EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1744-1770 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Rank-k perturbation of Hamiltonian systems with periodic coefficients and applications Mouhamadou Dosso1,∗, Traoré G. Y. Arouna1, Jean-Claude Koua Brou1 1 UFR Mathématiques et Informatique, Université Félix Houphouët-Boigny, Côte d’Ivoire Abstract. Jordan canonical forms of a rank-k perturbation of symplectic matrices and the fun- damental solutions of Hamiltonian systems are presented on the basis of work done by C. Mehl et, al.. Small rank-k perturbations of Mathieu systems are analyzed. More precisely, it is shown that the rank-k perturbations of coupled or non-coupled double pendulums and the motion of an ion through a quadrupole analyzer slightly perturb the behavior of their spectra and their stabilities. 2010 Mathematics Subject Classifications: 35E05, 39A30, 70H05, 70H09, 93C15 Key Words and Phrases: Symplectic matrices, Isotropic subspaces, Hamiltonian systems, Fun- damental solutions, rang-k perturbation, Stability(strong), Mathieu systems 1. Introduction Let J,W ∈ R2N×2N such that J is a skew-symmetric matrix. We say that the matrix W is J-symplectic or J-orthogonal if and only if W TJW = J [4, 18]. These types of matrices generally appear in control theory [3, 11, 15, 18], especially in optimal control [11] and in parametric resonance theory [15]. The spectra of the symplectic matrices is generally composed of three groups with respect to the unit circle (see e.g. [7, 8, 18]) : N0 eigenvalues outside the unit circle, N0 = N∞ eigenvalues inside the unit circle and 2N1 = 2(N −N0) eigenvalues on the unit circle. A symplectic matrix W is stable if all its powers are bounded. In other words, if the eigenvalues of W lie on the unit circle and are semi-simple. Some classifications of eigenvalues of W are given by the following definitions [4, 7, 10, 18] Definition 1. Let λ be a semi-simple eigenvalue of W lying on the unit circle. (i) Then λ is called an eigenvalue of the first (second) kind if the quadratic form (iJx, x) is positive (negative) on the eigenspace associated with λ. When (Jx, x) = 0, then λ is of mixed kind. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3574 Email addresses: mouhamadou.dosso@univ-fhb.edu.ci (M. Dosso), k brou@hotmail.com (J.-C. Koua Brou) http://www.ejpam.com 1744 c© 2019 EJPAM All rights reserved. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1745 (ii) Then λ is an eigenvalue of a red (green) color or in short r-eigenvalue (g-eigenvalue) if (S(0)x, x) is positive (negative) on the eigenspace associated with λ where S(0) = 1 2 ( JW + (JW )T ) . This leads us to the characterization of the strong stability by the following theorem (see [6, 7]) Theorem 1. A symplectic matrix W is strongly stable if all the eigenvalues are on the unit circle and verifies one of the following assertion : (i) the eigenvalues are either of the first or second kind and there is a sufficient gap between the eigenvalues of first and second kind. In other words, the quantity δKGL(W ) = min { |eiθk − eiθl | such that eiθk , eiθl are eigenvalues of W of different kinds } should not be close to zero. (ii) the eigenvalues are either of red color or green color and there is a sufficient gap between the eigenvalues of red and green color. In other words, the quantity δS(W ) = min { |eiθk − eiθl | such that eiθk , eiθl are r- and g-eigenvalues of W } should not be close to zero. These symplectic matrices are often obtained as solutions of Hamiltonian systems with periodic coefficients i.e. the differential systems of the form J dx(t) dt = H(t)x(t), t ∈ R (1) where H(t) ∈ R2N×2N is symmetric and P -periodic (i.e. H(t + P ) = H(t) = (H(t))T ). We know that the fundamental solution X(t) of (1), in other words, the solution of the system  J dX(t) dt = H(t)X(t), t ∈ R X(0) = I , (2) satisfies the relationship X(t + nP ) = X(t)Xn(P )(6= Xn(P )X(t)), ∀ (t, n) ∈ R × N and is J-symplectic [18, Vol. 1, chap. 2]. Regarding stability (strong stability), we have the following definitions [6, 18] Definition 2. (i) System (1) is stable if each of its solutions x(t) remains bounded for t ∈ R. (ii) System (1) is strongly stable if any Hamiltonian system with P -periodic coefficients to sufficiently close to (1), is stable. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1746 Specifically, system (1) (or system (2)) is strongly stable if there exists ε > 0 such that any Hamiltonian system with P -periodic coefficients of the form J dx(t) dt = H̃(t)x(t) and satisfying ‖H − H̃‖ = ∫ T 0 ‖H(t)− H̃(t)‖dt < ε, is stable. We also have the following theorem [18, p. 196] Theorem 2. System 1 is strongly stable if and only if the J-symplectic matrix X(P ) is strongly stable. Since the strong stability analysis of the Hamiltonian systems with P -periodic coef- ficients is related to the study of their perturbation, we will dwell on a type of the per- turbation which we call rank-k perturbation. Thus, we present in section 2, preliminaries necessary to the study of the isotropic subspaces, on the rank-k perturbation of a symplec- tic matrix and rank-k perturbation of a Hamiltonian system with P-periodic coefficients. In sections 3 and 4, we give respectively the Jordan canonical forms of a rank-k perturba- tion of a symplectic matrix and of a rang-k perturbation of the fundamental solution of (1). Finally in section 5, we present some applications for some systems of Mathieu: Namely systems that describe the movement of a double pendulum with oscillating support and those that describe the motion of an ion through a quadrupole analyzer. 