• Aucun résultat trouvé

On the distance between separatrices for the discretized pendulum equation

N/A
N/A
Protected

Academic year: 2021

Partager "On the distance between separatrices for the discretized pendulum equation"

Copied!
46
0
0

Texte intégral

(1)

HAL Id: hal-00287701

https://hal.archives-ouvertes.fr/hal-00287701

Preprint submitted on 12 Jun 2008

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

On the distance between separatrices for the discretized pendulum equation

Hocine Sellama

To cite this version:

Hocine Sellama. On the distance between separatrices for the discretized pendulum equation. 2008.

�hal-00287701�

(2)

disretized pendulum equation

IRMA -UMR7501CNRS/ULP

7rueRenéDesartes-67084StrasbourgCedex, Frane

email: sellamamath.u-strasbg.fr

abstrat

Weonsiderthedisretization

q(t+ε) +q(t−ε)−2q(t) =ε2sin q(t) ,

ε > 0 asmall parameter, of thependulum equation q′′ = sin(q); in system

form,wehavethedisretization

q(t+ε)−q(t) =εp(t+ε), p(t+ε)−p(t) =εsin q(t) .

ofthesystem

q=p, p= sin(q).

Thelattersystemofordinarydierentialequationshastwosaddlepoints

atA= (0,0),B = (2π,0)andnearboth,thereexiststableandunstableman-

ifolds. Italsoadmits aheteroliniorbitonnetingthestationary pointsB

andA parametrisedbyq0(t) = 4 arctan et

andwhihontainsthestable

manifoldofthissystematAaswellasitsunstablemanifoldatB. Weprove

thatthestablemanifoldofthepointAandtheunstablemanifoldofthepoint B do notoinide for the disretization. More preisely, we show that the vertialdistanebetweenthesetwomanifoldsisexponentiallysmallbut not

zeroand in partiular wegiveanasymptoti estimateof this distane. For

this purpose we usea method adapted from the artileof Shäfke-Volkmer

[10℄usingformalseriesandaurateestimatesoftheoeients. Ourresult

issimilartothatofLazutkinet.al.[9℄;ourmethodofproof,however,isquite

dierent.

Keywords: Dierene equation; Manifolds; Linear operator; Formal solu-

tion;Gevreyasymptoti;Quasi-solution

1 Introdution

We onsiderthe following dierene equation

q(t + ε) + q(t − ε) − 2q(t) = ε

2

sin q(t)

.

(1.1)

(3)

This seond order equation is a disretization of the pendulum equation

q

′′

= sin(q)

. It isequivalent to the following systemof rstorder diereneequations

(

q(t + ε) = q(t) + εp(t + ε), p(t + ε) = p(t) + ε sin q(t)

.

(1.2)

whih an beonsidered asadisretization ofthesystem

(

q

= p, p

= sin(q).

(1.3)

Thelattersystemhastwo saddlepointsat

A = (0, 0)

,

B = (2π, 0)

and thereexist

stableand unstable manifolds. For the disretized equation (1.2) and suiently

small

ε > 0

,these manifoldsstill exist.

The system (1.3) has

q

0

(t), q

0

(t)

, where

q

0

(t) = 4 arctan e

−t

,as a hetero-

lini orbit onneting the stationary points

B

and

A

; it is a parametrisation of the urve

p = − 2 sin(q/2)

and ontains thestable manifold of (1.3) at the point

A

aswellasits unstablemanifold at

B

. Thisurve,together with

p = 2 sin(q/2)

,

separates regions with periodi orbits from regions with non-periodi orbits and

is therefore often alled a separatrix. Our purpose is study the behavior of this

separatrixunderdisretizationoftheequationitturnsoutthatthereisnolonger

aheterolini orbit for system (1.2) and its thestable manifoldat

A

and the un-

stablemanifold at

B

no longer oinide. More preisely, we want to estimate the

distanebetween thestablemanifold

W

s,ε of(1.2)at

A

andtheunstablemanifold

W

u,ε+ of(1.2) at

B

asa funtionof the parameter

ε

.

