Coupled Mathieu Equations: γ -Hamiltonian and μ -Symplectic

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.


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].

Preliminaries on matrices 2.1 Symplectic matrices
Definition 1 The matrix A ∈ R 2nÂ2n is called symplectic if it satisfies with and I n is the n Â n identity matrix.
Note that for J the following relations hold: J T ¼ ÀJ, J À1 ¼ J T , J 2 ¼ ÀI 2n , and det J ð Þ ¼ 1. The determinant of a symplectic matrix is 1 ([9]), and I 2n and J are symplectic matrices themselves. If A and B are of the same dimensions and symplectic, then AB is also symplectic because AB ð Þ T J AB ð Þ ¼ B T A T JAB ¼ B T JB ¼ J. Finally and importantly, the inverse of a symplectic matrix always exists and is also symplectic: The set of the symplectic matrices of dimension 2n Â 2n forms a group. The corresponding characteristic polynomial of a symplectic matrix A ∈ R 2nÂ2n is a reciprocal polynomial: This is equivalent to stating that the coefficients of P A λ ð Þ satisfy the relation a k ¼ a 2nÀk or rewriting as a matrix product where λ ∈ σ A ð Þ. This transforms the characteristic polynomial P A λ ð Þ of degree 2n to an auxiliary polynomial Q A δ ð Þ of degree n, while keeping all pertinent information of the original polynomial [12].

Hamiltonian matrices
Definition 2 The matrix A ∈ R 2nÂ2n (A ∈ C 2nÂ2n ) is said to be Hamiltonian if and only if A T J þ JA ¼ 0: Let P A s ð Þ be the characteristic polynomial of A, then P A s ð Þ is an even polynomial, and it only has even powers. Thus, the eigenvalues of A are symmetric with respect to the imaginary axis, i.e., if s is an eigenvalue of A, then Às is an eigenvalue, too. Furthermore, if the matrix A is real, s and Às 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].

μ-symplectic matrices
The next definitions and properties attempt to generalize the classical definitions above.
is satisfied for μ ∈ 0; 1 ð . Lemma 4 The determinant of a μ-symplectic matrix M ∈ R 2nÂ2n is μ n . To see the proof of the last lemma, see Appendix A. If M is a μ-symplectic matrix, M 2 is a μ 2 -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 M ∈ R 2nÂ2n satisfies The eigenvalues of a μ-symplectic matrix M satisfy the symmetry The product of each pair of eigenvalues contributes with μ to det M ð Þ, and there are n of these pairs; therefore, det M ð Þ ¼ μ n . If all eigenvalues have the same magnitude, i.e., From this we find that r ¼ ffiffiffi μ p , independent of n. This may be interpreted as a "symmetry" with respect to a circle of radius r ¼ ffiffiffi μ p . Since M is real if λ is an eigenvalue of M, then λ, μ λ , and μ λ 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).
Remark 8 Due to Eq. (9), the characteristic polynomial P M λ ð Þ ¼ m 2n λ 2n þ …m 1 λ þ m 0 of the μ-symplectic matrix M satisfies the following relations: rewritten as a product of matrices yields  Remark 9 By applying the transformation the characteristic polynomial P M λ ð Þ of degree 2n, associated to a μ-symplectic matrix, is reduced to an auxiliary polynomial Q M δ ð Þ of degree n. For instance, Note that the property of the characteristic polynomial of a μ-symplectic matrix in Eq. (9) reduces to Eq. (4) at μ ¼ 1. Then Eq. (12) represents the "symmetry" of the characteristic polynomial for all μ ∈ 0; 1 ð . 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.

