On rank one perturbations of Hamiltonian system with periodic coefficients

1.6 MB·

by  Mouhamadou Dosso

On rank one perturbations of Hamiltonian system with periodic
coefficients
6 MOUHAMADOUDOSSO AROUNAG.Y.TRAORE
1 UniversitéFHBdeCocody-Abidjan UniversitéFHBdeCocody-Abidjan
0
UFRMaths-Info., UFRMaths-Info.
2
22BP582Abidjan22, 22BP582Abidjan22,

n
CÔTED’IVOIRE CÔTED’IVOIRE
a
J [email protected] [email protected]

5
2 JEAN-CLAUDEKOUABROU

UniversitéFHBdeCocody-Abidjan
]
A UFRMaths-Info.
N 22BP582Abidjan22,
CÔTED’IVOIRE
.
h k [email protected]
t −
a
m Abstract: From a theory developed by C. Mehl, et al., a theory of the rank one perturbation of Hamiltonian
[ systems with periodic coefficients is proposed. It is showed that the rank one perturbation of the fundamental

solution ofHamiltonian systemwithperiodic coefficients issolution ofitsrankoneperturbation. Someresults on
1
v theconsequences ofthestrong stability ofthese typesofsystems ontheirrank oneperturbation isproposed. Two
9 numericalexamplesaregiventoillustrate thistheory.
2
7
2010Mathematics SubjectClassification : 15A63,15A21,47A55,93B10,93C73.
6
0
. Key–Words: Eigenvalue, symplectic matrix,Hamiltoniansystem,Fundamentalsolutions, Perturbation.
1
0
6 1 Introduction isJ-symplectic [2,7,3,15]andsatisfiestherelation-
1
: shipX(t+nP)= X(t)Xn(P), t Rand n N.
v Let J,W R2N 2N be two matrices such that J ∀ ∈ ∀ ∈
× The solution of the system evaluated at the period
i ∈
X is nonsingular and skew-symmetric matrix. We say is called the monodromy matrix of the system. The
that the matrix W is J-symplectic (or J-orthogonal
r eigenvalues of this monodromy matrix are called the
a ) if WTJW = J. These types of matrices (so-
multipliers of the system (2). The following defini-
called structured) usually appear in control theory
tion permits to classify the multiplies of Hamiltonian
[15,12,11,1]: morepreciselyinoptimalcontrol[11]
system
and in the parametric resonance theory [10, 15]. In
these areas, these types of matrices are obtained as Definition1 Let ρ be a semi-simple multiplier of (2)
solutionsofHamiltoniansystemswithperiodiccoeffi- lyingontheunitcircle. Thenρiscalledamultiplierof
cients. Aboutthesesystems,thataredifferentialequa- thefirst(second)kindifthequadraticform(iJx,x)is
tionswithP-periodic coefficientsofthebelowform positive (negative) on the eigenspace associated with
ρ. When(Jx,x) = 0,thenρisofmixedkind.
dX(t)
J = H(t)X(t), t R (1) In this definition, the notation (iJx,x) stands for the
dt ∈ Euclideanscalarproductandi = √ 1.
−
Thisother definition proposed by S.K.Godunov
where JT = J, (H(t))T = H(t) = H(t + P).
− [4,5,8,9]givesanotherclassification ofthemultipli-
The fundamental solution X(t) of (1) i.e. the matrix
ersof(2)
satisfying
Definition2 Let ρ be a semi-simple multiplier of (2)
dX(t) lying on the unit circle. We say that ρ is of the red
J = H(t)X(t), t R
dt ∈ ∗+ (2) (green) color or in short r-multiplier ( g-multiplier)

X(0) = I if (S x,x) > 0 ( respectively (S x,x) < 0 )
 2N 0 0

on the eigenspace associated with ρ where S = 4. the multipliers of the system (2)are either ofthe
0
(1/2) (JX(P))T +(JX(P)) . If(S x,x) = 0, we red color and either of the green color. The r-
0
saythatρisofmixedcolor. multipliers and g-multipliers of the monodromy
(cid:0) (cid:1)
matrixshouldbewellseparated i.e. thequantity
From Definition 2, Dosso and Sadkane obtained a
result of strong stability of symplectic matrix (see δ (X(P)) = min eiθk eiθl ;eiθk,eiθl
S
[2,6,4]) | − |
arer multpliersanndg multipliers of (2)
− − }
Theorem3 A symplectic matrix is strong stability if (4)
andonlyif
shouldnotbeclosetozero.