Lazutkinet.al.[9 ℄,Gelfreih[4℄,(seealsoLazutkin[7℄[8℄) hadgivenanasymp-

totiestimateofthesplittinganglebetween themanifolds. Startingfromahetero-

linisolution of the dierential equation, they study thebehaviorof analyti so-

lutionsofthediereneequationintheneighbourhoodofitssingularities

t = ±

π2

i

.

We showthatthe distanebetween these twomanifolds isexponentiallysmall

but not zero and we give an asymptoti estimate of this distane. This result

is similar to that of Lazutkin et. al. [9℄; our method of proof, however, is quite

dierent.

Weuseamethodadaptedfromtheartile ofShäfke-Volkmer [10 ℄usingafor-

malpowerseries solutionand aurate estimates of theoeients. Thismethod

wasadapted for thelogisti equation in Sellama[11℄. It turns out that theadap-

tationof thismethodfor thependulum equationismorediult thaninthease

ofthe logistiequation.

We will show

Theorem1.1. Givenany positive

t

0,it isknownthat for suientlysmal

ε

0

> 0

andall

t ∈ ] − ∞ , t

0

]

there is exatly oneone point

w

u,ε+

(t) = (q

0

(t), p ˜

+u,ε

(t))

on the

stableunstable manifoldhavingrst oordinate

q

0

(t)

. There existonstants

α 6 = 0

,

suhthat for any positive

t

0

distv

w

s,ε+

(t), W

s,ε

= 4πα

ε

2

cosh(t) sin 2πt ε

e

π

2 ε

+ O

1 ε e

π

2 ε

,

as

ε ց 0,

(4)

uniformly for

− t

0

< t < t

0,where distv

P, W

u,ε

denotesthe vertial distane of a

point

P

from the unstable manifolds

W

u,ε.

ThisresultorrespondstotheresultofLazutkinet.al.[9℄astheanglebetween

themanifolds at anintersetionpoint isasymptotially equivalent to

1 q0(t)

d

dt distv

w

+s,ε

(t), W

u,ε

,but we do not want to give anydetail here.

Our proof uses the following steps. First, we onstrut a formal solution for

the dierene equation (1.1) in the form of a power series in

d = 2

arsinh

(ε/2)

,

whose oeients are polynomials in

u = tanh(dt/ε)

. This is done in setion 2;

theintrodution of

d

is neessarybeause polynomials aredesired asoeients.

Then, we give asymptoti approximations of these oeients using appropriate

norms onspaesof polynomials. To thatpurposeweintrodueoperatorsonpoly-

nomials series. In setion 6 we use the trunated Laplae transform to onstrut

a funtion whih satises(1.1) exeptfor an exponentially small error. The next

andlaststepistogiveanasymptotiestimateforthedistaneofsomepointofthe

stablemanifoldfromthe unstablemanifold. Aalulation showsthat

α = 89.0334

and therefore

4πα = 1118.8267

(See Remark5.4); theorresponding onstantsof Lazutkin have already been alulated with high preision (See Lazutkin et. al.

[9℄). A proof that

α 6 = 0

asin [10℄ or [11 ℄ would be possible. Y.B. Suris [12 ℄ had

shownthat

α 6 = 0

.

2 Formal solutions

The purpose of this setion is to nd a onvenient formal solution for equation

(1.1). First,we need some preparations. Weput

u : = tanh d

ε t

, q

0d

(t) : = 4 arctan exp

− d ε t

!

, q

d

(t) =

p

1 − u

2

A

d

(u) + q

0d

(t), A

d

(u) =

X

n=1

A

n

(u)d

n

for a formalsolution of (1.1), where

d = ε +

P

n=3

d

n

ε

n is aformal powers series

in

ε

to be determined.

Remark. Thelinearization ofequation(1.1)atthepoint

A

givesthefollowing

equation

Z(t + ε) + Z(t − ε) − 2Z (t) = ε

2

Z(t).

The parameter

d

issuh that

Z(t) = e

−dt is a solution ofthis equation, therefore

ε

and

d

are oupledbytherelation

d = 2

arsinh

(ε/2).

