2. Preliminaries on matrices

#### 2.1 Symplectic matrices

Definition 1 The matrix A ∈ R<sup>2</sup>n�2<sup>n</sup> is called symplectic if it satisfies

$$\mathbf{A}^T \mathbf{J} \mathbf{A} = \mathbf{J},\tag{1}$$

with

$$J = \begin{bmatrix} \mathbf{0} & I\_n \\ -I\_n & \mathbf{0} \end{bmatrix} \tag{2}$$

and In is the n � n identity matrix.

Note that for J the following relations hold: J <sup>T</sup> ¼ �J, <sup>J</sup> �<sup>1</sup> <sup>¼</sup> <sup>J</sup> <sup>T</sup>, J <sup>2</sup> ¼ �I2<sup>n</sup>, and detð Þ¼ J 1. The determinant of a symplectic matrix is 1 ([9]), and I2<sup>n</sup> and J are symplectic matrices themselves. If A and B are of the same dimensions and symplectic, then AB is also symplectic because ð Þ AB TJ AB ð Þ¼ BTATJAB <sup>¼</sup> BTJB <sup>¼</sup> <sup>J</sup>. Finally and importantly, the inverse of a symplectic matrix always exists and is also symplectic:

$$A^{-1} = \mathbf{J}^{-1} A^T \mathbf{J} : \qquad \left( \mathbf{J}^{-1} A^T \mathbf{J} \right)^T \mathbf{J} \left( \mathbf{J}^{-1} A^T \mathbf{J} \right) = \mathbf{J}^T A \mathbf{J} A^T \mathbf{J} = \mathbf{J}. \tag{3}$$

The set of the symplectic matrices of dimension 2n � 2n forms a group. The corresponding characteristic polynomial of a symplectic matrix A ∈ R<sup>2</sup>n�2<sup>n</sup>

$$P\_A(\lambda) = \det(\lambda I\_{2n} - A) = \lambda^{2n} + a\_{2n-1}\lambda^{2n-1} + \dots + a\_1\lambda + 1$$

is a reciprocal polynomial:

$$P\_A(\lambda) = \lambda^{2n} P\_A\left(\frac{1}{\lambda}\right) \tag{4}$$

This is equivalent to stating that the coefficients of PAð Þλ satisfy the relation ak ¼ a2n�<sup>k</sup> or rewriting as a matrix product

Coupled Mathieu Equations: γ-Hamiltonian and μ-Symplectic DOI: http://dx.doi.org/10.5772/intechopen.88635

$$
\begin{bmatrix} a\_0 \\ a\_1 \\ \vdots \\ a\_{2n-1} \\ a\_{2n} \end{bmatrix} = \begin{bmatrix} 0 & 0 & \cdots & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 0 & \cdots & 1 & 0 \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots \\ 0 & 1 & \cdots & 0 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & \cdots & 0 & 0 \end{bmatrix} \begin{bmatrix} a\_0 \\ a\_1 \\ \vdots \\ a\_{2n-1} \\ a\_{2n} \end{bmatrix} \tag{5}
$$

Since A is real, if λ is an eigenvalue of A, then so are λ�<sup>1</sup> , 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]:

$$
\delta = \lambda + \frac{1}{\lambda},
\tag{6}
$$

where λ∈σð Þ A . This transforms the characteristic polynomial PAð Þλ of degree 2n to an auxiliary polynomial QAð Þδ of degree n, while keeping all pertinent information of the original polynomial [12].

#### 2.2 Hamiltonian matrices

Definition 2 The matrix A ∈ R<sup>2</sup>n�2<sup>n</sup> (A ∈ C<sup>2</sup>n�2<sup>n</sup> ) is said to be Hamiltonian if and only if

$$\mathbf{A}^T \mathbf{J} + \mathbf{J} \mathbf{A} = \mathbf{0}.\tag{7}$$

Let PAð Þs be the characteristic polynomial of A, then PAð Þs 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 �s is an eigenvalue, too. Furthermore, if the matrix A is 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 M ∈ R<sup>2</sup>n�2<sup>n</sup> is called μ-symplectic matrix if

$$\mathbf{M}^T \mathbf{J} \mathbf{M} = \mu \mathbf{J} \tag{8}$$

is satisfied for μ∈ð � 0; 1 .