2. Preliminaries 2.1. Isotropic subspaces Definition 3. A subspace X ⊆ R2N is called isotropic if X ⊥ JX . A maximal isotropic subspace is called Lagrangian. The maximum isotropic subspaces containing X are of dimension N . Hence the fol- lowing definition (see [9]) Definition 4. A subspace L of RN is called a Lagrangian subspace if it has the dimension N and xTJy = 0, ∀x, y ∈ L. In other words, we say that a subspace L is Lagrangian if and only if every matrix L whose columns span L satisfies rankL = N and LTJL = 0. We list a set of properties on the isotropic subspaces in the following proposition Proposition 1. (i) Let X be an isotropic subspace. Then the dimension of X is less than or equal to N . (ii) Every isotropic subspace is contained in a Lagrangian subspaces. (iii) Let S = [S1 S2] ∈ R2N×2N be a symplectic matrix with Si ∈ R2N×N , i = 1, 2 ; then the columns of S1 and S2 span isotropic subspaces. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1747 Recall the two lemmas below (see [13]) Lemma 1. Let XS ⊆ R2N be a subspace that is invariant under a Hamiltonian matrix S which has all its eigenvalues associated with XS having their real part negative. Then XS is isotropic. Lemma 2. Let S ∈ R2N×2N be a skew-Hamiltonian matrix and X ∈ R2N×k(k ≤ N) with orthogonal columns. Then the columns of X span an isotropic invariant subspace of S if and only if there exists an orthogonal symplectic matrix U = [X,Z, JTX, JTZ] with some Z ∈ R2N×(N−k) so that UTSU = k N − k k N − k k N − k k N − k  A11 A12 G11 G12 0 A22 −GT12 G22 0 0 AT11 0 0 H22 AT12 AT22  On the other hand, if we consider the Krylov subspace defined below Km ≡ Km(A, v) = span { v,Av,A2v, . . . , Am−1v } , where A ∈ Rn×m (with n > 1) and v ∈ Rm. Then we have the following proposition which shows that we can construct isotropic invariant subspaces from Krylov process (see [17, p. 399]) Proposition 2. Let S ∈ R2N×2N be a skew-Hamiltonian matrix and u ∈ R2N be an arbitrary nonzero vector. Then the Krylov subspace Kj(S, u) is isotropic for all j. 2.2. Rank-k perturbation of symplectic matrices Let W ∈ R2N×2N and L be respectively a symplectic matrix and a J-Lagrangian subspace. Consider k vectors u1, · · · , uk of L, where k ≤ N . Setting U = [u1; . . . ;uk], and W̃ = ( I + UUTJ ) W, we have the following proposition Proposition 3. The matrix W̃ is J-symplectic. Proof. For the proof, see [2]. Definition 5. We call rank-k perturbation of W, any matrix of the form W̃ = (I + UUTJ)W, (3) where U is a matrix of rank k whose columns belong to a J-Lagrangian subspace. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1748 The matrix W̃ can be put in the form W̃ = (I + k∑ j=1 uju T j J)W. More specially, this shows that any rank-k perturbation of W is k rank-one perturbations of the symplectic matrix W . We have k∏ j=1 ( I + uju T j J )W = I + k∑ j=1 uju T j J W. Consider a symplectic matrix of function (X(t))t∈R ; we can consider for example the solution of system (2) which is J-symplectic. We have the following definition Definition 6. We call rank-k perturbation of X(t) any matrix function of the form X̃(t) = (I + UUTJ)X(t), (4) where rank(U) = k and the columns of U belong in a J-Lagrangian subspace. Remark 1. Since the matrix function (X(t))t∈R is J-symplectic, its rank-k perturbation will be J−symplectic. 2.3. Rank-k perturbation of Hamiltonian system with periodic coeffi- cients Let U ∈ R2N×k (with k ≤ N) be a constant matrix of rank k such that its columns belong to a J-Lagrangian subspace and (X(t))t≥0 be the fundamental solution of (2). We have the following proposition Proposition 4. a Consider the following perturbed Hamiltonian system J dX̃(t) dt = [H(t) + E(t)] X̃(t), (5) where E(t) = (JUUTH(t))T + JUUTH(t) + (UUTJ)TH(t)(UUTJ). (i) Then X̃(t) = (I + UUTJ)X(t) is a solution of system (5). (ii) Equation (5) can be put in the form J dX̃(t) dt = ( I − UUTJ )T H(t) ( I − UUTJ ) X̃(t), t ∈ R+, X̃(0) = I + UUTJ (6) M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1749 (iii) Any solution (X̃(t))t≥0 of perturbed system (5) of system (2), is of the form X̃(t) = (I + UUTJ)X(t), where (X(t))t≥0 is the fundamental solution of system (2). Proof. For the proof, see [2]. System (6) can be written as below J dX̃(t) dt = ( I − ∑k j=1 uju T j J )T H(t) ( I − ∑k j=1 uju T j J ) X̃(t) X̃(0) = (I + ∑k j=1 uju T j J) , (7) where each vector (uj)1≤j≤k ⊂ R2N belongs to a same