(5)

q

0d

(t + ε) + q

0d

(t − ε) − 2q

0d

(t) = 2

+∞

X

n=1

1

(2n) ! q

0d(2n)

(t) ε

2n

,

(2.1)

where

2

(2n)! q

(2n)0d

(t)ε

2n

/d

2n is anoddpolynomial

I

2n−1

(u)

multiplied by

√ 1 − u

2;

we nd

I

2n−1

(1) = 4/(2n)!

.

Using

cos(q

0d

) = 2u

2

− 1, sin(q

0d

) = 2u √

1 − u

2,we an express our equation

(1.1)in theform

A

d

(T

+

)

r

1 − (T

+

)

2

1 − u

2

+ A

d

(T

)

r

1 − (T

)

2

1 − u

2

− 2A

d

(u) = f ε, u, A

d

(u)

(2.2)

or equivalently

A

d

(T

+

)

cosh(d) + u sinh(d) + A

d

(T

)

cosh(d) − u sinh(d) − 2A

d

(u) = f ε, u, A

d

(u)

(2.3)

where

f ε, u, A

d

(u)

= ε

2

2u cos

A

d

(u)

p

1 − u

2

+ 2u

2

− 1

√ 1 − u

2

sin

A

d

(u)

p

1 − u

2

+∞

X

n=1

I

2n−1

(u) d

2n

,

T

+

= T

+

(d, u) = u + tanh(d)

1 + u tanh(d) = tanh d

ε (t + ε) , T

= T

(d, u) = u − tanh(d)

1 − u tanh(d) = tanh d

ε (t − ε) .

As

u → 1,

the expressions

T

+ and

T

redue to 1, the denominators in (2.3) simplifyto

e

±dand heneequation (2.3)redues to

(e

−d

+ e

d

− 2)A

d

(1) = ε

2

(2 + A

d

(1)) − 4(cosh(d) − 1).

This is equivalent to

(2 cosh(d) − 2 − ε

2

)(2 + A

d

(1)) = 0

and hene we have

neessarily

ε = 2 sinh(d/2)

ifwe want a formalsolution suh that theoeients

have limitsas

u → 1

.

Theorem 2.1. (On the formal solution) If

ε = 2 sinh(d/2)

, then equation

(2.2)has a unique formalsolution of the form

A

d

(u) =

+∞

X

n=1

A

2n−1

(u)d

2n

,

(2.4)

where

A

2n−1

(u)

are oddpolynomials of degree

≤ 2n − 1

.

(6)

Proof. We will use theIndution Priniple to show that there exist unique odd

polynomials

A

1

, A

3

, A

5

...A

2n−1 suhthat

Z

n

(d, u) =

n

X

k=1

A

2k−1

(u)d

2k (2.5)

satisfy

R

n

(d, u) = O(d

2n+4

)

(2.6)

where

R

n

(d, u) = Z

n,d

T

+

r

1 − (T

+

)

2

1 − u

2

+ Z

n,d

T

r

1 − (T

)

2

1 − u

2

− 2Z

n,d

(u) − f ε, u, Z

n,d

(u)

(2.7)

For

n = 1

,a short alulation shows that we must have

A

1

(u) = −

14

u

and hene

Z

1,d

(u) = −

14

ud

2. We obtain

R

1

(d, u) = ( − 91

48 u

5

+ 137

48 u

3

− 23

24 )d

6

+ O(d

8

).

Suppose now thatthereexists

A

1

, A

3

, A

5

...A

2n−1 suhthat

Z

n

(d, u) =

n

X

k=1

A

2k−1

(u)d

2k (2.8)

satises (2.6), (2.7). We show that there is a unique polynomial

A

2n+1

(u)

suh

that

Z

n+1

(d, u) = Z

n

(d, u) + A

2n+1

(u)d

2n+2 (2.9)

satises(2.6). We put

R

n

(d, u) = R

2n+3

(u)d

2n+4

+ O d

2n+6

(2.10)

where

R

2n+3

(u)

isoddand deg

(R

2n+3

(u)) ≤ 2n + 3

.