Lemma 4 The determinant of a μ-symplectic matrix M ∈ R<sup>2</sup>n�2<sup>n</sup> is μn.

To see the proof of the last lemma, see Appendix A. If M is a μ-symplectic matrix, M<sup>2</sup> is 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 M ∈ R2n�2<sup>n</sup> satisfies

$$P\_M\left(\frac{\mu}{\lambda}\right) = \frac{\mu^n}{\lambda^{2n}} P(\lambda). \tag{9}$$

Proof 6 PMð Þ¼ <sup>λ</sup> det <sup>λ</sup>I2<sup>n</sup> � <sup>M</sup><sup>T</sup> � � <sup>¼</sup> det <sup>λ</sup>I2<sup>n</sup> � <sup>μ</sup>JM�<sup>1</sup> J �1 � �

$$\begin{split} \lambda &= \det(J) \det\big(\lambda I\_{2n} - \mu \mathcal{M}^{-1}\big) \det\big(\mathcal{J}^{-1}\big) = \det\left(\frac{\lambda}{\mu}\mathcal{M} - I\_{2n}\right) \det\big(\mu \mathcal{M}^{-1}\big) \\ &= \mu^n \det\left(\left(-\frac{\lambda}{\mu}\right)\left(\frac{\mu}{\lambda}I\_{2n} - \mathcal{M}\right)\right) = \mu^n \left(-\frac{\lambda}{\mu}\right)^{2n} \det\left(\frac{\lambda}{\mu}I\_{2n} - \mathcal{M}\right) = \frac{\lambda^{2n}}{\mu^n} P\_{\mathcal{M}}\left(\frac{\mu}{\lambda}\right) \end{split}$$

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

$$
\lambda \in \sigma(M) \Rightarrow \left(\frac{\mu}{\lambda}\right) \in \sigma(M). \tag{10}
$$

The product of each pair of eigenvalues contributes with μ to detð Þ M , and there are n of these pairs; therefore, detð Þ¼ <sup>M</sup> <sup>μ</sup>n. If all eigenvalues have the same magnitude, i.e., <sup>λ</sup><sup>i</sup> <sup>¼</sup> <sup>r</sup> exp ð Þ <sup>θ</sup><sup>i</sup> , then <sup>Q</sup><sup>2</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup> j j <sup>λ</sup> <sup>¼</sup> <sup>Q</sup><sup>2</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup> reθ<sup>i</sup> � � � � <sup>¼</sup> <sup>r</sup><sup>2</sup><sup>n</sup> <sup>¼</sup> detð Þ¼ <sup>M</sup> <sup>μ</sup>n. 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, <sup>μ</sup> λ , and <sup>μ</sup> <sup>λ</sup> 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 PMð Þ¼ <sup>λ</sup> <sup>m</sup>2<sup>n</sup>λ<sup>2</sup><sup>n</sup> <sup>þ</sup> …m1<sup>λ</sup> <sup>þ</sup> <sup>m</sup><sup>0</sup> of the <sup>μ</sup>-symplectic matrix M satisfies the following relations:

$$\begin{aligned} m\_0 &= m\_{2\pi}\mu^n\\ m\_1 &= m\_{2\pi - 1}\mu^{n - 1} \\ &\vdots\\ m\_n &= m\_n \\ &\vdots\\ m\_{2n - 1} &= m\_1\mu^{1 - n} \\ m\_{2n} &= 1 = m\_0\mu^{-n} \end{aligned} \tag{11}$$