J-Lagrangian subspace. We can immediately see that the rank-k perturbation of (2) can be interpreted as k rank-one perturbations of (2). In fact, since I − UUTJ = I − k∑ j=1 uju T j J = k∏ j=1 ( I − ujuTj J ) , we easily see that system (7) can be put in the following form J dX̃(t) dt = (∏k j=1 ( I − ujuTj J ))T H(t) (∏k j=1 ( I − ujuTj J )) X̃(t) X̃(0) = ∏k j=1 ( I + uju T j J ) (8) which is the same as the bellow system, for all p ∈ {1, 2, ..., k − 1} :  J dX̃(t) dt = (∏k j=p+1(I − ujuTj J) )T H(p)(t) (∏k j=p+1(I − ujuTj J) ) X̃(t) X̃(0) = (∏k j=p+1(I + u(k+p−j+1)u T (k+p−j+1)J) ) X (p) (0) , (9) where H(p)(t) =  p∏ j=1 (I − ujuTj J) T H(t)  p∏ j=1 (I − ujuTj J)  and X (p) (0) = p∏ j=1 ( I + u(p−j+1)u T (p−j+1)J ) . M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1750 3. Jordan canonical form of rank-k perturbation of a symplectic matrix Let W,J ∈ R2N×2N be two matrices and λ ∈ C such that J is skew-symmetric, W is J-symplectic and λ an eigenvalue of W . We have the following theorem Theorem 3. Suppose that W has the following Jordan canonical form : l1⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J , where n1 > · · · > nm, m ∈ N∗ such that the algebraic multiplicity a of λ is of the form a = m∑ j=1 ljnj and J contains all the forms in Jordan blocks associated with eigenvalues of W that are different from λ. Moreover let B = UUTJW where U ∈ R2N×k is such that its columns generate an isotropic subspace. (1) If λ 6∈ {−1, 1}, then generally with respect to the components of U , the matrix W+B has the Jordan canonical form  l1−k⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , if k < l1 li−ki⊕ j=1 Jni(λ) ⊕  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , if  k = i−1∑ s=1 ls + ki, with ki < li and i > 1 where J̃ contains all the forms in Jordan blocks of W+B associated with eigenvalues different from λ. (2) If λ ∈ {−1, 1}, then (2a) if k = i−1∑ s=1 ls + ki where the n1, n2, . . . , ni are even and ki < li, then generally with respect to the components of U , then matrix W +B has the Jordan canon- ical formli−ki⊕ j=1 Jni(λ) ⊕  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ contains all the forms in Jordan blocks of W + B associated with eigenvalues different from λ. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1751 (2b) if k = i−1∑ s=1 ls + 2ki − 1 with 2ki ≤ li and ni is odd, then li is even and gener- ally with respect to the components of U , then matrix W + B has the Jordan canonical form Jni+1(λ)⊕ li−2ki⊕ j=1 Jni(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ contains all the forms in Jordan blocks of W + B associated with eigenvalues, different from λ. Proof. We know that the rang-k perturbation W̃ = W +B of W can be written in the form W̃ =  k∏ j=1 ( I + uk−j+1u T k−j+1J )W, where each vector uj is a column of U . Therefore 1) If λ /∈ {−1, 1}, then • For k < l1; – Set W̃1 = ( I + u1u T 1 J ) W . According to 1) of Theorem 7.1 of [16], W̃1 has the Jordan canonicall1−1⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃1, where J̃1 contains all the forms in blocks of Jordan of W̃1 associated with eigenvalues different from λ. – Set W̃2 = ( I + u2u T 2 J ) W̃1. According to 1) of Theorem 7.1 of [16], W̃2 has the Jordan canonical forml1−2⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃2, with J̃2 containing all the forms in Jordan blocks of W̃2 associated with eigenvalues different from λ. – On the other hand, W̃k = W +B = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u3u T 3 J ) W̃2, M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1752 by applying (k − 2)-times 1) of Theorem 7.1 of [16] to the matrix W̃2, we get that W̃k has the following Jordan canonical form :l1−k⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k contains all the forms in Jordan blocks of W +B associated with eigenvalues different from λ. • For k = i−1∑ s=1 ls + ki, with ki < li : – i = 2, we have k = l1 + k2 with k2 < l2. We know that W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + ul1+1u T l1+1J ) ×( I + ul1u T l1J ) ( I + ul1−1u T l1−1J ) × ...× ( I + u1u T 1 J ) W︸ ︷︷ ︸ W̃l1 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + ul1+1u T l1+1J ) W̃l1 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uk−k2+1u T k−k2+1J ) W̃l1 , because l1 = k − k2. As W̃l1 has the following Jordan canonical form : l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃l1 , where J̃l1 contains all the forms in Jordan blocks of W̃l1 associated with eigenvalues different from λ because W̃k is a rang-k2 perturbation of W̃l1 , then W̃k has the following Jordan canonical form :l2−k2⊕ j=1 Jn2(λ)  l3⊕ j=1 Jn3(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃k2 , where J̃k2 contains all the forms in Jordan blocks of W̃k = W+B associated with eigenvalues different from λ. – i > 2, k = i−1∑ s=1 ls + ki, with ki < li. Set γ(i) = i∑ s=1 ls, ∀ i ≥ 1. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1753 We have W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × · · · × ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × · · · × ( I + uγ(3)+1u T γ(3)+1J ) ( I + uγ(3)u T γ(3)J ) × · · · × ( I + uγ(2)+1u T γ(2)+1J ) ( I + uγ(2)u T γ(2)J ) × · · · × ( I + u1u T 1 J ) W︸ ︷︷ ︸ W̃γ(2) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × · · · × ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × · · · × ( I + uγ(3)+1u T γ(3)+1J ) × ( I + uγ(3)u T γ(3)J ) × · · · × ( I + uγ(2)+1u T γ(2)+1J ) W̃γ(2)︸ ︷︷ ︸ W̃γ(3) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × · · · × ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × · · · × ( I + uγ(3)+1u T γ(3)+1J ) W̃γ(3)︸ ︷︷ ︸ W̃γ(i−1) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × · · · × ( I + uγ(i−1)+1u T γ(i−1)+1J ) W̃γ(i−1) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × · · · × ( I + uk−ki+1u T k−ki+1J ) W̃γ(i−1), where W̃γ(i−1) is γ(i− 1) rank-one perturbations of the symplectic matrix W . Then W̃γ(i−1) has the following Jordan canonical form : li⊕ j=1 Jni(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃γ(i−1), where J̃γ(i−1) contains all the forms in Jordan blocks of W̃γ(i−1) associated with eigenvalues different from λ. Finally, using the fact that W̃k is ki rank one perturbation of W̃γ(i−1), according to 1) of Theorem 7.1 of [16] and it follows that the Jordan canonical form of W̃ = W̃k is given byli−ki⊕ j=1 Jni(λ)  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k contains all the form in Jordan blocks of W̃k associated with eigenvalues different from λ. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1754 2) If λ ∈ {−1, 1}, then : 2a) if k = i−1∑ s=1 ls + ki, where the n1, n2, . . . , ni are even and ki < li, then – i = 1, we have k = k1 and n1 is even. W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u2u T 2 J ) ( I + u1u T 1 J ) W︸ ︷︷ ︸ =W̃1 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u2u T 2 J ) W̃1︸ ︷︷ ︸ =W̃2 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u3u T 3 J ) W̃2. As W̃1 is a rank-one perturbation of W , according to 2a) of Theorem 7.1 of [16], it has the following Jordan canonical form :l1−1⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃1, where J̃1 contains all the forms in Jordan blocks of W̃1 associated with eigenvalues different from λ. Using the fact that W̃2 is a rank-one pertur- bation of W̃1, 2a) of Theorem 7.1 of [16] implies that the Jordan canonical form of W̃2 is given byl1−2⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃2, where J̃2 contains all the forms in Jordan blocks of W̃2 associated with eigenvalues different from λ. Hence, applying (k1−2)-times 2a) of Theorem 7.1 of [16] to the matrix W̃2, we have the following Jordan canonical form of W̃k1l1−k1⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃k1 , where J̃k1 contains all the forms in Jordan blocks of W̃k1 associated with eigenvalues different from λ. – For i > 1, we have k = i−1∑ s=1 ls + ki, with ki < li and n1, n2, ..., ni are even. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1755 With γ(i) = i∑ s=1 ls, ∀ i > 1, we have : W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × ...× ( I + uγ(3)+1u T γ(3)+1J )( I + uγ(3)u T γ(3)J ) ×...× ( I + uγ(2)+1u T γ(2)+1J )( I + uγ(2)u T γ(2)J ) × ...× ( I + u1u T 1 J ) W︸ ︷︷ ︸ W̃γ(2) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × ...× ( I + uγ(3)+1u T γ(3)+1J ) × ( I + uγ(3)u T γ(3)J ) × ...× ( I + uγ(2)+1u T γ(2)+1J ) W̃γ(2)︸ ︷︷ ︸ W̃γ(3) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × ...× ( I + uγ(3)+1u T γ(3)+1J ) W̃γ(3)︸ ︷︷ ︸ W̃γ(i−1) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) W̃γ(i−1) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uk−ki+1u T k−ki+1J ) W̃γ(i−1). Knowing that W̃γ(i−1) is γ(i− 1) rank-one perturbations of the symplectic matrixW , according to 2a) of Theorem 7.1 of [16], W̃γ(i−1) has the following Jordan canonical form li⊕ j=1 Jni(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃γ(i−1), where J̃γ(i−1) contains all the forms in Jordan blocks of W̃γ(i−1) associated with eigenvalues different from λ. Finally, as W̃k is ki rank-one pertur- bations of W̃γ(i−1), according to 2a) of Theorem 7.1 of [16] the Jordan canonical form of W̃k is given by :li−ki⊕ j=1 Jni(λ)  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k contains all the form in Jordan blocks from W̃k associated with eigenvalues different from λ. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1756 2b) if k = i−1∑ s=1 ls + 2ki − 1, with 2ki ≤ li and ni is even. – For i = 1, we have k = 2k1 − 1 with 2k1 ≤ l1 and n1 is odd. According to the property 2b) of Theorem 10 of [5], l1 is even and we have W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u3u T 3 J ) ( I + u2u T 2 J ) × ( I + u1u T 1 J ) W︸ ︷︷ ︸ W̃1 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u3u T 3 J ) ( I + u2u T 2 J ) W̃1︸ ︷︷ ︸ W̃2 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u4u T 4 J ) ( I + u3u T 3 J ) W̃2︸ ︷︷ ︸ W̃3 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + u4u T 4 J ) W̃3. We know that W̃1 is a rank-one perturbation of W and its Jordan canonical form in block of Jordan is given by (see [16, Theorem 7.1,2b)]) Jn1+1(λ)⊕ l1−2⊕ j=1 Jn1(λ)  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃1, where J̃1 contains all the form in Jordan blocks of W̃1 associated with eigenvalues different from λ. So W̃2 has the following Jordan canonical form (see [16, 2b) Theorem 7.1]) :l1−2⊕ j=1 Jn1(λ)  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃2, where J̃2 contains all the forms in Jordan blocks of W̃2 associated with eigenvalues different from λ, because it is a rank-one perturbation of W̃1. Similarly the Jordan canonical form of W̃3 [16, part 2b) du theorem 7.1] is given by Jn1+1(λ)⊕ l1−2×2⊕ j=1 Jn1(λ)  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃3, where J̃3 contains all the forms in Jordan blocks of W̃3 associated with eigenvalues different from λ. Hence, applying (2k1 − 4)-times this process M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1757 to matrix W̃3, we obtain the canonical form of Jordan below Jn1+1(λ)⊕ l1−2k1⊕ j=1 Jn1(λ)  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k contains all the forms in Jordan blocks of W +B associated with eigenvalues different from λ. – For i = 2, we have k = l1 + 2k2 − 1 with 2k2 ≤ l2 and n2 is odd. W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + ul1+1u T l1+1J ) × ( I + ul1u T l1J ) × ...× ( I + u1u T 1 J ) W︸ ︷︷ ︸ W̃l1 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + ul1+1u T l1+1J ) W̃l1 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uk−2k2+2u T k−2k2+2J ) W̃l1 , knowing that l1 = k − 2k2 + 1. i) If n1 is even, according to the property 2a) of Theorem 7.1 of [16], W̃l1 has the Jordan canonical form bellow l2⊕ j=1 Jn2(λ) ⊕ ...⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃l1 , where J̃l1 contains all the forms in Jordan blocks of W̃l1 associated with the eigenvalues different from λ. Finally, as W̃k is (2k2 − 1) rank-one perturbations of W̃l1 and n2 is odd, according to 2b) of Theorem 7.1 of [16], l2 is even and the Jordan canonical form of W̃k is given by : Jn2+1(λ)⊕ l2−2k2⊕ j=1 Jn2(λ) ⊕  l3⊕ j=1 Jn3(λ) ⊕...⊕  lm⊕ j=1 Jnm(λ) ⊕J̃k, where J̃k contains all the forms in Jordan blocks of W̃k associated to eigenvalues different from λ. ii) If n1 is odd, in this case l1 is even [16, Theorem 7.1,2b]. Applying l1- times the 2b) of Theorem 7.1 of [16] to W , we have the Jordan canonical form of W̃l1  l2⊕ j=1 Jn2(λ) ⊕ ...⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃l1 , (10) M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1758 where J̃l1 contains all the forms in Jordan blocks of W̃l1 associated with the eigenvalues different from λ. Since n2 is odd and W̃k is (2k2 − 1) rank-one perturbations of the symplectic matrix W̃l1 , according the property 2b) of Theorem 7.1 of [16], l2 is even and W̃k has the following Jordan canonical form : Jn2+1(λ)⊕ l2−2k2⊕ j=1 Jn2(λ) ⊕ ...⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃k, where J̃k contains all the forms in Jordan blocks of W̃k associated with eigenvalues different from λ. • For i > 2, we have k = i−1∑ s=1 ls + 2ki − 1, with 2ki ≤ li and ni is odd. Set γ(i) = i∑ s=1 ls, ∀ i > 2. W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × ...× ( I + u1u T 1 J ) W︸ ︷︷ ︸ W̃γ(i−1) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) ×W̃γ(i−1). from 2a) and 2b) of Theorem 7.1 of [16],we deduce that W̃γ(i−1) has the following Jordan canonical form : li⊕ j=1 Jni(λ) ⊕  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃γ(i−1), where J̃γ(i−1) contains all the forms in Jordan blocks of W̃γ(i−1) associated with the eigenvalues different from λ. Thus W̃k has the following Jordan canonical form : Jni+1(λ)⊕ li−2ki⊕ j=1 Jni(λ) ⊕  li+1⊕ j=1 Jni+1(λ) ⊕ · · ·⊕  lm⊕ j=1 Jnm(λ) ⊕J̃ , where J̃ = J̃k contains all the forms in Jordan blocks of W̃k associated with eigenvalues different from λ because W̃k is (2ki− 1) rank-one perturbation of W̃γ(i−1). M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1759 Remark 2. In the property (2) of Theorem 3, if k = i−1∑ j=1 ls + 2ki, with 2ki < li and the ni is odd, then the li are even and generally with respect to the components of U , the rank-k perturbation W̃ = W +B of W has the following Jordan canonical form :li−2ki⊕ j=1 Jni(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃k, where J̃ = J̃k contains all the forms in Jordan blocks of W̃ associated with the eigenvalues different from λ. Using (2a) and (2b) of Theorem 3, we have the following corollary : Corollary 1. Suppose that λ ∈ {−1, 1}. If k = i−1∑ s=1 ls + ki, with ki < li and only ni is even, then generally with respect to the components of U , the W + B has the Jordan canonical formli−ki⊕ j=1 Jni(λ) ⊕  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ contains all the forms in Jordan blocks of W + B associated with the eigen- values different from λ. Proof. • If i = 1, then k = k1 and n1 is even. According to (2a) of Theorem 3, W̃k has the following Jordan canonical forml1−k1⊕ j=1 Jn1(λ) ⊕  l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k1 contains all the forms in Jordan blocks of W̃k1 associated with eigenvalues different from λ. • If i = 2, then k = l1 + k2, with (k2 < l2) and n2 is even. We know that W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + ul1+1u T l1+1J ) × ( I + ul1u T l1J ) × ...