Open access peer-reviewed chapter

Coupled Mathieu Equations: γ-Hamiltonian and μ-Symplectic

By Miguel Ramírez Barrios, Joaquín Collado and Fadi Dohnal

Submitted: April 23rd 2019Reviewed: July 17th 2019Published: August 23rd 2019

DOI: 10.5772/intechopen.88635

Downloaded: 118

Abstract

Several theoretical studies deal with the stability transition curves of coupled and damped Mathieu equations utilizing numerical and asymptotic methods. In this contribution, we exploit the fact that symplectic maps describe the dynamics of Hamiltonian systems. Starting with a Hamiltonian system, a particular dissipation is introduced, which allows the extension of Hamiltonian and symplectic matrices to more general γ -Hamiltonian and μ -symplectic matrices. A proof is given that the state transition matrix of any γ -Hamiltonian system is μ -symplectic. Combined with Floquet theory, the symmetry of the Floquet multipliers with respect to a μ -circle, which is different from the unit circle, is highlighted. An attempt is made for generalizing the particular dissipation to a more general form. The methodology is applied for calculation of the stability transition curves of an example system of two coupled and damped Mathieu equations.

Keywords

  • Hamiltonian systems
  • periodic systems
  • Mathieu equation
  • parametric excitation
  • parametric resonance
  • symplectic maps

1. Introduction

Dynamical systems represented by nonlinear or linear ordinary differential equations with periodic coefficients occur in many engineer problems (see for instance [1, 2]). The simplest example of such a system is the Mathieu equation. Most investigations in literature deal with the corresponding stability transition curves [3]. Some works analyze the stability of two coupled Mathieu equations [4, 5, 6]. In general, an asymptotic or a numerical analysis method is required for analyzing this class of systems. Perturbation techniques may lead to cumbersome expression, at least for second-order perturbation [7], and a numerical analysis may require considerable computation time. In this contribution, an extension of the theory developed in [8] is exposed in which coupled Mathieu equations are analyzed in the context of a Hamiltonian system.

The literature on Hamiltonian systems is vast. We focus on the two main references [9, 10] that are relevant for the present work. The latter focuses on linear periodic Hamiltonian systems. Although every periodic mechanical system possesses at least a small amount of dissipation, the main literature on linear Hamiltonian systems does not incorporate a dissipation. The dynamics of Hamiltonian systems can be described by symplectic maps [11]. A key fact here is that a symplectic transformation preserves the Hamiltonian structure of the underlying dynamic system. In this work we attempt to derive an appropriate formalism for linear Hamiltonian systems incorporating a very particular dissipation. For this purpose we redefine and develop the properties of the so-called γ-Hamiltonian and μ-symplectic matrices. With the last definitions, we prove that the state transition matrix of any γ-Hamiltonian system is μ-symplectic. The relevance of the symplectic matrices or symplectic maps lies on their symmetry which allows simplifying many computations and analysis [12]. The formalism is benchmarked for two coupled and damped Mathieu equations highlighting its advantages. Due to the symmetry of the symplectic matrices, the parametric resonance zones are characterized, which allows faster computations, and with higher accuracy, of the stability transition curves. This work is an extension of the contribution presented in [8, 13].

2. Preliminaries on matrices

2.1 Symplectic matrices

Definition 1 The matrix AR2n×2nis called symplectic if it satisfies

ATJA=J,E1

with

J=0InIn0E2

and Inis the n×nidentity matrix.

Note that for Jthe following relations hold: JT=J, J1=JT, J2=I2n, and detJ=1. The determinant of a symplectic matrix is 1 ([9]), and I2nand Jare symplectic matrices themselves. If Aand Bare of the same dimensions and symplectic, then ABis also symplectic because ABTJAB=BTATJAB=BTJB=J. Finally and importantly, the inverse of a symplectic matrix always exists and is also symplectic:

A1=J1ATJ:J1ATJTJJ1ATJ=JTAJATJ=J.E3

The set of the symplectic matrices of dimension 2n×2nforms a group. The corresponding characteristic polynomial of a symplectic matrix AR2n×2n

PAλ=detλI2nA=λ2n+a2n1λ2n1++a1λ+1

is a reciprocal polynomial:

PAλ=λ2nPA1λE4

This is equivalent to stating that the coefficients of PAλsatisfy the relation ak=a2nkor rewriting as a matrix product

a0a1a2n1a2n=0000100010001000100010000a0a1a2n1a2nE5

Since Ais real, if λis an eigenvalue of A, then so are λ1, λ¯, and λ¯1, where the bar indicates the complex conjugate. Equivalently, the eigenvalues of a symplectic matrix are reciprocal pairs. This property is called reflexivity [11]. Consequently, the eigenvalues are symmetric with respect to the unit circle, namely, if there is an eigenvalue inside of the unit circle, then there must be a corresponding eigenvalue outside of the unit circle. As a result of the coefficient symmetry of a symplectic matrix A, the following transformation is proposed in [12]:

δ=λ+1λ,E6

where λσA. This transforms the characteristic polynomial PAλof degree 2nto an auxiliary polynomial QAδof degree n, while keeping all pertinent information of the original polynomial [12].

2.2 Hamiltonian matrices

Definition 2 The matrix AR2n×2n( AC2n×2n) is said to be Hamiltonian if and only if

ATJ+JA=0.E7

Let PAsbe the characteristic polynomial of A, then PAsis an even polynomial, and it only has even powers. Thus, the eigenvalues of Aare symmetric with respect to the imaginary axis, i.e., if sis an eigenvalue of A, then sis an eigenvalue, too. Furthermore, if the matrix Ais real, s¯and s¯are eigenvalues as well. Then the eigenvalues of the Hamiltonian matrix are located symmetrically with respect to both real and imaginary axis. The eigenvalues appear in real pairs, purely imaginary pairs, or complex quadruples [9, 14].

2.3 μ-symplectic matrices

The next definitions and properties attempt to generalize the classical definitions above.

Definition 3 MR2n×2nis called μ-symplectic matrix if

MTJM=μJE8

is satisfied for μ01.

Lemma 4 The determinant of a μ-symplectic matrix MR2n×2nis μn.

To see the proof of the last lemma, see Appendix A. If Mis a μ-symplectic matrix, M2is a μ2-symplectic matrix, and the set of μ-symplectic matrix matrices does not form a group.

Lemma 5 The characteristic polynomial of a μ-symplectic matrix MR2n×2nsatisfies

PMμλ=μnλ2nPλ.E9

Proof 6 PMλ=detλI2nMT=detλI2nμJM1J1

=detJdetλI2nμM1detJ1=detλμMI2ndetμM1=μndetλμμλI2nM=μnλμ2ndetλμI2nM=λ2nμnPMμλ

Corollary 7 The eigenvalues of a μ-symplectic matrix Msatisfy the symmetry

λσMμλσM.E10

The product of each pair of eigenvalues contributes with μto detM, and there are nof these pairs; therefore, detM=μn. If all eigenvalues have the same magnitude, i.e., λi=rexpθi, then i=12nλ=i=12nreθi=r2n=detM=μn. From this we find that r=μ, independent of n. This may be interpreted as asymmetry” with respect to a circle of radius r=μ. Since Mis real if λis an eigenvalue of M, then λ¯,μλ, and μλ¯are also eigenvalues of M. Moreover, the eigenvalues are symmetric with respect to the μ-circle: if there is an eigenvalue inside of the μ-circle, then there must be another eigenvalue outside (see Figure 1a for a visualization).

Figure 1.

Symmetries in the spectra in the complex plane: (a) μ -symplectic matrix and (b) γ -Hamiltonian matrix.

Remark 8 Due to Eq. (9) , the characteristic polynomial PMλ=m2nλ2n+m1λ+m0of the μ-symplectic matrix Msatisfies the following relations:

m0=m2nμnm1=m2n1μn1mn=mnm2n1=m1μ1nm2n=1=m0μnE11

rewritten as a product of matrices yields

m0m1m2n1m2n=0000μn000μn+10001000μn1000μn0000m0m1m2n1m2nE12

For μ=1, the relations in Eq. (12) reduce to Eq. (5) .

