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Under some assumptions on the parities of Fx, t and ex, t, by a small twist theorem of reversible mapping we obtain the existence of quasiperiodic solutions and boundedness of all the so

Volume 2011, Article ID 845413, 18 pages

doi:10.1155/2011/845413

Research Article

Lagrangian Stability of a Class of

Second-Order Periodic Systems

Shunjun Jiang, Junxiang Xu, and Fubao Zhang

Department of Mathematics, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Junxiang Xu,xujun@seu.edu.cn

Received 24 November 2010; Accepted 5 January 2011

Academic Editor: Claudianor O Alves

Copyrightq 2011 Shunjun Jiang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We study the following second-order differential equation: Φp xFx, txω pΦp xα|x| l x

e x, t  0, where Φ p s  |s| p−2 s p > 1, α > 0 and ω > 0 are positive constants, and l

satisfies−1 < l < p − 2 Under some assumptions on the parities of Fx, t and ex, t, by a

small twist theorem of reversible mapping we obtain the existence of quasiperiodic solutions and boundedness of all the solutions

1 Introduction and Main Result

equation

are bounded for all time, that is, whether there are resonances that might cause the amplitude

of the oscillations to increase without bound

superlinear function gx, t.

When gx satisfies

where ϕx  ox as |x| → ∞ and et is a 2π-periodic function After introducing

Under some suitable assumptions on ϕx and et, by using a variant of Moser’s small twist

and the boundedness of all solutions

considered the following nonlinear differential equation





x



x−

also to change the original problem to Hamiltonian system and then use a twist theorem of area-preserving mapping to the Pioncar´e map

The above differential equation essentially possess Hamiltonian structure It is well known that the Hamiltonian structure and reversible structure have many similar property Especially, they have similar KAM theorem

where a is a positive constant and ex, t is 2π-periodic in t Under some assumption of F, ϕ

0 ϕ tdt and 0 < γ < 1 < α < 2 Moreover,





x k

d k Φx

dx k





where c is a constant Note that here and below we always use c to indicate some constants.





x k

∂ kl F x, t

∂x k ∂t l





 ≤c · |x| σ , 



x k

∂ kl e x, t

∂x k ∂t l





Then, the following conclusions hold true





x

Then a KAM Theorem for reversible mapping can be applied to the Poincar´e mapping of this nearly integrable reversible system and some desired result can be obtained

Our main result is the following theorem

Theorem 1.1 Suppose that e and F are of class C6in their arguments and 2π-periodic with respect

to t such that

Moreover, suppose that there exists σ < l such that





x k

∂ km F x, t

∂x k ∂t m





 ≤c · |x| σ ,





x k

∂ km e x, t

∂x k ∂t m





for all x /  0, for all 0 ≤ k ≤ 6, 0 ≤ m ≤ 6 Then every solution of 1.9 is bounded.

Remark 1.2 Our main nonlinearity α |x| l x in 1.9 corresponds to ϕ in 1.5 Although it is

more special than ϕ, it makes no essential difference of proof and can simplify our proof

greatly It is easy to see from the proof that this main nonlinearity is used to guarantee the small twist condition

2 The Proof of Theorem

mainly consists of two steps The first one is to find an equivalent system, which has a nearly integrable form of a reversible system The second one is to show that Pincar´e map of the equivalent system satisfies some twist theorem for reversible mapping

2.1 Action-Angle Variables

n → Ê

is called reversible with respect to G, if

with DG denoting the Jacobian matrix of G.

Z1



x, y, t

,

Z2





x, y, t

x Φq



y

,



y

.

