On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability

In this paper, we establish a KAM-theorem about the existenceof invariant tori in non-conservative dynamical systems with finitely differentiable vector fields and multiple degeneracies under the assumption that theintegrable part is finitely differentiable with respect to parameters, instead ofthe usual assumption of analyticity. We prove these results by constructingapproximation and inverse approximation lemmas in which all functions arefinitely differentiable in parameters.

The integrable system of (1.1) has an invariant n 2 -dimensional torus I = 0 with frequencies ω = (ε q5 ω 1 , ω 2 ). The aim of the present paper is to examine the persistence of this family of rotational tori I = 0 under small perturbations (i.e., g j = 0, j = 1, · · · , 4). System (1.1) has the following features: the perturbation terms g j , j = 1, · · · , 4 are only finitely differentiable, there exist small frequencies, small twist and higher-order degeneracy and the number of parameter variables is possibly less than the dimension of tori.
To tackle these difficulties we establish a new KAM theorem for (1.1), which is a extension of Pöschel [27] for finitely differentiable Hamiltonian systems and Han, Li and Yi [18] for multiple-degenerate and analytic Hamiltonian systems to nonconservative dynamical systems, respectively. In the context of finitely differentiable perturbations, the study on the persistence of quasi-periodic invariant tori has originated from the work of Moser [24] on area-preserving mappings of an annulus, which was extended to dissipative vector fields in [6] based on smoothing operator technique. Another important method, which can relax the requirement for regularity of perturbations, is to approximate a differentiable function by real analytic ones [25,30,40,27,35,13,1,38]. Rüssmann [30] proved an optimal estimate result on approximating a differentiable function by analytic ones. Following this approach Zehnder [40] established a generalized implicit function theorem and applied it to the existence of parameterized invariant tori of nearly integrable Hamiltonian systems in finitely differentiable case, Pöschel [27] showed that on a Cantor set, invariant tori of the perturbed Hamiltonian system form a differentiable family in the sense of Whitney. The results and ideas of Moser and Pöschel are extended to the case of symplectic mappings by Shang [35] and to the case of lower dimensional elliptic tori by Chierchia and Qian [13], respectively. Wagener [38] extended the modifying terms theorem of Moser [26] (i.e., introducing additional parameters) to finitely differentiable and Gevrey regular vector fields.
The results mentioned above, except for [13], were restricted to the case where the integrable part is analytic in coordinate variables as well as in parameters. The integrable part in [13] is assumed to be Lipschitz with respect to parameters and the frequency map to be a Lipschitz homeomorphism. Of course, if the unperturbed (integrable) part and the perturbation are both of class C l with l > 2n (n is the number of degrees of freedom), it is reduced to the case where the integrable part is analytic and the perturbation is of class C l by regarding the initial values of action variables as parameters for Hamiltonian systems. In this paper, we shall extend the result and method of Pöschel [27] to the non-conservative system (1.1) with degeneracies, removing the restriction that the integrable part is analytic in parameters.
The perturbation was assumed to be C 333 originally in the work of Moser [24] on area-preserving mappings of an annulus, and then was weakened to C 5 by Rüssmann [30] and to C l (l > 3) (meaning that the perturbation is of class C 3 and the derivatives of order 3 are Hölder continuous) by Rüssmann [32] and Herman [19], where a counterexample for l < 3 was given. For improvements on weakening the regularity of perturbations in the Hamiltonian case we refer to [1] and references therein.