Remark 9 By applying the transformation

δ=λ+μλE13

the characteristic polynomial PMλof degree 2n, associated to a μ-symplectic matrix, is reduced to an auxiliary polynomial QMδof degree n. For instance,

n=2PMλ=λ4+m3λ3+m2λ2+m3μλ+μ2QMδ=δ2+m3δ+m22μn=3PMλ=λ6+m5λ5+m4λ4+m3λ3+μm4λ2+μ2m5λ+μ3QMδ=δ3+m52δ+m43μδ+m32m5μn=4PMλ=λ8+m7λ7+m6λ6+m5λ5+m4λ4+μm5λ3+μ2m6λ2+μ3m7λ+μ4QMδ=δ4+m73δ+m64μδ2+m53m7μδ+m42m6μ+2μ2n=5PMλ=λ10+m9λ9+m8λ8+m7λ7+m6λ6+m5λ5+μm6λ4+μ2m7λ3+μ3m8λ2+μ4m9λ+μ5QMδ=δ5+m9δ4+m85μδ3+m74m9μδ2+m63m8μ+5μ2δ+m52μm7+2m9μ2

Note that the property of the characteristic polynomial of a μ-symplectic matrix in Eq. (9) reduces to Eq. (4) at μ=1. Then Eq. (12) represents the “symmetry” of the characteristic polynomial for all μ01. Although the definition of μ-symplectic matrices appears in [9], no further properties were developed within this reference. In the next section, we reveal their relationship as a generalized definition of Hamiltonian matrices, the so-called γ-Hamiltonian matrices.

2.4 γ-Hamiltonian matrices

Definition 10 A matrix AR2n×2n( AC2n×2n) is called γ-Hamiltonian matrix if for some γ0,

ATJ+JA=2γJE14

Lemma 11 Ais γ-Hamiltonian if and only if A+γI2nis Hamiltonian.

Proof 12 If Ais γ-Hamiltonian, then ATJ+JA=2γJwhich can be rewritten as A+γI2nTJ+JA+γI2n=0. Hence, A+γI2nis Hamiltonian.

Lemma 13 If Ais γ-Hamiltonian and if s+γσA, then s+γσA.

Proof 14 Recall that if σR=r1r2n, then σR+γI2n=r1+γr2n+γ. Then if s+γσA, then sσA+γI2n, since A+γI2nis Hamiltonian and consequently sσA+γI2nwhich is equivalent to s+γσA.

Remark 15 If in the last lemma all the eigenvalues of the Hamiltonian matrix A+γI2nhave zero real parts, then the real parts of the eigenvalues of the γ-Hamiltonian matrix Aare identical to γ. Thus, the eigenvalues of the γ-Hamiltonian matrix Aare symmetric with respect to the vertical line γin the complex plane (see Figure 1b for a visualization).

Figure 2.

All configurations of multiplier positions with respect to the unit (in solid line) and μ -circle (dashed line).

Notice that real Hamiltonian matrices have their spectrum symmetric with respect to the real and imaginary axes, whereas the spectrum of real γ-Hamiltonian matrices is symmetric with respect to the real axis and a vertical line at Res=γ. Then the eigenvalues of a real γ-Hamiltonian matrix are placed: (i) in quadruples symmetrically with respect the real axis and the line Res=γ, (ii) pairs on the line Res=γand symmetric with the real axis, and (iii) real pairs symmetric with the line Res=γ. All cases are shown in Figure 1b .

By the last lemma, the characteristic polynomial of the γ-Hamiltonian Asatisfies

PAs+γ=PAγs

with

PAγs=γs2n+a2n1γs2n1++a1γs+a0PAs+γ=s+γ2n+a2n1s+γ2n1++a1s+γ+a0

Thus, PAsdepends only on ncoefficients. For instance, for n=1,

s+γ2+a1s+γ+a0=γs2+a1γs+a0. Equating the coefficients leads to a1=2γ, a0=a0, and finally to

PAs=s22γs+a0.

Similarly, the polynomials for the lowest values of nread

n=2:
PAs=s44γs3+a2s2+8γ32γa2+a0
n=3:
PAs=s66γs5+a4s4+40γ34γa4s3+a2s2+96γ5+8γ3a42γa2+a0
n=4:
PAs=s88γs7+a6s6+112γ36γa6s5+a4s4+896γ5+40γ3a64γa4s3
+a2s2+2176γ796γ5a6+8γ3a42γa2+a0.

Furthermore, by applying the transformation

ϕ=sγ,E15

the polynomial PAscan be written as an auxiliary polynomial QAϕwhich only has even coefficients, namely,

PAs=s2n+a2n1s2n1+a2n2s2n2++a2s2+a1s+a0QAϕ=ϕ2n+q2n2ϕ2n2++q2ϕ2+q0

For instance,

n=1:
QAϕ=ϕ2+a0γ2
n=2:
QAϕ=ϕ4+a26γ2ϕ2+5γ4a2γ2+a0
n=3:
QAϕ=ϕ6+a415γ2ϕ4+a26a4γ2+75γ4ϕ261γ6+5a4γ4a2γ2+a0
n=4:
QAϕ=ϕ8+a628γ2ϕ6+a415a6γ2+350γ4ϕ4
+a26a4γ2+75a6γ41708γ6ϕ2+1385γ861a6γ6+5a4γ4a2γ2+a0.