2.4

Consider the homogeneous differential equation:





u

This equation takes as an integrable part of1.9 We will use its solutions to construct a pair

π p 2

p−1 1/p

0

ds



1− s p /

wt

0

ds



1− s p /

The function wt will be extended to the whole real axis R as explained below, and the

i sinp0  0, sin

ii p − 1|sin

p t| p |sinp t| p  p − 1;





x

x  r 2/psinp ωθ,

y  r 2/qΦp

ω sinp ωθ

2.9

It is easy to see that

∂

x, y

2

q ω

r f1t, θ, r  N1t, θ, r  P1t, θ, r,

2

1

ω p−1 r

4/p−12/plsinp θsinl

p θsin

p θ,

P1t, θ, r  − q

2

1

ω p−1 r

1−2/qsinp θF r 2/psinp θ, t Φq

r 2/qΦp

ω sinp θ

2

1

ω p−1 r

1−2/qsinp θe r 2/psinp θ, t ,

N2t, θ, r  α q

p

1

ω p r 4/p−22/plsinl

p θsin2

p θ,

P2t, θ, r  q

p

1

ω p r −2/qsinp θF r 2/psinp θ, t Φq

r 2/qΦp

ω sinp θ

p

1

ω p r −2/qsinp θe r 2/psinp θ, t ,

2.12

2.2 Some Lemmas

Lemma 2.1 Let Ft, θ, r  Fr 2/psinp θ, t , et, θ, r  er 2/psinp θ, t  If Fx, t and ex, t satisfy

1.11, then





r k ∂

ks F t, θ, r

∂r k ∂t s





 ≤c · r 2/pσ ,





r k ∂

ks e t, θ, r

∂r k ∂t s





 ≤c · r 2/pσ1 , 2.13

Proof We only prove the second inequality since the first one can be proved similarly.





r k

∂ ks e t, θ, r

∂r k ∂t s





 





r k

∂ ks e x, t

∂x k ∂t s

∂x

∂r

k

 · · ·  r k ∂1se x, t

∂x∂t s

∂ k x

∂r k











c1



p

r k ∂

ks e x, t

∂x k ∂t s

r 2/p−1 ksink θ  · · ·  c k



p

r k ∂

1se x, t

∂x∂t s r 2/p−ksinp θ











cx k ∂

ks e x, t

∂x k ∂t s  · · ·  cx ∂1se x, t

∂x∂t s







≤ c · |x| σ1 ≤ c · r 2/pσ1

2.14

To describe the estimates inLemma 2.1, we introduce function space M nΨ, where Ψ

is a function of r.

Definition 2.2 Let n  n1, n2 ∈ N2 We say f ∈ M n Ψ, if for 0 < j ≤ n1, 0 < s ≤ n2, there

exist r0> 0 and c > 0 such that

r jD j

r D s t f t, θ, r ≤ c · Ψr, ∀r ≥ r

Lemma 2.3 see 6  The following conclusions hold true:

r f ∈ M n−0,j r −j Ψ and D s

t f ∈ M n−s,0 Ψ;

ii if f1∈ M nΨ1 and f2∈ M nΨ2, then f1f2∈ M nΨ1Ψ2;

Ψξ ≤ cΨr,

lim

f

t  g1, θ, r  g2



∈ M nΨ, nn1, n2

with n1 n

Moreover,

f

t  g1, θ, r

− ft, θ, r ∈ M n1−1,min{n1,n2 }Ψ · Ψ1,

f

t, θ, r  g2



− ft, θ, r ∈ M min{n1,n2},n2 −1

Proof This lemma was proved in6 , but we give the proof here for reader’s convenience



