ALMOST-PERIODIC BIFURCATIONS FOR 2-DIMENSIONAL DEGENERATE HAMILTONIAN VECTOR FIELDS

In this paper, we develop almost-periodic tori bifurcation theory for 2-dimensional degenerate Hamiltonian vector ﬁelds. With KAM theory and singularity theory, we show that the universal unfolding of completely degenerate Hamiltonian N ( x, y ) = x 2 y + y l and partially degenerate Hamiltonian M ( x, y ) = x 2 + y l , respectively, can persist under any small almost-periodic time-dependent perturbation and some appropriate non-resonant conditions on almost-periodic frequency ω = ( · · · , ω i , · · · ) i ∈ Z ∈ R Z . We extend the analysis about almost-periodic bifurcations of one-dimensional degenerate vector ﬁelds considered in [21] to 2-dimensional degenerate vector ﬁelds. Our main results (Theorem 2.1 and Theorem 2.2) imply inﬁnite-dimensional degenerate umbilical tori or normally parabolic tori bifurcate according to a generalised umbilical catastrophe or generalised cuspoid catastrophe under any small almost-periodic perturbation. For the proof in this paper we use the overall strategy of [21], which however has to be substantially developed to deal with the equations considered here.


Introduction
It is well known that the qualitative structure of a dynamical system can be characterized by its invariant subsets, for example, the equilibria, periodic orbits, invariant tori and the stable and unstable manifolds of all these.These invariant subsets form the theoretical framework of the dynamics, and one is interested in the properties that are persistent under small perturbations.If we assume that a dynamical system depends on external parameters, then bifurcation theory describes changes in the qualitative structure of the dynamical system with small changes in these parame-ters.In the early 1970s, Meyer [16,17] or Broer et al [3,4] established the theory of bifurcations for equilibria and periodic solutions to some Hamiltonian systems and dissipative systems.Subsequently, many authors have been devoted to studies of quasi-periodic bifurcation for conservative and dissipative dynamical systems using KAM skill, see ( [2, 7, 13]).
As we know, in quasi-periodic bifurcation theory, normally degenerate bifurcation is difficult to handle because standard KAM theory is not directly applicable.Generally speaking, it is necessary to establish a new KAM theory in order to study quasi-periodic bifurcation of normally degenerate dynamical systems.In normally one dimensional cases, Broer et al. ( [8]) considered the quasi-periodic saddle-node bifurcation by investigating the perturbation of the vector field X ω,λ tr (x, y) = (ω + a(ω)y) where (x, y) ∈ T n × R, ω ∈ R n and λ ∈ R is parameter.They proved an adapted KAM-theorem, to obtain a conjugacy sending the above vector field to Xω,λ tr (x, y) = (ω + a(ω)y + O(|y| 2 + |λ| 2 )) when ω satisfies Diophantine condition.Later, Wagener ( [22]) considered the case of diffeomorphisms.Furthermore, higher order degenerate quasi-periodic bifurcation in one-dimensional systems can been seen in [20].In the last years, the quasiperiodic bifurcation theory has been extended to higher dimension and higher order normally degenerate cases, and quasi-periodic analogues of cuspoid bifurcations of equilibria are also developed in normally higher dimensional cases.In 1998, Hanßmann ( [12]) considered the quasi-periodic center-saddle bifurcation by investigating the perturbation of following Hamiltonian X(x, y, p, q, ω, λ) = ω, y + a(ω)p 2 + b(ω) on (x, y, p, q) ∈ T n ×R n ×R 2 , and proved the persistence of universal unfolding (1.1) under small perturbations provided that ω satisfies Diophantine condition.Broer et al. ( [5,6]) considered the following Hamiltonian universal unfolding and where (x, y, p, q) ∈ T n × R n × R 2 , respectively.They proved universal unfolding (1.2) and (1.3) can persist under any small quasi-periodic perturbation provided that the frequencies satisfy Diophantine condition.Most direct results are the tori in the unperturbed system bifurcate according to a generalized cuspoid catastrophe and a generalised umbilical catastrophe, respectively.
We notice that many perturbations tend to be more irregular in nature, such as almost-periodic perturbations.Recently, many works focus on the existence of almost-periodic invariant tori for the finite-dimensional ( [14,15,24]) and infinitedimensional ( [1,10,18]) almost-periodic forced systems.Thus, the our aim is to investigate the almost-periodic bifurcations phenomena.In 2020, W. Si, X. Xu and J. Si ( [21]) studied the almost-periodic time-dependent perturbations of universal unfolding of one-dimensional vector field ẋ = x l .More precisely, the authors considered where analytic in all variables and parameters and almostperiodic in t with frequency vector ω = (. . ., ω i , . ..) i∈Z ∈ R Z , and proved that all the bifurcation scenario in ẋ = M (x, λ) can persist under a small almost-periodic perturbation when the almost-periodic frequency ω satisfies some non-resonant conditions.This shows the infinite-dimensional tori bifurcate from the almost-periodic perturbation.
The perturbed tori are the most degenerate ones corresponding to the central singularity at λ = 0 or λ = 0, and the remaining part of our perturbation problem asks what happens to the invariant tori of N and M that occur in the unfolding for λ = 0.When it is quasi-periodic case and according to [5,6], there are (quasi-periodic) centre-saddle bifurcations, (quasi-perodic) Hamiltonian pitchfork bifurcation in universal unfoldings N and M .Degenerate umbilical tori or normally parabolic tori bifurcate according to a generalised umbilical catastrophe or generalised cuspoid catastrophe under any small quasi-periodic perturbation.Different from quasi-periodic case, the persistence of above universal unfolding becomes more difficult to handle in almost-periodic case because it is not only necessary to control the normal form with vanished linear part but also necessary to deal with the small divisor produced by the integer combination of infinite many frequencies.
In this paper, we will prove both infinite-dimensional invariant tori and the bifurcation scenario in H = N (x, y, λ) and H = M (x, y, λ) can persist under a small almost-periodic perturbation when the almost-periodic frequency ω satisfies some non-resonant conditions.
The method we adopt is first introduced by Pöschel( [19]), which deals with small divisors with spatial structure in infinite dimensional Hamiltonian systems.We generalize it to deal with normally degenerate problems.Although in our proof we use the overall strategy of [17], however it has to be substantially developed to deal with the equations considered here.
The rest of this paper is organized as follows.In Section 2, we introduce some notations and definitions applied in the sequent and give the main results.In Section 3, we give the proof of Theorem 2.1.In section 4, we give an outline of proof of Theorem 2.2.