γ-Hamiltonian matrices
If in the last lemma all the eigenvalues of the Hamiltonian matrix A þ γI 2n have zero real parts, then the real parts of the eigenvalues of the γ-Hamiltonian matrix A are identical to Àγ. Thus, the eigenvalues of the γ-Hamiltonian matrix A are symmetric with respect to the vertical line Àγ in the complex plane (see Figure 1b for a visualization).
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 Re s ð Þ ¼ Àγ. Then the eigenvalues of a real γ-Hamiltonian matrix are placed: (i) in quadruples symmetrically with respect the real axis and the line Re s ð Þ ¼ Àγ, (ii) pairs on the line Re s ð Þ ¼ Àγ and symmetric with the real axis, and (iii) real pairs symmetric with the line Re s ð Þ ¼ Àγ. All cases are shown in Figure 1b. By the last lemma, the characteristic polynomial of the γ-Hamiltonian A satisfies Thus, P A s ð Þ depends only on n coefficients. For instance, for n ¼ 1, Equating the coefficients leads to a 1 ¼ À2γ, a 0 ¼ a 0 , and finally to Similarly, the polynomials for the lowest values of n read n ¼ 2 : Furthermore, by applying the transformation the polynomial P A s ð Þ can be written as an auxiliary polynomial Q A ϕ ð Þ which only has even coefficients, namely, For instance, n ¼ 1 : Coupled Mathieu Equations: γ-Hamiltonian and μ-Symplectic DOI: http://dx.doi.org/10.5772/intechopen.88635 n ¼ 3 : 3. Linear γ-Hamiltonian systems Definition 16 If there is a differentiable function called Hamiltonian function then it is called a Hamiltonian system. If H t; x; y ð Þis a quadratic function with respect to x and y, then the system is a linear Hamiltonian system.
It is easy to prove that if H does not depend on t, H x; y ð Þ is a first integral. However, this is no longer true in the time-periodic case. In the time-periodic case, even for n ¼ 1, the integration of the equations is not possible. Any linear Hamiltonian system can be written as where Herein, the variables used in the definition satisfy z ¼ x T ; y T Â Ã T . Therefore, the dimension of real Hamiltonian systems is always even. Finally, note that the product JH satisfies 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 A is γ-Hamiltonian matrix, or equivalently, A þ γI 2n is a Hamiltonian matrix for some γ . 0; then it follows from Eq. (16) that Any γ-Hamiltonian matrix A may be written as in Eq. (17), which motivates the next definition.
Definition 17 Any linear system that can be written as with x ∈ R 2n , H T t ð Þ ¼ H t ð Þ, and γ ≥ 0 is 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 ¼ e À2γt . Proof 19 Let be N t ð Þ ¼ Φ t; 0 ð Þ be the state transition of Eq. (17), and then Differentiating the product N T JN gives Since Therefore, N is μ-symplectic. Lemma 20 Consider the transformation with S t ð Þ a symplectic matrix for all t. Then the transformation in Eq. (20) preserves the γ-Hamiltonian form of the system, Eq. (18).
3.1 Mechanical, linear γ-Hamiltonian system Consider any mechanical system described by the equatioñ where y t ð Þ ∈ R n ,K t ð Þ ¼K T t ð Þ ∈ R nÂn , and the constant matricesM andD ∈ R nÂn such thatM ¼M T . 0 andD ¼D T . Then there always exists a linear transformation T such that (e.g., see [15]). Therefore, applying the transformation y ¼ Tz yields where K t ð Þ ¼ T TK t ð ÞT. Eq. (22) can be rewritten as a first-order system by introducing the state vector be an orthogonal matrix satisfying QQ T ¼ Q T Q ¼ I 2n , and also JQ ¼ Q J, one can introduce the transformation w ¼ Q T x, and Eq. (23) gives is also symmetric H t ð Þ ¼ H t ð Þ T . Therefore, Eq. (25) can be cast into the γ-Hamiltonian linear system form _ γ is given exactly given by γ ¼ d 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: where x ∈ R n , B ∈ R nÂn , and Ω are the fundamental periods. Theorem 22 (Floquet) The state transition matrix Φ t; t 0 ð Þof the system in Eq. (26) may be factorized as where In addition P À1 t ð Þ ¼ P À1 t þ Ω ð Þis a periodic matrix of the same period Ω, and R is in general a complex constant matrix [18].
Definition 23 We define the monodromy matrix M associated to the Eq. (26) as The monodromy matrix may be defined as M t 0 ¼ Φ Ω; Ω þ t 0 ð Þ , but we use only the spectrum of the monodromy matrix, σ M ð Þ. From.
ð Þe RW P t 0 ð Þ, because P and P À1 are Ω-periodic. This last relation shows that M and M t 0 are similar matrices and possess the same spectrum. Moreover, if t 0 ¼ 0 in the Floquet theorem, then Φ t; 0 Definition 24 The eigenvalues λ i of the monodromy matrix are called characteristic multipliers or multipliers. The numbers ρ i , not unique, defined as λ i ¼ e ρ i Ω , are called characteristic exponents or Floquet exponents.
Corollary 25 (Lyapunov-Floquet Transformation) If we define the change of coordinates where P fulfills Eq. (28), then the periodic linear system in Eq. (26) can be transformed into a linear time-invariant system where R is 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 coefficients 2 ( [16]). However, the matrix R is not always real (e.g., see [10,20]). In the present discussion, we only use its spectrum σ R ð Þ. For analyzing x t ð Þ as t ! ∞, we assume that the initial conditions are given at t 0 ¼ 0. Then for any t . 0, t may be expressed as t ¼ kΩ þ τ, where k ∈ Z þ and τ ∈ ½0, ΩÞ. Applying the well-known properties of the state transition matrix, the solution can be written as Analyzing the last expression, the terms Φ τ; 0 ð Þ and x 0 are bounded; the following three cases can be distinguished: g , and if λ ∈ σ M ð Þ and λ j j ¼ 1, λ is a simple root of the minimal polynomial of M.