1. alleigenvalues areontheunitcircle;
5. S 0,S 0andS S > 0
r g r g
≥ ≤ −
2. the eigenvalues are either red color or green 6. P +P = I andPTS P = 0.
r g r 0 g
color;
The paper is organized as follows. In Section
3. thesubspaces associated ofthesedeuxgroupsof 2 we give some preliminaries and useful results to
theeigenvalues arewellseparated. introduce the rank one perturbations of Hamiltonian
systems with periodic coefficients. More specifically,
Denote by Pr and Pg the spectral projectors as- this section explains what led us to rank one pertur-
sociated with the r eigenvalues and r eigenvalues bations of Hamiltonian system with periodic coeffi-
− −
of the monodromy matrix X(P) of (2) and let’s cients. Section3explainstheconceptofrankoneper-
put Sr := PTrS0Pr = SrT 0 and turbationofHamiltoniansystemswithcoefficients. In
Sg := PTgS0Pg = SgT 0 wh≥ere S0 = Section 4 we analyze the consequences of strongly
≤
(1/2) (X(P)J)+(X(P)J)T . Wegivethefollow- stable of Hamiltonian systems with periodic coeffi-
ingtheoremwhichgathersallassertionsonthestrong cientsonitsrankonperturbation. Section5isdevoted
(cid:0) (cid:1)
stability ofHamiltonian systems withperiodic coeffi- to numerical tests. Finally some concluding remarks
cients[15,6,2]. aresummarizedinSection6
Throughout this paper, we denoted the identity
Theorem4 The Hamiltonian system (2) is strongly andzeromatricesoforderkbyIk and0k respectively
stableifoneofthefollowing conditions issatisfied: or just I and 0 whenever it is clear from the context.
The 2-norm of a matrix A is denoted by A . The
1. If there exists ǫ > 0 such that any Hamiltonian transposeofamatrix(orvector)U isdenotkedkbyUT.
system with P-periodic coefficients of the form
dx(t)
J = H(t)x(t)andsatisfying 2 Rank one perturbation of sym-
dt
plecticmatricesdepending onapa-
e T
H H H(t) H(t) dt < ǫ rameter
k − k≡ k − k
Z0
Let W R2N 2N be a J-symplectic matrix where
isstable. e e J R2∈N 2N i×s skew-symmetric matrix (i.e. JT =
×
∈
J)[13,14].
2. The monodromy matrix W = X(P) of the sys- −
tem(2)isstrongly stable Definition5 We call a rank one perturbation of the
symplectic matrix W any matrix of the form W =
3. (KGLcriterion) the multipliers of the system (2)
(I +uuTJ)W whereu R2N.
are either of the first kind and either of second ∈
f
kind. The multipliers of the first kind and sec- Werecallinthefollowingpropositionsomeproperties
ondkindofthemonodromymatrixshouldbewell of rank one perturbations of symplectic matrices (see
separated i.e. thequantity [16]).
Proposition 6 LetW beaJ-sympectic matrix.
δ (X(P)) = min eiθk eiθl ;eiθk,eiθl
KGL
| − |
1. AnyrankoneperturbationofW isJ-symplectic.
aremultipliers of(2)nofdifferentkinds (3)
}
2. The invertible of a rank one perturbation I +
shouldnotbeclosetozero. uuTJ ofidentitymatrixI isthematrixI uuTJ.
−
Proof: See[13,16]fortheproof. ifρisaneigenvalue ofmixedcolor,
⊓⊔ •
LetubeavectorofR2N. Considerthefollowing
lemma (S y,y) = 0 = (S y,y) = ϕ(y).