Some definitions and notations
Some of the notations and definitions given in this section are also given in [21].For convenience of readers and the integrity of this paper, we restate them here.9,11]).A function f : R → R is called an almost periodic function, if it is continuous and, for any > 0, the -translation set of f , is a relative dense set on R (i.e., there exists l > 0, such that for any a ∈ R, [a, a + l] ∩ T (f, ) = 0).We denote by AP the set of analytic almost periodic functions.If f ∈ AP , then f exp(−il•) ∈ AP for any real number l. Define is called the set of Fourier exponents of f , the numbers a(l, f ), l ∈ Λ(f ), the Fourier coefficients and Fourier series of f is designated by Throughout the paper, we study a class of special Bohr almost periodic function with the frequency ω that is we always assume that the analytic almost periodic function f has the Fourier exponents ∪ i∈Z Ω i , where each Ω i is a real number set which is expanded integrally by ω Here Elm(•) represents the elements of set •. Thus, almost periodic function f can be expressed as (2.1) Next, we will give more general definition of this kind of special almost-periodic function.Let S is a family of finite subsets A of Z which has the following spatial structure: Z ⊂ ∪ A∈S A and the union of any two sets in S is again in S, i.e., there exists a real analytic function which admits a uniformly converging Fourier series expansion where S has spatial structure (2.2) and with k, θ = i∈Z k i θ i , such that f (t) = F (ωt) for all t ∈ R, where F is bounded in a complex neighborhood for some s.Here F (θ) is called the shell function of f (t).
Thus, f (t) can be represented as According to Definition 2.3, we can observe that