× ( I + u1u T 1 J ) W︸ ︷︷ ︸ W̃l1 = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + ul1+1u T l1+1J ) W̃l1 . So M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1760 (a) if n1 is even, according to the property 2a) of Theorem 3, W̃l1 has the following Jordan canonical form l2⊕ j=1 Jn2(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃l1 , (11) where J̃ = J̃l1 contains all the forms in Jordan blocks of W̃l1 associated with all eigenvalues different from λ. W̃k being k2 rank-one perturbations of W̃l1 , according to 2a) of Theorem 3, the Jordan canonical form of W̃k is given byl2−k2⊕ j=1 Jn2(λ) ⊕  l3⊕ j=1 Jn3(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k contains all the forms in Jordan blocks of W̃k associated with eigenvalues different from λ. (b) If n1 is odd, according to 2b) of Theorem 3, The Jordan canonical form of W̃l1 is given by (11). Knowing that n2 is even and W̃k is k2 rank-one perturbations of W̃l1 , we obtain from 2a) of Theorem 3 that W̃k has the following Jordan canonical forml2−k2⊕ j=1 Jn2(λ) ⊕  l3⊕ j=1 Jn3(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k contains all the forms in Jordan blocks of W̃k associated with eigenvalues different from λ. • For i > 2, we have k = i−1∑ s=1 ls + ki, with ki < li and ni is even. Let’s put γ(i) = ∑i s=1 ls. we know that W̃k = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) × ( I + uγ(i−1)u T γ(i−1)J ) × ...× ( I + u1u T 1 J ) W︸ ︷︷ ︸ =W̃γ(i−1) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uγ(i−1)+1u T γ(i−1)+1J ) W̃γ(i−1) = ( I + uku T k J ) ( I + uk−1u T k−1J ) × ...× ( I + uk−ki+1u T k−ki+1J ) W̃γ(i−1), because γ(i− 1) = k − ki. So W̃γ(i−1) has the following Jordan canonical form li⊕ j=1 Jni(λ) ⊕  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃γ(i−1), M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1761 where J̃γ(i−1) contains all the forms in Jordan blocks of W̃γ(i−1) associated with eigenvalues different from λ. Applying thereafter ki times 2a) of Theorem 3 to the matrix W̃γ(i−1), we obtain the Jordan canonical form of W̃kli−ki⊕ j=1 Jni(λ) ⊕  li+1⊕ j=1 Jni+1(λ) ⊕ · · · ⊕  lm⊕ j=1 Jnm(λ) ⊕ J̃ , where J̃ = J̃k contains all the forms in Jordan blocks of W + B associated with eigenvalues different from λ. 4. Jordan canonical form of (X̃(t))t>0 Theorem 4. Let t > 0, J ∈ R2N×2N be a skew-symmtric and invertible matrix, (X(t))t>0 be the fundamental solution of the system (2) and λ(t) ∈ C be an eigenvalue of (X(t))t>0. Suppose that X(t) has the following Jordan canonical form l1⊕ j=1 Jn1(λ(t)) ⊕  l2⊕ j=1 Jn2(λ(t)) ⊕ · · · ⊕ lm(t)⊕ j=1 Jnm(t) (λ(t)) ⊕ J (t), where n1 > · · · > nm(t) et m : R −→ N∗ is a index function such that the algebraic multiplicity a(t) of λ(t) is of the form a(t) = m(t)∑ j=1 ljnj and J (t) contains all the forms in Jordan blocks associated with eigenvalues of X(t) different from λ(t). Moreover, Set B(t) = UUTJX(t) where U ∈ R2N×k is such that its columns generate an isotropic subspace. (1) If λ(t) 6∈ {−1, 1}, then generally with respect to the components of U , X(t) + B(t) has the following Jordan canonical form  l1−k⊕ j=1 Jn1(λ(t)) ⊕  l2⊕ j=1 Jn2(λ(t)) ⊕ · · · ⊕ lm(t)⊕ j=1 Jnm(t) (λ(t)) ⊕ J̃ (t), if k < l1 li−ki⊕ j=1 Jni(λ(t)) ⊕  li+1⊕ j=1 Jni+1(λ(t)) ⊕ · · · ⊕ lm(t)⊕ j=1 Jnm(t) (λ(t)) ⊕ J̃ (t), if  k = i−1∑ s=1 ls + ki, with ki < li et i > 1 M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1762 where J̃ (t) contains all the form in Jordan blocks of X(t) + B(t) associated with eigenvalues different from λ(t). (2) if ∃ t0 > 0 such that λ(t0) ∈ {−1, 1}, then (2a) if k = i−1∑ s=1 ls + ki where the n1, n2, . . . , ni are even and ki < li, then generally with respect to the components of U , X(t) + B(t) has the following Jordan canonical formli−ki⊕ j=1 Jni(λ(t0)) ⊕  li+1⊕ j=1 Jni+1(λ(t0)) ⊕· · ·⊕ lm(t0)⊕ j=1 Jnm(t0) (λ(t0)) ⊕J̃ (t0), where J̃ (t0) contains all the form in Jordan blocks of X(t0) +B(t0) associated with eigenvalues different from λ(t0). (2b) If k = i−1∑ s=1 ls + 2ki − 1 with 2ki ≤ li and ni is odd, then li is even and generally with respect to the components of U , X(t0) + B(t0) has the following Jordan canonical form Jni+1(λ(t0))⊕ li−2ki⊕ j=1 Jni(λ(t0)) ⊕ · · · ⊕ lm(t0)⊕ j=1 Jnm(t0) (λ(t0)) ⊕ J̃ (t0), where J̃ (t0) contains all the form in Jordan blocks of X(t0) +B(t0) associated with eigenvalues different from λ(t0). Proof. It suffices to adapt the proof of Theorem 3 to the solution (X(t))t≥0 of (2) and to its rank-k perturbation ( X̃(t) ) t≥0 (which is the fundamental solution of (6)). 5. Application to some Mathieu systems 5.1. Double pendulum with oscillating supports Consider two identical simple pendulums attached to the same support. When the support of each pendulum is subjected to an oscillatory movement f(t) of amplitude α and of a pulsation Ω, defined by f(t) = α cos(Ωt) [14], we present two cases: M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1763 Figure 1: Model of the uncoupled double pendulum with oscillating supports. 