The above mentioned results were proved under the so-called non-degeneracy conditions. In the context of degenerate KAM theory, i.e., if Kolmogorov's nondegeneracy or Arnold's iso-energetic non-degeneracy condition is violated, Arnol'd [3] established a properly degenerate KAM theorem (refined by [14,12]) to deal with quasi-periodic motions in the planetary many body problem. In this case the integrable part does not depend on the full set of action variables, and the nondegeneracy conditions are imposed additionally on the averaged perturbation. The ideas of Arnol'd [3] were extended to the resonant torus case in [10,23] and the normal zero-frequency case in [16,17,15] for Hamiltonian systems and in [5,22,20] for non-conservative systems. Another method is to search for weaker nondegeneracy conditions concerning frequency maps, which have been studied in a series of papers, for example, by Bruno [8], Cheng and Sun [9], Rüssmann [33,34], Han, Li and Yi [18] for finite dimensional Hamiltonian systems, and Bambusi, Berti and Magistrelli [4] for infinite dimensional case. The weaker non-degeneracy condition in [9] is that the image of the frequency map in an open set includes a curved C n+2 one-dimensional submanifold. Rüssmann [33,34] pointed out that the weaker non-degeneracy condition means that the image of the frequency map does not lie in an (n − 1)-dimensional linear subspace of R n (this condition is also necessary in the analytic case). An interesting and real analytic Hamiltonian of the form with the degeneracy involving several time scales was considered in [18]. The degeneracy in (1.1) is somewhat similar to the one in [18].
We will construct approximation and inverse approximation lemmas in which approximating and approximated functions are finitely differentiable in parameters and then prove the existence of invariant tori in (1.1) for the most of parameter values.
The KAM theorems in [7,38] can be applied to quasi-periodic bifurcations (i.e., bifurcations of quasi-periodic invariant tori). Our results in this paper can be applied to investigating the persistence of quasi-periodic invariant tori in bifurcation theory of equilibrium points. To study the bifurcations of equilibria of a system of differential equations (ODEs, PDEs and functional differential equations), one usually reduces such a system to a lower-dimensional one on the center manifold by the Center Manifold Theorem. However the reduced system need not be analytic even if the original system is analytic. When the equilibrium is partially elliptic and the reduced subsystem has a normal form of Birkhoff type on the center manifold, then the truncated normal form may possess quasi-periodic invariant tori. In this case, a question arises naturally: does the original system (equivalently, the reduced system on the center manifold) have quasi-periodic invariant tori with the same dimension ? This problem can be discussed by KAM theory and a careful study leads us to consider the existence of quasi-periodic tori of a system similar to (1.1). We will give the derivation and special forms of (1.1) corresponding to various bifurcations in a forthcoming paper.

Statement of results.
Let Ω 1 and Ω 2 be convex open neighbourhoods of the origin in R n11 and R n12 , respectively, Ω = Ω 1 × Ω 2 , the parameter set Π be a convex bounded open set of positive Lebesgue measure in R n3 . Let |x| denote the maximum norm and |x| p the p-norm. In the following, l and α represent the differentiability orders of functions in the space variables (I, ϕ) and the parameter variables ξ, respectively. Definition 2.1. Let α be a positive integer and l > 0, C l,α (Ω × T n2 , Π) be the class of all functions f on Ω × T n2 × Π whose partial derivatives ∂ β ξ f with respect to the parameter variable ξ ∈ Π (which means the Whitney derivative if Π is a closed set) for all β, 0 ≤ |β| 1 ≤ α are of class C l in the space variable x = (I, ϕ) ∈ Ω×T n2 , that is, there is some positive constant M such that the partial derivatives D k ∂ β ξ f of ∂ β ξ f with respect to the space variable x = (I, ϕ) ∈ Ω × T n2 satisfy for all x, y ∈ Ω × T n2 , ξ ∈ Π and all β, In addition, define a norm ||f || l,α;Ω×T n 2 ,Π = inf M, it is the smallest M for which the inequalities (2.1) and (2.2) hold. C l,α (Ω × T n2 , Π) is a Banach space with respect to the norm ||·|| l,α;Ω×T n 2 ,Π , which is a generalization of the Hölder space to a parameter-depending case. The norms || · || l,α;T n 2 ,Π and || · || α;Π are defined in a similar way, which means that the associated function only depends on ϕ ∈ T n2 , ξ ∈ Π and ξ ∈ Π, respectively.
Here, we drop ε from functions, the continuous differentiability of functions ω ν i and Λ ν i (i = 1, 2) on the closed set Π ν means that they are continuously differentiable in some neighbourhood of Π ν . Here and in the sequel, we also regard the Λ as a column vector of its diagonal elements when Λ is a diagonal matrix.
b) The Cantor set Π γ is not empty and indeed the measure meas(Π \ Π γ ) → 0 as γ → 0 as long as we impose proper non-degeneracy conditions on frequencies.