0 0
⇒
Lemma7 Considertherankoneperturbations W = e
(I+uuTJ)W oftheJ-symplecticmatrixW. Thenfor
⊔⊓
any y R2N, the quadratic form (S0y,y) is deffined We consider the following rank one perturbation
∈
by ofthefundamental solution X(t)of(2)
(S y,y) = (S y,y) ϕ(y) (5)
0 0
− X(t) = (I +uuT)X(t) (6)
where
e
thenwehavethefollowinglemma
S = (1/2) (JW)+(JW)T e
0
and (cid:16) (cid:17) Lemma9 If X(t) is a J-symplectic matrix function
e f f
ϕ(y) = (1/2) ((JuuTJW)+(JuuTJW))T)y,y . issucahvtehcatotrrafnukn(ceXtio(nt)u−(tX) (t)C)2=N1, t∀>t >00su,cthhetnhathtere
∈ ∀
(cid:0) (cid:1) e
Proof: DevelopingS ,wehave
0 X(t) = (I +u(t)u(t)TJ)X(t), t R
∀ ∈
S0 =(1/2) (JWe )+(JW)T + Converesely, for any vector u(t) C2N, the matrix
(1/2)(cid:0)(JuuTJW)+(J(cid:1)uuTJW)T . function X(t)isJ-symplectic. ∈
e
wededuct (cid:2) (cid:3)
Proof: Aeccording toLemma7.1of[13, Section 7,p.
18], for all t > 0, there exists a vector u(t) C2N
(S0y,y) = (S0y,y)+ ∈
suchthat
(1/2) (JuuTJW)+(JuuTJW)T y,y
e
X(t) = (I +u(t)u(t)TJ)X(t).
(cid:0)(cid:2) ϕ(y) (cid:3) (cid:1)
| =(S y,y)+ϕ{(yz) }
0 Moreover, eif X(t) is J-symplectic, X(t) is also J-
symplectic.
⊔⊓
⊓⊔ This Lemmaleads us to introduceethe concept of
rank one perturbation of Hamiltonian systems with
Corollary 8 Letρbeaneigenvalue ofW ofmodulus
periodic coefficients.
1 and y an eigenvector associated with ρ. Then ρ is
Nowconsider, inthefollow, thatthevector func-
aneigenvalue ofredcolor (respectively eigenvalue of
tion is a vector constant. We give the following the-
green ) if and only if S y,y > ϕ(y) (respectively
0 orem which extend Theorem 7.2 of [13, Section 7, p.
S y,y < ϕ(y)). (cid:16) (cid:17) 19]tomatrizantofsystem(2).
0 e
H(cid:16)oweve(cid:17)rif S y,y = ϕ(y),thenρisofmixedcolor.
e 0 Theorem10 Let J C2N 2N be skew-symmetric
×
∈
(cid:16) (cid:17) and nonsingular matrix, (X(t)) fondamental so-
t>0
Proof: Accoerdingtolemma7,weget lution of system (2) and λ(t) C an eigenvalue of
∈
X(t) for all t > 0. Assume that X(t) has the Jordan
(S y,y) = (S y,y) ϕ(y) canonical form
0 0
−
FromifDρefiisnaitnioenig2e,nwvaeluheavoeefredcolor, jMl=11Jn1(λ(t))⊕jMl=21Jn2(λ(t))⊕···⊕lmjM=(1t)Jnm(t)(λ(t))⊕J(t),
•
where n > > n with m : R N
(S0y,y)> 0 =⇒ (S0y,y) > ϕ(y) ; afunction1ofin·d·e·x suchmth(ta)t the algebraic mu−l→tiplici∗-
ties is a(t) = l n + +l n and (t) with
1 1 m(t) m(t)
ifρisaneigenvalue ofgreeencolor, σ( (t)) C λ(t)··c·ontains allJordan Jblocks as-
•
J ⊆ \{ }
sociated with eigenvalues different from λ(t). Fur-
(S0y,y)< 0 = (S0y,y) < ϕ(y) ; thermore,letu C2N andB(t) =uuTJX(t).
⇒ ∈
e
(1) If t > 0, λ(t) 1,1 , then generically with we considered the integers l and n constant k
k k
∀ 6∈ {− } ∀ ∈
respecttothecomponentsofu,thematrixX(t)+ 1,...,m(t) for an index m(t) given. When t = 0,
{ }
B(t)hastheJordancanonical form λ(0) = 1 with m(0) = 2N and l = 1, k. All
k
∀
Jordanblocksarereducedto1.
