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Existence of quasi-periodic solutions of the real pendulum equation
Chaos, Solitons & Fractals 62-63 (2014) 23–33
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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Existence of quasi-periodic solutions of the real pendulum equation Lin Lu, Xuemei Li ⇑ Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
a r t i c l e
i n f o
Article history: Received 26 February 2014 Accepted 26 March 2014 Available online 22 April 2014
a b s t r a c t x ¼ ax dx_ ð1 þ f0 cos x1 tÞ sin x þ f1 sin x2 t is considered in The pendulum equation € this paper, where f0 ; f1 and d are small real parameters, the ratio of x1 and x2 is irrational, and frequencies x1 and x2 satisfy the Diophantine condition. The unperturbed system ðf0 ¼ f1 ¼ d ¼ 0Þ has several fixed points for different parameter a. We use KAM theory to prove that the perturbed system possesses quasi-periodic solutions in neighborhoods of those fixed points. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Consider the real pendulum equation
€x ¼ ax dx_ ð1 þ f0 cos x1 tÞ sin x þ f1 sin x2 t;
ð1Þ
where x represents the angular displacement with respect to the vertical direction, a 2 R1 corresponds to spring modulus, d > 0 corresponds to damping, f0 cos x1 t and f1 sin x2 t correspond to the parametric excitations and external excitations, respectively. Meanwhile, f0 ; f1 and d are small real parameters, the ratio of x1 and x2 is irrational. The pendulum equation is a very famous physical model frequently appeared in many textbooks and papers, and it is important in modeling engineer systems such as tall and slender structures in earthquakes [12] and Josephson junction circuit [30]. For a ¼ f0 ¼ 0, D’humieres et al. [10] use numerical simulation to study the behavior of chaos and the path to chaos of the system (1), and show there exist symmetrybreaking of periodic orbits, breaking behavior of periodic orbits and period-doubling-3 bifurcations in the system. ⇑ Corresponding author. Tel.: +86 073188838037. E-mail addresses: [email protected] (L. Lu), lixuemei_1@sina. com (X. Li). http://dx.doi.org/10.1016/j.chaos.2014.03.003 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
For the case with a ¼ f1 ¼ 0, the Eq. (1) has been extensively studied. The rotating periodic solutions and the behavior of chaos of the system have been discussed in [5,6,11]. Clifford and Bishop [5] employ two approximate techniques to predict the escape zone in which there is no major stable non-rotating orbits for the system (1). Garira and Bishop [11] divide the steady state rotating orbits of the Eq. (1) into four categories, i.e., purely rotating orbits, oscillating rotating orbits, straddling rotating orbits and large amplitude rotating orbits, and they analyse the complex properties of these orbits. Leven and Koch [22] investigate the temporal behavior of the Eq. (1) by the numerical solution. Later, using Melnikov method and averaging methods, Koch and Leven [21] discuss the boundaries of subharmonic and homoclinic bifurcations of the system, and verify some theoretical analysis by numerical simulations. In [23], the authors present an experimental study of periodic and chaotic type aperiodic motions of the system (1), and demonstrate the coexistence of periodic solutions and chaotic solutions and various transitions between them. Recently, Jing and Yang [16,17] discuss bifurcation and chaos of the system (1) under periodic perturbations and quasi-periodic perturbations, respectively. Later, by Melnikov method and numerical simulations, Chen et al. [8] investigate the Eq. (1) with a phase shift and obtain criteria
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on the existence of chaos under periodic and quasiperiodic perturbations, and expose complex dynamical behaviors, including the sudden conversion of chaos to period orbits, jumping behaviors of periodic orbits, interior crisis, and so on. Numerical simulations in [17] reveal that the Eq. (1), under quasi-periodic perturbations, possesses the following dynamical phenomena: interleaving occurrence of chaotic behaviors and quasi-periodic behaviors, jumping behavior of quasi-periodic sets, different nice quasi-periodic attractors, and the coexistence of three quasi-periodic sets. These phenomena show that the Eq. (1) should have quasi-periodic solutions for some parameter values. Our aim in this paper is theoretically to verify the speculation above. More precisely, since f0 ; f1 and d are small real parameters in (1), we assume that they are comparable, and set d ¼ ~ d; f0 ¼ f~0 and f1 ¼ f~1 . With ~ d; f~0 and f~1 again denoted by d; f0 and f1 , respectively, letting x_ ¼ y, we can easily write (1) as
8 > < x_ ¼ y; y_ ¼ ða 1Þx dy ðf0 cos x1 tÞx þ f1 sin x2 t > : þð1 þ f0 cos x1 tÞðx sin xÞ:
ð2Þ
We shall prove that for sufficiently small the system (2) has invariant tori in a neighborhood of each fixed point of the unperturbed system for most parameter values a in the sense of Lebesgue measure. To end this, we shall write (2) formed
z_ ¼ AðaÞz þ gðhÞ þ Q ðhÞz þ hðz; h; Þ;
h_ ¼ x;
ð3Þ
2
where z 2 R ; x ¼ ðx1 ; x2 Þ; AðaÞ is an 2 2 matrix depending on a parameter a, h ¼ Oðz2 Þ; gðhÞ; Q ðhÞ and hðz; h; Þ are real analytic with respect to h on a strip of the torus T2 with width r > 0; hðz; h; Þ is real analytic with respect to z on the domain jzj 6 b; b > 0. By Newton iteration and KAM techniques, the system (3) can be transformed into
~z_ ¼ A1 ðaÞ~z þ h1 ð~z; h; aÞ;
h_ ¼ x;
where h1 ¼ Oð~z2 Þ. Obviously, ~z ¼ 0 is a solution of the equation, and this shows that the original system (3) possesses a quasi-periodic solution with the basic frequency x. It is well known that the KAM theory is a powerful tool for discussing the existence and linear stability of invariant tori of both Hamiltonian Systems [1,3,14,20,27,28,32] and dissipative systems [4,7,9,18,25]. Cong et al. [7] discuss the existence of quasi-periodic solutions of (3) when the eigenvalues of A are fixed and frequency x varies in some set. Using the method of accelerated convergence and asymptotic solution technique, they show for most x 2 P (in the sense of Lebesgue measure), the system (3) possesses a positive quasi-periodic solution with basic frequency x. Jorba and Simo [18] study the system (3) in the case where the eigenvalues of A and frequency x are fixed, and the size of small perturbations varies in some set. They show that under appropriate hypotheses, if the eigenvalues of A and frequency x satisfy the non-resonance conditions, then there exists a Cantorian set e ða; bÞ with positive Lebesgue measure such that for sufficiently small
belonging to the Cantorian set e, (3) possesses a quasiperiodic solution. Jorba and Simo [19] give a positive measure reducibility result for the linear differential system
z_ ¼ Az þ Q ðhÞz;
h_ ¼ x:
ð4Þ
They prove that if Q is real analytic with respect to h on a strip of the torus T2 with width r > 0, under suitable hypotheses on non-resonance and non-degeneracy with respect to sufficiently small , there exists a Cantorian set e ða; bÞ with positive Lebesgue measure such that for all 2 e, (4) can be reduced into a linear system with constant coefficients. Li and Zhu [26] also consider the reducibility of this system in finitely differentiable case. For the delay differential equation
z_ ðtÞ ¼ AzðtÞ þ Bzðt sÞ þ f ðzðtÞ; zðt sÞ; xt; aÞ; where eigenvalues are fixed, Li and Llave [24] obtain quasiperiodic solutions of this equation by KAM theory. Obviously, in [7,18,19,24,26], there are few discussion about the situation when the constant matrix A depends on parameters. Her and You [15] analyse the reducibility of the system
z_ ¼ AðaÞz þ Q ðhÞz;
h_ ¼ x;
where the matrix AðaÞ depends on a parameter a. They mainly contribute to improve the measure of the reducible set. Yuan [33] constructs the quasi-periodic breathers for networks of weakly coupled oscillators via KAM techniques, where both frequencies x and perturbed terms depend on parameters. Li and Yuan [25] obtain quasiperiodic solutions for autonomous delay differential equations by KAM theory, where both frequencies x and eigenvalues rely on parameters. In this paper, the system (2) is a dissipative system and can be written in a normal form, where the constant matrix depends on the parameter a, so that we obtain quasi-periodic solutions by KAM techniques. Some of ideas seeking the quasi-periodic solutions in this paper could be found in [9,13,25,28,33], although they deal with slightly different problems and models. This paper is arranged as follows. In Section 2, we give some main results and outline ideas of proofs. In Section 3, we construct and prove an iterative lemma which is important in the proofs of our results. In Section 4, we give a KAM theorem which is used to obtain the quasi-periodic solutions of (2). In Appendix A, we list some lemmas used in this paper. 2. Main results For the unperturbed system with d ¼ f0 ¼ f1 ¼ 0, we can calculate directly and get the following for x 2 ð2p; 2pÞ [16]. (i) if a < 1, then the system (2) possesses a unique fixed point Oð0; 0Þ which is a saddle point. (ii) if 1 < a < 0, then the system (2) possesses three fixed points: a center point Oð0; 0Þ and two saddle points C 1 ða1 ; 0Þ; C 2 ða1 ; 0Þ, where a1 is the positive root of the equation sin x ¼ ax in the interval ð0; p2 Þ.
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(iii) if 0 < a < 0:217234, then the system (2) possesses five fixed points: three center points Oð0; 0Þ; C 3 ða2 ; 0Þ; C 4 ða2 ; 0Þ and two saddle points C 5 ða3 ; 0Þ; C 6 ða3 ; 0Þ, where a3 and a2 are the positive roots of the equation sin x ¼ ax in the interval ðp; 32pÞ and ð32p ; 2pÞ, respectively. (iv) if a > 0:217234, then the system (2) possesses a unique fixed point Oð0; 0Þ which is a center.
N i ðhÞði ¼ 1; 2; 3Þ are real analytic in h 2 UðrÞ and sufficiently smooth in a 2 P, and hðz; hÞ; pi ðw; hÞði ¼ 1; 2; 3Þ are real analytic in the domain Wðb; rÞ (its definition given in Section 3) and sufficiently smooth in a 2 P. Notice that 0; a1 ; a2 ; a3 are roots of the equation sin x ¼ ax, where a1 2 ð0; p2 Þ, a3 2 ðp; 32pÞ; a2 2 ð32p ; 2pÞ, the eigenvalues of matrices AðaÞ and Bi ðaÞ ði ¼ 1; 2; 3Þ are as follows: (1) for a < 1,
For convenience, letting
x ; z¼ y
h¼
h1 ; h2
x¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ a 1; a 1 ;
x1 ;
x2
(2) for 1 < a < 0,
the Eq. (2) can be written as
(
n pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ 1 a þ 1; 1 a þ 1 ;
h_ ¼ x; z_ ¼ AðaÞz þ gðhÞ þ Q ðhÞz þ hðz; hÞ;
ð5Þ SpecB1 ðaÞ ¼ SpecB1 ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a1 a; cos a1 a ; ¼
where
AðaÞ ¼
0
1
a 1 0
Q ðhÞ ¼ hðz; hÞ ¼
;
gðhÞ ¼
0
0
f0 cos h1
d 0
0
f1 sin h2
;
(3) for 0 < a < 0:217234,
ð6Þ
n pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ 1 a þ 1; 1 a þ 1 ;
;
SpecB3 ðaÞ ¼ SpecB3 ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a3 a; cos a3 a ; ¼
ð1 þ f0 cos h1 Þðx sin xÞ
ð7Þ
:
SpecB2 ðaÞ ¼ SpecB2 ðaÞ npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio ¼ 1 cos a2 þ a; 1 cos a2 þ a ;
For the fixed point C 1 ða1 ; 0Þ, taking a translational transformation x ¼ a1 þ u; y ¼ v , setting w ¼ ðu v ÞT (here T represents transpose), using Taylor’s formula to sinða1 þ uÞ and the fact sin a1 ¼ aa1 , we can rewrite (2) in the form
(
h_ ¼ x; _ ¼ B1 ðaÞw þ j1 ðhÞ þ N 1 ðhÞw þ p1 ðw; hÞ; w 1 ; 0
0 B1 ðaÞ ¼ a cos a1 0 j1 ðhÞ ¼ ; f1 sin h2 f0 cos h1 sin a1 N1 ðhÞ ¼
0 f0 cos h1 cos a1
0 ; d
n pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ 1 a þ 1; 1 a þ 1 :
ð8Þ
where
(4) for a > 0:217234,
ð9Þ
When the eigenvalues of the matrix are complex and different from each other, we can make a complex linear transformation such that the coefficient matrix is diagonal. Taking the matrix AðaÞ as an example, for a > 1, by means of a complex linear transformation z ¼ SðaÞz0 , the system (5) is changed into
( 2
p1 ðw; hÞ ¼ Oðw Þ: ð10Þ
Similarly, for the fixed points C 2 ða1 ; 0Þ; C 3 ða2 ; 0Þ; C 4 ða2 ; 0Þ; C 5 ða3 ; 0Þ and C 6 ða3 ; 0Þ, we take translational transformations and write (2) in the form of (8), with a1 ; a2 ; a2 ; a3 and a3 instead of a1 , and denote the constant matrix by B1 ðaÞ, B2 ðaÞ; B2 ðaÞ; B3 ðaÞ and B3 ðaÞ, respectively. Definition 2.1. We say a function f ðhÞ is real analytic in a strip of width r, if f ðhÞ is analytic in the complex strip b 2 : jImhj :¼ maxi¼1;2 jImhi j 6 rg and real for UðrÞ ¼ fh 2 T the real argument h. It is easy to see that AðaÞ and Bi ðaÞði ¼ 1; 2; 3 corresponding to a1 ; a2 ; a3 instead of a1 in (8)) are sufficiently smooth in a in some bounded closed set P (which we will consider), the functions gðhÞ; Q ðhÞ; ji ðhÞ;
h_ ¼ x; z_0 ¼ A0 ðaÞz0 þ g 0 ðh; aÞ þ Q 0 ðh; aÞz0 þ h0 ðz0 ; h; aÞ;
ð11Þ
npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi where A ðaÞ ¼ diag k01 ðaÞ; k02 ðaÞ :¼ diag 1 a þ 1; pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi0 1 a þ 1g,
SðaÞ ¼
pffiffiffiffiffi 1ffi pffiffiffiffiffiffi aþ1
1
pffiffiffiffiffi ! 1ffi pffiffiffiffiffiffi aþ1 ;
1
g 0 ðh; aÞ ¼
1 2
f1 sin h2 f1 sin h2
;
ð12Þ
0 1 pffiffiffiffiffi pffiffiffiffiffi 1ffi 1ffi d f0 cos h1 pffiffiffiffiffiffi d 1 @ f0 cos h1 pffiffiffiffiffiffi aþ1 aþ1 A; Q 0 ðh; aÞ ¼ pffiffiffiffiffi pffiffiffiffiffi 1ffi 1ffi 2 f0 cos h1 pffiffiffiffiffiffi d f0 cos h1 pffiffiffiffiffiffi d aþ1 aþ1 jjg 0 ðh; aÞjj 6 C ; jjQ 0 ðh; aÞjj 6 C ; h0 ðz0 ; h; aÞ ¼ Oðz20 Þ:
ð13Þ
For a < 1, we take a real linear transformation such that the coefficient matrix is diagonal. For convenience, we sometimes omit the parameter a in some functions. In order to consider the reality condition, we make some notations and remarks.