5.1.1. Uncoupled double pendulums with oscillating supports According to [14], the differential equation of the movement of the two pendulums will be the same. Thus, the equation of motion is given by: d2xi dt2 + cg k20 ( 1− 1 g d2f dt2 ) xi = 0, i = 1, 2, (12) where k0 is the radius of gyration of the pendulum around its point of suspension, and c is the distance between the point of suspension and the center of the pendulum. Since f(t) = α cos(Ωt), then system (12) becomes: d2xi dt2 + cg k20 ( 1 + αΩ2 g cos(Ωt) ) xi = 0, i = 1, 2, (13) and by the change of variable τ = Ωt [14], equation (13) becomes: d2xi dτ2 + (δ + ε cos(τ))xi = 0, i = 1, 2, (14) where ε = cα k20 and δ = cg k20Ω2 . Finally, using the following change of variables X(τ) = [ x(τ) dx dτ (τ) ] , J = [ 02 −I2 I2 02 ] and H(τ, δ, ε) = [ P (τ, δ, ε) 02 02 I2 ] , (15) with x(τ) = [ x1(τ) x2(τ) ] and P (τ, δ, ε) = (δ + ε cos(τ))I2, we obtain equation (1). Now, consider the rank-2 perturbation of the fundamental solution X(τ, δ, ε) of its corresponding Hamiltonian system by the following matrix of rank 2 Ea(τ, δ, ε) = UaU T a JX(τ, δ, ε), (16) where Ua = a  1 0 0 1 0 0 0 0  and a ∈ [0, 1[. (17) According to [2, 5], its rank-2 perturbation X̃a(τ, δ, ε) = ( I + UaU T a J ) X(τ, δ, ε). (18) M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1764 is the solution of the following rank-k perturbation Hamiltonian system J dX̃a(τ, δ, ε) dτ = (I − UaUTa J)TH(τ, δ, ε)(I − UaUTa J)︸ ︷︷ ︸ H̃(τ,δ, ε,a) X̃a(τ, δ, ε), X̃a(0, δ, ε) = I + UaU T a J (19) Figure 2 represents the movement of eigenvalues of X̃a(τ, δ, ε) for (δ, ε) ∈ {(1, 0.8) , (1.93, 1.93)} and a ∈ {0, 0.35}, with τ ∈ [0, 2π]. These figures show that small rank-k −1 0 1 −1 0 1 0 2 4 ℜ(λ(τ)) δ=1 and ε=0.8, with a=0.35 ℑ(λ(τ)) τ −1 0 1 −1 0 1 0 2 4 ℜ(λ(τ)) δ=1 and ε=0.8, with a=0 ℑ(λ(τ)) τ −1 0 1 −1 0 1 0 2 4 ℜ(λ(τ)) δ=1.93 and ε=1.93, with a=0.35 ℑ(λ(τ)) τ −1 0 1 −1 0 1 0 2 4 ℜ(λ(τ)) δ=1.93 and ε=1.93, with a=0 ℑ(λ(τ)) τ Figure 2: spectral portrait of X̃a(τ, δ, ε) for τ ∈ [0, 2π] and (δ, ε) ∈ {(1, 0.8) , (1.93, 1.93)} with a ∈ {0, 0.35}. perturbations on the movement of pendulums do not change the nature of the spectral portrait of X̃a(τ, δ, ε). For all (δ, ε) ∈ [0, 1.98] × [0, 2], Figure 3 shows the stability region of X̃a(2π, δ, ε). The first figure (left) shows areas of stability in blue and of instability in red when our system is subject to a rank-2 perturbation (with a = 0.35). The second figure (right) also shows the zone of stability in blue and of instability in red of the unperturbed system (i.e. a = 0). Thus we notice a slight difference between the two figures due to the small rank-k perturbation of the system described by our two uncoupled pendulums. 5.1.2. Coupled double pendulums with oscillating supports In this part, the two simple pendulums are coupled by a spring of constant stiffness k (see Figure 4). According to [14], the motion of the system is governed by the following differential system : d2x dt2 + ( B0 − c k20 d2f dt2 I2 ) x = 0, (20) where x = ∣∣∣∣ x1x2 ] and B0 =  cg k20 + kb2 mk20 − kb2 mk20 − kb2 mk20 cg k20 + kb2 mk20  , M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1765 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 δ ε a=0.35 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 δ ε a=0 Figure 3: Stability zone of matrix X̃a(2π, δ, ε), ∀ (δ, ε) ∈ [0, 1.98]× [0, 2] and a ∈ {0, 0.35}. Figure 4: Model of the coupled double pendulum with oscillating supports. with m is the mass of each pendulum and b is the distance between the point of suspension and the point of attachment of the coupling spring. Replacing f(t) by its expression in (20), equation (20) becomes: d2x dt2 + ( B0 + cαΩ2 k20 cos(Ωt)I2 ) x = 0. (21) Using successively the change of variables z = ( z1 z2 ) = ( x1 + x2 x1 − x2 ) and τ = Ωt, the equation of motion of the system can be reduced as (see [14]) d2zi dτ2 + (δi + εi cos(τ)) zi = 0, i = 1, 2 (22) where δ1 = δ = cg k20Ω2 , ε1 = ε = cα k20 and ε2 = ε+ 2e, with e = kb2 mk20Ω2 . M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1766 Finally, using the change of variables given in (15) with N = 2, it is easy to see that the equation of motion of the coupled system can be reduce to form (1) with H(τ, δ, ε, e) = ( P (τ, δ, ε, e) 02 02 I2 ) and P (τ, δ, ε, e) =  δ + ε cos(τ) 0 0 δ + 2e+ ε cos(τ)  . Consider the rank-2 perturbation of the fundamental solution X̃a(τ, δ, ε, e) of its cor- responding Hamiltonian system Ea(τ, δ, ε, e) = UaU T a JX(τ, δ, ε, e), (23) where Ua is defined in (17). In this case, it is easy to see that the equation of motion is of the form (19) [1, 2, 5]. Figure 5 represents the spectral portrait of the matrix X̃a(τ, δ, ε, e), ∀ (δ, ε, e) ∈ {(1, 0.8, 0.5) , (1.93, 1.93, 0.5)} and a ∈ {0, 0.35}, ∀τ ∈ [0, 2π]. The figures also show that the