Since in applications the non-degeneracy conditions on frequencies are different, Theorem 2.2 does not involve the measure estimate of Π γ so that it can be used more widely. In the following theorems, we give some conditions to ensure that the Cantor set Π γ is not empty.
Usually the normal form (integrable part) of (1.1) related bifurcation problems of actual models is only finitely differentiable, not analytic in the parameter ξ, and the frequency map is possibly degenerate so that we need the higher-order derivatives of the frequency map to estimate the Lebesgue measure of Π γ and obtain Π γ is the most part of Π. Hence, we need to establish an approximation lemma and the corresponding inverse approximation lemma in which a finitely differentiable function is approximated by a sequence of functions being analytic in space variables, but finitely differentiable in parameter variables. These comprise Section 3. The proofs of Theorems 2.2-2.4 are given in Sections 4 and 5, respectively.
3. Approximation lemmas. Zehnder [40] established the approximation and inverse approximation Lemmas on a finitely differentiable real function approximated by a sequence of real analytic functions, which was generalized to the anisotropic case by Pöschel [27], and was sharpened to covering the finitely differentiable and Gevrey regular cases by Wagener [38], respectively. Here, we give generalized versions of Zehnder's approximation and inverse approximation lemmas finitesmoothly depending on parameters, and obtain estimates of higher-order regularity. a) We first introduce some notations. Let m, n and α be positive integers, U ⊂ C m and Π ⊂ R n be open sets, A α (U, Π) be the class of all functions of (z, ξ) on U × Π which are analytic in z ∈ U and α-times continuously differentiable in ξ ∈ Π.
In particular, for U = {z ∈ C m : |Imz| ≡ sup 1≤j≤m |Imz j | < r}, we denote |g| U ,α;Π by |g| r,α;Π . Take an even function u 0 ∈ C ∞ 0 (R), vanishing outside the interval [−1, 1] and identically equal to 1 in a neighbourhood of 0 (see [38] for the construction of such a function). For x ∈ R m , let u(x) = u 0 (|x| 2 2 ) andũ be the inverse Fourier transform of uũ for x ∈ C m . We list some properties of the analytic smoothing operator S r in Section A.3 of the appendix, which will be used in the proof of the next lemma.
for some real number l > 0 and α ∈ N, where Π ⊂ R n is an open set. Then for every r ∈ (0, 1], the function f r (x, ξ) is α-times continuously differentiable in ξ ∈ Π, entire real analytic in x ∈ C m together with derivatives up to order α with respect to ξ, and satisfies are constants depending on l, p and the dimension m. Moreover, f r is ω-periodic in some variable if in which f is ω-periodic.
Proof. From (3.1) it is clear that f r (x, ξ) is analytic in x ∈ C m , and α-times continuously differentiable in ξ ∈ Π, taking real values on real variables x, and if f is periodic in some variable, then so is f r . As differentiation may commute with integration in (3.1) for functions with bounded derivatives, we obtain ∂ β Hence we only need to prove the estimates (i)-(iii) in the case without parameterdependence. In the following, we will use C to denote some constant depending l, p and m.
(i) The case where p is a integer, is proved by Chierchia [11], see Lemma 6.6 (f) in Appendix. Hence we only give the proof for the case p = q+µ ≤ l, µ ∈ (0, 1), q ∈ Z + . Denote g(x) = D β f, |β| 1 = q. Then by (a) and (b) in Lemma 6.6, we have for Case I. q = [l], the integer part of l. For |x − y| ≥ r, by g ∈ C l−q and Lemma 6.6 (d), we have For |x − y| < r, we also have Hence, ||g − S r g|| µ;R m ≤ Cr l−p ||f || l;R m , which, combining with Lemma 6.6 (f) for the case of integers, implies (i) for the case q = [l].