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For a matrix B ¼ ðbij Þ22 and a 2-dimensional vector u ¼ ðu1 ; u2 ÞT , we define B ¼ ðbli lj Þ, B ¼ ðbilj Þ and u ¼ ðu2 ; u1 ÞT (i.e., u1 ¼ u2 and u2 ¼ u1 ), where ðl1 ; l2 Þ ¼ ð2; 1Þ; 1 6 i; j 2. The complex conjugation F of a function F means taking complex conjugation of the coefficients in the power series of F. Obviously, g 0 ðhÞ; Q 0 ðhÞ; SðaÞ and h0 ðz0 ; hÞ in (12) and (13), because of k01 ¼ k02 , satisfy the following reality condition (see Sections 15 and 27 in [31]). Reality condition S ¼ S; g 0 ðhÞ ¼ g 0 ðhÞ; h0 ðz0 ; hÞ ¼ h0 ðz0 ; hÞ; Q 0 ðhÞ ¼ Q 0 ðhÞ. 2.1. Outline of scheme We only discuss the fixed point O here, and for other fixed points, the discussion is similar. To obtain quasi-periodic solutions of (11) in a neighborhood of the fixed point O, we make some changes to simplify the system (11) by Newton iteration and KAM techniques. Firstly, by means of a family of quasi-periodic changes of variables, we want to eliminate the terms g 0 ðhÞ and Q 0 ðhÞ. More exactly, taking the change of variables formed z0 ¼ z1 þ /ðh; aÞþ wðh; aÞz1 into (11) (such changes of variables have been considered in [9,25]), it implies
ðE þ wÞz_1 ¼ A0 z1 þ A0 / þ A0 wz1 þ g 0 þ Q 0 z1 þ Q 0 / þ Q 0 wz1 @/ @w þ h0 ðz1 þ / þ wz1 ; hÞ x xz1 : ð14Þ @h @h
ð15Þ
and
@w b 0 ð0Þ; Q b 0 ð0Þ ; x ¼ A0 w wA0 þ CK Q 0 diag Q 11 22 @h
ð16Þ
where CK denotes a truncation operator for the Fourier serb 0 ðkÞ denotes the kth-Fourier ies of functions (see [25]), Q ii coefficient of Q 0ii ðhÞ; Q 0ij ðhÞ is the element of the matrix Q 0 ðhÞ; 1 6 i; j 2. Once Eqs. (15) and (16) are solved, the system (14) can be written as h b 0 ð0Þ; Q b 0 ð0Þ z1 z_1 ¼ A1 z1 þ ðE þ wÞ1 Q 0 / þ Q 0 wz1 wdiag Q 11 22 i þðId CK Þg 0 þ ðId CK ÞQ 0 z1 þ h0 ðz1 þ / þ wz1 ;hÞ ; ð17Þ b 0 ð0Þ; Q b 0 ð0ÞÞ. where A1 ¼ A0 þ diagð Q Using the 11 22 Taylor’s formula to h0 ðz1 þ / þ wz1 ; hÞ, we write (17) as follows
z_1 ¼ A1 ðaÞz1 þ g 1 ðh; aÞ þ Q 1 ðh; aÞz1 þ h1 ðz1 ; h; aÞ;
ð18Þ
where 1
g 1 ðh; aÞ ¼ ðE þ wÞ ðQ 0 / þ ðId CK Þg 0 þ h0 ð/; h; aÞÞ; b 0 ð0Þ; Q b 0 ð0ÞÞ Q 1 ðh; aÞ ¼ ðE þ wÞ1 Q 0 w wdiagð Q 11 22
@h0 ð/; hÞ ðE þ wÞ ; þðId CK ÞQ 0 þ @z0
In this way, after n steps of iteration, the Eq. (11) will be formed as
(
h_ ¼ x; z_n ¼ An ðaÞzn þ g n ðh; aÞ þ Q n ðh; aÞzn þ hn ðzn ; h; aÞ:
If the norms of g n and Q n tend to zero with a superexponential velocity, then the composition of transformations is convergent, and the above equation converges to the form
(
h_ ¼ x; ~z_ ¼ A1 ðaÞ~z þ h1 ð~z; h; aÞ;
where h1 ð~z; h; aÞ ¼ Oð~z2 Þ. Obviously, ~z ¼ 0 is a solution of the equation, and this shows that for the original system (5), there exists a quasi-periodic solution with the basic frequency x. 2.2. The small divisor It is clear that in the process of solving (15) and (16), we meet small divisors and need the small divisor (or nonresonance) conditions
pffiffiffiffiffiffiffi c 1ðk; xÞj P s ; jkj pffiffiffiffiffiffiffi c jkj ðaÞ ki ðaÞ þ 1ðk; xÞj P s ; jkj jki ðaÞ
Thus, we derive the homological equations
@/ x ¼ A0 / þ C K g 0 @h
h1 ðz1 ; h; aÞ ¼ ðE þ wÞ1 h0 ðz1 þ / þ wz1 ; h; aÞ h0 ð/; h; aÞ
@h0 ð/; h; aÞ ðE þ wÞz1 : ð21Þ @z0
ð19Þ
ð20Þ
1 6 i; j 2;
where 0 – k 2 Z2 ; s > 2 is a constant. In order to solve equations like (15), we need the first inequality above, and to solve equations like (16), we need the second one. Noting that for a 2 ½a; b; a > 1, both k01 ðaÞ and k02 ðaÞ are pure imaginary, the non-resonance conditions could not be true for all a. Hence, we want to take out some small (in the sense of Lebesgue measure) open sets, such that the non-resonance conditions hold for the remaining set. On each step of the iteration, we need to remove the resonant set and control its measure. To bound the measure of the resonant set, we introduce the non-degeneracy conditions. Definition 2.2. We say a 2 2 matrix AðaÞ satisfies the non-degeneracy conditions on the interval K, if there exists a constant v > 0 such that
dðki ðaÞÞ da P v;
dðk1 ðaÞ k2 ðaÞÞ P v; da
uniformly hold for a 2 K, where ki ðaÞ is the eigenvalue of AðaÞ; i ¼ 1; 2. Using the above method, one important thing is that on each step when we bound the measure of the resonant set, we have to verify the non-degeneracy conditions. Therefore, we need a bit of words to illustrate that the nondegeneracy conditions are maintained along the iteration although the proof is simple.
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
In the following, we suppose the frequency x ¼ ðx1 ; x2 Þ is fixed and satisfies the Diophantine condition
jðk; xÞj P
c jkjs
;
80 – k 2 Z2 ;
where s > 2, and c > 0 is a sufficiently small constant. Based on the above description, we state the main results in this paper. Theorem 2.1. If a 2 ½a; b and b < 1, then there is a sufficiently small positive number 0 such that for all a 2 ½a; b and 0 < < 0 , the Eq. (2) possesses a real quasiperiodic solution in a neighborhood of the fixed point O.
m
1. m ¼ ð1þqÞ ; 0 ¼ , where q ¼ 13; P 2 2. m0 ¼ 0; mm ¼ ð12 þ þ m2 Þ=ð2 1 q¼1 q Þ; 3. rm ¼ ð1 mm Þr0 ; 4. bm ¼ ð1 mm Þb0 ; 5. jm ¼ rm rmþ1 ; 6. cm ¼ c=ðm þ 1Þ2 ; c0 ¼ c ¼ Oðs Þð0 < s 6 16Þ; 1 P v0 2 if is sufficiently 7. vm ¼ v0 C m1 q¼0 q (vm P 2 small); 1 P 2 8. fm ¼ f0 þ C m1 q¼0 q (fm 2f0 if is sufficiently small); 9. K m ¼ ðm þ 1Þ2 2m ln ; 10. gm ¼ 4fcm1 s; 0 Km 2 2 b 2 ¼ C2 =ð2pZÞ2 ; 11. T ¼ R =ð2pZÞ2 and T 12. For
Theorem 2.2. Suppose a 2 ½a; b and a > 1; b < 0. Then there is a sufficiently small positive number 0 such that for all a 2 ½a; b and 0 < < 0 , the Eq. (2) possesses two real quasi-periodic solutions which are in neighborhoods of fixed points C 1 and C 2 , respectively. Moreover, for given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ, there is a sufficiently small positive number 0 such that if 0 < < 0 , then there is a Cantorian subset P1 ½a; b with Lebesgue measure larger than b a OðcÞ, and for any a 2 P1 , the Eq. (2) possesses a real quasi-periodic solution in a neighborhood of the fixed point O. Theorem 2.3. Suppose a 2 ½a; b, where a > 0; b < a0 < 0:217234, or a > a0 > 0; b < 0:217234, and a0 satisfies the equality cos a2 ða0 Þ þ a0 þ a2 ða0 Þ sin a2 ða0 Þ ¼ 0. Then there is a sufficiently small positive number 0 such that for all a 2 ½a; b and 0 < < 0 , the Eq. (2) possesses two real quasi-periodic solutions which are in neighborhoods of fixed points C 5 and C 6 , respectively. Moreover, for given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ, there is a sufficiently small positive number 0 such that if 0 < < 0 , then there is a Cantorian subset P1 ½a; b with Lebesgue measure larger than b a OðcÞ, and for any a 2 P1 , the Eq. (2) possesses three real quasi-periodic solutions which are in neighborhoods of the fixed points O; C 3 and C 4 , respectively. Theorem 2.4. Suppose a 2 ½a; b and a > 0:217234. Then for given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ, there is a sufficiently small positive number 0 such that if 0 < < 0 , then there is a Cantorian subset P1 ½a; b with Lebesgue measure larger than b a OðcÞ, and for any a 2 P1 , the Eq. (2) possesses a real quasi-periodic solution which are in a neighborhood of the fixed points O.
3. Iterative lemma Theorems 2.1, 2.2, 2.3, 2.4 are proved by an iterative procedure. Therefore, let us introduce some iterative constants and notations in the following. Let r0 ; b0 ; v0 and f0 be positive constants. To simplify the notation, we shall denote by C a universal positive constant which is independent of the KAM iteration and may be different in different places. For all m P 1,
r > 0; b > 0, let
UðrÞ ¼
b 2 : jImhj :¼ maxjImhi j 6 r h2T i¼1;2
and
b 2 : jjzjj 6 b; jImhj 6 rg; Wðb; rÞ ¼ fðz; hÞ 2 C2 T 13. Pg ¼ fa 2 R1 : distða; PÞ < gg;
Remark 3.1. As we want to drop some resonant sets which are opened, and the remaining parameter sets are closed, in order to use the derivative to estimate the measure of resonant sets, we need to extend the remaining closed sets to an its small neighborhood, Li and de la Llave [24], Yuan [33].
14. If P # R1 is a parameter set with positive measure, given 0 < 1; F : Wðb; rÞ Pg ! C2 is analytic in ðz; hÞ 2 Wðb; rÞ and C 1 -smooth in a 2 Pg , and satisfies
jjFjjb;r;g :¼ max sup j@ sa Fðz; h; aÞj 6 C ; s¼0;1 Wðb;rÞPg
we set F ¼ Ob;r;g ðÞ, where the notation @ a denotes the partial derivative with respect to a; 15. If H : Wðb; rÞ Pg ! C2 is analytic in ðz; hÞ 2 Wðb; rÞ and C 1 -smooth in a 2 Pg , vanishes with its zderivative for z ¼ 0, and satisfies
sup j@ jz @ sa Hðz; h; aÞj 6 C;
Wðb;rÞPg
s ¼ 0; 1;
j ¼ 0; 1; 2;
we write H ¼ Ob;r;g ðz2 Þ.
Lemma 3.1. Suppose that there is a sequence of closed parameter sets R1 ½a; b ¼ P0 P1 Pl and a family of equations ðEqÞm ðm ¼ 0; 1; . . . ; lÞ defined in Wðbm ; rm Þ Pgmm ( h_ ¼ x; ðEqÞm : ð22Þ z_ ¼ Am ðaÞz þ g m ðh; aÞ þ Q m ðh; aÞz þ hm ðz;h; aÞ; m where Am ¼ diagfkm Assume that for 1 ðaÞ; k2 ðaÞg. 0; 1; 2; ; l, the following conditions are satisfied:
m¼
28
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
m (a.1) for a 2 Pgmm , the eigenvalues km 1 ðaÞ; k2 ðaÞ satisfy nondegeneracy conditions
m dðki ðaÞÞ v0 da P vm P 2 ; m dðk1 ðaÞ km v0 2 ðaÞÞ P vm P 2 ; da
i ¼ 1; 2
ð23Þ
m dðki ðaÞÞ da 6 fm 2f0 ; m dðk1 ðaÞ km 2 ðaÞÞ 6 fm 2f0 ; da
ð24Þ
m k ðaÞ km ðaÞ P v P v0 ; 1 2 m 2 m v k ðaÞ P v P 0 ; i ¼ 1; 2; i m 2
1
ð25Þ
(a.3) the term hm ðz; h; aÞ is analytic in ðz; hÞ 2 Wðbm ; rm Þ and C 1 -smooth in a 2 Pgmm , and the following estimate holds
hm ¼ Obm ;rm ;gm ðz2 Þ;
ð26Þ
(a.4) there is a constant C 0 > 0 such that the Lebesgue measure of Pm satisfies
MeasPm P MeasPm1 ð1 C 0 cm1 Þ; 1 6 m 6 l;
aÞ ¼
k02 ð
aÞ, then
km 1ð
in the form
h ¼ h;
i b l ð0Þ; Q b l ð0ÞÞz1 þ hl ðz1 þ / þ w z1 ; hÞ : wl diagð Q l l 11 22
b l ð0Þ; Q b l ð0ÞÞ and using Letting Alþ1 :¼ Al þ diagð Q 11 22 Taylor’s formula to hl ðz1 þ /l þ wl z1 ; hÞ, above equality is written as
where
Wðblþ1 ; rlþ1 Þ Plþ1 ! Wðbl ; rl Þ Pl T l : z ¼ z1 þ /l ðh; aÞ þ wl ðh; aÞz1 ;
h i b l ð0Þ; Q b l ð0Þ z1 þ ðE þ w Þ1 z_1 ¼ Al þ diag Q l 11 22 h ðId CK lþ1 Þg l þ ðId CK lþ1 ÞQ l z1 þ Q l /l þ Q l wl z1
aÞ ¼
gl
a ¼ a; ð28Þ
g lþ1 ðhÞ ¼ ðE þ wl Þ1 Q l /l þ ðId CK lþ1 Þg l þ hl ð/l ; hÞ ; b l ð0Þ; Q b l ð0ÞÞ Q lþ1 ðhÞ ¼ ðE þ wl Þ1 Q l wl wl diagð Q 11 22
@hl ð/l ; hÞ ðE þ wl Þ ; þðId CK lþ1 ÞQ l þ @z
ð29Þ
and the following estimates 2ðsþ1Þ
/l ¼ Orlþ1 ;glþ1 ðl c2 l jl 1 2
Þ;
2ðsþ1Þ Þ; l
wl ¼ Orlþ1 ;glþ1 ðl c2 l j
ð30Þ
such that by the change of variable T l , the ðEqÞl is transformed into
ðEqÞlþ1
8 _ > < h ¼ x; : z_1 ¼ Alþ1 ðaÞz1 þ g lþ1 ðh; aÞ þ Q lþ1 ðh; aÞz1 > : þhlþ1 ðz1 ; h; aÞ ð31Þ
ð35Þ
ð36Þ
hlþ1 ðz1 ; hÞ ¼ ðE þ wl Þ1 hl ðz1 þ /l þ wl z1 ; hÞ
where z1 is a new variable, and /l ðh; aÞ; wl ðh; aÞ are glþ1 analytic in h 2 Uðrlþ1 Þ and C 1 -smooth in a 2 Plþ1 , satisfy the reality condition
wl ¼ wl
ð34Þ
Inserting (33) and (34) into (32), we get
z_1 ¼ Alþ1 z1 þ g lþ1 ðhÞ þ Q lþ1 ðhÞz1 þ hlþ1 ðz1 ; hÞ;
Then there exists a closed subset Plþ1 Pl and a change of variable glþ1
@wl b l ð0Þ; Q b l ð0ÞÞ: x ¼ Al wl wl Al þ CK lþ1 Q l diagð Q 11 22 @h
ð27Þ
km 2 ðaÞ, g m ¼ g m ; Q m ¼ Q m , and hm ðz; h; aÞ ¼ hm ðz; h; aÞ.