∂u ∂t ≤ c · Ψ2, 

∂ js u

∂r j ∂t s  ∂ js g2

∂r j ∂t s , ∂ js v

∂r j ∂t s  ∂ js g1

Since g1∈ M nΨ1, g2∈ M nΨ2, it follows that

∂ js u

∂r j ∂t s ∈ M n

r −jΨ2 , ∂ js v

∂r j ∂t s ∈ M n

large



∂ ks g

∂r b ∂t m · ∂ j1j



1u

∂r j1∂t j1 · · · ∂ j b j



b u

∂r j b ∂t jb

· ∂ i1i



1v

∂r i1∂t i1· · · ∂ i m i



m v

where the sum is found for the indices satisfying

j1 · · ·  j b  i1 · · ·  i m  k, j1 · · ·  j

b  i

1 · · ·  i

Without loss of generality, we assume that

j1 j

1 1, , j b1 j

b1  1,

i1 i

1 1, , i m1 i

Since

∂ ks g

∂r b ∂t m · ∂ j1j



1u

∂r j1∂t j1· · · ∂ j b2 j



b2 u

∂r j b2 ∂t j b2

· ∂ j b21 j



b21 u

∂r j b21 ∂t j b21

· · · ∂ j b1 j



b1 u

∂r j b1 ∂t j b1

· ∂ j b11 j



b11 u

∂r j b11 ∂t j b11

· · · ∂ j b j



b u

∂r j b ∂t jb

· ∂ i1i



1v

∂r i1∂t i1· · · ∂ i m2 i



m2 v

∂r i m2 ∂t im2 · ∂ i m21 i



m21 v

∂r i m21 ∂t im21

· · · ∂ i m1 i



m1 v

∂r i m1 ∂t im1

· ∂ i m11 i



m11 v

∂r i m11 ∂t im11

· · · ∂ i m i



m v

∂r i m ∂t im ,

2.28

we have

∂ ks g

∂r k ∂t s ≤c · r −b Ψr −j b11 ···j br m2−m1Ψb−b1b2

1 r −i m11 ···i mΨm−m2m2−m1 

2

≤ c · r b2−b1−j b11 ···j b m2−m1−i m11 ···i m

r −bb2−b1 Ψbb2−b1

1 Ψm−m1

2

≤ c · r −k Ψ,

2.29

and then,

f

t  g1, θ, r  g2



Obviously

f

t  g1, θ, r

− ft, θ, r 

1

0

∂f

∂t



t  ηg1, θ, r

Since

∂f

∂t ∈ M n−1,0 Ψ, lim

r→ ∞



ηg1



f

t  g1, θ, r

2.3 Some Estimates

Lemma 2.4 f1t, θ, r ∈ M 5,5 r β1 , f2t, θ, r ∈ M 5,5 r β , where β  22 − p  l/p.

Proof Since f1t, θ, r  P1t, θ, r  N1t, θ, r, we first consider P1t, θ, r and N1t, θ, r By

Lemma 2.1, Ft, θ, r ∈ M 5,5 r 2/pσ Again Φq r 2/qΦp ω sin

p θ  r 2/pΦqΦp ω sin

p θ ∈

M 5,5 r 2/p, using the conclusion iii ofLemma 2.3, we have P1t, θ, r ∈ M 5,5 r β1, where

M 5,5 r β1  In the same way we can prove f2t, θ, r ∈ M 5,5 r β ThusLemma 2.4is proved

Since−1 < l < p − 2, we get β < 0 So |f2| ≤ r β

dr

dθ  f1t, θ, r1 f2t, θ, r−1, dt

dθ 1 f2t, θ, r−1.

2.34

We will prove that the Poincar´e mapping can be a small perturbation of integrable

dr

dθ  f1t, θ, r  h1t, θ, r  N1t, θ, r  P1t, θ, r  h1t, θ, r,

dt

dθ  1 − f2t, θ, r  h2t, θ, r  1 − N2t, θ, r  −P2t, θ, r  h2t, θ, r,

2.35

where h1t, θ, r  −f1f2/ 1f2, h2t, θ, r  f2

2/ 1f2, with f1t, θ, r and f2t, θ, r defined

Lemma 2.5 h1t, θ, r ∈ M 5,5 r 2β1 , h2t, θ, r ∈ M 5,5 r 2β .

Proof If r0 is sufficiently large, then |f2t, θ, r| < 1/2 and so 1/1  f2t, θ, r 

s0−1s f s

h1t, θ, r ∞

s0

It is easy to verify that

∂ km

∂r k ∂t m f2s1 f1 

|i|k,|j|m,

c i,j ∂

i1j1

∂r i1∂t j2f1 ∂

i2j2

∂r i2∂t j2f2· · · ∂ i s2j s2

where i  i1, , i l2 , |i|  i1 · · ·  i s2 , and j and |j| are defined in the same way as i and |i|.