3. Linear γ-Hamiltonian systems

Definition 16 If there is a differentiable function called Hamiltonian function (energy) Htxy,H:R×Rn×RnR, which satisfies

ẋ=HyTandẏ=Hy,

then it is called a Hamiltonian system. If Htxyis a quadratic function with respect to xand y, then the system is a linear Hamiltonian system.

It is easy to prove that if Hdoes not depend on t, Hxyis a first integral. However, this is no longer true in the time-periodic case. In the time-periodic case, even for n=1, the integration of the equations is not possible. Any linear Hamiltonian system can be written as

ż=JHtzE16

where HTt=Htis a symmetric matrix (Hermitian in the complex case). Herein, the variables used in the definition satisfy z=xTyTT. Therefore, the dimension of real Hamiltonian systems is always even. Finally, note that the product JHsatisfies the condition for a Hamiltonian matrix. The fundamental property of any linear Hamiltonian system is that the state transition matrix of the system in Eq. (16) is a symplectic matrix (see [9] for more details).

If Ais γ-Hamiltonian matrix, or equivalently, A+γI2nis a Hamiltonian matrix for some γ>0; then it follows from Eq. (16) that

ẋ=A+I2nx=JHx

for some matrix H=HT. From the last equation A+I2n=JH, we obtain

A=JH+γI2n.E17

Any γ-Hamiltonian matrix Amay be written as in Eq. (17), which motivates the next definition.

Definition 17 Any linear system that can be written as

ẋ=Atx=JHt+γJxE18

with xR2n, HTt=Ht, and γ0is called a linear γ-Hamiltonian system.

Lemma 18 The state transition matrix of a linear γ-Hamiltonian system in Eq. (18) is μ-symplectic with μ=e2γt.

Proof 19 Let be Nt=Φt0be the state transition of Eq. (17) , and then

Ṅt=AtNt.

Differentiating the product NTJNgives

ddtNTJN=ṄTJN+NTJṄ=ANTJN+NTJAN=NTATJ+JAN=NTJH+γJTJ+JJH+γJN=2γNTJNE19

Since NT0JN0=J, we get 1

NTtJNt=e2γtJ=μJ.

Therefore, Nis μ-symplectic.

Lemma 20 Consider the transformation

x=StzE20

with Sta symplectic matrix for all t. Then the transformation in Eq. (20) preserves the γ-Hamiltonian form of the system, Eq.(18).

Proof 21 From the definition STJS=0ṠTJS+STJṠ=0, thus ṠTJS=STJṠ, and from Eq.(20)

ẋ=Sż+ṠzS1ẋ=ż+S1Ṡz

then applying the transformation Eq.(20) into Eq.(18) it is obtained as ż+S1Ṡz=S1JH+γJSz; then from the symplectic definition matrix S1=J1STJ,

ż=S1JH+γJSzS1Ṡz=J1STJJHSzγIzJ1STJṠz=JSTHSzγIz+JSTJṠz=JSTHS+STJṠ+γJz=JH˜+γJz

where H˜=STHS+STJṠ, but STJṠT=ṠTJTS=ṠTJS=STJṠ=STJṠ; therefore H˜=H˜T.

3.1 Mechanical, linear γ-Hamiltonian system

Consider any mechanical system described by the equation

M˜y¨+D˜ẏ+K˜ty=0E21

where ytRn, K˜t=K˜TtRn×n, and the constant matrices M˜and D˜Rn×nsuch that M˜=M˜T>0and D˜=D˜T. Then there always exists a linear transformation Tsuch that

TTM˜T=InTTD˜T=D=diagd1d2dnσM˜1D˜=d1d2dn

(e.g., see [15]). Therefore, applying the transformation y=Tzyields

z¨+Dż+Ktz=0,E22

where Kt=TTK˜tT. Eq. (22) can be rewritten as a first-order system by introducing the state vector x=zTżTT:

ẋ=0n×nInKtDxE23

where xR2n×2n. Let

Q=12InInInInE24

be an orthogonal matrix satisfying QQT=QTQ=I2n, and also JQ=QJ, one can introduce the transformation w=QTx, and Eq. (23) gives

ẇ=QT0n×nInKtDQw=12KtInDKt+In+DKt+DInKt+InDw,

or equivalently,

ẇ=J12Kt+InDKtInKtInKt+In+D+12D0n×n0n×nDJw.E25

Since D=diagd1d2dnand K=KT, the matrix

Ht=12Kt+InDKtInKtInKt+In+D

is also symmetric Ht=HtT. Therefore, Eq. (25) can be cast into the γ-Hamiltonian linear system form ẇ=JH+γJwif γis approximated as γ12ni=1ndi. In the special case d=d1=d2==dn, γis given exactly given by γ=d2.