We substitute

Z

n+1

(d, u)

in equation (2.7). Using Taylor expansion, (2.9),

(2.10) and

ε = 2 sinh(d/2)

, we obtain

Z

n+1,d

T

+

r

1 − (T

+

)

2

1 − u

2

− Z

n+1,d

T

r

1 − (T

)

2

1 − u

2

− 2Z

n+1,d

(u) − f ε, u, Z

n+1,d

=

(u

4

− 2u

2

+ 1)A

′′2n+1

(u) + (4u

3

− 4u)A

2n+1

(u) + R

2n+3

(u)

d

2n+4

+ O d

2n+6

(7)

(1 − u

2

)

2

A

2n+1

(u)

+ R

2n+3

(u) = 0

(2.11)

This dierential equation has a unique solution vanishing at

u = 0

without

singularityat

u = 1

,namely

A

2n+1

(u) = −

Z u

0

Rt

1

R

2n+3

(s)ds

(1 − t

2

)

2

dt.

(2.12)

We now show that this solution is an odd polynomial of

u

. It is lear that Rt

1

R

2n+3

(s)ds

vanishes for

t = 1

and as

R

2n+3

(s)

is odd, it also vanishes for

t = − 1

. It suesto show that

R

2n+3

(s)

also vanishesat

t = ± 1

. Indeed,taking

the limit of(2.7) as

u → 1

aswe didfor (2.3) and using

u→1

lim f ε, u, Z(d, u)

= ε

2

Z(d, 1)

we obtain

R

2n+3

(1)d

2n+4

= e

d

+ e

−d

− 2 − ε

2

!

Z(d, 1) + O(d

2n+6

).

By our hoie of

ε = 2 sinh(d/2)

, we obtain

R

2n+3

(1)d

2n+4

= O(d

2n+6

)

. Conse-

quently

R

2n+3

(1) = 0

. As

R

2n+3

(u)

isodd,wealsohave

R

2n+3

( − 1) = − R

2n+3

(1) = 0

. Thisprovesthat

A

2n+1

(u)

isan oddpolynomialof degree

A

2n+1

(u)

≤ 2n + 1

and

A

2n+1

(0) = 0.

Therst polynomials

A

2n−1

(u)

with

n > 0

an be alulated usingMaple.

n 1 2 3

A

2n−1

(u) −

14

u

91

864

u

3

57647

u

2880319

u

5

+

1152185

u

3

691203703

u

Now, wentrodue theoperators

C

2

, C , S

2

, S

dened by

C (Z )(d, u) =

12

Z (d, T

+12

) + Z(d, T

12

) S (Z)(d, u) =

12

Z(d, T

+12

) − Z (d, T

12

) C

1

(Z)(d, u) =

12

Z(d, T

+

) + Z(d, T

) S

1

(Z)(d, u) =

12

Z(d, T

+

) − Z (d, T

)

(2.13)

where

T

+12

= T

+

(

d2

, u)

,

T

12

= T

(

d2

, u)

and

Z(d, u)

is aformalpower series in

d

whose oeientsarepolynomials. We an showthat

C

1

= 2 S

2

+ Id

S

1

= 2 SC

(2.14)

(8)

C

1

(Q · G) = C

1

(Q) C

1

(G) + S

1

(Q) S

1

(G) S

1

(Q · G) = C

1

(Q) S

1

(G) + S

1

(Q) C

1

(G)) C (Q · G) = C (Q) C (G) + S (Q) S (G) S (Q · G) = C (Q) S (G) + S (Q) C (G)

(2.15)

if

Q, G

areformal powerseries in

d

whoseoeients arepolynomials of

u

.

3 Norms for polynomials and basis

In this setion we reall some denitions and results of [10℄. Using a ertain

suquene of polynomials. we dene onvenient norms on spaes of polynomials

whihsatisessome useful proprieties. We denoteby

• P

the setof allpolynomial whose oeentsare omplex

• P

n the spaes ofall polynomials ofdegree lessthan or equalto n

Proposition 3.1. [10℄. We dene the sequene of polynomials

τ

n

(u)

by

τ

0

(u) = 1, τ

1

(u) = u, τ

n+1

(u) = 1

n Dτ

n

(u)

for

n ≥ 1,

where the operator

D

is dened by

D := (1 − u

2

) ∂

∂u .