So, we have

∂ km

∂r k ∂t m h1 

|i|k,|j|m,n≥2

c i,j ∂ i1j1

∂r i1∂t j1f1 ∂ i2j2

∂r i2∂t j2f2· · · ∂ i n j n

∂ i τ j τ

So







∂ km

∂r k ∂t m h1





 ≤c i,j r β1−i1r β−i2· · · r β−i n

≤ c1r β1 r β

r β n−2 r −i1···i n

≤ cr −k r 2β1

2.40

dr

dθ  N1t, θ, r  g1t, θ, r,

dt

dθ  1 − N2t, θ, r  g2t, θ, r,

2.41

where g1t, θ, r  P1t, θ, r  h1t, θ, r and g2t, θ, r  −P2t, θ, r  h2t, θ, r By the proof of

Lemma 2.4, we know P1 ∈ M 5,5 r β1 and P2 ∈ M 5,5 r β Thus, g1t, θ, r ∈ M 5,5 r β1−σ

and g2t, θ, r ∈ M 5,5 r β−σ where

σ  min





with σ < l < p − 2, −1 < l.

2.4 Coordination Transformation

Lemma 2.6 There exists a transformation of the form

such that the system2.41 is changed into the form

dλ

dθ  g1t, θ, λ,

dt

dθ  1 − N2t, θ, λ  g2t, θ, λ,

2.44

where g1, g2satisfy:

g1∈ M 5,5

λ β1−σ , g2∈ M 5,5

Moreover, the system2.44 is reversible with respect to the involution G: λ, −t → λ, t.

Proof Let

θ

0

2

α

ω p−1

1

l 2sinl2

then

S r, θ  Sr, θ  2π p



It is easy to see that

where

L

λ, θ  2π p



g1t, θ, λ  g1t, θ, λ  L, g2t, θ, λ  N2t, θ, λ − N2t, θ, λ  L  g2t, θ, λ  L.

2.51

simpler

Obviously,

lim

λ→ ∞

λ−1λ 4/p−12/pl  lim

λ→ ∞

Note that

By the third conclusion ofLemma 2.3, we have that

g2t, θ, λ  L ∈ M 5,5

In the same way as the above, we have

and so

N2t, θ, r − N2t, θ, λ  N2t, θ, λ  L − N2t, θ, λ ∈ M 5,5

λ−1λ β λ 4/p−12/pσ

 M 5,5 λ ββ .

2.56

g2t, θ, λ ∈ M 5,5

2.44

Lemma 2.7 There exists a transformation of the form

which changes2.44 to the form

dλ

where N2  α · λ β with α  1/2π p q/pα/ω p 2/p2π p /ω

0 |sinl

p θ| l2 d θ and the new perturbations H1λ, τ, θ, H2λ, τ, θ satisfy:





λ k

∂ ks

∂λ k ∂t s H1λ, τ, θ



,





λ k1

∂ ks

∂λ k ∂t s H2λ, τ, θ



 ≤C · λ β1−σ . 2.60

Moreover, the system2.59 is reversible with respect to the involution G: λ, τ → λ, −τ.

Proof We choose

0

Then

Defined a transformation by

dλ

where

H1λ, τ, θ  g1

H2λ, τ, θ  g2

∂λ g1

2.65

It is easy to very that





λ k

∂ ks

∂λ k ∂t s H1λ, τ, θ



,





λ k1

∂ ks

∂λ k ∂t s H2λ, τ, θ



 ≤C · λ β1−σ . 2.67 ThusLemma 2.7is proved

of small perturbation of an integrable

Let

then

Now, we define a transformation by

λ ρ α

1/β

dρ

dθ  g1



ρ, τ, θ, 

, dτ



ρ, τ, θ, 

where

g1



ρ, τ, θ, 

 ε−1d N2

dλ H1



λ

, ρ

, τ, θ

, g2



ρ, τ, θ, 



λ

, ρ

, τ, θ

Lemma 2.8 The perturbations g1and g2satisfy the following estimates:







∂ ks

∂ρ k ∂τ s g1





 ≤c · 1σ0,







∂ ks

∂ρ k ∂τ s g2





 ≤c · 1σ0, σ0 −σ

β > 0. 2.74

Proof By2.73, 2.60 and noting that λ  ρ/α 1/β, it follows that

g1 N 

 H1



 ≤c·−1λ β1 H1

≤ c · −1λ β−1 λ β1−σ ≤ c · −1λ 2β−σ ≤ c · 1σ 0 .