l1M−1Jn1(λ(t))⊕Ml2 Jn2(λ(t))⊕···⊕
j=1  j=1 
3 Rank one perturbations of Hamil-
lm(t) 
jM=1 Jnm(t)(λ(t))⊕Je(t), (7) tonian system with periodic coeffi-
cients
where (t) contains all the Jordan blocks of
J
X(t)+B(t)associated with eigenvalues differ-
Let u be a constant vector of R2N. (X(t)) the
entfromλ(t). t 0
fundamental solution of system 2. We have∈th≥e fol-
(2) If t > 0,verifying λ(t ) +1,1 ,wehave lowingproposition
0 0
∃ ∈ { }
(2a) if n is even, then generically with respect Proposition 11 Consider the perturbed Hamiltonian
1
tothecomponents ofu,thematrix system
X(t ) + B(t ) has the Jordan canonical
0 0
form dX(t)
J = [H(t)+E(t)]X(t) (8)
dt
lj1M=−11Jn1(λ(t0))⊕jMl=21Jn2(λ(t0))⊕···⊕ where e e
lm(t)  E(t) = (JuuTH(t))T+JuuTH(t)+(uuTJ)TH(t)(uuTJ).
jM=1 Jnm(t)(λ(t0))⊕Je(t0),
ThenX(t) = (I+uuTJ)X(t)isasolutionofsystem
where (t) contains all the Jordan of (8).
J
X(t) + B(t) associated with eigenvalues e
Proof: Byderivation ofX(t),weobtain:
different fromλ(t).
(2b) icfalnly1 wisithodrde,sptehcetntol1thise ecvoemnpoannedntgsenoefrui-, JdX(t) =J(I +uuTJ)Je−1JdX(t)
dt dt
the matrix X(t ) + B(t ) has the Jordan
0 0 e =J(I +uuTJ)J 1H(t)X(t),
canonical form −
according fromsystem (2)
Jn1+1(λ(t0))⊕lj1M=−12Jn1(λ(t0))⊕···⊕ ==[[HH((tt))++JJuuuuTTHH((tt))]]X(I(+t)uuTJ)−1X(t)
lm(t)  =[H(t)+JuuTH(t)](I uuTJ)X(t)
jM=1 Jnm(t)(λ(t0))⊕Je(t0), because (I +uuTJ) 1 = (I −uuTJ) (seee[16])
−
− e
where (t ) contains all the blocks of = H(t) H(t)uuTJ +JuuTH(t)
J 0 − −
X(t0)+B(t0)associatedwitheigenvalues (cid:2)JuuTH(t)uuTJ X(t)
different fromλ(t ).
0
 (cid:3) 
dPercooomf:poFsoitrioanll t(7>) a0c,ciofrλd(int)g6∈to{[−131,,1T}h,eworeehmav7e.2th])e. =H(t)+(|JuuTH(t))T+JeuuTHE({(tzt))+(uuTJ)TH(t)(uuTJ})Xe(t)
Other hand, the number of Jordan blocks depend on HencethefollowingperturbedHamiltonianequa-
the variation of t. Thus, this number is a function of tion(8)where
indexm : R+ N .
∗
For the other tw−o→points (2a) and (2b), they show in E(t) =(JuuTH(t))T +JuuTH(t)+
thesamewaythatitems(2)and(3)ofTheorem7.2of (uuTJ)TH(t)(uuTJ) (9)
[13,Theorem7.2])sinceX(t )+B(t )isaconstant
0 0
matrix.
⊓⊔ ⊔⊓
In reality, the integers l ,...,l (t) and indexes We note that E(t) is symmetric and P-periodic
1 m
n ,...,n (t) are not constant when t varies. The i.e. E(t)T = E(t) and E(t + P) = E(t) for all
1 m
number of Jordan blocks and their sizes can varied t 0. Thefollowing corollary givesusasimplified
∈≥
in function of the variation of t. In Theorem 10, formofsystem(8)
Corollary 12 Thesystem(8)canbeputattheform Definition14 We call rank one perturbations of
Hamiltonian system with periodic coefficients, any
dX(t) perturbation oftheform(10)of(2).
J = (I uuTJ)TH(t)(I uuTJ)X(t),
dt − −
 Consider the following canonical perturbed system
 e
 X(0) = I +uuTJ e takingI2N att = 0.
(10)

 dW(t)
 e J = (I uuTJ)TH(t)(I uuTJ)W(t),
Proof: Indeed,developing dt − −

(I uuTJ)TH(t)(I uuTJ)andweget  f
− −  f
W(0) = I
(I uuTJ)TH(t)(I uuTJ) = H(t)+ (11)

− − 
(JTuuTH(t))T +JTuuTH(t)+(uuTJ)TH(t)(uuTJ) f
4 Consequence of the strong stabil-
E(t)
ity on rank one perturbations
| {z }
andX(0) = (I +uuTJ)X(0) = I +uuTJ.