/l ¼ /l ;
ð33Þ
and
Q m ¼ Orm ;gm ðm 2 Þ;
(a.5) (Reality condition) if
Suppose that /l and wl are the solution of the following homological equations
@/l x ¼ Al /l þ CK lþ1 g l @h
(a.2) the terms g m ðh; aÞ and Q m ðh; aÞ are analytic in h 2 Uðrm Þ and C 1 -smooth in a 2 Pgmm , and the following estimates hold
k01 ð
Proof. To simplify notations, sometimes we shall drop the parameter a from functions whenever there is no confusion. Let E be the identity matrix. Substituting (28) into (22) with m ¼ l, we obtain the following equation
ðE þ wl Þz_1 ¼ Al z1 þ Al /l þ Al wl z1 þ g l þ Q l z1 þ Q l /l þ Q l wl z1 @/ @w þ hl ðz1 þ /l þ wl z1 ; hÞ l x l xz1 : @h @h ð32Þ
and conditions
g m ¼ Orm ;gm ðm Þ;
and the conditions (a.1)–(a.5) are satisfied with m being replaced by l þ 1 and ðz; hÞ by ðz1 ; hÞ, respectively.
hl ð/l ; hÞ
@hl ð/l ; hÞ ðE þ wl Þz1 : @z
ð37Þ
In order to estimate g lþ1 ; Q lþ1 and hlþ1 , we first solve the homological Eqs. (33) and (34), and estimate their solutions. Let /il and g il be the ith element of /l and g l , respectively, i ¼ 1; 2. Expanding g il ; /il into Fourier series in h and truncating them by operator CK lþ1
X
CK lþ1 g il ¼
b g il ðkÞe
pffiffiffiffiffi 1ðk;hÞ
;
jkj6K lþ1
CK lþ1 /il ¼
X
pffiffiffiffiffi 1ðk;hÞ
b i ðkÞe / l
;
jkj6K lþ1
b i ðkÞ depend on where the Fourier coefficients b g il ðkÞ and / l the parameter a, the Eq. (33) implies
b i ðkÞ ¼ / l
kli ð
b g il ðkÞ pffiffiffiffiffiffiffi ; aÞ 1ðk; xÞ
jkj 6 K lþ1 ;
i ¼ 1; 2:
ð38Þ
29
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
Since g l ðhÞ is analytic in h 2 Uðrl Þ and C 1 -smooth in
a 2 Pgl l ,
j@ sa b g il ðkÞj 6 max s¼0;1
sup j@ sa g il jejkjrl ;
s ¼ 0; 1:
gl
ð39Þ
Uðrl ÞP
ðE þ wl Þ1 ¼ E wl þ w2l w3l þ ;
l
Set
pffiffiffiffiffiffiffi
Plþ ¼ a 2 Pl : jkli ðaÞ 1ðk; xÞj P
cl jkj
s;
0 < jkj 6 K lþ1 ; i ¼ 1; 2 :
g
Then for a 2 Plþlþ1 , there is a0 2 Plþ ja a0 j < glþ1 , and for 0 < jkj 6 K lþ1 , we get
jkli ðaÞ
such that
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1ðk; xÞj ¼ jkli ða0 Þ 1ðk; xÞ þ kli ðaÞ kli ða0 Þj P
cl
jkjs
2f0 glþ1 P
cl
2jkjs
i ¼ 1; 2
;
2s i 2 b i ðkÞj 6 C maxj@ s b maxj@ sa / l a g l ðkÞjjkj =cl s¼0;1
6
c
q¼0
for l 2 N. So it is easy to see that hlþ1 satisfies the condition (a.3) with m ¼ l þ 1 by the equality (37). From (26) with m ¼ l and (30) which has just been proved, it follows that
max s¼0;1
j@ sa /j 6
sup g
Uðrlþ1 ÞPlþlþ1
a 2 Plþ :
6C
X
s¼0;1
CK lþ1 Þg l jjrlþ1 ;glþ1 þ jjhl ð/l ;hÞjjrlþ1 ;glþ1 Þ
g
jjg l jjrl ;gl ejkjðrl rlþ1 Þ jkj2s =c2l 6 C
3 2
l c2l j2l sþ2
elements of Q l and wl , respectively. Expanding Q lij ; wlij into Fourier series in h and truncating them, the Eq. (34) implies
b l ðkÞ ¼ w ij
b Q l ðkÞ
pffiffiffiffiffi ; klj ðaÞkli ðaÞþ 1ðk;xÞ ij
ð41Þ if k – 0; i ¼ j
for jkj 6 K lþ1 . Set
pffiffiffiffiffiffiffi
Plþ1 ¼ fa 2 Plþ : jklj ðaÞ kli ðaÞ þ 1ðk; xÞj P
cl
jkjs
;
0 < jkj 6 K lþ1 ; 1 6 i; j 6 2g: g
lþ1 Then for a 2 Plþ1 ; 0 < jkj 6 K lþ1 , similarly, by Condition (a.1) we have
pffiffiffiffiffiffiffi c l kj ðaÞ kli ðaÞ þ 1ðk; xÞ P l s ; 2jkj
1 6 i; j 6 2:
bl s bl 2s 2 max @ sa w ij ðkÞ 6 Cmax @ a Q ij ðkÞ jkj =cl s¼0;1 s¼0;1 6 C Q lij ejkjrl jkj2s =c2l ;
lþ1 a 2 Pglþ1 :
Hence, by Condition (a.2) and Lemma A.1 again, it implies
jjwl jjrlþ1 ;glþ1
X b l ðkÞj ejkjrlþ1 max sup j@ sa w 6 jkj6K lþ1
6C
X k2Z2
s¼0;1
jjQ l jjrl ;gl ejkjðrl rlþ1 Þ jkj2s =c2l 6 C
¼ Orlþ1 ;glþ1 ðjj/l jjrlþ1 ;glþ1 Þ:
1 2
l : c2l j2l sþ2 ð43Þ
!
ð45Þ
By (36), (30), (45) and Lemma A.2, it implies
jjQ lþ1 jjrlþ1 ;glþ1 6 jjðE þ wl Þ1 jjrlþ1 ;glþ1 b l ð0Þ; Q b l ð0Þ jj jjQ l wl wl diag Q rlþ1 ;glþ1 11 22 @hl ð/l ; hÞ þ jjðId CK lþ1 ÞQ l jjrlþ1 ;glþ1 þ @z rlþ1 ;glþ1 jjE þ wl jjrlþ1 ;glþ1 6 C jjQ l jjrlþ1 ;glþ1 jjwl jjrlþ1 ;glþ1 þjjQ l jjrl ;glþ1 K 2lþ1 ejl K lþ1 þ jj/l jjrlþ1 ;glþ1
l c2l j2l sþ2
1
6 C 2lþ1 :
Thus, we verify (a.2) with m ¼ l þ 1. In the following, we verify that the non-degeneracy conditions are maintained in the iterative process. Noting b l ð0Þ; Q b l ð0ÞÞ, we have that Alþ1 ¼ Al þ diagð Q 11 22 lþ1 l l bl bl klþ1 1 ðaÞ k2 ðaÞ ¼ k1 ðaÞ k2 ðaÞ þ Q 11 ð0Þ Q 22 ð0Þ:
Since
g
lþ1 a2Plþ1
2 þ 4 4l sþ4 cl jl
Using (a.3) with m ¼ l, we obtain
@hl ð/l ; hÞ @z
6C
ð42Þ
K 2lþ1 ejl K lþ1 lþ1
6 C lþ1 :
Differentiating (41) in a and using (24), we obtain
rl ;gl
l
rlþ1 ;glþ1
if i – j
> b > Q l ðkÞ > : pffiffiffiffiffiij ; 1ðk;xÞ
l þ jjg l jjr ;g c2l jl2sþ2
6C
ð40Þ
:
Next we solve (34). Let Q lij and wlij ð1 6 i; j 6 2Þ be the
8 > > > <
ð44Þ
:
6 CðjjQ l jjrlþ1 ;glþ1 jj/l jjrlþ1 ;glþ1 þ jjðId
a2Plþlþ1
k2Z2
!
þ hl ð/l ; hÞjjrlþ1 ;glþ1
b l ðkÞjÞejkjrlþ1 ðmax sup j@ sa /
jkj6K lþ1
2l 4 4sþ4 cl jl
Now we estimate g lþ1 ðh; aÞ and Q lþ1 ðh; aÞ. By (35), (44) and Lemma A.2, we have
Hence, by Condition (a.2) and Lemma A.1 (see Appendix A), we obtain
X
1 X jjwl jjqrlþ1 ;glþ1 6 C
jjðE þ wl Þ1 jjrlþ1 ;glþ1 6
jjg lþ1 jjrlþ1 ;glþ1 6 jjðE þ wl Þ1 jjrlþ1 ;glþ1 jjQ l /l þ ðId CK lþ1 Þg l
glþ1
Cjjg il jjrl ;gl ejkjrl jkj2s = 2l ;
we get
hl ð/l ; hÞ ¼ Orlþ1 ;glþ1
by Condition (a.1). Differentiating (38) in a and using (24), we have s¼0;1
Thus, noting Plþ1 Plþ , (40) and (43) imply the estimates (30). Since
bm bm max Q 11 ð0Þ ; Q 22 ð0Þ gm
gm
1 2
6 jjQ m jjrm ;gm 6 C m ;
m ¼ 0; 1; . . . ; l;
30
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33 1
we obtain
g lþ1 ðhÞ ¼ ðE þ wl ðhÞÞ ðQ l ðhÞ/l ðhÞ þ ðId CK lþ1 Þg l ðhÞ þ hl ð/l ; hÞÞ ¼ ðE þ wl ðhÞÞ1 Q l ðhÞ/l ðhÞ þðId CK lþ1 Þg l ðhÞ þ hl ð/l ; hÞ :
d klþ1 ðaÞ klþ1 ðaÞ 0 0 l X 1 1 2 P d k1 ðaÞ k2 ðaÞ C 2q da da q¼0 P vlþ1 P
1 v 2 0
glþ1
for a 2 Plþ1 . Similarly, we can prove other inequalities in Condition (a.1) with m ¼ l þ 1. To estimate the measure of Plþ1 , we introduce some sets. Let
c pffiffiffiffiffiffiffi Rlijk ¼ a 2 Pl : klj ðaÞ kli ðaÞ þ 1ðk; xÞ < ls ; 0 < jkj 6 K lþ1 ; jkj ð46Þ
[
Rlk ¼
Rlijk ;
16i;j2
V lik ¼
[
Rl ¼
Rlk ;
0–jkj6K lþ1
pffiffiffiffiffiffiffi
a 2 Pl : kli ðaÞ 1ðk; xÞ <
cl jkj
; 0 < jkj 6 K lþ1 ; s
By direct calculation, we can easily verify g lþ1 ðhÞ ¼ g lþ1 ðhÞ. The proof of Lemma 3.1 is completed. h Remark 3.2. If the eigenvalues of the initial matrix A0 ðaÞ of (22) with m ¼ 0 are nonzero, real and different from each other, and the terms g 0 ðh; aÞ; Q 0 ðh; aÞ; h0 ðz; h; aÞ are real analytic in h, then the iterative Lemma 3.1, removing Conditions (a.1), (a.4) and (a.5), can be true for all a 2 ½a; b as there is no small divisor problem and we do not need to drop any a. The proof is trivial (see [2]), and we only point out the main ideas here. Since the frequency vector x satisfies the Diophantine condition jðk; xÞj P jkjcs ; 80 – k 2 Z2 , the inequalities
pffiffiffiffiffiffiffi c 0 ki ðaÞ 1ðk; xÞ P s ; jkj
ð47Þ V lk
¼
[
V lik ;
[
l
V ¼
i¼1;2
V lk :
Then
[
min V l Þ;
l ¼ 0; 1; 2; . . .