rewritten as a product of matrices yields

$$
\begin{bmatrix} m\_0 \\ m\_1 \\ \vdots \\ m\_{2n-1} \\ m\_{2n} \end{bmatrix} = \begin{bmatrix} 0 & 0 & \cdots & 0 & \cdots & 0 & \mu^{-n} \\ 0 & 0 & \cdots & 0 & \cdots & \mu^{-n+1} & 0 \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots \\ 0 & \mu^{n-1} & \cdots & 0 & \cdots & 0 & 0 \\ \mu^n & 0 & \cdots & 0 & \cdots & 0 & 0 \end{bmatrix} \begin{bmatrix} m\_0 \\ m\_1 \\ \vdots \\ m\_{2n-1} \\ m\_{2n} \end{bmatrix} \tag{12}
$$

For μ ¼ 1, the relations in Eq. (12) reduce to Eq. (5).

Coupled Mathieu Equations: γ-Hamiltonian and μ-Symplectic DOI: http://dx.doi.org/10.5772/intechopen.88635

Figure 1.

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

Remark 9 By applying the transformation

$$
\delta = \lambda + \frac{\mu}{\lambda} \tag{13}
$$

the characteristic polynomial PMð Þλ of degree 2n, associated to a μ-symplectic matrix, is reduced to an auxiliary polynomial QMð Þδ of degree n. For instance,

$$\begin{aligned} n=2 \quad \left\{ \begin{aligned} P\_M(\lambda) &= \lambda^4 + m\_3 \lambda^3 + m\_2 \lambda^2 + m\_3 \mu \lambda + \mu^2 \\ Q\_M(\delta) &= \delta^2 + m\_3 \delta + m\_2 - 2\mu \\ P\_M(\lambda) &= \lambda^6 + m\_5 \lambda^5 + m\_4 \lambda^4 + m\_3 \lambda^3 + \mu m\_4 \lambda^2 + \mu^2 m\_5 \lambda + \mu^3 \\ Q\_M(\delta) &= \delta^3 + m\_2^2 \delta + (m\_4 - 3\mu)\delta + m\_3 - 2m\_5 \mu \\ P\_M(\lambda) &= \delta^3 + m\_7 \lambda^7 + m\_6 \delta^6 + m\_5 \lambda^5 + m\_4 \lambda^4 + \mu m\_5 \lambda^3 \\ &+ \mu^2 m\_6 \lambda^2 + \mu^3 m\_7 \lambda + \mu^4 \\ Q\_M(\delta) &= \delta^4 + m\_7^2 \delta + (m\_6 - 4\mu)\delta^2 + (m\_5 - 3m\_7 \mu)\delta + m\_4 \\ &- 2m\_6 \mu + 2\mu^2 \\ P\_M(\lambda) &= \lambda^{10} + m\_3 \delta^9 + m\_8 \lambda^8 + m\_7 \lambda^7 + m\_6 \lambda^6 + m\_5 \lambda^5 \\ &+ \mu m\_6 \lambda^4 + \mu^2 m\_7 \lambda^3 + \mu^3 m\_8 \lambda^2 + \mu^4 m\_9 \lambda + \mu^5 \\ Q\_M(\delta) &= \delta^7 + m\_8 \delta^4 + (m\_8 - 5\mu)\delta^3 + (m\_7 - 4m\_9 \mu)\delta^2 \\ &+ (m\_6 - 3m\_8 \mu + \mu^2)\delta + m\_5 - 2\mu m\_7 + 2m\_9 \mu^2 \end{aligned}$$