⊓⊔
We give the following proposition which is a conse-
Wegivethefollowingcorollary
quentofCorollary8
e
Corollary 13 Let (X(t)) be the fundamental so-
t 0
lutionofsystem(2). ≥ Proposition 15 If a symplectic matrix W is strongly
stable, then there exists a positif constant δ such that
All solution X(t) of perturbed system (10) of system
any vector u R2N verifying uuTJW < δ, we
(2),isoftheformX(t) = (I +uuTJ)X(t). ∈ k k
have S y,y = ϕ(y) for any eigenvector y of W
e 0
6
Proof: From Proposition 8 if X(t) is a solution de
(2),theperturbed meatrixW(t)= (I +uuTJ)X(t)is where(cid:16)Se0 = ((cid:17)1/2) (JW)+(JW) withW = (I +
asolution of(10). uuT)W. (cid:16) (cid:17)
e f f f
Reciprocally, foranysolution X(t)de(10),Let’sput
Proof: The strong stability of symplectic matrix W
X(t) = (I uuTeJ)X(t) implies that the eigenvalues of W are either of red
− color either of green color i.e. for any eigenvector y
ofW,wehave
whereuisthevectordefinedinsystem(10)
e
= X(t) = (I +uuTJ)X(t) (S0y,y) 6= 0=⇒ (S0y,y)6= ϕ(y)
⇒
because(I +uuTJ)isinverseofthematrix usingCorollary8. e
(I uuTJ) (seee[16]). By replacing this expression This following Proposition gives us another con⊔⊓-
−
X(t)in(10),weobtain sequenceofthestrongstabilityofW undersmallper-
turbation thatpreservesymplecticity.
d
eJ(I +uuTJ) X(t) = (I uuTJ)TH(t)X(t) Proposition 16 If a symplectic matrix W is strongly
dt −
stable, then there exists a positif constant δ such that
d
J(I +uuTJ) X(t) = (I uuTJ)TH(t)X(t) anyvectoru R2N verifies uuTJW < δ,wehave
dt − ∈ k k
d W = (I +uutJ)W isstable.
(I uuTJ) TJ(I +uuTJ) X(t) =H(t)X(t)
−
− dt
Proof: If W is strongly stable, then there exists
d f
(I +uuTJ)TJ(I +uuTJ) X(t) =H(t)X(t) a positif constant δ such that any small perturba-
dt
tion W of W preserving its symplecticity verifying
=J
W W δ,isstable. Inparticulary, ifthepertur-
d
| {z J} X(t) =H(t)X(t) bkatio−nfis akr≤ank one perturbation with W of the form
dt
Wf+uuTJW,anyvectoruverifying uuTJW δ
k k≤
and X(0) = (I uuTJ)X(0) = (I uuTJ)(I + givesW stable. f
uuTJ) = I. Con−sequently, X(t)issolu−tion of(2). Hencewehavethisfollowingresultonthestron⊔⊓g
⊓⊔
Fromtheforegoing, weegivethefollowingdefini- stabiliftyoftheHamiltoniansystemswithperiodicco-
tion: efficients
Proposition 17 If Hamiltonian system with periodic For a = 7 and b = 4, consider the vector u =
•
coefficients(2)isstronglystable,thenthereexistsε> 0.8913
. InFigure1,weconsiderarandom
0suchthatforanyvectoruverifying 0.7621
(cid:18) (cid:19)
vector uwhich permits to disrupt system (13)by
E(t) ε the vectors u,10 1u,10 2u and 10 3u. In this
k k ≤ − − −
first figure, wenote that ψ(t) 1.5 10 14. This
where E(t) is defined in (9), rank one perturbation ≤ −
shows that X (t) = X (t) for all t [0,π] i.e.