Thenwe have

1.

T

+

(d, u) =

P

n=0

τ

n+1

(u)d

n,

2.

τ

n

(u)

has exatly degree

n

and hene

τ

0

(u), ..., τ

n

(u)

form a basisof

P

n,

3.

τ

n

(tanh(z)) =

(n−1) !1 dzdn−1

tanh(z)

Denition 3.2. Let

p ∈ P

n. As

τ

0

(u), ..., τ

n

(u)

form a basisof

P

n, we an write

p ∈ P

n as

p =

n

X

k=0

a

k

τ

k

(u).

Thenwe dene the norms

k p k

n

=

n

X

i=0

| a

i

| π 2

n−i

.

(3.1)

(9)

Theorem3.3. [10℄. Letn,mbepositiveintegers and

p ∈ P

n,

q ∈ P

m

.

The norms

(3.1)have the followingproperties:

1.

k Dp k

n+1

≤ n k p k

n.

2. If the onstant term of

p

in the basis

{ τ

0

, τ

1

.., τ

n

}

iszero, we have

k p k

n

≤ k Dp k

n+1

.

3. There exists a onstant

M

2 suh that

k pq k

n+m

≤ M

2

k p k

n

k q k

m.

4. There isa onstant

M

3 suh that that for all

n > 1

,

| p(u) | ≤ M

3 π2n

k p k

n

( − 1 ≤ u ≤ 1)

.

5. There is a onstant

M

4 suh that for all

n > 1

with

p(1) = p( − 1) = 0

p τ

2

n−2

≤ M

4

k p k

n

4 Operators

Inthis setionwewillusesomedenitionsofShäfke-Volkmer[10℄andadapt their

resultson operatorsonpolynomialseries to our ontext. Let

Q :=

(

Q(d, u) =

X

n=0

Q

n

(u)d

n

,

where

Q

n

(u) ∈ P

n

,

for all

n ∈

N )

.

Byabuseof notation, let

k Q k

n

= k Q

n

k

n for a polynomial series

Q(d, u) =

X

n=0

Q

n

(u)d

n

.

Denition 4.1. Let

f

be formalpower series of

z

whose oeients are omplex.

We dene a linear operator

f (dD)

on

Q

by

f (dD)Q(d, u) =

X

n=0

Xn

i=0

f

i

D

i

Q

n−i

(u)

d

n (4.1)

where

f (z) =

P

i=0

f

i

z

i and

Q ∈ Q

.

BytheaboveDenition and (1)ofProposition 3.1we an showthat

Q(d, T

+

θd, u)

= exp(θdD)Q

(d, u)

for

Q ∈ Q

and all

θ ∈

C

(10)

C (Q) = cosh(

d2

D)Q, S (Q) = sinh(

d2

D)Q

C

1

(Q) = cosh(dD)Q, S

1

(Q) = sinh(dD)Q

(4.2)

for polynomial series

Q

in

Q

.

Remark. Aording to the denitionof norms in(3.1),we have

If

Q ∈ Q ,

then

dQ ∈ Q

and

k dQ k

n

= π

2 k Q k

n−1 for all

n ≥ 1.

(4.3)

Theorem4.2. [10℄Let

f (z)

be formalpowerserieshavingaradiusofonvergene

greater than

and letk be apositve integer. There isa onstant

K

suhthat: If

Q

isa polynomialseries havingthe following property

k Q k

n

0

for

n < k

M(n − k) !(2π)

−n for

n ≥ k

where

M

is independent of

n

and

Q ∈ Q

then the polynomial series

f (dD)Q

satises

k f (dD)Q k

n

0

for

n < k

M K(n − k) !(2π)

−n for

n ≥ k

Nowwedene on

Q

the following operator

J = S

dD .

(4.4)

where the notation

S

dD means simply

F (dD)

with

F(z) = 1

z sinh(

z2

)

.