By Lemma A.3 and the non-degeneracy conditions (23), we obtain
MeasRlijk 6
4cl MeasPl : v0 jkjs
l
MeasR 6
16cl MeasPl ; v0 jkjs 16
v0
X
cl MeasPl
;
6 i; j 2;
min jk01 ðaÞ k02 ðaÞj; min jk0i ðaÞj; i ¼ 1; 2 :¼ r0 > 0
a2½a;b
a2½a;b
and
b 0 ð0Þ; Q b 0 ð0Þ þ diag Q b 1 ð0Þ; Q b 1 ð0Þ Alþ1 ¼ A0 þ diag Q 11 22 11 22 b l ð0Þ; Q b l ð0Þ ; þ þ diag Q 11 22 for each positive integer l, we can conclude that kl1 ðaÞ and kl2 ðaÞ are real, and satisfy
Hence,
MeasRlk 6
c jkjs
obviously hold for all a 2 ½a; b. Since
0–jkj6K lþ1
Plþ1 ¼ Pl n ðRl
P
pffiffiffiffiffiffiffi 0 kj ðaÞ k0i ðaÞ þ 1ðk; xÞ
min
jkj
s
6 C 0 cl MeasPl :
0–k2Z2
Similarly, we get
MeasV l 6 C 0 cl MeasPl : Thus,
MeasPlþ1 P MeasPl MeasRl MeasV l P MeasPl ð1 C 0 cl Þ and (a.4) is available with m ¼ l þ 1. Finally, we consider the reality condition by induction. Without loss of generality, we only verify the reality condition for g lþ1 ðh; aÞ as for the terms Q lþ1 ðh; aÞ and hlþ1 ðz1 ; h; aÞ, the proof is similar. We have proved (a.5) with m ¼ 0 in Section 2. Assuming that the reality condition holds for m 6 l, we shall verify that the reality condition is satisfied when m ¼ l þ 1. By (35), the uniqueness of the solution of (33) and (34), and the reality condition (a.5) with m ¼ l, we have
min kl1 ðaÞ kl2 ðaÞ ; min kli ðaÞ ; i ¼ 1; 2
a2½a;b
P r0 C
a2½a;b
l1 X
1 2
q P
q¼0
r0 ; 2
by (25). Hence,
pffiffiffiffiffiffiffi c l ki ðaÞ 1ðk; xÞ P ls jkj and
pffiffiffiffiffiffiffi c l kj ðaÞ kli ðaÞ þ 1ðk; xÞ P ls ; jkj hold for all a 2 ½a; b; 0 – k 2 Z2 ; 1 6 i; j 2, and the nonresonance conditions are true along the iteration.
4. A KAM theorem and Proofs of Theorems 2.1–2.4 Theorem 4.1. Suppose that the equation
( ðEqÞ0 :
h_ ¼ x; z_0 ¼ A0 ðaÞz0 þ g 0 ðh; aÞ þ Q 0 ðh; aÞz0 þ h0 ðz0 ; h; aÞ; ð48Þ
31
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
satisfies Conditions (a.1)–(a.5) in Lemma 3.1 with l ¼ 0. Then for any given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ and sufficiently small > 0, there exists a Cantorian closed subset P1 ½a; b with MeasP1 ¼ b a OðcÞ such that for any a 2 P1 , by a transformation
T : z0 ¼ ez þ Uðh; aÞ þ Wðh; aÞez ;
2
1
Moreover, h ¼ h0 þ xt; z0 ¼ Uðh0 þ xt; aÞ is a quasi-periodic solution of (48). The terms Uðh; aÞ; Wðh; aÞ satisfy the reality condition
U ¼ U;
X
wi þ
T2 P1
wi1 wi2 wil2 wil1 þ w0 w1 wl2 wl1 A
zl :¼ zl þ Ul1 þ Wl1 zl : By (30), it is easy to see that fUl g and fWl g are convergent and 1 2
X i i i jjUl1 jj 6 þ 2 2sþ2 2 2sþ2 2 2sþ2 cj c ji ci ji 06i6l1 i i 06i
1
1
Remark 4.1. If the initial matrix A0 ðaÞ of (48) and the terms g 0 ðh; aÞ; Q 0 ðh; aÞ; h0 ðz0 ; h; aÞ satisfy the conditions in Remark 3.2, and Conditions (a.2), (a.3) in Lemma 3.1 with l ¼ 0 are fulfilled for all a 2 ½a; b, then the conclusions of Theorem 4.1 hold for all a because in this case there is no small divisor problem.
6C
þ
b 1 ð0Þ; diagð Q 11
b 1 ð0ÞÞ Q 22
þ
b l ð0Þ; diagð Q 11
b l ð0ÞÞ; Q 22
T2 P1
1 2
i
i¼0
1
c20 j20sþ2 c21 j12sþ2
i
c2 j2sþ2 06i6l1 i i
l2
1 2
l2 sþ2 c2l2 j2l2
:
1 2
X
i : 2 2sþ2 c j 06i6l1 i i
P1 ¼
1 \
~z ¼ lim zl ; l!þ1
A1 ðaÞ ¼ lim Al ;
Pm ;
l!þ1
m¼0
W ¼ lim Wl : l!þ1
1
sup jjA0 ðaÞ A1 ðaÞjj 6 C 2 ;
a2P1
1 X
1 2
i
sup jjUjj 6 C 12s ; T2 P 1
1
sup jjWjj 6 C 22s : T2 P1
1
6 C 2 :
i¼0
On the lth step of the iteration process in Lemma 3.1, we denote the variables z by zl and z1 by zlþ1 , respectively. By (28) and using iteration, it implies
z0 ¼ z1 þ /0 þ w0 z1 ; z0 ¼ z2 þ ð/0 þ /1 þ w0 /1 Þ þ ðw0 þ w1 þ w0 w1 Þz2 ;
l2
We have
þ
6C
1
1 2
Let
l!þ1
from (25), it follows that fAl ðaÞg is convergent and l X
0
l2
l1 X
sup jjWl1 jj 6 C
U ¼ lim Ul ;
b 0 ð0Þ; Q b 0 ð0ÞÞ Alþ1 ðaÞ ¼ A0 ðaÞ þ diagð Q 11 22
2
1 2
2sþ2 c2l1 jl1
l!þ1
and a family of change of variables T l such that ðEqÞl T l ¼ ðEqÞlþ1 . Since for each positive integer l,
þ
2
Similarly, we obtain
½a; b ¼ P0 P1 Pl
a2P1
l1
T ¼ lim T 0 T 1 T l ;
a series of closed subsets
sup jjA0 ðaÞ Alþ1 ðaÞjj 6 C
i
1
1 2
sþ2 c2il1 j2il1
1
i i 2 2sþ2 2 2sþ2 c ji ci ji 6l1 i 1
06i1
T2 P1
1
2
2
1 2
X
þ þ
1
sup jjWjj 6 C 22s :
b r0 ; Wðb0 ; r0 Þ Wðb1 ; r1 Þ Wðbl ; rl Þ W 0 ; 2 2
1
06i1
T2 P1
Proof. Since the conditions (a.1)–(a.5) with l ¼ 0 in Lemma 3.1 are satisfied for Eq. (48), inductively we get a series of domains
wi1 wi2 wi3 þ
06i1
and estimates
sup jjUjj 6 C 12s ;
X
wi1 wi2 þ
06i1
X
þ
sup
W¼W
1
wi1 wi2 wil2 /il1 þ w0 w1 wl2 /l1 A
06i6l1
1
2
X
þ
ð49Þ
z 2 Þ and supa2P1 jjA0 ðaÞ A1 ðaÞjj 6 C 2 . where h1 ¼ Ob0 ;r0 ;P ðe
wi1 wi2 /i3 þ
06i1
06i1
in which U and W are analytic in h 2 r0 Þ and Lipschitz in a 2 P1 , the Eq. (48) can be transformed into
h_ ¼ x; ez_ ¼ A1 ðaÞez þ h1 ðez ; h; aÞ;
X
wi1 /i2 þ
06i1
X
þ
Uð12
(
X
/i þ
06i6l1
a¼a
h ¼ h;
X
z0 ¼ zl þ
From (29), it easily deduces that U and W satisfy the reality condition
U ¼ U;
W ¼ W:
By (a.4) in Lemma 3.1, we get
MeasP1 ¼ lim MeasPl ¼ b a OðcÞ: l!þ1
Hence, by the coordinate change T , Eq. (48) can be transformed into
32
(
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
h_ ¼ x ez_ ¼ A1 ðaÞez þ h1 ðez ; h; aÞ;
which is defined in the domain W h1 ¼ Ob0 ;r0 ;P ðe z 2 Þ. h 2
2
b0 r0 ; 2 P1 , where 2
1
z ¼ 0 is a solution of (49), returning to (48), Particularly, e and it is obvious that for every a 2 P1 , Eq. (48) possesses a quasi-periodic solution h ¼ h0 þ xt; z0 ¼ Uðh0 þ xt; aÞ. Proofs of Theorems 2.1–2.4. We first prove Theorem 2.3. For the fixed point O, let us return to the Eq. (11). For 8a 2 ½a; b, where a > 0; b < a0 < 0:217234, or a > a0 > 0; b < 0:217234, and a0 satisfies the equality cos a2 þ a0 þ a2 sin a2 ¼ 0, we have
SpecA0 ðaÞ ¼ SpecAðaÞ ¼
npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio 1 a þ 1; 1 a þ 1
and
dk0 ðaÞ 1 1 1 i pffiffiffiffiffiffiffiffiffiffiffiffi P ¼ pffiffiffiffiffiffiffiffiffiffiffiffi P pffiffiffiffiffiffiffiffiffiffiffiffi ; d a 2 aþ1 2 aþ1 2 bþ1
i ¼ 1; 2;
dðk0 ðaÞ k0 ðaÞÞ 1 1 1 1 2 pffiffiffiffiffiffiffiffiffiffiffiffi P ¼ pffiffiffiffiffiffiffiffiffiffiffiffi P pffiffiffiffiffiffiffiffiffiffiffiffi : da aþ1 aþ1 bþ1 Obviously, the system (11) satisfies all conditions (a.1)– (a.5) in Lemma 3.1 with m ¼ 0. Using Theorem 4.1, we only drop a small open set D1 with measure OðcÞ such that for a 2 ½a; b n D1 , the system (11) has a quasi-periodic solution formed h ¼ h0 þ xt; z0 ¼ Uðh0 þ xt; aÞ in a neighborhood of the fixed point O. Returning to (5), h ¼ h0 þ xt; z ¼ SðaÞUðh0 þ xt; aÞ is a real analytic quasi-periodic solution of (5), where SðaÞ is the matrix defined in (12). For the fixed points C 3 and C 4 , we only need to analyse the system (8) with B2 and B2 instead of B1 ; a2 and a2 instead of a1 , respectively. Noting that
SpecA0 ðaÞ ¼ SpecB2 ðaÞ ¼ SpecB2 ðaÞ npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio ¼ 1 cos a2 þ a; 1 cos a2 þ a ; since sin a2 ¼ aa2 ; a2 2
3p 2
i ¼ 1; 2;
SpecA0 ðaÞ ¼ SpecB3 ðaÞ ¼ SpecB3 ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a3 a; cos a3 a ; ¼ the constant matrices belong to the case in Remark 3.2. By Remark 4.1 we obtain two real analytic quasi-periodic solutions in neighborhoods of C 5 and C 6 for all a 2 ½a; b, respectively. This completes the proof of Theorem 2.3. The proofs of Theorems 2.1, 2.2 and 2.4 are similar to that of Theorem 2.3, we omit the detail. h Acknowledgements This work was supported by the NNSF grants (11371132, 11071066) of China, by Hunan Provincial Innovation Foundation For Postgraduate, by Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China) and by the Construct Program of the Key Discipline in Hunan Province. We would like to thank the reviewers for helpful comments and suggestions which led to significant improvements of the present paper. Appendix A Lemma A.1. [2,33] For d > 0; l > 0, the following inequality holds true:
X
e2jkjd jkjl 6
k2Zn
ll 1 ð1 þ eÞn : e dlþn
Lemma A.2. [29] Denoting As as the space of all functions on Tn bounded and analytic in the strip fh : jImhj 6 sg. If v 2 As and K r P 1, then
0 6 r 6 s;
where the constant C only depends on n.
ð50Þ
2
dðk0 ðaÞ k0 ðaÞÞ cos a þ a þ a sin a 1 2 2 2 2 ¼ : 3 da ðcos a2 þ aÞ2
For the fixed points C 5 and C 6 ,
jjðId CK Þv jjsr 6 CK n eK r jjv jjs ;
; 2p , we obtain
dk0 ðaÞ cos a þ a þ a sin a i 2 2 2 ¼ ; 3 da 2ðcos a þ aÞ2
S S Let D ¼ D1 D2 D3 . Obviously, MeasD ¼ OðcÞ, and for a 2 ½a; b n D there are real analytic quasi-periodic solutions in neighborhoods of C 3 ; C 4 and O simultaneously.
ð51Þ
By calculating, there exists a unique a0 2 ð0; 0:217234Þ such that the equation cos a2 þ a þ a2 sin a2 ¼ 0 holds. Notice that the functions of (50) and (51) are continuous in a on the close set ½a; b, we can obtain that there exists a maximum value f0 and a minimum value v0 , and both of them are positive constants. Similarly, by Theorem 4.1, we also take out small open sets D2 and D3 with measure OðcÞ such that for a 2 ½a; b n D2 there is a real analytic quasi-periodic solution in a neighborhood of the fixed point C 3 , and for a 2 ½a; b n D3 there is a real analytic quasi-periodic solution in a neighborhood of the fixed point C 4 .
Lemma A.3. [32] Suppose that g (u) is a C m function on the closure I, where I # R1 is an interval of length L. Let Ih ¼ fu : jgðuÞj 6 hg; h > 0. Assume for some constant 1 d > 0; jg m ðuÞj P d for 8u 2 I, then MeasIh 6 cLhm , where 1 c ¼ 2ð2 þ 3 þ þ m þ d Þ. References [1] Bourgain J. Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Int Math Res Notices 1994:475–97. [2] Bogoljubov NN, Mitropoliskii JUA, Samoilenko AM. Method of accelerated convergence in nonlinear mechanics. New York, Berlin: Springer; 1976. [3] Broer HW, Hoo J, Naudot V. Normal linear stability of quasi-periodic tori. J Differ Equ. 2007;232:355–418. [4] Celletti A, Ruzza SD, Lhotka C, Stefanelli L. Nearly-integrable dissipative systems and celestial mechanics. Eur Phys J Spec Top 2010;186:33–66.
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33 [5] Clifford MJ, Bishop SR. Approximating the escape zone for the parametrically excited pendulum. J Sound Vib 1994;172:572–6. [6] Clifford MJ, Bishop SR. Rotating periodic orbits of the parametrically excited pendulum. Phys Lett A 1995;201:191–6. [7] Cong H, Mi L, Yuan X. Positive quasi-periodic solutions of Lotka– Volterra system. Sci China Math 2010;53(5):1151–60. [8] Chen X, Fu X, Jing Z. Complex dynamics in a pendulum equation with a phase shift. Int J Bifurcation Chaos 2012;22(12): 125030:1–125030:40. [9] Chung K, Yuan X. Periodic and quasi-periodic solutions for the complex Ginzburg–Landau equation. Nonlinearity 2008;21:435–51. [10] D’Humieres D, Beasley MR, Huberman BA, libchaber AF. Chaotic states and routes to chaos in the forced pendulum. Phys Rev A 1982;26:3483–92. [11] Garira W, Bishop SR. Rotating solutions of the parametrically excited pendulum. J Sound Vib 2003;263:233–9. [12] Housner George W. The behaviour of inverted pendulum structures during earthquakes. Bull Seismol Soc Am 1963;53(2):403–17. [13] Haro A, de la Llave R. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms. Discrete Cont Dyn B 2006;6:1261–300. [14] Huang D, Liu Z. On the persistence of lower-dimensional invariant hyperbolic tori for smooth Hamiltonian systems. Nonlinearity 2000;13:189–202. [15] Her H-L, You J. Full measure reducibility for generic one-parameter family of quasi-periodic linear systems. J Dyn Differ Equ 2008;20:831–66. [16] Jing Z, Yang J. Complex dynamics in pendulum equation with parametric and external excitations I. Int J Bifurcation Chaos 2006;16:2887–902. [17] Jing Z, Yang J. Complex dynamics in pendulum equation with parametric and external excitations II. Int J Bifurcation Chaos 2006;16:3053–78. [18] Jorba A, Simo C. On quasi-periodic perturbations of elliptic equilibrium points. SIAM J Math Anal 1996;27(6):1704–37. [19] Jorba A, Simo C. On the reducibility of linear differential equations with quasi-periodic coefficients. J Differ Equ. 1992;98:111–24.