1).
From the definitions of the support suppk and the set A, we know that f A (t) is a real analytic quasi-periodic function with the frequency ω A = {ω i : i ∈ A}.Therefore, the almost periodic function f (t) can be represented as the sum of countably many quasi-periodic functions f A (t) formally.
) is a quasihomogeneous polynomial of order d with weight (α x , α y , α 1 , . . ., α k ), then F can write as and the definition of (i, j, h) is as follows: (i, j, h) Let M be a compact set in C k+2 and define D = T Z s × M .If F (θ, x, y, λ) is an analytic function defined on D, then we can expand F (θ, x, y, λ) into a Fourier-Taylor series Furthermore, we write and denote the norm of F by Here and below, • M stands for the supremum norm on the set M. For all above bounded real analytic functions, we can define a Banach space For F (θ, x, y, λ) ∈ C ω m (D), we denote the average of F (θ, x, y, λ) by [F (θ, x, y, λ)] = A∈S [F A (θ, x, y, λ)] where [F A (θ, x, y, λ)] represents the average of quasi-periodic function F A (θ, x, y, λ).Define a nonnegative weight function where ρ > 2 is a constant.Next, we define the strongly non-resonant conditions of the almost-periodic frequency ω For k ∈ Z Z S , we define the weight of its support Then the non-resonant conditions read where γ > 0, |k| = i |k i | and ∆ is some fixed approximation function.A function In the following we will give a criterion for the existence of strongly non-resonant frequencies.It is based on growth conditions on the distribution function for i ≥ 1 and t ≥ 0.
Lemma 2.1.There exist a constant N 0 and an approximation function Φ such that with a sequence of real numbers t i satisfying i log ρ−1 i ≤ t i ∼ i log ρ i for i large with some exponent ρ − 1 > 1.Here we say a i ∼ b i , if there are two constants c, C such that ca i ≤ b i ≤ Ca i and c, C are independent of i.
For a rigorous proof of Lemma 2.1, the reader is referred to Pöschel ( [19]) and Huang et al. ( [15]), we omit it here.According to Lemma 2.1, there exist an approximation function ∆ and a probability measure U on the parameter space R Z with support at any prescribed point such that the measure of the set of ω satisfying the following inequalities is positive for a suitably small γ, the proof can be found in Pöschel ( [19]), 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 a quasi-periodic function h(t) by h(ωt), for the sake of simplicity.

Main results
We work in extended phase space T Z × R Z × R 2 .The general question we focus on is what remains of the integrable bifurcations when perturbing to nearly integrable Hamiltonian families H(θ, J, x, y, λ) = N (J, x, y, λ) + P (θ, x, y, λ, ) and Hamiltonian families H(θ, J, x, y, λ) = M (J, x, y, λ) + P (θ, x, y, λ, ), ( where (θ, J, x, y) , Λ and Λ are closed and bounded neighbourhood of zero, with A, B, a and b being fixed constants.
We have the following theorems.
Theorem 2.1.Consider the Hamiltonian system (2.5).Suppose that (i) ω satisfies the non-resonant conditions (2.3); (ii) P (θ, x, y, λ, ) = O( ); Then for sufficiently small positive constant , there exists a close to identity transformation Φ which transforms the Hamiltonian system (2.5) into the following form such that (i) Φ is symplectic, real analytic for θ and C ∞ -smooth for (x, ỹ, λ); (ii) The new normal form N ∞ (J, x, ỹ, λ) has the same form as N (J, x, y, λ); Theorem 2.2.Consider the Hamiltonian system (2.6).Suppose that (i) ω satisfies the non-resonant condition (2.3); (ii) P (θ, x, y, λ, ) = O( ); Then for sufficiently small positive constant , there exists a close to identity transformation Φ which transforms the Hamiltonian system (2.6) into the following form such that (i) Φ is symplectic, real analytic for θ and C ∞ -smooth for (x, ỹ, λ); (ii) The new normal form M ∞ (J, x, ỹ, λ) has the same form as M (J, x, y, λ); Remark 2.1.The formulations of Theorems 2.1 and 2.2 are based on KAM theory on infinite dimensional invariant tori for almost-periodic forced Hamiltonian systems.For the proofs we use the overall strategy of [21], which however has to be substanially developed in techniques to deal with (2.5) and (2.6).In addition, Theorem 2.1 can be regarded as a generalization of the work in [6] from the case of quasi-periodic perturbations to the case of almost-periodic perturbations and Theorem 2.2 can be regarded as a generalization of the work in [21] from 1-dimensional almost-periodic systems to 2-dimensional almost-periodic systems.
Remark 2.3.A motivating example in practical application background besides its mathematical interest is the almost-periodic forced Duffing-van der Pol oscillators.For example, the nonlinear forced oscillator where x, t ∈ R and > 0 is a small parameter, f (t, x, ) is real analytic with regard to all variables and almost-periodic in t with frequency vector ω where P (θ, x, ) = − x 0 f (θ, u, )du.According to Theorem 2.2, almost-periodic response solutions exist when e P ( ) = 0.

