R n with the standard symplectic structure n j=1 dp j ∧ dq j .
As = 0, system (1) is said to be integrable, and its general solution is with ω(p 0 ) = h p (p 0 ), which forms an invariant torus T p0 = {p 0 } × T n .
To state our results, we need some concepts. Throughout this paper we use Euclidean norm and the supremum norm, and denoted by | · | and · , respectively. For an m × n matrix function A(u) defined on some set D, let A = sup u∈D sup |z|=1 A(u)z . Definition 1. ( [14]) System (1) is said to be effective stable in E ×T n , if there exist positive constants a, b, c and 0 such that, as 0 ≤ ≤ 0 , for all (p 0 , q 0 ) ∈ E × T n , one has |p(t) − p 0 | ≤ c b with (p(0), q(0)) = (p 0 , q 0 ), provided |t| ≤ exp(c −a ). Here a and b are called stable exponents, T ( ) = exp(c −a ) stable time, R( ) = c b stable radius.
Definition 2. An orbit (p(t), q(t)) starting from (p 0 , q 0 ) of system (1) is said to be of near-invariant tori on exponentially long time, if there exist positive constants a, b, c, 0 and constant d ≥ 0, and the function ω * * defined on E × T n such that Definition 2 is a notion of stability of orbits. This definition is established by Morbidelli and Giorgilli [10,11,12], and Perry and Wiggins ( [15]), and Delshams and Gutiérrez [4], respectively. They deal with two different cases of invariant tori of the integrable system. In [15] and [11], the property of near-invariant tori are expressed in terms of the distance to a given KAM torus. In [10] and [12] and [4], the invariant tori are considered only under the frequency vector satisfying the finite inequalities of small denominators. This paper concerns the above two cases. (2) For all (p 0 , q 0 ) ∈ E × T n , the orbit (p(t), q(t)) starting from (p 0 , q 0 ) satisfies the estimate provided |t| ≤ exp(c −a ). Here a and b are called stable exponents of the system, T ( ) = exp(c −a ) stable time, R( ) = c b stable radius.
It directly follows from the above definitions that the effective stability implies quasi-effective stability.
Note that real analytic property of Hamiltonian H(p, q) implies that there exists a positive constant δ such that it is analytic in (D×T n )+δ. Moreover, on (D×T n )+δ, for with 0 ≤ ≤ 1, for some positive constant M . Here ω(p) = h p (p). Assume that ω(p) satisfies Rüssmann's nondegenerate condition as follows (H1) where Z n + denotes the subset of Z n with nonnegative integer components; Now we describe the main result of this paper.
Recently, many achievements have been made in studying KAM theory and the effective stability. For examples, Guzzo, Chierchia and Benettin have announced that they obtained optimal stability exponents under the steepness [6]; Bounemoura and Fischler make use of geometry of numbers to relate two dual Diophantine problems which correspond to the situations of KAM and Nekhoroshev theorems, respectively [2]. For the others, see [3,5,8,19].
The paper is divided into five sections. In section 2 the stickiness of Diophantine invariant tori is considered and the theorem on property of near-invariant tori is described. Section 3 proposes an auxiliary proposition which plays a fundamental role in the proofs of theorems. Finally, the proofs of the theorems are placed in section 4 and section 5.
For a given p 0 ∈ D, if ω(p 0 ) satisfies the following inequalities for some positive constants α and τ , then T p0 is said to be Diophantine. According to KAM theory there is a nearly identity transformation Φ which changes Diophantine invariant torus T p0 of a integrable system into the the invariant torus Φ (T p0 ) of perturbed system (1) ( is sufficiently small), and Φ 0 (T p0 ) = T p0 .
3. An auxiliary proposition. We first construct a rapidly converging iteration scheme with two small parameters. This design is important to prove theorems. Notice that we only need finite iterations in the proof of theorems. Hence, instead of the Diophantine condition, we employ another weaker condition. Take a fixed p 0 ∈ D and a given sufficiently small positive constant κ. Define two integers L(κ) and J(κ) depending on κ, where [ · ] denotes the integer part of a real number. Let For the sake of convenience, by c 1 , c 2 , · · · , denote the positive constants depending only on M, n, K 1 and τ . We continue to assume (H2) For constants α > 0 and τ > 0, ω(p 0 ) suits the inequalities for any k ∈ Z n with 0 < |k| ≤ L(κ).
Here ω 0 suits inequality (7). For a real analytic function f , its Fourier's expansion is

FUZHONG CONG, JIALIN HONG AND HONGTIAN LI
We need the following lemmas.

5.
Proof of Theorem A. Now we prove Theorem A. To this end we regard κ and α as a function in , respectively, in this section. Simply, let By using (46) we determine Here τ > n(n − 1). This condition is a requirement of the measure estimate. Let By assumption (H1) and Lemma 2.1 in [20], one has Then, by KAM theory, measD = measD − O(α 1 n ), and D * is a set of full measure in R n , for any τ > n(n − 1).