From Lemma 3.1, it follows the approximation lemma. j=0 be a monotonically decreasing sequence of positive numbers with r 0 ≤ 1 and tend to zero. Then there exists a sequence of functions {f j (z, ξ)} ∞ j=0 , being of class C α in ξ ∈ Π, and entire, real analytic in z ∈ C m together with derivatives up to order α with respect to ξ, starting with f 0 ≡ 0, such that where the constant C 0 depends on l and the dimension m. Moreover, the f j is ω-periodic in each variable in which f is ω-periodic.
b) Now, we want to apply the approximation lemma to the proof of Theorem 2.2 and obtain sequences of real analytic functions approximating g i (i = 1, · · · , 4) in the equation (1.1).
We first expand the definition domain Ω×T n2 ×Π of g i (i = 1, · · · , 4) to R n1 ×T n2 ×Π in the following manner: we multiply g i by a C ∞ -function on R n1 which identical 1 on Ω * and vanishes outside Ω. The obtained function belongs to C l,α (R n1 × T n2 , Π) and is equal to g i on Ω * × T n2 × Π, its norm is bounded by c l ||g i || l,α;Ω×T n 2 ,Π , where c l is a constant depending l, n 1 and the chosen C ∞ -function. Then by the approximation lemma (Lemma 3.2) we have the following corollary.
c) Let Ω ⊂ R m be an open convex set, and Π 0 ⊂ R n be a closed set, where r j = r 0 θ j , 0 < θ < 1 and {s j } ∞ j=0 is a monotonically decreasing sequence of positive numbers with s 0 ≤ 1 and tend to zero.
j=0 be a sequence of functions such that f 0 ≡ 0, f j (x, ξ) is of class C α in ξ ∈ Π j , real analytic in x ∈ W j together with derivatives up to order α with respect to ξ, and for every j ≥ 1 and some constant M . If there exists a constant c 0 > 0 such that r l j ≤ c 0 s α j , j = 1, 2, · · · , then there is a unique function f (x, ξ) being of class C α in ξ ∈ Π 0 in the sense of Whitney (see Appendix A.1), and of classĈ l in x ∈ Ω together with derivatives up to order α − 1 with respect to ξ such that ||f ||Ĉ l ,α−1;Ω,Π ≤ C 0 M and lim j→∞ ||f − f j || p,α−1;Ω,Π = 0 for all 0 ≤ p < l.
If s j0+1 ≤ |ξ − ζ| < s j0 for some positive integer j 0 , then the line segment L connecting ξ to ζ is contained in Π j with 1 ≤ j ≤ j 0 , and the Taylor expansion implies sup If |ξ − ζ| ≥ s 1 , then we also have Thus, we prove the compatibility conditions (3.9) and (3.10), and obtain ∂ β Then the (3.11) implies By the Cauchy inequality and (3.7), we have where C(k) is a constant depending only on k. By a similar proof to one for the compatibility and replacing (3.7) with (3.13), (3.12) implies The proof of the lemma is complete.
We want to look for a transformation T 1 to eliminate the lower-degree terms of P G 1 such that in new coordinates the lower-degree terms of analytic part in (4.4) are much smaller than the old ones. Assume that at the ν-th step of the process, we have already found a coordinate transformation T ν (ν ≥ 0 with T 0 = Id, the identity map) such that the system (4.1) is transformed into ,H ν 2 = O(I), D ν = P −1 (DT ν ) −1 P , the circle "•" indicates composition of functions and DT the Jacobian matrix of T with respect to coordinate variables. Then we replace G ν with G ν+1 which is closer to G, and the above equation is rewritten as We want to construct a coordinate change T ν+1 to eliminate the lower-degree terms in (4.8) such that the lower-degree terms of the next step are much smaller. Repetition of this process leads to a sequence of transformation T ν = T ν−1 • T ν with T 0 = Id, ν = 1, 2, · · · , the limit transformation of which , if converges, reduces (4.1) into a system without the lower-degree terms. Thus, we can obtain the quasiperiodic solution of (4.1). The proof of convergence is due to the following iteration lemma which describes quantitatively the KAM iteration process. b) Iteration Lemma Before stating the iteration lemma we first introduce the iterative sequences and notations used at each iteration step. Set ε 0 = ε q2 , ||G|| l.α;Ω×T n 2 ,Π = M ε 0 , Ω = {I ∈ R n1 : |I| < 3r}, Ω * = {I ∈ R n1 : |I| ≤ 2r}, Ω 0 = {I ∈ R n1 : |I| <r} with some constant 0 <r ≤ 1. For ν ≥ 1, let (i) r 0 =r, r ν =r3 −ν , (vi) f (I, ϕ, ξ) = O U ,α;Π (I k ) denotes a map which is real analytic in coordinate variables (I, ϕ) ∈ U, continuously differentiable up to order α in parameter ξ ∈ Π, and vanishes with I-derivatives up to order k − 1 ≥ 0, and f and its ξ-derivatives up to order α are bounded on U × Π.