Proof of Theorem 2.1
In this section, we give the proof of Theorem 2.1 detailedly.Taylor expansion of each function in (2.5) is regard as the infinite sum of quasi-homogeneous polynomials in (x, y, λ) with weight (l − 1, 2; 2l − 2, • • • , 2, l + 1).For given s, r > 0, we define the set We define the norm where 0 < w ≤ m * is a fixed constant, and the weights at the individual lattice are defined by such that defined on D(s ν+1 , r ν+1 ) and conditions ( l.1) and ( l.2) are fulfilled by replacing l by ν + 1. Moreover , Proof.We divide the proof into several parts.A. Solving linear homological equations.P ν (θ, x, y, λ ν ) admits spatial expansion Then, we truncate P ν (θ, x, y, λ ν ) = P ν (θ, x, y, λ ν ) + P ν (θ, x, y, λ ν ) with and Then, we have the following estimate In the following, we will look for a coordinate transformation φ ν defined in a domain D ⊂ D(s ν , r ν ), which is written as the time-1-map of the flow X t Fν of a Hamiltonian vector field X Fν : Fν | t=1 with respect to t at 0 using Taylor's formula.Recall that the Poisson bracket of G and F ν evaluated at X t Fν .Thus, we may write The point is to find F ν such that N ν + <2l {[ P ν ]} = N * ν is again a normal form.This amounts to solving the linear equation for F ν .Then the Hamiltonian H ν becomes where Now we solve homological equation (3.1).For each A ∈ S, we have For each suppk ⊂ A, the same order of quasi-homogeneous polynomial of function F ν A can be defined inductively by where Nν = N ν − ω, J .Thus, we obtain According to the above computation, one can check that F ν is independent of J.By Cauchy's estimate ν ), we have ν ) e mν [A] ≤ C 0 σ ν δ ν q (2l+ α 2 )ν .
as long as |h| ≤ 1 and |h| + i + j ≤ 1.In particular, we can conclude that these all vanish for weighted order (i, j, h) ≤ 2l.This completes the proof.

Outline of proof of Theorem 2.2
In this section, we only give the main points on the proof of Theorem 2.2, the details of which are similar to the proof of Theorem 2.1.In other words, this section is a road map through technical aspects of proof.In the proof of Theorem 2.2, the weight of quasi-homogeneous polynomials is different from that in proof of Theorem 2.1.
In this proof, Taylor expansion of each function in (2.6) is regard as the infinite sum of quasi-homogeneous polynomials in (x, y, λ) with weight (l, 2; 2l − 2, • • • , 4).For given s, r > 0, we define the set D(s, r) = T We use KAM iteration to prove Theorem 2.2.At the ν-th step in KAM, Hamiltonian (2.6) becomes where One can use a coordinate transformation defined in a domain D ⊂ D(s ν , r ν ), which is written as the time-1-map of the flow X t Fν of a Hamiltonian vector field X Fν : In what follows, we give how to apply singular theory to renormalize the normal form M * ν .Used transformation ϕ ν 1 , ϕ ν 2 defined in j=1 Q ν j ( λν ) aν +2 A∈S P ν A,0200 y j−1 , and ϕ ν 2 :    y 2 = y 1 + (l−1)!bν+1 G ν l−1 ( λν ), ν can be renormalized into M ν+1 .In concretely, readers can see the proof in [5].