3.2 Periodic linear systems

This section summarizes the main results on periodic linear systems. The proofs and details are omitted and can be found in [16, 17]. Consider the linear periodic system:

ẋ=BtxwithBt=Bt+ΩE26

where xRn, BRn×n, and Ωare the fundamental periods.

Theorem 22 (Floquet) The state transition matrix Φtt0of the system in Eq. (26) may be factorized as

Φtt0=P1teRtt0Pt0E27

where

P1t=Φt0eRt.E28

In addition P1t=P1t+Ωis a periodic matrix of the same period Ω, and Ris in general a complex constant matrix [ 18 ].

Definition 23 We define the monodromy matrix Massociated to the Eq. (26) as

M=ΦΩ0.E29

The monodromy matrix may be defined as Mt0=ΦΩΩ+t0, but we use only the spectrum of the monodromy matrix, σM. From.

Φtt0=P1teRtt0Pt0t=t0+Ω=ΦΩΩ+t0=P1t0+ΩeRWPt0=P1t0eRWPt0, because Pand P1are Ω-periodic. This last relation shows that Mand Mt0are similar matrices and possess the same spectrum. Moreover, if t0=0in the Floquet theorem, then Φt0=QteRtbased on Qt=Qt+Ωand Q0=In; we have

M=ΦΩ0=QΩeRΩ=Q0eRΩ=eRΩ.E30

Definition 24 The eigenvalues λiof the monodromy matrix are called characteristic multipliers or multipliers. The numbers ρi, not unique, defined as λi=eρiΩ, are called characteristic exponents or Floquet exponents.

Corollary 25 (Lyapunov-Floquet Transformation) If we define the change of coordinates

zt=PtxtE31

where Pfulfills Eq. (28), then the periodic linear system in Eq. (26) can be transformed into a linear time-invariant system

żt=RztE32

where Ris a constant matrix as introduced in the Floquet theorem.

The transformation in Eq. (31) is a Lyapunov transformation which means that the stability properties of the linear system in Eq. (26) are preserved. Therefore any periodic system as in Eq. (26) is reducible to a system in Eq. (32) with constant coefficients2 ([16]). However, the matrix Ris not always real (e.g., see [10, 20]). In the present discussion, we only use its spectrum σR.

For analyzing xtas t, we assume that the initial conditions are given at t0=0. Then for any t>0, tmay be expressed as t=kΩ+τ, where kZ+and τ[0,Ω). Applying the well-known properties of the state transition matrix, the solution can be written as

xt=Φt0x0=ΦkΩ+τ0x0=ΦkΩ+τkΩΦkΩk1ΩΦk1Ωk2ΩΦΩ0ktermsx0=Φτ0ΦΩ0kx0=Φτ0Mkx0

Analyzing the last expression, the terms Φτ0and x0are bounded; the following three cases can be distinguished:

xt0limkMk=0σMD=zC:z<1.

  1. the solution xtis bounded limkMk=0is boundedσMD¯=zC:z1, and if λσMand λ=1, λis a simple root of the minimal polynomial of M.

  2. xtλσM:λ>1or λσM:λ=1and λis a multiple root of the minimum polynomial of M.

Theorem 26 (Lyapunov-Floquet) Considering the linear periodic system in Eq. (26) , then the system is (a) asymptotic stable if and only if Eq. (1) is satisfied, (b) stable if and only if Eq. (2) is satisfied, and (c) Unstable if and only if Eq. (3) is satisfied.

Due to the Lyapunov-Floquet transformation in Eq. (31), the stability of the periodic linear system in Eq. (26) can be determined by analyzing the system in Eq. (32).