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[20] Jorba A, Villanueva J. On the persistence of lower-dimensional invariant tori under quasi-periodic perturbations. J Nonlinear Sci 1997;7(5):427–73. [21] Koch BP, Leven RW. Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. Physica D 1985;16(1):1–13. [22] Leven RW, Koch BP. Chaotic behaviour of a parametrically excited damped pendulum. Phys Lett A 1981;86(2):71–4. [23] Leven RW, Pompe B, Wilke C, Koch BP. Experiments on periodic and chaotic motions of a parametrically forced pendulum. Physica D 1985;16(3):371–84. [24] Li X, de la Llave R. Construction of quasi-periodic solutions of delay differential equations via KAM techniques. J Differ Equ. 2009;247:822–65. [25] Li X, Yuan X. Quasi-periodic solutions for perturbed autonomous delay differential equations. J Differ Equ. 2012;252:3752–96. [26] Li J, Zhu C. On the reducibility of a class of finitely differentiable quasi-periodic linear systems. J Math Anal Appl 2014. Available from:. [27] Liu B, Zhu W, Han Y. Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems. J Math Anal Appl 2006;322:251–75. [28] Luque A, Villanueva J. A KAM theorem without action-angle variables for elliptic lower dimensional tori. Nonlinearity 2011;24: 1033–88. [29] Pöschel J. A lecture on the classical KAM theorem. Proc Symp Pure Math 2001;69:707–32. [30] Salam F, Sastry SS. Dynamics of the forced Josephson junction circuit: the regions of chaos. IEEE Trans Circuits Syst 1985;32(8): 784–96. [31] Siegel CL, Moser JK. Lectures on Celestial Mechanics. New York, Berlin: Springer; 1971. [32] You J. A KAM Theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems. Commun Math Phys 1998;192:145–68. [33] Yuan X. Construction of quasi-periodic breathers via KAM technique. Commun Math Phys 2002;226:61–100.
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Existence of quasi-periodic solutions of the real pendulum equation Lin Lu, Xuemei Li ⇑ Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
a r t i c l e
i n f o
Article history: Received 26 February 2014 Accepted 26 March 2014 Available online 22 April 2014
a b s t r a c t x ¼ ax dx_ ð1 þ f0 cos x1 tÞ sin x þ f1 sin x2 t is considered in The pendulum equation € this paper, where f0 ; f1 and d are small real parameters, the ratio of x1 and x2 is irrational, and frequencies x1 and x2 satisfy the Diophantine condition. The unperturbed system ðf0 ¼ f1 ¼ d ¼ 0Þ has several fixed points for different parameter a. We use KAM theory to prove that the perturbed system possesses quasi-periodic solutions in neighborhoods of those fixed points. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Consider the real pendulum equation
€x ¼ ax dx_ ð1 þ f0 cos x1 tÞ sin x þ f1 sin x2 t;
ð1Þ
where x represents the angular displacement with respect to the vertical direction, a 2 R1 corresponds to spring modulus, d > 0 corresponds to damping, f0 cos x1 t and f1 sin x2 t correspond to the parametric excitations and external excitations, respectively. Meanwhile, f0 ; f1 and d are small real parameters, the ratio of x1 and x2 is irrational. The pendulum equation is a very famous physical model frequently appeared in many textbooks and papers, and it is important in modeling engineer systems such as tall and slender structures in earthquakes [12] and Josephson junction circuit [30]. For a ¼ f0 ¼ 0, D’humieres et al. [10] use numerical simulation to study the behavior of chaos and the path to chaos of the system (1), and show there exist symmetrybreaking of periodic orbits, breaking behavior of periodic orbits and period-doubling-3 bifurcations in the system. ⇑ Corresponding author. Tel.: +86 073188838037. E-mail addresses: [email protected] (L. Lu), lixuemei_1@sina. com (X. Li). http://dx.doi.org/10.1016/j.chaos.2014.03.003 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.
For the case with a ¼ f1 ¼ 0, the Eq. (1) has been extensively studied. The rotating periodic solutions and the behavior of chaos of the system have been discussed in [5,6,11]. Clifford and Bishop [5] employ two approximate techniques to predict the escape zone in which there is no major stable non-rotating orbits for the system (1). Garira and Bishop [11] divide the steady state rotating orbits of the Eq. (1) into four categories, i.e., purely rotating orbits, oscillating rotating orbits, straddling rotating orbits and large amplitude rotating orbits, and they analyse the complex properties of these orbits. Leven and Koch [22] investigate the temporal behavior of the Eq. (1) by the numerical solution. Later, using Melnikov method and averaging methods, Koch and Leven [21] discuss the boundaries of subharmonic and homoclinic bifurcations of the system, and verify some theoretical analysis by numerical simulations. In [23], the authors present an experimental study of periodic and chaotic type aperiodic motions of the system (1), and demonstrate the coexistence of periodic solutions and chaotic solutions and various transitions between them. Recently, Jing and Yang [16,17] discuss bifurcation and chaos of the system (1) under periodic perturbations and quasi-periodic perturbations, respectively. Later, by Melnikov method and numerical simulations, Chen et al. [8] investigate the Eq. (1) with a phase shift and obtain criteria
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
on the existence of chaos under periodic and quasiperiodic perturbations, and expose complex dynamical behaviors, including the sudden conversion of chaos to period orbits, jumping behaviors of periodic orbits, interior crisis, and so on. Numerical simulations in [17] reveal that the Eq. (1), under quasi-periodic perturbations, possesses the following dynamical phenomena: interleaving occurrence of chaotic behaviors and quasi-periodic behaviors, jumping behavior of quasi-periodic sets, different nice quasi-periodic attractors, and the coexistence of three quasi-periodic sets. These phenomena show that the Eq. (1) should have quasi-periodic solutions for some parameter values. Our aim in this paper is theoretically to verify the speculation above. More precisely, since f0 ; f1 and d are small real parameters in (1), we assume that they are comparable, and set d ¼ ~ d; f0 ¼ f~0 and f1 ¼ f~1 . With ~ d; f~0 and f~1 again denoted by d; f0 and f1 , respectively, letting x_ ¼ y, we can easily write (1) as
8 > < x_ ¼ y; y_ ¼ ða 1Þx dy ðf0 cos x1 tÞx þ f1 sin x2 t > : þð1 þ f0 cos x1 tÞðx sin xÞ:
ð2Þ
We shall prove that for sufficiently small the system (2) has invariant tori in a neighborhood of each fixed point of the unperturbed system for most parameter values a in the sense of Lebesgue measure. To end this, we shall write (2) formed
z_ ¼ AðaÞz þ gðhÞ þ Q ðhÞz þ hðz; h; Þ;
h_ ¼ x;
ð3Þ
2
where z 2 R ; x ¼ ðx1 ; x2 Þ; AðaÞ is an 2 2 matrix depending on a parameter a, h ¼ Oðz2 Þ; gðhÞ; Q ðhÞ and hðz; h; Þ are real analytic with respect to h on a strip of the torus T2 with width r > 0; hðz; h; Þ is real analytic with respect to z on the domain jzj 6 b; b > 0. By Newton iteration and KAM techniques, the system (3) can be transformed into
~z_ ¼ A1 ðaÞ~z þ h1 ð~z; h; aÞ;
h_ ¼ x;
where h1 ¼ Oð~z2 Þ. Obviously, ~z ¼ 0 is a solution of the equation, and this shows that the original system (3) possesses a quasi-periodic solution with the basic frequency x. It is well known that the KAM theory is a powerful tool for discussing the existence and linear stability of invariant tori of both Hamiltonian Systems [1,3,14,20,27,28,32] and dissipative systems [4,7,9,18,25]. Cong et al. [7] discuss the existence of quasi-periodic solutions of (3) when the eigenvalues of A are fixed and frequency x varies in some set. Using the method of accelerated convergence and asymptotic solution technique, they show for most x 2 P (in the sense of Lebesgue measure), the system (3) possesses a positive quasi-periodic solution with basic frequency x. Jorba and Simo [18] study the system (3) in the case where the eigenvalues of A and frequency x are fixed, and the size of small perturbations varies in some set. They show that under appropriate hypotheses, if the eigenvalues of A and frequency x satisfy the non-resonance conditions, then there exists a Cantorian set e ða; bÞ with positive Lebesgue measure such that for sufficiently small
belonging to the Cantorian set e, (3) possesses a quasiperiodic solution. Jorba and Simo [19] give a positive measure reducibility result for the linear differential system
z_ ¼ Az þ Q ðhÞz;
h_ ¼ x:
ð4Þ
They prove that if Q is real analytic with respect to h on a strip of the torus T2 with width r > 0, under suitable hypotheses on non-resonance and non-degeneracy with respect to sufficiently small , there exists a Cantorian set e ða; bÞ with positive Lebesgue measure such that for all 2 e, (4) can be reduced into a linear system with constant coefficients. Li and Zhu [26] also consider the reducibility of this system in finitely differentiable case. For the delay differential equation
z_ ðtÞ ¼ AzðtÞ þ Bzðt sÞ þ f ðzðtÞ; zðt sÞ; xt; aÞ; where eigenvalues are fixed, Li and Llave [24] obtain quasiperiodic solutions of this equation by KAM theory. Obviously, in [7,18,19,24,26], there are few discussion about the situation when the constant matrix A depends on parameters. Her and You [15] analyse the reducibility of the system
z_ ¼ AðaÞz þ Q ðhÞz;
h_ ¼ x;
where the matrix AðaÞ depends on a parameter a. They mainly contribute to improve the measure of the reducible set. Yuan [33] constructs the quasi-periodic breathers for networks of weakly coupled oscillators via KAM techniques, where both frequencies x and perturbed terms depend on parameters. Li and Yuan [25] obtain quasiperiodic solutions for autonomous delay differential equations by KAM theory, where both frequencies x and eigenvalues rely on parameters. In this paper, the system (2) is a dissipative system and can be written in a normal form, where the constant matrix depends on the parameter a, so that we obtain quasi-periodic solutions by KAM techniques. Some of ideas seeking the quasi-periodic solutions in this paper could be found in [9,13,25,28,33], although they deal with slightly different problems and models. This paper is arranged as follows. In Section 2, we give some main results and outline ideas of proofs. In Section 3, we construct and prove an iterative lemma which is important in the proofs of our results. In Section 4, we give a KAM theorem which is used to obtain the quasi-periodic solutions of (2). In Appendix A, we list some lemmas used in this paper. 2. Main results For the unperturbed system with d ¼ f0 ¼ f1 ¼ 0, we can calculate directly and get the following for x 2 ð2p; 2pÞ [16]. (i) if a < 1, then the system (2) possesses a unique fixed point Oð0; 0Þ which is a saddle point. (ii) if 1 < a < 0, then the system (2) possesses three fixed points: a center point Oð0; 0Þ and two saddle points C 1 ða1 ; 0Þ; C 2 ða1 ; 0Þ, where a1 is the positive root of the equation sin x ¼ ax in the interval ð0; p2 Þ.
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
(iii) if 0 < a < 0:217234, then the system (2) possesses five fixed points: three center points Oð0; 0Þ; C 3 ða2 ; 0Þ; C 4 ða2 ; 0Þ and two saddle points C 5 ða3 ; 0Þ; C 6 ða3 ; 0Þ, where a3 and a2 are the positive roots of the equation sin x ¼ ax in the interval ðp; 32pÞ and ð32p ; 2pÞ, respectively. (iv) if a > 0:217234, then the system (2) possesses a unique fixed point Oð0; 0Þ which is a center.
N i ðhÞði ¼ 1; 2; 3Þ are real analytic in h 2 UðrÞ and sufficiently smooth in a 2 P, and hðz; hÞ; pi ðw; hÞði ¼ 1; 2; 3Þ are real analytic in the domain Wðb; rÞ (its definition given in Section 3) and sufficiently smooth in a 2 P. Notice that 0; a1 ; a2 ; a3 are roots of the equation sin x ¼ ax, where a1 2 ð0; p2 Þ, a3 2 ðp; 32pÞ; a2 2 ð32p ; 2pÞ, the eigenvalues of matrices AðaÞ and Bi ðaÞ ði ¼ 1; 2; 3Þ are as follows: (1) for a < 1,
For convenience, letting
x ; z¼ y
h¼
h1 ; h2
x¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ a 1; a 1 ;
x1 ;
x2
(2) for 1 < a < 0,
the Eq. (2) can be written as
(
n pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ 1 a þ 1; 1 a þ 1 ;
h_ ¼ x; z_ ¼ AðaÞz þ gðhÞ þ Q ðhÞz þ hðz; hÞ;
ð5Þ SpecB1 ðaÞ ¼ SpecB1 ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a1 a; cos a1 a ; ¼
where
AðaÞ ¼
0
1
a 1 0
Q ðhÞ ¼ hðz; hÞ ¼
;
gðhÞ ¼
0
0
f0 cos h1
d 0
0
f1 sin h2
;
(3) for 0 < a < 0:217234,
ð6Þ
n pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ 1 a þ 1; 1 a þ 1 ;
;
SpecB3 ðaÞ ¼ SpecB3 ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a3 a; cos a3 a ; ¼
ð1 þ f0 cos h1 Þðx sin xÞ
ð7Þ
:
SpecB2 ðaÞ ¼ SpecB2 ðaÞ npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio ¼ 1 cos a2 þ a; 1 cos a2 þ a ;
For the fixed point C 1 ða1 ; 0Þ, taking a translational transformation x ¼ a1 þ u; y ¼ v , setting w ¼ ðu v ÞT (here T represents transpose), using Taylor’s formula to sinða1 þ uÞ and the fact sin a1 ¼ aa1 , we can rewrite (2) in the form
(
h_ ¼ x; _ ¼ B1 ðaÞw þ j1 ðhÞ þ N 1 ðhÞw þ p1 ðw; hÞ; w 1 ; 0
0 B1 ðaÞ ¼ a cos a1 0 j1 ðhÞ ¼ ; f1 sin h2 f0 cos h1 sin a1 N1 ðhÞ ¼
0 f0 cos h1 cos a1
0 ; d
n pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio SpecAðaÞ ¼ 1 a þ 1; 1 a þ 1 :
ð8Þ
where
(4) for a > 0:217234,
ð9Þ
When the eigenvalues of the matrix are complex and different from each other, we can make a complex linear transformation such that the coefficient matrix is diagonal. Taking the matrix AðaÞ as an example, for a > 1, by means of a complex linear transformation z ¼ SðaÞz0 , the system (5) is changed into
( 2
p1 ðw; hÞ ¼ Oðw Þ: ð10Þ
Similarly, for the fixed points C 2 ða1 ; 0Þ; C 3 ða2 ; 0Þ; C 4 ða2 ; 0Þ; C 5 ða3 ; 0Þ and C 6 ða3 ; 0Þ, we take translational transformations and write (2) in the form of (8), with a1 ; a2 ; a2 ; a3 and a3 instead of a1 , and denote the constant matrix by B1 ðaÞ, B2 ðaÞ; B2 ðaÞ; B3 ðaÞ and B3 ðaÞ, respectively. Definition 2.1. We say a function f ðhÞ is real analytic in a strip of width r, if f ðhÞ is analytic in the complex strip b 2 : jImhj :¼ maxi¼1;2 jImhi j 6 rg and real for UðrÞ ¼ fh 2 T the real argument h. It is easy to see that AðaÞ and Bi ðaÞði ¼ 1; 2; 3 corresponding to a1 ; a2 ; a3 instead of a1 in (8)) are sufficiently smooth in a in some bounded closed set P (which we will consider), the functions gðhÞ; Q ðhÞ; ji ðhÞ;
h_ ¼ x; z_0 ¼ A0 ðaÞz0 þ g 0 ðh; aÞ þ Q 0 ðh; aÞz0 þ h0 ðz0 ; h; aÞ;
ð11Þ
npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi where A ðaÞ ¼ diag k01 ðaÞ; k02 ðaÞ :¼ diag 1 a þ 1; pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi0 1 a þ 1g,
SðaÞ ¼
pffiffiffiffiffi 1ffi pffiffiffiffiffiffi aþ1
1
pffiffiffiffiffi ! 1ffi pffiffiffiffiffiffi aþ1 ;
1
g 0 ðh; aÞ ¼
1 2
f1 sin h2 f1 sin h2
;
ð12Þ
0 1 pffiffiffiffiffi pffiffiffiffiffi 1ffi 1ffi d f0 cos h1 pffiffiffiffiffiffi d 1 @ f0 cos h1 pffiffiffiffiffiffi aþ1 aþ1 A; Q 0 ðh; aÞ ¼ pffiffiffiffiffi pffiffiffiffiffi 1ffi 1ffi 2 f0 cos h1 pffiffiffiffiffiffi d f0 cos h1 pffiffiffiffiffiffi d aþ1 aþ1 jjg 0 ðh; aÞjj 6 C ; jjQ 0 ðh; aÞjj 6 C ; h0 ðz0 ; h; aÞ ¼ Oðz20 Þ:
ð13Þ
For a < 1, we take a real linear transformation such that the coefficient matrix is diagonal. For convenience, we sometimes omit the parameter a in some functions. In order to consider the reality condition, we make some notations and remarks.