Then there exists a closed set Π ν+1 ⊂ Π ν for all m ∈ m, k ∈ Z n2 , K ν < |k| 2 ≤ K ν+1 (see Theorem 2.2 and (H3) for definitions of m and ι, respectively) and a coordinate transformation 13) where ρ and φ are new coordinate variables, and all terms in the transformation are real analytic in φ and continuously differentiable in ξ up to order α, satisfy the estimates , (4.14) and and the conditions (v.1)-(v.3) are satisfied by replacing ν by ν + 1 and (I, ϕ) by c) Proof of Theorem 2.2 Theorem 2.2 is easy to be proven by the Iteration Lemma and Inverse Approximation Lemma.
It is easy to see that when the ε 0 is sufficiently small, the transformation T + maps V + into V * + ⊂ V and V * + into V, respectively, and (4.32) d2) Proof of (4.17). Corresponding to the transformation T + , we have its Jacobian matrix 4.33) and the inverse (4.34) Thus, (4.29)-(4.31) imply Noting that the derivatives of DT + with respect to the parameter ξ is sufficiently small and that for a matrix M (ξ) with a small norm, differentiating the left-and right-hand sides of (E +M (ξ)) −1 (E +M (ξ)) = E and using the Leibniz formula, we find , E is the identity matrix, the estimates (4.30) and (4.31) imply ;s+ (4.39) for 1 ≤ |β| 1 ≤ α and sufficiently small ε 0 . Hence, (4.17) follows from (4.34), (4.29), (4.30) and (4.36)-(4.39). Moreover, we have for suitable F which is real analytic in V + and continuously differentiable up to order α in ξ ∈ Π s+ + . d3) We proceed to verify (v.3) with ν + 1 replacing ν. As the transformation is real analytic in coordinate variables and continuously differentiable up to order α in parameter ξ, so is T + = T • T + .

Appendix.
A.1. Whitney extension theorem. Let Ω ⊂ R n be a closed set, p be a nonnegative integer, p < l ≤ p + 1. C l W (Ω) is the class of all collections f = {f (k) } |k|1≤p of functions defined on Ω which satisfy, for some finite M , for all x, y ∈ Ω and |k| 1 ≤ p, where is the analogue of the k-th Taylor polynomial. f is called C l Whitney in Ω with Whitney derivatives D k f = f (k) for |k| 1 ≤ p. Define a norm ||f || C l W (Ω) = inf M is the smallest M for which both inequalities in (6.1) hold. Then C l W (Ω) with the norm is a Banach space.
The following extension theorem indicates that a Whitney differentiable function has an extension to R n which is differentiable in the standard sense. Lemma 6.3. (Whitney extension theorem, [39,37,27]) Let Ω be a closed set in R n , p ∈ Z + and p < l ≤ p + 1. Then there exists a linear extension operator E : C l W (Ω) → C l (R n ), f = {f (k) } |k|1≤p → F = Ef such that D k F | Ω = f (k) , |k| 1 ≤ p and ||F || l;R n ≤ C||f || C l W (Ω) , where the constant C depends only on l and the dimension n, but not on Ω. Moreover, if Ω = Ω 1 × T n2 ⊂ R n1 × T n2 , then the extension can be chosen to be defined on R n1 × T n2 , so that the periodicity is preserved. for some α ∈ N and a constant c > 0. Then we have the measure estimate