Corollary 27 The system in Eq. (26) is:

  1. Asymptotically stable σRZ=zC:Rez<0.

  2. Stable σRZ¯=zC:Rez0, if Rezi=0are simple roots of the minimum polynomial of R.

  3. Unstable ρiσR:Rez>0or σRZ¯&Rezi=0which is a multiple root of minimum polynomial of R.

4. Periodic γ-Hamiltonian systems

Once the linear Hamiltonian systems become periodic, i.e., the matrix Htof the system in Eq. (18) possesses a periodically time-varying Ht=Ht+Ω, the underlying monodromy matrix becomes μ-symplectic and γ-Hamiltonian.

Definition 28 Any linear periodic system that can be written as

ẋ=JHt+γJxE33

with Ht=Ht+Ωwill be named linear periodic γ-Hamiltonian system, where xR2nand HTt=Htare a 2n×2nmatrix and γ0.

Remark 29 According to Lemma 18, the state transition matrix Φtt0of Eq. (33) is μ-symplectic, in particular, the state transition matrix evaluated over one period Ω.

Corollary 30 The monodromy matrix M=eRΩand the matrix Rof the periodic system in Eq. (33) are μ-symplectic and γ-Hamiltonian matrices, respectively, with μ=e2γΩ.

Proof 31 From the definition of a μ-symplectic matrix MTJM=eRΩTJeRΩ=μJ, we obtain eRTΩ=μJeRΩJ1=μJI2nRΩ+RRΩ22RRRΩ33!++RkΩ4k!+J1=μeJRJ1Ω=e2γΩeJRJ1Ωthus eRTΩ=e2γΩJRJ1ΩRTJ+JR=2γJ.

This corollary states the main relation in our analysis. The symmetry of the μ-symplectic matrix will be utilized for obtaining the stability conditions of the system in Eq. (33). Furthermore, by applying the Lyapunov transformation

zt=PtxtE34

we conclude that any linear periodic γ-Hamiltonian system can be reduced to a linear time-invariant γ-Hamiltonian system

żt=Rzt.E35

The next two subsections are based on [12] and are adapted for characteristic polynomials of μ-symplectic matrices.

4.1 Stability of a system with one degree of freedom

For n=1, the characteristic polynomial of the monodromy matrix Massociated with the system in Eq. (33) becomes PMλ=λ2++μwith a=trM. According to the Lemma 18, Mis μ-symplectic. Then, there are two multipliers symmetric to the circle of radius rand the real axis. Therefore, the multipliers only can leave the unit circle at the coordinates 10or 10(see Figure 2 ). Note that the term ais equal to the transformation in Eq. (13):

δ=λ+μλ=trM=a

Theorem 32 For n=1, the system in Eq. (33) is asymptotically stable if and only if the inequality

a<1+μ

is satisfied.

Proof 33 Since the multipliers only leave the unit circle on the points λ=1or λ=1, the stability boundaries are given by

PM1=12+a1+μ=a+μ+1PM1=12+a1+μ=a+μ+1.

This means that a+μ+1>0and a+μ+1>0must be fulfilled; thus, a<1+μ.

4.2 Stability of a system with two degrees of freedom

For n=2, the characteristic polynomial of the monodromy matrix Massociated with the system in Eq. (33) reads

PMλ=λ4+aλ3+bλ2+aμλ+μ2E36

where a=trMand 2b=trM2trM2. There are four multipliers, and due to the symmetry with respect to the μ-circle, they can be categorized in the position configurations depicted in Figure 2 .

Respecting that the characteristic polynomial is associated with a μ-symplectic matrix, we can use the transformation

δ=λ+μλE37

to obtain the auxiliary polynomial

QMδ=δ2++b2μ.E38

The symmetry of the eigenvalues yield

a=trM=λ1+μλ1+λ3+μλ3=δ1+δ2.

The transition boundaries are characterized by having at least one eigenvalue at λ=1. The simplest cases are if λ=1(δ=1+μ) or λ=1(δ=1μ). These points overlay if a real-valued multiplier leaves the unit circle at the point 10or 01(see the cases c, d, e, f, or g in Figure 2 ). Substituting these two values into Eq. (36) gives

b=a1+μ1+μ2E39

and

b=a1+μ1+μ2.E40

Considering the case λC, we search the transition boundary line when two complex multipliers leave the unit circle at points different to 10and 01(see cases a or b in Figure 2 ). Then the transition boundary line can be obtained by considering the symmetry of the multipliers with respect to the real axis and the circle of the radius r=μ. Here, λ1=x+iy, λ2=μλ1, λ3=xiy, and λ4=μλ3. At λ=1x2+y2=1, it follows that

λ1=x+iy,λ2=μxiy,λ3=yiy,λ4=μx+iy.