Lemma 4.3. For eah integer

k

there exist a positive onstant

K

suh that: If

Q

is a polynomial series with odd

Q

n of degree at most

n

,

k Q k

n

= 0

for

n < k

in

ase of positive

k

and

k dDQ k

n

≤ M(n − k) !(2π)

−n for

n ≥ max(0, k),

where

M

is independent of

n

,then the polynomialseries

J

−1

(Q)

satises

kJ

−1

(Q) k

n

≤ M K(2π)

−n

(n − k + 1) !

for

k ≤ 1

(n − 1) ! log(n)

for

k = 2

(n − 1) !

for

k ≥ 3

(11)

Proof. We an see easily that

J

−1

= π C ˜

−1

+ g(dD)

, where

C ˜ = cosh(

14

dD)

and

g(z)

isanalyti for

| z | < 4π

,and usethe proofof [10 ℄.

We have

S = dD J = J dD

, but using this relation for the inversion of

S

wouldgive aninsuient result. Usingof theformula

1 = 2 z sinh( z

2 ) + F (z)z,

where

F(z) = z

−2

(z − 2 sinh( z 2 ))

isan entire funtion,we obtaintherelation

Q = 2 J Q + F(dD) dDQ

(4.5)

forpolynomialseries

Q ∈ Q

. Thiswill beessential intheproof of

Theorem 4.4. For eah integer

k

there exist a positive onstant

K

suh that: If

Q

is a polynomialseries with odd

Q

n of degree atmost

n

,

k Q k

n

= 0

for

n < k

in

ase of positive

k

, and

kS (Q) k

n

≤ M (n − k) !(2π)

−n for

n ≥ max(0, k),

where

M

isindependentof

n

, then the polynomial series

Q

satises

k Q k

n

≤ M K(2π)

−n





(n − k + 1) !

for

k ≤ 1 (n − 1) ! log(n)

for

k = 2 (n − 1) !

for

k ≥ 3

Proof. Bythepreedingtheorem,wehavethewantedinequalitiesfor

dDQ = J

−1

S Q

in theplae of

Q

. Herewe used again

k dDZ k

n

≤ (n − 1) k Z k

n−1 for any

polynomialseries

Z ∈ Q

. Usingtheorem4.2impliesthesamefor

F (dD)dDQ

with

the entire funtion

F

of (4.5) As

k Z k

n

≤ k dDZ k

n+1 bytheorem 3.3, we nd the

wanted inequalities (and even something better in the ases

k ≥ 2

) also for

J Q

beause

dD J = S

. Thus formula(4.5) yields theresult

Inordertoobtainanasymptotiapproximationfortheoeientsoftheformal

solution, we will need to reverse some operators. This is not possible for the

operators

S

and

dD

on the set

Q

,but we an dene a subset

Q

of

Q

on whih

theseoperatorshave aright inverses.

Ifwe dene

Q

:=

(

Q(d, u) =

X

n=1

P

n

(u)d

n

,

where

P

n

(u) ∈ P

n

,

for all

n ≥ 1

)

.

where

P

n isthe subspae of

P

n dened by

P

n

:=

n

X

i=0

α

i

τ

i

∈ P

n

, | α

0

= 0

Références

Documents relatifs

The absolute magnitude M and apparent magnitude m of a Cepheid star can be also related by the distance modulus equation (1), and its distance d can be

The main new ingredient in the proofs of Theorems 1.1, 1.2 and 1.4 is a bilinear estimate in the context of Bourgain’s spaces (see for instance the work of Molinet, Saut and

Here we take the logarithmic Schr¨ odinger equation (LogSE) as a prototype model. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which

Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds..

a function that satisfies the equation (1.5) except for an expo- nentially small error, then we show that this quasi-solution and the exact solution of equation (1.5) are

&amp; Hall/CRC Research Notes in Mathematics. The Navier-Stokes problem in the 21st century. From the Boltzmann equation to an incompressible Navier- Stokes-Fourier system. From

The main new ingredient in the proofs of Theorems 1.1, 1.2 and 1.4 is a bilinear estimate in the context of Bourgain’s spaces (see for instance the work of Molinet, Saut and

VELO, The global Cauchy problem for the nonlinear Klein-Gordon