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
For a matrix B ¼ ðbij Þ22 and a 2-dimensional vector u ¼ ðu1 ; u2 ÞT , we define B ¼ ðbli lj Þ, B ¼ ðbilj Þ and u ¼ ðu2 ; u1 ÞT (i.e., u1 ¼ u2 and u2 ¼ u1 ), where ðl1 ; l2 Þ ¼ ð2; 1Þ; 1 6 i; j 2. The complex conjugation F of a function F means taking complex conjugation of the coefficients in the power series of F. Obviously, g 0 ðhÞ; Q 0 ðhÞ; SðaÞ and h0 ðz0 ; hÞ in (12) and (13), because of k01 ¼ k02 , satisfy the following reality condition (see Sections 15 and 27 in [31]). Reality condition S ¼ S; g 0 ðhÞ ¼ g 0 ðhÞ; h0 ðz0 ; hÞ ¼ h0 ðz0 ; hÞ; Q 0 ðhÞ ¼ Q 0 ðhÞ. 2.1. Outline of scheme We only discuss the fixed point O here, and for other fixed points, the discussion is similar. To obtain quasi-periodic solutions of (11) in a neighborhood of the fixed point O, we make some changes to simplify the system (11) by Newton iteration and KAM techniques. Firstly, by means of a family of quasi-periodic changes of variables, we want to eliminate the terms g 0 ðhÞ and Q 0 ðhÞ. More exactly, taking the change of variables formed z0 ¼ z1 þ /ðh; aÞþ wðh; aÞz1 into (11) (such changes of variables have been considered in [9,25]), it implies
ðE þ wÞz_1 ¼ A0 z1 þ A0 / þ A0 wz1 þ g 0 þ Q 0 z1 þ Q 0 / þ Q 0 wz1 @/ @w þ h0 ðz1 þ / þ wz1 ; hÞ x xz1 : ð14Þ @h @h
ð15Þ
and
@w b 0 ð0Þ; Q b 0 ð0Þ ; x ¼ A0 w wA0 þ CK Q 0 diag Q 11 22 @h
ð16Þ
where CK denotes a truncation operator for the Fourier serb 0 ðkÞ denotes the kth-Fourier ies of functions (see [25]), Q ii coefficient of Q 0ii ðhÞ; Q 0ij ðhÞ is the element of the matrix Q 0 ðhÞ; 1 6 i; j 2. Once Eqs. (15) and (16) are solved, the system (14) can be written as h b 0 ð0Þ; Q b 0 ð0Þ z1 z_1 ¼ A1 z1 þ ðE þ wÞ1 Q 0 / þ Q 0 wz1 wdiag Q 11 22 i þðId CK Þg 0 þ ðId CK ÞQ 0 z1 þ h0 ðz1 þ / þ wz1 ;hÞ ; ð17Þ b 0 ð0Þ; Q b 0 ð0ÞÞ. where A1 ¼ A0 þ diagð Q Using the 11 22 Taylor’s formula to h0 ðz1 þ / þ wz1 ; hÞ, we write (17) as follows
z_1 ¼ A1 ðaÞz1 þ g 1 ðh; aÞ þ Q 1 ðh; aÞz1 þ h1 ðz1 ; h; aÞ;
ð18Þ
where 1
g 1 ðh; aÞ ¼ ðE þ wÞ ðQ 0 / þ ðId CK Þg 0 þ h0 ð/; h; aÞÞ; b 0 ð0Þ; Q b 0 ð0ÞÞ Q 1 ðh; aÞ ¼ ðE þ wÞ1 Q 0 w wdiagð Q 11 22
@h0 ð/; hÞ ðE þ wÞ ; þðId CK ÞQ 0 þ @z0
In this way, after n steps of iteration, the Eq. (11) will be formed as
(
h_ ¼ x; z_n ¼ An ðaÞzn þ g n ðh; aÞ þ Q n ðh; aÞzn þ hn ðzn ; h; aÞ:
If the norms of g n and Q n tend to zero with a superexponential velocity, then the composition of transformations is convergent, and the above equation converges to the form
(
h_ ¼ x; ~z_ ¼ A1 ðaÞ~z þ h1 ð~z; h; aÞ;
where h1 ð~z; h; aÞ ¼ Oð~z2 Þ. Obviously, ~z ¼ 0 is a solution of the equation, and this shows that for the original system (5), there exists a quasi-periodic solution with the basic frequency x. 2.2. The small divisor It is clear that in the process of solving (15) and (16), we meet small divisors and need the small divisor (or nonresonance) conditions
pffiffiffiffiffiffiffi c 1ðk; xÞj P s ; jkj pffiffiffiffiffiffiffi c jkj ðaÞ ki ðaÞ þ 1ðk; xÞj P s ; jkj jki ðaÞ
Thus, we derive the homological equations
@/ x ¼ A0 / þ C K g 0 @h
h1 ðz1 ; h; aÞ ¼ ðE þ wÞ1 h0 ðz1 þ / þ wz1 ; h; aÞ h0 ð/; h; aÞ
@h0 ð/; h; aÞ ðE þ wÞz1 : ð21Þ @z0
ð19Þ
ð20Þ
1 6 i; j 2;
where 0 – k 2 Z2 ; s > 2 is a constant. In order to solve equations like (15), we need the first inequality above, and to solve equations like (16), we need the second one. Noting that for a 2 ½a; b; a > 1, both k01 ðaÞ and k02 ðaÞ are pure imaginary, the non-resonance conditions could not be true for all a. Hence, we want to take out some small (in the sense of Lebesgue measure) open sets, such that the non-resonance conditions hold for the remaining set. On each step of the iteration, we need to remove the resonant set and control its measure. To bound the measure of the resonant set, we introduce the non-degeneracy conditions. Definition 2.2. We say a 2 2 matrix AðaÞ satisfies the non-degeneracy conditions on the interval K, if there exists a constant v > 0 such that
dðki ðaÞÞ da P v;
dðk1 ðaÞ k2 ðaÞÞ P v; da
uniformly hold for a 2 K, where ki ðaÞ is the eigenvalue of AðaÞ; i ¼ 1; 2. Using the above method, one important thing is that on each step when we bound the measure of the resonant set, we have to verify the non-degeneracy conditions. Therefore, we need a bit of words to illustrate that the nondegeneracy conditions are maintained along the iteration although the proof is simple.
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
In the following, we suppose the frequency x ¼ ðx1 ; x2 Þ is fixed and satisfies the Diophantine condition
jðk; xÞj P
c jkjs
;
80 – k 2 Z2 ;
where s > 2, and c > 0 is a sufficiently small constant. Based on the above description, we state the main results in this paper. Theorem 2.1. If a 2 ½a; b and b < 1, then there is a sufficiently small positive number 0 such that for all a 2 ½a; b and 0 < < 0 , the Eq. (2) possesses a real quasiperiodic solution in a neighborhood of the fixed point O.
m
1. m ¼ ð1þqÞ ; 0 ¼ , where q ¼ 13; P 2 2. m0 ¼ 0; mm ¼ ð12 þ þ m2 Þ=ð2 1 q¼1 q Þ; 3. rm ¼ ð1 mm Þr0 ; 4. bm ¼ ð1 mm Þb0 ; 5. jm ¼ rm rmþ1 ; 6. cm ¼ c=ðm þ 1Þ2 ; c0 ¼ c ¼ Oðs Þð0 < s 6 16Þ; 1 P v0 2 if is sufficiently 7. vm ¼ v0 C m1 q¼0 q (vm P 2 small); 1 P 2 8. fm ¼ f0 þ C m1 q¼0 q (fm 2f0 if is sufficiently small); 9. K m ¼ ðm þ 1Þ2 2m ln ; 10. gm ¼ 4fcm1 s; 0 Km 2 2 b 2 ¼ C2 =ð2pZÞ2 ; 11. T ¼ R =ð2pZÞ2 and T 12. For
Theorem 2.2. Suppose a 2 ½a; b and a > 1; b < 0. Then there is a sufficiently small positive number 0 such that for all a 2 ½a; b and 0 < < 0 , the Eq. (2) possesses two real quasi-periodic solutions which are in neighborhoods of fixed points C 1 and C 2 , respectively. Moreover, for given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ, there is a sufficiently small positive number 0 such that if 0 < < 0 , then there is a Cantorian subset P1 ½a; b with Lebesgue measure larger than b a OðcÞ, and for any a 2 P1 , the Eq. (2) possesses a real quasi-periodic solution in a neighborhood of the fixed point O. Theorem 2.3. Suppose a 2 ½a; b, where a > 0; b < a0 < 0:217234, or a > a0 > 0; b < 0:217234, and a0 satisfies the equality cos a2 ða0 Þ þ a0 þ a2 ða0 Þ sin a2 ða0 Þ ¼ 0. Then there is a sufficiently small positive number 0 such that for all a 2 ½a; b and 0 < < 0 , the Eq. (2) possesses two real quasi-periodic solutions which are in neighborhoods of fixed points C 5 and C 6 , respectively. Moreover, for given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ, there is a sufficiently small positive number 0 such that if 0 < < 0 , then there is a Cantorian subset P1 ½a; b with Lebesgue measure larger than b a OðcÞ, and for any a 2 P1 , the Eq. (2) possesses three real quasi-periodic solutions which are in neighborhoods of the fixed points O; C 3 and C 4 , respectively. Theorem 2.4. Suppose a 2 ½a; b and a > 0:217234. Then for given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ, there is a sufficiently small positive number 0 such that if 0 < < 0 , then there is a Cantorian subset P1 ½a; b with Lebesgue measure larger than b a OðcÞ, and for any a 2 P1 , the Eq. (2) possesses a real quasi-periodic solution which are in a neighborhood of the fixed points O.
3. Iterative lemma Theorems 2.1, 2.2, 2.3, 2.4 are proved by an iterative procedure. Therefore, let us introduce some iterative constants and notations in the following. Let r0 ; b0 ; v0 and f0 be positive constants. To simplify the notation, we shall denote by C a universal positive constant which is independent of the KAM iteration and may be different in different places. For all m P 1,
r > 0; b > 0, let
UðrÞ ¼
b 2 : jImhj :¼ maxjImhi j 6 r h2T i¼1;2
and
b 2 : jjzjj 6 b; jImhj 6 rg; Wðb; rÞ ¼ fðz; hÞ 2 C2 T 13. Pg ¼ fa 2 R1 : distða; PÞ < gg;
Remark 3.1. As we want to drop some resonant sets which are opened, and the remaining parameter sets are closed, in order to use the derivative to estimate the measure of resonant sets, we need to extend the remaining closed sets to an its small neighborhood, Li and de la Llave [24], Yuan [33].
14. If P # R1 is a parameter set with positive measure, given 0 < 1; F : Wðb; rÞ Pg ! C2 is analytic in ðz; hÞ 2 Wðb; rÞ and C 1 -smooth in a 2 Pg , and satisfies
jjFjjb;r;g :¼ max sup j@ sa Fðz; h; aÞj 6 C ; s¼0;1 Wðb;rÞPg
we set F ¼ Ob;r;g ðÞ, where the notation @ a denotes the partial derivative with respect to a; 15. If H : Wðb; rÞ Pg ! C2 is analytic in ðz; hÞ 2 Wðb; rÞ and C 1 -smooth in a 2 Pg , vanishes with its zderivative for z ¼ 0, and satisfies
sup j@ jz @ sa Hðz; h; aÞj 6 C;
Wðb;rÞPg
s ¼ 0; 1;
j ¼ 0; 1; 2;
we write H ¼ Ob;r;g ðz2 Þ.