Periodic γ-Hamiltonian systems
Once the linear Hamiltonian systems become periodic, i.e., the matrix H t ð Þ of the system in Eq. (18) possesses a periodically time-varying H t ð Þ ¼ H t þ Ω ð Þ, the underlying monodromy matrix becomes μ-symplectic and γ-Hamiltonian.
Definition 28 Any linear periodic system that can be written as Þwill be named linear periodic γ-Hamiltonian system, where x ∈ R 2n and H T t ð Þ ¼ H t ð Þ are a 2n Â 2n matrix and γ ≥ 0. Remark 29 According to Lemma 18, the state transition matrix Φ t; t 0 ð Þof Eq. (33) is μ-symplectic, in particular, the state transition matrix evaluated over one period Ω.
Corollary 30 The monodromy matrix M ¼ e RΩ and the matrix R of the periodic system in Eq. (33) are μ-symplectic and γ-Hamiltonian matrices, respectively, with μ ¼ e À2γΩ .
Proof 31 From the definition of a μ-symplectic matrix M T JM ¼ e RΩ À Á T J e RΩ À Á ¼ μJ, 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 we conclude that any linear periodic γ-Hamiltonian system can be reduced to a linear time-invariant γ-Hamiltonian system The next two subsections are based on [12] and are adapted for characteristic polynomials of μ-symplectic matrices.

Stability of a system with one degree of freedom
For n ¼ 1, the characteristic polynomial of the monodromy matrix M associated with the system in Eq. (33) becomes P M λ ð Þ ¼ λ 2 þ aλ þ μ with a ¼ Àtr M ð Þ. According to the Lemma 18, M is μ-symplectic. Then, there are two multipliers symmetric to the circle of radius r and the real axis. Therefore, the multipliers only can leave the unit circle at the coordinates 1; 0 ð Þor À1; 0 ð Þ(see Figure 2). Note that the term Àa is equal to the transformation in Eq. (13): Theorem 32 For n ¼ 1, the system in Eq. (33) is asymptotically stable if and only if the inequality Proof 33 Since the multipliers only leave the unit circle on the points λ ¼ 1 or λ ¼ À1, the stability boundaries are given by This means that a þ μ þ 1 ð Þ. 0 and Àa þ μ þ 1 ð Þ. 0 must be fulfilled; thus, a j j , 1 þ μ ð Þ.

Stability of a system with two degrees of freedom
For n ¼ 2, the characteristic polynomial of the monodromy matrix M associated with the system in Eq. (33) reads 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 to obtain the auxiliary polynomial 13 Coupled Mathieu Equations: γ-Hamiltonian and μ-Symplectic DOI: http://dx.doi.org/10.5772/intechopen.88635 The symmetry of the eigenvalues yield The transition boundaries are characterized by having at least one eigenvalue at λ j j ¼ 1. The simplest cases are if λ ¼ 1 (δ ¼ 1 þ μ) or λ ¼ À1 (δ ¼ À1 À μ). These points overlay if a real-valued multiplier leaves the unit circle at the point 1; 0 ð Þ or 0; À1 ð Þ(see the cases c, d, e, f, or g in Figure 2). Substituting these two values into Eq. (36) gives Considering the case λ ∈ C, we search the transition boundary line when two complex multipliers leave the unit circle at points different to 1; 0 ð Þand 0; À1 ð Þ(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 ¼ ffiffiffi Hence, the transformation in Eq. (13) follows: Adding δ 1 and δ 2 gives From Eq. (38) we obtain : Note that for δ 1 and δ 2 to become complex, the inequality 4b . a 2 þ 8μ must be fulfilled. Adding δ 1 and δ 2 , one obtains Equating Eqs. (41) and (43) yields The real part x of the eigenvalues results from Eq. (37) Substituting Eq. (42) into Eq. (45) and choosing only the positive signs gives . Consequently, the real part of λ is and substituting into Eq. (44) results in which can be solved for b to obtain the transition boundary curve Two intersection points exist on each line in Eqs. (39) and (40) with the curve defined by Eq. (46). These points are and are highlighted in From this analysis, the multipliers position in relation to the unit circle and μcircle are defined by inequalities. These split the complex plane into four regions as it is shown in the Figure 3.  Additionally, parametric primary resonances occur at parametric excitation frequencies ν ¼ 2ω i =k, with k ∈ ℕ þ , and parametric combination resonances of summation type occur at ν ¼ ω 1 þ ω 2 ð Þ =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 ¼ Àtr M ð Þ and 2b ¼ tr M ð Þ ð Þ 2 À tr M 2 À Á . 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.