small −1 0 1 −1 0 1 0 1 2 3 4 5 ℜ(λ(τ)) δ=1, ε=0.8 and e=0.5, with a=0.35 ℑ(λ(τ)) τ −1 0 1 −1 0 1 0 1 2 3 4 5 ℜ(λ(τ)) δ=1, ε=0.8 and e=0.5, with a=0 ℑ(λ(τ)) τ −1 0 1 −1 0 1 0 1 2 3 4 5 ℜ(λ(τ)) δ=1.93, ε=1.93 and e=0.5, with a=0.35 ℑ(λ(τ)) τ −1 0 1 −1 0 1 0 1 2 3 4 5 ℜ(λ(τ)) δ=1.93, ε=1.93 and e=0.5, with a=0 ℑ(λ(τ)) τ Figure 5: spectral portrait of X̃a(τ, δ, ε, e), ∀τ ∈ [0, 2π] and (δ, ε, e) ∈ {(1, 0.8, 0.5) , (1.93, 1.93, 0.5)}, with a ∈ {0, 0.35}. perturbation of rank-2 on the movement of pendulums do not change the nature of the spectral portrait of the fundamental solution X̃(τ, δ, ε, e). For all (δ, ε) ∈ [0, 1.98]×[0, 2], Figure 6 shows the stability(strong) region of X̃a(2π, δ, ε, e). The first figure (left) shows the zone of strong stability in white and instability in red when our system is subject to a rank-2 perturbation with a = 0.35. The second figure (right) also shows the zone of strong stability in white and instability in red of the unperturbed system (a = 0). However, we observe some points of stability in blue. Thus we notice a slight difference between the two figures due to the small rank-2 perturbations of the system described by our two coupled pendulums. M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1767 Figure 6: Stability(strong) of the matrix X̃a(2π, δ, ε, e), ∀ (δ, ε) ∈ [0, 1.98]× [0, 2], a ∈ {0, 0.35} and e = 0.5. 5.2. Motion of an ion through a quadrupole analyser Consider an ion of mass m and of electric charge |Ze| which moves with a velocity v through a quadrupole analyzer of potential Φ0 = U −V cos(ωt). Within the analyzer, the ion experiences a force f(x, y, t) = −Ze∇V(x, y, t), where Z is the number of protons and e is the charge of a proton. We assume that the component of the electric field along the axis Oz is zero, and the component z of the velocity remains constant. Figure 7: Model of a quadrupole analyzer. According to [12], the motion of the ion through the analyzer is governed by the following equation  d2x dξ2 + (α− 2q cos(2ξ))x = 0 d2y dξ2 − (α− 2q cos(2ξ)) y = 0 (24) where α = 8ZeU r20mω 2 , q = 4ZeV r20mω 2 and ξ = ωt 2 . M. Dosso, T. G. Y. Arouna, J.-C. Koua Brou / Eur. J. Pure Appl. Math, 12 (4) (2019), 1744-1770 1768 This equation was proposed in 1866 by physicist Mathieu to describe the propagation of waves in membranes. We apply the rank-k perturbation to this system in view of comparing the spectral portraits and the stability zones of the perturbed and unperturbed systems. Using the change of variable given in (15) with N = 2, we obtain Hamiltonian system (1) with H(ξ) = ( P (ξ) 02 02 I2 ) and P (ξ) =  α− 2q cos(2ξ) 0 0 −α+ 2q cos(2ξ)  . Considering that the motion of the ion is subjected to a perturbation of the type (16), the equation of the motion of the ion then becomes [2, 5] J dX̃a(ξ, α, q) dξ = (I − UUTJ)TH(ξ, α, q)(I − UUTJ)︸ ︷︷ ︸ H̃(ξ, α, q) X̃a(ξ, α, q), X̃a(0, α, q) = I + UUTJ (25) where U = a  1 0 0 1 0 0 0 0  and a ∈ [0, 1[. Figure 8 represents the spectral portrait of the matrix X̃a(α, q, ξ), ∀ (α, q) ∈ {(0, 0.025) , (0.1, 0.7)}, and a = 0, 0.3003, ∀ξ ∈ [0, π]. Once again, these figures show that the small 0.8 1 1.2 1.4 −0.4 −0.2 0 0.2 0 1 2 3 4 5 ℜ(λ(ξ)) α=0 and q=0.025, with a=0.3003 ℑ(λ(ξ)) ξ 0.8 1 1.2 1.4 −0.4 −0.2 0 0.2 0 1 2 3 4 5 ℜ(λ(ξ)) α=0 and q=0.025, with a=0 ℑ(λ(ξ)) ξ 0 2 4 −1 0 1 0 1 2 3 4 5 ℜ(λ(ξ)) α=0.1 and q=0.7, with a=0.3003 ℑ(λ(ξ)) ξ 0 2 4 −1 0 1 0 1 2 3 4 5 ℜ(λ(ξ)) α=0.1 and q=0.7, with a=0 ℑ(λ(ξ)) ξ Figure 8: Spectral porttait of the matrix X̃a(ξ, α, q), ∀ξ ∈ [0, π] and (α, q) ∈ {(0, 0.025) , (0.1, 0.7)} with a ∈ {0, 0.3003}. rang-2 perturbation on the movement of an ion through a quadrupole analyzer do not change the nature of the spectral portrait of X̃a(ξ, α, q). ∀ (α, q) ∈ [0, 0.2] × [0, 0.9], Figure 9 shows the stability(strong) zone of the matrix X̃a(π, α, q). The first figure (to the left) shows the zone of strong stability in white color REFERENCES 1769 and the zone of instability in red color when our system is subject to a rank-2 perturbation with a = 0.35. The second figure (to the right) also shows the zone of strong stability in white color and instability in red color of the unperturbed system (a = 0). However, we also see points of stability visible in blue on the first figure compared to the second. Thus we notice a slight difference between the two figures due to the small rank-2 perturbation of the system described by the movement of an ion through a quadrupole analyzer. 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α q a=0.3003 0 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α q a=0 Figure 9: Stability(strong) zone of the matrix X̃a(π, α, q), ∀ (α, q) ∈ [0, 0.2]× [0, 0.9] and a ∈ {0, 0.3003}. 6. Concluding remarks From works by C. Mehl, et al.[16] on the perturbation theory of structured matrices, we presented Jordan canonical forms of rank-k perturbations of symplectic matrices and fundamental solutions of Hamiltonian system with periodic coefficients. 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