Lemma 3.1. Suppose that there is a sequence of closed parameter sets R1 ½a; b ¼ P0 P1 Pl and a family of equations ðEqÞm ðm ¼ 0; 1; . . . ; lÞ defined in Wðbm ; rm Þ Pgmm ( h_ ¼ x; ðEqÞm : ð22Þ z_ ¼ Am ðaÞz þ g m ðh; aÞ þ Q m ðh; aÞz þ hm ðz;h; aÞ; m where Am ¼ diagfkm Assume that for 1 ðaÞ; k2 ðaÞg. 0; 1; 2; ; l, the following conditions are satisfied:
m¼
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
m (a.1) for a 2 Pgmm , the eigenvalues km 1 ðaÞ; k2 ðaÞ satisfy nondegeneracy conditions
m dðki ðaÞÞ v0 da P vm P 2 ; m dðk1 ðaÞ km v0 2 ðaÞÞ P vm P 2 ; da
i ¼ 1; 2
ð23Þ
m dðki ðaÞÞ da 6 fm 2f0 ; m dðk1 ðaÞ km 2 ðaÞÞ 6 fm 2f0 ; da
ð24Þ
m k ðaÞ km ðaÞ P v P v0 ; 1 2 m 2 m v k ðaÞ P v P 0 ; i ¼ 1; 2; i m 2
1
ð25Þ
(a.3) the term hm ðz; h; aÞ is analytic in ðz; hÞ 2 Wðbm ; rm Þ and C 1 -smooth in a 2 Pgmm , and the following estimate holds
hm ¼ Obm ;rm ;gm ðz2 Þ;
ð26Þ
(a.4) there is a constant C 0 > 0 such that the Lebesgue measure of Pm satisfies
MeasPm P MeasPm1 ð1 C 0 cm1 Þ; 1 6 m 6 l;
aÞ ¼
k02 ð
aÞ, then
km 1ð
in the form
h ¼ h;
i b l ð0Þ; Q b l ð0ÞÞz1 þ hl ðz1 þ / þ w z1 ; hÞ : wl diagð Q l l 11 22
b l ð0Þ; Q b l ð0ÞÞ and using Letting Alþ1 :¼ Al þ diagð Q 11 22 Taylor’s formula to hl ðz1 þ /l þ wl z1 ; hÞ, above equality is written as
where
Wðblþ1 ; rlþ1 Þ Plþ1 ! Wðbl ; rl Þ Pl T l : z ¼ z1 þ /l ðh; aÞ þ wl ðh; aÞz1 ;
h i b l ð0Þ; Q b l ð0Þ z1 þ ðE þ w Þ1 z_1 ¼ Al þ diag Q l 11 22 h ðId CK lþ1 Þg l þ ðId CK lþ1 ÞQ l z1 þ Q l /l þ Q l wl z1
aÞ ¼
gl
a ¼ a; ð28Þ
g lþ1 ðhÞ ¼ ðE þ wl Þ1 Q l /l þ ðId CK lþ1 Þg l þ hl ð/l ; hÞ ; b l ð0Þ; Q b l ð0ÞÞ Q lþ1 ðhÞ ¼ ðE þ wl Þ1 Q l wl wl diagð Q 11 22
@hl ð/l ; hÞ ðE þ wl Þ ; þðId CK lþ1 ÞQ l þ @z
ð29Þ
and the following estimates 2ðsþ1Þ
/l ¼ Orlþ1 ;glþ1 ðl c2 l jl 1 2
Þ;
2ðsþ1Þ Þ; l
wl ¼ Orlþ1 ;glþ1 ðl c2 l j
ð30Þ
such that by the change of variable T l , the ðEqÞl is transformed into
ðEqÞlþ1
8 _ > < h ¼ x; : z_1 ¼ Alþ1 ðaÞz1 þ g lþ1 ðh; aÞ þ Q lþ1 ðh; aÞz1 > : þhlþ1 ðz1 ; h; aÞ ð31Þ
ð35Þ
ð36Þ
hlþ1 ðz1 ; hÞ ¼ ðE þ wl Þ1 hl ðz1 þ /l þ wl z1 ; hÞ
where z1 is a new variable, and /l ðh; aÞ; wl ðh; aÞ are glþ1 analytic in h 2 Uðrlþ1 Þ and C 1 -smooth in a 2 Plþ1 , satisfy the reality condition
wl ¼ wl
ð34Þ
Inserting (33) and (34) into (32), we get
z_1 ¼ Alþ1 z1 þ g lþ1 ðhÞ þ Q lþ1 ðhÞz1 þ hlþ1 ðz1 ; hÞ;
Then there exists a closed subset Plþ1 Pl and a change of variable glþ1
@wl b l ð0Þ; Q b l ð0ÞÞ: x ¼ Al wl wl Al þ CK lþ1 Q l diagð Q 11 22 @h
ð27Þ
km 2 ðaÞ, g m ¼ g m ; Q m ¼ Q m , and hm ðz; h; aÞ ¼ hm ðz; h; aÞ.
/l ¼ /l ;
ð33Þ
and
Q m ¼ Orm ;gm ðm 2 Þ;
(a.5) (Reality condition) if
Suppose that /l and wl are the solution of the following homological equations
@/l x ¼ Al /l þ CK lþ1 g l @h
(a.2) the terms g m ðh; aÞ and Q m ðh; aÞ are analytic in h 2 Uðrm Þ and C 1 -smooth in a 2 Pgmm , and the following estimates hold
k01 ð
Proof. To simplify notations, sometimes we shall drop the parameter a from functions whenever there is no confusion. Let E be the identity matrix. Substituting (28) into (22) with m ¼ l, we obtain the following equation
ðE þ wl Þz_1 ¼ Al z1 þ Al /l þ Al wl z1 þ g l þ Q l z1 þ Q l /l þ Q l wl z1 @/ @w þ hl ðz1 þ /l þ wl z1 ; hÞ l x l xz1 : @h @h ð32Þ
and conditions
g m ¼ Orm ;gm ðm Þ;
and the conditions (a.1)–(a.5) are satisfied with m being replaced by l þ 1 and ðz; hÞ by ðz1 ; hÞ, respectively.
hl ð/l ; hÞ
@hl ð/l ; hÞ ðE þ wl Þz1 : @z
ð37Þ
In order to estimate g lþ1 ; Q lþ1 and hlþ1 , we first solve the homological Eqs. (33) and (34), and estimate their solutions. Let /il and g il be the ith element of /l and g l , respectively, i ¼ 1; 2. Expanding g il ; /il into Fourier series in h and truncating them by operator CK lþ1
X
CK lþ1 g il ¼
b g il ðkÞe
pffiffiffiffiffi 1ðk;hÞ
;
jkj6K lþ1
CK lþ1 /il ¼
X
pffiffiffiffiffi 1ðk;hÞ
b i ðkÞe / l
;
jkj6K lþ1
b i ðkÞ depend on where the Fourier coefficients b g il ðkÞ and / l the parameter a, the Eq. (33) implies
b i ðkÞ ¼ / l
kli ð
b g il ðkÞ pffiffiffiffiffiffiffi ; aÞ 1ðk; xÞ
jkj 6 K lþ1 ;
i ¼ 1; 2:
ð38Þ
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
Since g l ðhÞ is analytic in h 2 Uðrl Þ and C 1 -smooth in
a 2 Pgl l ,
j@ sa b g il ðkÞj 6 max s¼0;1
sup j@ sa g il jejkjrl ;
s ¼ 0; 1:
gl
ð39Þ
Uðrl ÞP
ðE þ wl Þ1 ¼ E wl þ w2l w3l þ ;
l
Set
pffiffiffiffiffiffiffi
Plþ ¼ a 2 Pl : jkli ðaÞ 1ðk; xÞj P
cl jkj
s;
0 < jkj 6 K lþ1 ; i ¼ 1; 2 :
g
Then for a 2 Plþlþ1 , there is a0 2 Plþ ja a0 j < glþ1 , and for 0 < jkj 6 K lþ1 , we get
jkli ðaÞ
such that
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1ðk; xÞj ¼ jkli ða0 Þ 1ðk; xÞ þ kli ðaÞ kli ða0 Þj P
cl
jkjs
2f0 glþ1 P
cl
2jkjs
i ¼ 1; 2
;
2s i 2 b i ðkÞj 6 C maxj@ s b maxj@ sa / l a g l ðkÞjjkj =cl s¼0;1
6
c
q¼0
for l 2 N. So it is easy to see that hlþ1 satisfies the condition (a.3) with m ¼ l þ 1 by the equality (37). From (26) with m ¼ l and (30) which has just been proved, it follows that
max s¼0;1
j@ sa /j 6
sup g
Uðrlþ1 ÞPlþlþ1
a 2 Plþ :
6C
X
s¼0;1
CK lþ1 Þg l jjrlþ1 ;glþ1 þ jjhl ð/l ;hÞjjrlþ1 ;glþ1 Þ
g
jjg l jjrl ;gl ejkjðrl rlþ1 Þ jkj2s =c2l 6 C
3 2
l c2l j2l sþ2
elements of Q l and wl , respectively. Expanding Q lij ; wlij into Fourier series in h and truncating them, the Eq. (34) implies
b l ðkÞ ¼ w ij
b Q l ðkÞ
pffiffiffiffiffi ; klj ðaÞkli ðaÞþ 1ðk;xÞ ij
ð41Þ if k – 0; i ¼ j
for jkj 6 K lþ1 . Set
pffiffiffiffiffiffiffi
Plþ1 ¼ fa 2 Plþ : jklj ðaÞ kli ðaÞ þ 1ðk; xÞj P
cl
jkjs
;
0 < jkj 6 K lþ1 ; 1 6 i; j 6 2g: g
lþ1 Then for a 2 Plþ1 ; 0 < jkj 6 K lþ1 , similarly, by Condition (a.1) we have
pffiffiffiffiffiffiffi c l kj ðaÞ kli ðaÞ þ 1ðk; xÞ P l s ; 2jkj
1 6 i; j 6 2:
bl s bl 2s 2 max @ sa w ij ðkÞ 6 Cmax @ a Q ij ðkÞ jkj =cl s¼0;1 s¼0;1 6 C Q lij ejkjrl jkj2s =c2l ;
lþ1 a 2 Pglþ1 :
Hence, by Condition (a.2) and Lemma A.1 again, it implies
jjwl jjrlþ1 ;glþ1
X b l ðkÞj ejkjrlþ1 max sup j@ sa w 6 jkj6K lþ1
6C
X k2Z2
s¼0;1
jjQ l jjrl ;gl ejkjðrl rlþ1 Þ jkj2s =c2l 6 C
¼ Orlþ1 ;glþ1 ðjj/l jjrlþ1 ;glþ1 Þ:
1 2
l : c2l j2l sþ2 ð43Þ
!
ð45Þ
By (36), (30), (45) and Lemma A.2, it implies
jjQ lþ1 jjrlþ1 ;glþ1 6 jjðE þ wl Þ1 jjrlþ1 ;glþ1 b l ð0Þ; Q b l ð0Þ jj jjQ l wl wl diag Q rlþ1 ;glþ1 11 22 @hl ð/l ; hÞ þ jjðId CK lþ1 ÞQ l jjrlþ1 ;glþ1 þ @z rlþ1 ;glþ1 jjE þ wl jjrlþ1 ;glþ1 6 C jjQ l jjrlþ1 ;glþ1 jjwl jjrlþ1 ;glþ1 þjjQ l jjrl ;glþ1 K 2lþ1 ejl K lþ1 þ jj/l jjrlþ1 ;glþ1
l c2l j2l sþ2
1
6 C 2lþ1 :
Thus, we verify (a.2) with m ¼ l þ 1. In the following, we verify that the non-degeneracy conditions are maintained in the iterative process. Noting b l ð0Þ; Q b l ð0ÞÞ, we have that Alþ1 ¼ Al þ diagð Q 11 22 lþ1 l l bl bl klþ1 1 ðaÞ k2 ðaÞ ¼ k1 ðaÞ k2 ðaÞ þ Q 11 ð0Þ Q 22 ð0Þ:
Since
g
lþ1 a2Plþ1
2 þ 4 4l sþ4 cl jl
Using (a.3) with m ¼ l, we obtain
@hl ð/l ; hÞ @z
6C
ð42Þ
K 2lþ1 ejl K lþ1 lþ1
6 C lþ1 :
Differentiating (41) in a and using (24), we obtain
rl ;gl
l
rlþ1 ;glþ1
if i – j
> b > Q l ðkÞ > : pffiffiffiffiffiij ; 1ðk;xÞ
l þ jjg l jjr ;g c2l jl2sþ2
6C
ð40Þ
:
Next we solve (34). Let Q lij and wlij ð1 6 i; j 6 2Þ be the
8 > > > <
ð44Þ
:
6 CðjjQ l jjrlþ1 ;glþ1 jj/l jjrlþ1 ;glþ1 þ jjðId
a2Plþlþ1
k2Z2
!
þ hl ð/l ; hÞjjrlþ1 ;glþ1
b l ðkÞjÞejkjrlþ1 ðmax sup j@ sa /
jkj6K lþ1
2l 4 4sþ4 cl jl
Now we estimate g lþ1 ðh; aÞ and Q lþ1 ðh; aÞ. By (35), (44) and Lemma A.2, we have
Hence, by Condition (a.2) and Lemma A.1 (see Appendix A), we obtain
X
1 X jjwl jjqrlþ1 ;glþ1 6 C
jjðE þ wl Þ1 jjrlþ1 ;glþ1 6
jjg lþ1 jjrlþ1 ;glþ1 6 jjðE þ wl Þ1 jjrlþ1 ;glþ1 jjQ l /l þ ðId CK lþ1 Þg l
glþ1
Cjjg il jjrl ;gl ejkjrl jkj2s = 2l ;
we get
hl ð/l ; hÞ ¼ Orlþ1 ;glþ1
by Condition (a.1). Differentiating (38) in a and using (24), we have s¼0;1
Thus, noting Plþ1 Plþ , (40) and (43) imply the estimates (30). Since
bm bm max Q 11 ð0Þ ; Q 22 ð0Þ gm
gm
1 2
6 jjQ m jjrm ;gm 6 C m ;
m ¼ 0; 1; . . . ; l;
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L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33 1
we obtain
g lþ1 ðhÞ ¼ ðE þ wl ðhÞÞ ðQ l ðhÞ/l ðhÞ þ ðId CK lþ1 Þg l ðhÞ þ hl ð/l ; hÞÞ ¼ ðE þ wl ðhÞÞ1 Q l ðhÞ/l ðhÞ þðId CK lþ1 Þg l ðhÞ þ hl ð/l ; hÞ :
d klþ1 ðaÞ klþ1 ðaÞ 0 0 l X 1 1 2 P d k1 ðaÞ k2 ðaÞ C 2q da da q¼0 P vlþ1 P
1 v 2 0
glþ1
for a 2 Plþ1 . Similarly, we can prove other inequalities in Condition (a.1) with m ¼ l þ 1. To estimate the measure of Plþ1 , we introduce some sets. Let
c pffiffiffiffiffiffiffi Rlijk ¼ a 2 Pl : klj ðaÞ kli ðaÞ þ 1ðk; xÞ < ls ; 0 < jkj 6 K lþ1 ; jkj ð46Þ
[
Rlk ¼
Rlijk ;
16i;j2
V lik ¼
[
Rl ¼
Rlk ;
0–jkj6K lþ1
pffiffiffiffiffiffiffi
a 2 Pl : kli ðaÞ 1ðk; xÞ <
cl jkj
; 0 < jkj 6 K lþ1 ; s
By direct calculation, we can easily verify g lþ1 ðhÞ ¼ g lþ1 ðhÞ. The proof of Lemma 3.1 is completed. h Remark 3.2. If the eigenvalues of the initial matrix A0 ðaÞ of (22) with m ¼ 0 are nonzero, real and different from each other, and the terms g 0 ðh; aÞ; Q 0 ðh; aÞ; h0 ðz; h; aÞ are real analytic in h, then the iterative Lemma 3.1, removing Conditions (a.1), (a.4) and (a.5), can be true for all a 2 ½a; b as there is no small divisor problem and we do not need to drop any a. The proof is trivial (see [2]), and we only point out the main ideas here. Since the frequency vector x satisfies the Diophantine condition jðk; xÞj P jkjcs ; 80 – k 2 Z2 , the inequalities
pffiffiffiffiffiffiffi c 0 ki ðaÞ 1ðk; xÞ P s ; jkj
ð47Þ V lk
¼
[
V lik ;
[
l
V ¼
i¼1;2
V lk :
Then
[
min V l Þ;
l ¼ 0; 1; 2; . . .