Hamiltoniansystem(10)associated isstable. 1 2
∈
the rank one perturbation (X (t)) of the
1 t [0,π]
Proof: ThispropositionisaconsequenceofTheorem fundamentalesolutioneof system (13)∈is equal to
4usingsystem(8)ofProposition 11.
thesolution (X (t)) oferank one perturba-
⊓⊔ 2 t [0,π]
On the other hand, if the unperturbed system is tionsystem(10). ∈
unstable,thereexitsaneighborhoodinwhichanyrank
e
oneperturbation ofsystem(2)remainsunstable.
x 10−14 a=7 and b=4 with u x 10−14 a=7 and b=4 with 10−1u
4 4
ψ(t) ψ(t)
Remark18 The stability of any small rank one per- y=1.5 10−14 y=1.5 10−14
3 3
turbation of a Hamiltonian system with periodic co-
ψ(t)2 ψ(t)2
efficients doesn’t implyitsstrongstability because we
1 1
areinaparticular caseoftheperturbation ofthesys-
tem. However it can permit to study the behavior of 0 0
0 1 2 3 0 1 2 3
t,times t,times
multipliers of Hamiltonian systems with periodic co-
x 10−14 a=7 and b=4 with 10−2u x 10−14 a=7 and b=4 with 10−3u
efficients. 4 ψ(t) 4 ψ(t)
y=1.5 10−14 y=1.5 10−14
3 3
5 Numerical examples ψ(t)2 ψ(t)2
1 1
Example19 ConsidertheMathieuequation
0 0
0 1 2 3 0 1 2 3
t,times t,times
d2y(t)
J = (a+bsin(2t))y(t) (12)
dt2 Figure1: Comparisonoftwosolutions
wherea,b R(see[15,vol. 2,p. 412],[4]). Putting
∈ However, unperturbed system (13) is strongly
stable. We remark that the rank one perturbed
y
0 1
systems (10) of (13) is strongly stable when the
x(t) = dy ,J = −
dt ! (cid:18) 1 0 (cid:19) vector u ∈ u,10−1u,10−2,10−3 . Therefore
theyarestable. ThisjustifiesProposition (17)
(cid:8) (cid:9)
and
b sin 2t+a 0
H(t) = , For a = 16.1916618724166685... and b = 5,
0 1 •
0.4565
(cid:18) (cid:19) consider the vector u = . In this
weobtainthefollowingcanonicalHamiltonianEqua- 0.0185
(cid:18) (cid:19)
tion anotherexampleillustrated byFigure2,wecon-
sider a random vector u which permits to dis-
JdX(t) = H(t)X(t), t R,X(0) = I , (13) rupt system (13) by the vectors u,10−1u,10−2u
2
dt ∀ ∈ and 10 3u. In figure 2, we note that ψ(t)
−
≤
wherethematrixH(t)isHamiltonianandπ-periodic. 1.5 10−14. This shows that X1(t) = X2(t) for
Let u R2N 2N be a random vector in a neighbor- allt [0,π].
∈ × ∈
hood of the zero vector. Consider perturbed system e e
(10) of (13). We show that the rank one perturbation
Inthisexample,theunperturbedsystembeingun-
ofthefundamental solution isasolution ofperturbed
stable, the rank one perturbation system is unstable
system(10). Consider when the vector u u,10 1u,10 2u,10 3 . This
− − −
∈
justifiestheexistence ofaneighborhood oftheunper-
ψ(t) = X (t) X (t) , t 0 (cid:8) (cid:9)
1 2 turbed system in which any rank one perturbation of
k − k ∀ ≥
thesystemisunstable.
where X1(t) = (Ie uuTeJ)X(t) and (X2(t))t 0
is the solution of sys−tem (10). We show by numer∈ic≥al
exampleesthat(ψ(t)) 1.5 10−14, t [e0,π]. Example20 Consider the system of differential
≤ ∀ ∈
4x 10−14a=16.191... and b=5 ψw(itt)h u 4x 10−1a4=16.191... and b=5 wiψth( t1)0−1u wthheesroeluXti1o(nt)of=the(Ira−nkuounTeJp)eXrt(utr)baatnidon(XHa2(mt)il)tto∈nRiains
y=1.5 10−14 y=1.5 10−14
3 3 system e(10) of (15). Figures 3 and 4 reepresent the
ψ(t)2 ψ(t)2 normofthedifference betweenX1 etX2.