Hence, the transformation in Eq. (13) follows:

δ1=λ1+μλ1=x1+μ+iy1μδ2=λ3+μλ3=x1+μiy1μ

Adding δ1and δ2gives

δ1+δ2=2x1+μ.E41

From Eq. (38) we obtain

δ1,2=a2±a2+8μ4b2.E42

Note that for δ1and δ2to become complex, the inequality

4b>a2+8μ

must be fulfilled. Adding δ1and δ2, one obtains

δ1+δ2=aE43

Equating Eqs. (41) and (43) yields

2x1+μ=a.E44

The real part xof the eigenvalues results from Eq. (37)

λ2λδ+μ=0

and

λ=δ±δ24μ2.E45

Substituting Eq. (42) into Eq. (45) and choosing only the positive signs gives

λ1=14a+w4μ2b+a2+i4w+4μ+2ba2+a2+8μ+4b

with the abbreviation w=24a2μ+b+2μ2. Consequently, the real part of λis

x=14a+w4μ2b+a2,

and substituting into Eq. (44) results in

12a+w4μ2b+a21+μ=a

which can be solved for bto obtain the transition boundary curve

b=μ4+2μ3+2μ2+2μ+a2μ+11+μ2.E46

Two intersection points exist on each line in Eqs. (39) and (40) with the curve defined by Eq. (46). These points are

b=1μμ4+μ3+2μ2+μ+1E47
b=μ2+4μ+1E48

and are highlighted in Figure 3 . The line in Eq. (47), dashed line in the figure, is a necessary condition for stability.

Figure 3.

Multiplier map in the case of n = 2 : Horizontal and vertical axes are the coefficients a and b of the characteristic polynomial of the monodromy matrix M in Eq. (36). The solid lines represent the borders of the inequalities in Theorem 34, Eqs. (49), (50), and (51). The dots indicate the position of multipliers and the unit circle, in solid line, associated with the system in Eq. (33) in the case of n = 2 . The dashed circle depicts the μ -circle.

Theorem 34 The Eq. (33) when n=2is asymptotically stable if and only if the inequalities are fulfilled:

ba1+μ1+μ2,E49
ba1+μ1+μ2,E50
bμ4+2μ3+2μ2+a2μ+2μ+11+μ2.E51

From this analysis, the multipliers position in relation to the unit circle and μ-circle are defined by inequalities. These split the complex plane into four regions as it is shown in the Figure 3 .

5. Coupled Mathieu equations

Figure 4.

Multiplier map and stability charts, for example, systems in Eq. (52). Multiplier map corresponds to Figure 3 but now with colored regions for the different unstable multiplier configurations. Stability charts are given for different values of damping. (a) Multiplier map: a and b are the coefficients of the corresponding characteristic polynomial of the monodromy matrix. All colored zones correspond to unstable positions multipliers configurations. (b) Stability chart of coupled Mathieu equations, Eq. (52), with small damping: Θ 12 = Θ 21 = 0 and Θ 11 = Θ 22 = 0.1 . Each color code is according to the position of the multipliers as in Figure 4a . (c) Stability chart of coupled Mathieu equations, Eq. (52), with damping: Θ 12 = Θ 21 = 0 and Θ 11 = Θ 22 = 0.3 .

Consider two coupled and damped Mathieu equations of the following form:

z¨1z¨2+Θ11Θ12Θ21Θ22ż1ż2+ω1200ω22+βQ11Q12Q21Q22z1z2cosνtz1z2=0.E52

Following the procedure presented in Section 3.1, the system in Eq. (52) can be cast into the γ-Hamiltonian form in Eq. (33) if Θ12=Θ21and Q12=Q21, i.e., the coefficient matrices Θand Qare symmetric. In this case, the coupled Mathieu equations present all the properties of the periodic γ-Hamiltonian system defined in Eq. (33) for n=2and Ω=2π/ν. Hence, all the above analysis on Hamiltonian systems can be applied. The monodromy matrix is computed by numerical methods, and the stability chart is obtained by applying the Theorem 34.