By Lemma A.3 and the non-degeneracy conditions (23), we obtain
MeasRlijk 6
4cl MeasPl : v0 jkjs
l
MeasR 6
16cl MeasPl ; v0 jkjs 16
v0
X
cl MeasPl
;
6 i; j 2;
min jk01 ðaÞ k02 ðaÞj; min jk0i ðaÞj; i ¼ 1; 2 :¼ r0 > 0
a2½a;b
a2½a;b
and
b 0 ð0Þ; Q b 0 ð0Þ þ diag Q b 1 ð0Þ; Q b 1 ð0Þ Alþ1 ¼ A0 þ diag Q 11 22 11 22 b l ð0Þ; Q b l ð0Þ ; þ þ diag Q 11 22 for each positive integer l, we can conclude that kl1 ðaÞ and kl2 ðaÞ are real, and satisfy
Hence,
MeasRlk 6
c jkjs
obviously hold for all a 2 ½a; b. Since
0–jkj6K lþ1
Plþ1 ¼ Pl n ðRl
P
pffiffiffiffiffiffiffi 0 kj ðaÞ k0i ðaÞ þ 1ðk; xÞ
min
jkj
s
6 C 0 cl MeasPl :
0–k2Z2
Similarly, we get
MeasV l 6 C 0 cl MeasPl : Thus,
MeasPlþ1 P MeasPl MeasRl MeasV l P MeasPl ð1 C 0 cl Þ and (a.4) is available with m ¼ l þ 1. Finally, we consider the reality condition by induction. Without loss of generality, we only verify the reality condition for g lþ1 ðh; aÞ as for the terms Q lþ1 ðh; aÞ and hlþ1 ðz1 ; h; aÞ, the proof is similar. We have proved (a.5) with m ¼ 0 in Section 2. Assuming that the reality condition holds for m 6 l, we shall verify that the reality condition is satisfied when m ¼ l þ 1. By (35), the uniqueness of the solution of (33) and (34), and the reality condition (a.5) with m ¼ l, we have
min kl1 ðaÞ kl2 ðaÞ ; min kli ðaÞ ; i ¼ 1; 2
a2½a;b
P r0 C
a2½a;b
l1 X
1 2
q P
q¼0
r0 ; 2
by (25). Hence,
pffiffiffiffiffiffiffi c l ki ðaÞ 1ðk; xÞ P ls jkj and
pffiffiffiffiffiffiffi c l kj ðaÞ kli ðaÞ þ 1ðk; xÞ P ls ; jkj hold for all a 2 ½a; b; 0 – k 2 Z2 ; 1 6 i; j 2, and the nonresonance conditions are true along the iteration.
4. A KAM theorem and Proofs of Theorems 2.1–2.4 Theorem 4.1. Suppose that the equation
( ðEqÞ0 :
h_ ¼ x; z_0 ¼ A0 ðaÞz0 þ g 0 ðh; aÞ þ Q 0 ðh; aÞz0 þ h0 ðz0 ; h; aÞ; ð48Þ
31
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
satisfies Conditions (a.1)–(a.5) in Lemma 3.1 with l ¼ 0. Then for any given 0 < c 1; c ¼ Oðs Þð0 < s 6 16Þ and sufficiently small > 0, there exists a Cantorian closed subset P1 ½a; b with MeasP1 ¼ b a OðcÞ such that for any a 2 P1 , by a transformation
T : z0 ¼ ez þ Uðh; aÞ þ Wðh; aÞez ;
2
1
Moreover, h ¼ h0 þ xt; z0 ¼ Uðh0 þ xt; aÞ is a quasi-periodic solution of (48). The terms Uðh; aÞ; Wðh; aÞ satisfy the reality condition
U ¼ U;
X
wi þ
T2 P1
wi1 wi2 wil2 wil1 þ w0 w1 wl2 wl1 A
zl :¼ zl þ Ul1 þ Wl1 zl : By (30), it is easy to see that fUl g and fWl g are convergent and 1 2
X i i i jjUl1 jj 6 þ 2 2sþ2 2 2sþ2 2 2sþ2 cj c ji ci ji 06i6l1 i i 06i
1
1
Remark 4.1. If the initial matrix A0 ðaÞ of (48) and the terms g 0 ðh; aÞ; Q 0 ðh; aÞ; h0 ðz0 ; h; aÞ satisfy the conditions in Remark 3.2, and Conditions (a.2), (a.3) in Lemma 3.1 with l ¼ 0 are fulfilled for all a 2 ½a; b, then the conclusions of Theorem 4.1 hold for all a because in this case there is no small divisor problem.
6C
þ
b 1 ð0Þ; diagð Q 11
b 1 ð0ÞÞ Q 22
þ
b l ð0Þ; diagð Q 11
b l ð0ÞÞ; Q 22
T2 P1
1 2
i
i¼0
1
c20 j20sþ2 c21 j12sþ2
i
c2 j2sþ2 06i6l1 i i
l2
1 2
l2 sþ2 c2l2 j2l2
:
1 2
X
i : 2 2sþ2 c j 06i6l1 i i
P1 ¼
1 \
~z ¼ lim zl ; l!þ1
A1 ðaÞ ¼ lim Al ;
Pm ;
l!þ1
m¼0
W ¼ lim Wl : l!þ1
1
sup jjA0 ðaÞ A1 ðaÞjj 6 C 2 ;
a2P1
1 X
1 2
i
sup jjUjj 6 C 12s ; T2 P 1
1
sup jjWjj 6 C 22s : T2 P1
1
6 C 2 :
i¼0
On the lth step of the iteration process in Lemma 3.1, we denote the variables z by zl and z1 by zlþ1 , respectively. By (28) and using iteration, it implies
z0 ¼ z1 þ /0 þ w0 z1 ; z0 ¼ z2 þ ð/0 þ /1 þ w0 /1 Þ þ ðw0 þ w1 þ w0 w1 Þz2 ;
l2
We have
þ
6C
1
1 2
Let
l!þ1
from (25), it follows that fAl ðaÞg is convergent and l X
0
l2
l1 X
sup jjWl1 jj 6 C
U ¼ lim Ul ;
b 0 ð0Þ; Q b 0 ð0ÞÞ Alþ1 ðaÞ ¼ A0 ðaÞ þ diagð Q 11 22
2
1 2
2sþ2 c2l1 jl1
l!þ1
and a family of change of variables T l such that ðEqÞl T l ¼ ðEqÞlþ1 . Since for each positive integer l,
þ
2
Similarly, we obtain
½a; b ¼ P0 P1 Pl
a2P1
l1
T ¼ lim T 0 T 1 T l ;
a series of closed subsets
sup jjA0 ðaÞ Alþ1 ðaÞjj 6 C
i
1
1 2
sþ2 c2il1 j2il1
1
i i 2 2sþ2 2 2sþ2 c ji ci ji 6l1 i 1
06i1
T2 P1
1
2
2
1 2
X
þ þ
1
sup jjWjj 6 C 22s :
b r0 ; Wðb0 ; r0 Þ Wðb1 ; r1 Þ Wðbl ; rl Þ W 0 ; 2 2
1
06i1
T2 P1
Proof. Since the conditions (a.1)–(a.5) with l ¼ 0 in Lemma 3.1 are satisfied for Eq. (48), inductively we get a series of domains
wi1 wi2 wi3 þ
06i1
and estimates
sup jjUjj 6 C 12s ;
X
wi1 wi2 þ
06i1
X
þ
sup
W¼W
1
wi1 wi2 wil2 /il1 þ w0 w1 wl2 /l1 A
06i6l1
1
2
X
þ
ð49Þ
z 2 Þ and supa2P1 jjA0 ðaÞ A1 ðaÞjj 6 C 2 . where h1 ¼ Ob0 ;r0 ;P ðe
wi1 wi2 /i3 þ
06i1
06i1
in which U and W are analytic in h 2 r0 Þ and Lipschitz in a 2 P1 , the Eq. (48) can be transformed into
h_ ¼ x; ez_ ¼ A1 ðaÞez þ h1 ðez ; h; aÞ;
X
wi1 /i2 þ
06i1
X
þ
Uð12
(
X
/i þ
06i6l1
a¼a
h ¼ h;
X
z0 ¼ zl þ
From (29), it easily deduces that U and W satisfy the reality condition
U ¼ U;
W ¼ W:
By (a.4) in Lemma 3.1, we get
MeasP1 ¼ lim MeasPl ¼ b a OðcÞ: l!þ1
Hence, by the coordinate change T , Eq. (48) can be transformed into
32
(
L. Lu, X. Li / Chaos, Solitons & Fractals 62-63 (2014) 23–33
h_ ¼ x ez_ ¼ A1 ðaÞez þ h1 ðez ; h; aÞ;
which is defined in the domain W h1 ¼ Ob0 ;r0 ;P ðe z 2 Þ. h 2
2
b0 r0 ; 2 P1 , where 2
1
z ¼ 0 is a solution of (49), returning to (48), Particularly, e and it is obvious that for every a 2 P1 , Eq. (48) possesses a quasi-periodic solution h ¼ h0 þ xt; z0 ¼ Uðh0 þ xt; aÞ. Proofs of Theorems 2.1–2.4. We first prove Theorem 2.3. For the fixed point O, let us return to the Eq. (11). For 8a 2 ½a; b, where a > 0; b < a0 < 0:217234, or a > a0 > 0; b < 0:217234, and a0 satisfies the equality cos a2 þ a0 þ a2 sin a2 ¼ 0, we have
SpecA0 ðaÞ ¼ SpecAðaÞ ¼
npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffio 1 a þ 1; 1 a þ 1
and
dk0 ðaÞ 1 1 1 i pffiffiffiffiffiffiffiffiffiffiffiffi P ¼ pffiffiffiffiffiffiffiffiffiffiffiffi P pffiffiffiffiffiffiffiffiffiffiffiffi ; d a 2 aþ1 2 aþ1 2 bþ1
i ¼ 1; 2;
dðk0 ðaÞ k0 ðaÞÞ 1 1 1 1 2 pffiffiffiffiffiffiffiffiffiffiffiffi P ¼ pffiffiffiffiffiffiffiffiffiffiffiffi P pffiffiffiffiffiffiffiffiffiffiffiffi : da aþ1 aþ1 bþ1 Obviously, the system (11) satisfies all conditions (a.1)– (a.5) in Lemma 3.1 with m ¼ 0. Using Theorem 4.1, we only drop a small open set D1 with measure OðcÞ such that for a 2 ½a; b n D1 , the system (11) has a quasi-periodic solution formed h ¼ h0 þ xt; z0 ¼ Uðh0 þ xt; aÞ in a neighborhood of the fixed point O. Returning to (5), h ¼ h0 þ xt; z ¼ SðaÞUðh0 þ xt; aÞ is a real analytic quasi-periodic solution of (5), where SðaÞ is the matrix defined in (12). For the fixed points C 3 and C 4 , we only need to analyse the system (8) with B2 and B2 instead of B1 ; a2 and a2 instead of a1 , respectively. Noting that
SpecA0 ðaÞ ¼ SpecB2 ðaÞ ¼ SpecB2 ðaÞ npffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio ¼ 1 cos a2 þ a; 1 cos a2 þ a ; since sin a2 ¼ aa2 ; a2 2
3p 2
i ¼ 1; 2;
SpecA0 ðaÞ ¼ SpecB3 ðaÞ ¼ SpecB3 ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a3 a; cos a3 a ; ¼ the constant matrices belong to the case in Remark 3.2. By Remark 4.1 we obtain two real analytic quasi-periodic solutions in neighborhoods of C 5 and C 6 for all a 2 ½a; b, respectively. This completes the proof of Theorem 2.3. The proofs of Theorems 2.1, 2.2 and 2.4 are similar to that of Theorem 2.3, we omit the detail. h Acknowledgements This work was supported by the NNSF grants (11371132, 11071066) of China, by Hunan Provincial Innovation Foundation For Postgraduate, by Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China) and by the Construct Program of the Key Discipline in Hunan Province. We would like to thank the reviewers for helpful comments and suggestions which led to significant improvements of the present paper. Appendix A Lemma A.1. [2,33] For d > 0; l > 0, the following inequality holds true:
X
e2jkjd jkjl 6
k2Zn
ll 1 ð1 þ eÞn : e dlþn
Lemma A.2. [29] Denoting As as the space of all functions on Tn bounded and analytic in the strip fh : jImhj 6 sg. If v 2 As and K r P 1, then
0 6 r 6 s;
where the constant C only depends on n.
ð50Þ
2
dðk0 ðaÞ k0 ðaÞÞ cos a þ a þ a sin a 1 2 2 2 2 ¼ : 3 da ðcos a2 þ aÞ2
For the fixed points C 5 and C 6 ,
jjðId CK Þv jjsr 6 CK n eK r jjv jjs ;
; 2p , we obtain
dk0 ðaÞ cos a þ a þ a sin a i 2 2 2 ¼ ; 3 da 2ðcos a þ aÞ2
S S Let D ¼ D1 D2 D3 . Obviously, MeasD ¼ OðcÞ, and for a 2 ½a; b n D there are real analytic quasi-periodic solutions in neighborhoods of C 3 ; C 4 and O simultaneously.
ð51Þ
By calculating, there exists a unique a0 2 ð0; 0:217234Þ such that the equation cos a2 þ a þ a2 sin a2 ¼ 0 holds. Notice that the functions of (50) and (51) are continuous in a on the close set ½a; b, we can obtain that there exists a maximum value f0 and a minimum value v0 , and both of them are positive constants. Similarly, by Theorem 4.1, we also take out small open sets D2 and D3 with measure OðcÞ such that for a 2 ½a; b n D2 there is a real analytic quasi-periodic solution in a neighborhood of the fixed point C 3 , and for a 2 ½a; b n D3 there is a real analytic quasi-periodic solution in a neighborhood of the fixed point C 4 .
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