2.75

2.5 Poincar ´e Map and Twist Theorems for Reversible Mapping

We can use a small twist theorem for reversible mapping to prove that the Pioncar´e map P has an invariant closed curve, if is sufficiently small The earlier result was due to Moser

Let A  a, b ×S1be a finite part of cylinder C  S1×R, where S1 R/2πZ, we denote



ρ, τ

 g1



ρ, τ, 

,

ρ1 ρ  l2



ρ, τ

 g2



ρ, τ, 

periodic

Lemma 2.9 see 14, Theorem 2  Let ω  2nπ with an integer n and the functions l1, l2, g1, and

g2satisfy

∂ρ > 0, ∀ρ, τ

∈ A,

l2·, ·, g1·, ·, , g2·, ·,  ∈ C5A.

2.78

In addition, we assume that there is a function I : A → R satisfying

I ∈ C6A, ∂I

∂ρ > 0, ∀ρ, τ

∈ A,

l1



ρ, τ

∂τ



ρ, τ

 l2



ρ, τ

·∂I

∂ρ



ρ, τ

∈ A.

2.79

Moreover, suppose that there are two numbers a, and b such that a < a < b < b and

I M a < I m a ≤ I M a < I m

b < I m b, 2.80

where

ρ∈S1 I

ρ, τ

ρ∈S1I

ρ, τ

Then there exist ς > 0 and Δ > 0 such that, if < Δ and

g1·, ·, 

C5Ag2·, ·, 

C5A < ς 2.82

the mapping f  has an invariant curve inΓA , the constant ς and Δ depend on a, a, b, b, l1, l2, and I.

In particular, ς is independent of

Remark 2.10 If −l1, l2, g1, g2satisfy all the conditions ofLemma 2.9, thenLemma 2.9still holds.

Lemma 2.11 see 14, Theorem 1  Assume that ω /∈ 2πQ and l1·, ·, l2·, ·g1·, ·,  and

g2·, ·,  ∈ C4A If

2π

0

∂l1

∂ρ



τ, ρ

dτ > 0, ∀ρ ∈ a, b 2.83

then there exist Δ > 0 and ς > 0 such that f  has an invariant curve inΓA if 0 < < Δ and

g1·, ·, 

C4Ag2·, ·, 

C4A < ς. 2.84

The constants ς and Δ depend on ω, l1, l2only.

2.86, l1 −2π p ρ, l2 0

2.6 Invariant Curves

g1





ρ, τ, θ, 

, g2





ρ, τ, θ, 

2.85

P :

⎧

⎨

⎩

τ1 τ  2π p − 2π p ρ  g1



ρ, τ, 

,

ρ1 ρ  g2



ρ, τ, 







∂ kl

∂ρ k ∂τ l g1





,







∂ kl

∂ρ k ∂τ l g2





Case 1 2πpis rational Let I  −l1 2π p ρ, it is easy to see that

l1



ρ, τ



ρ, τ

 −2π p ρ < 0, ∂l1



ρ, τ

∂ρ < 0,

I

ρ, τ

∂ρ



ρ, τ

> 0, l2

ρ, τ

 0,

l1



ρ, τ  ∂I

∂τ



ρ, τ

 l2



ρ, τ  ∂I

∂ρ



ρ, τ

 0.

2.88

Case 2 2π pis irrational Since

2π p

0

∂l1

∂ρ



τ, ρ

dτ  −2π p

2

Thus, in the both cases, the Poincare mapping P always have invariant curves for

being su

curve of the Poincare mapping, which guarantees the boundedness of solutions of the system

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of small perturbation of an integrable

Let

then

Now, we define a transformation...

Proof We choose

0

Then

Defined a transformation...

By the third conclusion ofLemma 2.3, we have that

g2t, θ, λ  L ∈ M 5,5