The following numerical values are chosen for the analysis of a specific system ω12=8, ω22=26, Q11=Q22=2, Q12=Q21=2. Figure 4a depicts the multiplier chart similar to Figure 3 . The unstable regions are colored and the stable regions are kept white. Each color depicts a specific configuration of the multiplier positions within the unit circle and the μ-circle according to the inequalities stated in Theorem 34 and visualized in Figures 3 and 4a . The description of each color is relevant because each color describes the parametric resonance phenomenon. Thus, yellow, magenta, and cyan colors refer to the configuration of four real-valued multipliers, two of them inside and two outside of the unit circle. These multipliers are either all negative (magenta region), all positive (yellow region), or two positive and two negative (cyan region). The blue and red regions indicate two complex conjugate multipliers on the μ-circle, while the other two are real with λ>1. The two real multipliers are either positive (blue) or negative (red). Then, all four multipliers are complex conjugate within the green region. In this case, two multipliers lie inside and two outside of the unit circle.

Additionally, parametric primary resonances occur at parametric excitation frequencies ν=2ωi/k, with k+, and parametric combination resonances of summation type occur at ν=ω1+ω2/k[7, 10]. These frequencies are also observed for the example system in Figure 4 . The green regions mark parametric combination resonances. The blue and red regions correspond to parametric primary resonances. The presented calculation technique can be categorized as a semi-analytical method. After rewriting the original system into the form in Eq. (33), the monodromy matrix is constructed by integrating the equations of motion using numerical methods.

Subsequently, the coefficients of the characteristic polynomial of the monodromy matrix can be computed as a=trMand 2b=trM2trM2. This technique avoids the computation of the eigenvalues itself. This has the main advantage that numerical problems on the computation of the eigenvalues are avoided, e.g., numerical sensitivity of multipliers [21].

The definitions of μ-symplectic and γ-Hamiltonian matrices allow the analysis of a linear periodic Hamiltonian system with a particular dissipation. The main result of the proposed theory lies in Corollary 30 which states that the state transition matrix of any γ-Hamiltonian system is μ-symplectic. The symmetry properties of the eigenvalues of μ-symplectic matrices lead to an efficient calculation of the stability boundaries of this type of system. The general framework is applied for the example analysis of two damped and coupled Mathieu equations confirming the faster and robust computation of the stability chart. The procedure can be extended to a higher number of coupled Mathieu equations as outlined above.

Acknowledgments

M.Ramírez is thankful to Conacyt for the provided support during his Ph.D. studies in México and to the Austrian EFRE/LEADER project LaZu-CLLD IWB TIROL OSTT 005 for his research stay at Campus Technik Lienz.

Proof 35 In [22], Rim proposed an elementary proof that real symplectic matrices have determinant one; following the same procedure, we prove that the symplectic matrices have determinant μn. From the definition detMTJM=detMTdetJdetM=detμJ=μ2ndetM=±μn, therefore it is necessary to prove that detM=μ2nis false. Considering the matrix S=MTM+μI2nsince MTM0and μ01, the matrix Shas real and greater than μeigenvalues:

detS=detMTM+μI2n>μ.E53

Now from the definition MT=μ1JMJ1and rewriting S,

S=MTM+μI2n=MTM+μMT=MTM+JMJ1

denotes the subblocks of Mas follows:

M=M11M12M21M21

with M11,M12,M21,M21Rn×n; thus M+JMJ1=M11+M22M12M21M21M12M11+M22if A=M11+M22and B=M12M21; then

M+JMJ1=ABBA.E54

We rewrite (54) with the unitary transformation T:

T=12InIniIniInT1=12IniInIniIn
M+JMJ1=ABBA=12InIniIniInA+iB00AiB12IniInIniIn

therefore

detS=detMTM+JMJ1=detMTdetM+JMJ1=detMdetA+iBdetAiB=detMdetA+iBdetA+iB¯

since A,Bare real, the complex conjugation commute with the determinant, and then

detS=detMdetA+iBdetAiB¯=detMdetA+iB2

and from Eq. (53)

0<μ<detS=detMdetA+iB2

then detA+iB2>0and detM>0; therefore detM=μn.

Notes

  • The matrix product d dt N T JN N T JN = N T JN d dt N T JN is commutative.
  • For applying the transformation in Eq. (31), the analytical solution of Eq. (26) is only available for special cases [19], and in general a numerical solution needs to be calculated.

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Miguel Ramírez Barrios, Joaquín Collado and Fadi Dohnal (August 23rd 2019). Coupled Mathieu Equations: <em>γ</em>-Hamiltonian and <em>μ</em>-Symplectic, Dynamical Systems Theory, Jan Awrejcewicz and Dariusz Grzelczyk, IntechOpen, DOI: 10.5772/intechopen.88635. Available from:

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