1 1
0 0 forǫ = 15.5andδ = 1,Lete’stakee
0 1 t,times 2 3 0 1 t,times 2 3 • 0.8214
x 10−1a4=16.191... and b=5 with 10−2u x 10−1a4=16.191... and b=5 with 10−3u 0.4447
4 ψy=(t1).5 10−14 4 ψy=(t1).5 10−14 u =  00..67195149 . Figure 3 is ob-
3 3 0.9218
0.7382
ψ(t)2 ψ(t)2 tained forvalues of the vector u taken in
u,10 1u,10 2u,10 3u . In figure 3, we
1 1 − − −
note that ψ(t) 5 10 13. This shows that
−
0 0 (cid:8) ≤ (cid:9)
0 1 2 3 0 1 2 3
t,times t,times X1(t)= X2(t)forallt [0,π]i.e. therankone
∈
perturbation (X (t)) of the fundamental
Figure2: Comparisonoftwosolutions 1 t [0,π]
seolution oef system (15∈) is equal to the solution
(X (t)) eof the rank one perturbation
2 t [0,π]
equations (see[9]and[15,Vil. 2,p. 412]) systemof∈(15).
e
q1dd2tη21 +p1η1+[aη1cos 2γt+(bcos 2γt+csin 2γt)η3]=0 x 10−13 ε=15.5 and δ=1 with u x 10−13ε=15.5 and δ=1 with 10−1u
 q2dd2tη22 +p2η2+gη3sin 5γt=0, 5 ψy=(t5) 10−13 5 ψy=(t2).5 10−13
 q3dd2tη23 +p3η3+[(bcos 2γt+csin 2γt)η1+gη2sin 5γt]=(01,4) ψ(t)34 ψ(t)34
 2 2
which can be reduced on the following canonical
1 1
Hamiltoniansystem 0 0
0 1 2 3 0 1 2 3
t,times t,times
dX(t) x 10−13ε=15.5 and δ=1 with 10−2u x 10−13ε=15.5 and δ=1 with 10−3u
J = H(t), X(0) = I (15) ψ(t) ψ(t)
dt 6 5 y=2.5 10−13 5 y=2.5 10−13
4 4
where ψ(t)3 ψ(t)3
2 2
η 1 1
0 I
x = dη , J = 3 − 3 , 00 1 2 3 00 1 2 3
! I3 03 t,times t,times
dt (cid:18) (cid:19)
Figure3: Comparisonoftwosolutions
P(t) 0
H(t) = 3 ,
0 I
(cid:18) 3 3 (cid:19) However, unperturbed system (15) is strongly
η1 stable. Wealso note that the rank oneperturbed
√ηq1 systems (10) of (15) is strongly stable when the
2
withη =  √q2 and vector u ∈ u,10−1u,10−2u,10−3 . Therefore
 eta3  theyarestab(cid:8)le. ThisjustifiesPropos(cid:9)ition (17)
 √q3  0.0272
  0.3127
P(t)= bcosp21γ+√2aγqqt0c11o+qs3c2sγint 2γt g√siqpqn0222q53γt bcos 2γg√√2sγiqqpnqt2133+q5q3γ3ctsin 2γt . • Tvǫeh=cetofo1rl5luoawtnaidknegδnfi=ignur2eu,s,Li1es0to’−sb1ttauai,kn1ee0du−f2o=ur,v1a0lu−0000e....30360su18892432o9018f.thIne.
figure 4,wealsonote thatψ(t) 2 10 13. This
Let u R2N be a random vector in a neighbor- (cid:8) ≤ − (cid:9)
hood of th∈e zero vector. Consider perturbed system showsthatX1(t) =X2(t)forallt ∈ [0,π].
(10) of (15). We show that the rank one perturbation In this latter example, the unperturbed system
e e
of the fundamental solution of 15 is a solution of its is unstable and the rank one perturbation sys-
rankoneperturbation system. Consider tems remain unstable when the vector u
u,10 1u,10 2u,10 3 . Thisjustifiestheexis∈-
− − −
ψ(t) = X (t) X (t) , t R tence of a neighborhood of the unperturbed sys-
1 2
k − k ∀ ∈ (cid:8) (cid:9)
e e
x 10−13 ε=15 and δ=2 with u x 10−13 ε=15 and δ=2 with 10−1u periodic. FarEastJournal ofMathematical Sci-
ψ(t) ψ(t) ences.Vol.99,Number3,2016, 301–322.
5 y=2 10−13 2.5 y=2 10−13
4 2 [4] M. Dosso, N. Coulibaly, Symplectic matrices
ψ(t)3 ψ(t)1.5 andstrongstabilityofHamiltoniansystemswith
2 1
periodic coefficients. Journal of Mathematical
1 0.5
Sciences : Advances and Applications. Vol. 28,
0 0
0 1 2 3 0 1 2 3
t,times t,times 2014,Pages 15–38.
x 10−13 ε=15 and δ=2 with 10−2u x 10−13 ε=15 and δ=2 with 10−3u [5] M. Dosso, N. Coulibaly and L. Samassi, Strong
ψ(t) ψ(t)
2.5 y=2 10−13 2.5 y=2 10−13 stability of symplectic matrices using a spectral
2 2
dichotomy method. FarEastJournal ofApplied
ψ(t)1.5 ψ(t)1.5
Mathematics. Vol. 79, Number 2, 2013, pp. 73–
1 1
0.5 0.5 110.
00 1 2 3 00 1 2 3 [6] M.DossoandM.Sadkane.Onthestrongstabil-
t,times t,times
ityofsymplecticmatrices.NumericalLinearAl-
Figure4: Comparisonsoftwosolutions gebrawithApplications,20(2)(2013),234–249.
[7] M. Dosso, M. Sadkane, A spectral trichotomy
method for symplectic matrices, Numer Algor.
tem in which any rank one perturbation of the
52(2009), 187–212
systemisunstable.
[8] S.K. Godunov, Verification of boundedness for
the powers of symplectic matrices withthe help
of averaging, Siber. Math. J. 33,(1992), 939–
6 Conclusion
949.
From a theory developed by C. Mehl, et al., on the [9] S.K.Godunov,M.Sadkane,Numericaldetermi-
rank one perturbation of symplectic matrices (see nation of a canonical form of a symplectic ma-
[13]),wedefined therank oneperturbation ofHamil- trix,SiberianMath.J.42(2001), 629–647.
tonian system ofperiodic coefficients. Afteranadap- [10] S.K. Godunov, M. Sadkane, Spectral analysis
tation of some results of [13] on symplectic matrices of symplectic matrices with application to the
when they depend on atime parameter, we show that theory of parametric resonance, SIAM J. Matrix
therankone perturbation ofthefundamental solution Anal.Appl.28(2006), 1083–1096.
of a Hamiltonian system with periodic coefficients is
[11] B. Hassibi, A. H. Sayed, T. Kailath, Indefinite-
solution of the rank one perturbation of the system.
Quadratic Estimation and Control, SIAM,
A result of this theory, we give a consequence of the Philadelphia, PA,1999.
strong stability on a small rank one perturbation of
[12] P. Lancaster, L. Rodman, Algebraic Riccati
these Hamiltonian systems. Twonumerical examples
Equations, ClarendonPress,1995.
aregiventoillustrate thistheory.
[13] C. Mehl, V. Mehrmann, A.C.M. Ran, and L.
In future work, we will look how to use the rank
Rodman. Eigenvalue perturbation theory un-
oneperturbation ofHamiltoniansystemwithperiodic
der generic rank one perturbations: Symplec-
coefficients to analyze the behavior of their multipli-
tic, orthogonal, and unitary matrices. BIT,
ers and also how this theory can analyze their strong
54(2014), 219–255.
stability ?
[14] C.Mehl,V.Mehrmann,A.C.M.RanandL.Rod-
man. Eigenvalue perturbation theory of classes
of structured matrices under generic structured
References:
rank one perturbations. Linear Algebra Appl.,
[1] C. Brezinski, Computational Aspercts of Linear
435(2011), 687–716.
Control,KluwerAcademicPublishers, 2002.
[15] V.A., Yakubovich, V.M. Starzhinskii, Linear
[2] M.Dosso,Surquelquesalgorithmsd’analysede differential equations with periodic coefficients,
stabilité forte de matrices symplectiques, PHD Vol.1&2.,Wiley,NewYork(1975)
Thesis (September 2006), Université de Bre-
tagne Occidentale. Ecole Doctorale SMIS, Lab- [16] YANQing-you. Theproperties ofakind ofran-
oratoire de Mathématiques, UFR Sciences et dom symplectic matrices. Applied mathematics
Techniques. andMechanics. Vol23,No5,May2002.
[3] M.Dosso, N.Coulibaly, AnAnalysis oftheBe-
haviorofMulpliersofHamiltonianSystemwith