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∈ð � 0; 1 . 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 A ∈ R2n�2<sup>n</sup> (A ∈ C2n�2<sup>n</sup> ) is called γ-Hamiltonian matrix if for some γ ≥0,

$$\mathbf{A}^T \mathbf{J} + \mathbf{J} \mathbf{A} = -2\eta \mathbf{J} \tag{14}$$

Lemma 11 A is γ-Hamiltonian if and only if A þ γI2<sup>n</sup> is Hamiltonian. Proof 12 If A is <sup>γ</sup>-Hamiltonian, then ATJ <sup>þ</sup> JA ¼ �2γJ which can be rewritten as ½ � <sup>A</sup> <sup>þ</sup> <sup>γ</sup>I2<sup>n</sup> TJ <sup>þ</sup> J A½ �¼ <sup>þ</sup> <sup>γ</sup>I2<sup>n</sup> <sup>0</sup>. Hence, A½ � <sup>þ</sup> <sup>γ</sup>I2<sup>n</sup> is Hamiltonian.

Lemma 13 If A is γ-Hamiltonian and if s þ γ ∈σð Þ A , then �s þ γ ∈σð Þ A .

Proof 14 Recall that if σð Þ¼ R f g r1; …r2<sup>n</sup> , then σð Þ¼ R þ γI2<sup>n</sup> f g r<sup>1</sup> þ γ; …;r2<sup>n</sup> þ γ . Then if s þ γ ∈σð Þ A , then s∈ σð Þ A þ γI2<sup>n</sup> , since A½ � þ γI2<sup>n</sup> is Hamiltonian and consequently �s∈ σð Þ A þ γI2<sup>n</sup> which is equivalent to �s þ γ ∈σð Þ A .

Remark 15 If in the last lemma all the eigenvalues of the Hamiltonian matrix A þ γI2<sup>n</sup> have zero real parts, then the real parts of the eigenvalues of the γ-Hamiltonian matrix A are identical to �γ. Thus, the eigenvalues of the γ-Hamiltonian matrix A are symmetric with respect to the vertical line �γ 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 ReðÞ¼� s γ. Then the eigenvalues of a real γ-Hamiltonian matrix are placed: (i) in quadruples symmetrically with respect the real axis and the line ReðÞ¼� s γ, (ii) pairs on the line ReðÞ¼� s γ and symmetric with the real axis, and (iii) real pairs symmetric with the line ReðÞ¼� s γ. All cases are shown in Figure 1b.

By the last lemma, the characteristic polynomial of the γ-Hamiltonian A satisfies

$$P\_A(s+\chi) = P\_A(\chi - s)$$

with

$$\begin{aligned} P\_A(\chi - s) &= (\chi - s)^{2n} + a\_{2n-1}(\chi - s)^{2n-1} + \dots + a\_1(\chi - s) + a\_0 \\ P\_A(s + \chi) &= (s + \chi)^{2n} + a\_{2n-1}(s + \chi)^{2n-1} + \dots + a\_1(s + \chi) + a\_0 \end{aligned}$$

Thus, PAð Þs depends only on n coefficients. For instance, for n ¼ 1,

ð Þ s þ γ <sup>2</sup> <sup>þ</sup> <sup>a</sup>1ð Þþ <sup>s</sup> <sup>þ</sup> <sup>γ</sup> <sup>a</sup><sup>0</sup> <sup>¼</sup> ð Þ <sup>γ</sup> � <sup>s</sup> <sup>2</sup> <sup>þ</sup> <sup>a</sup>1ð Þþ <sup>γ</sup> � <sup>s</sup> <sup>a</sup>0. Equating the coefficients leads to a<sup>1</sup> ¼ �2γ, a<sup>0</sup> ¼ a0, and finally to

$$P\_A(\mathfrak{s}) = \mathfrak{s}^2 - 2\mathfrak{yr} + a\_0.$$

Similarly, the polynomials for the lowest values of n read

$$\begin{aligned} n &= 2: \\ P\_A(s) &= s^4 - 4\eta s^3 + a\_2 s^2 + \left( 8\eta^3 - 2\eta a\_2 \right) + a\_0 \\ n &= 3: \\ P\_A(s) &= s^6 - 6\eta s^5 + a\_4 s^4 + \left( 40\eta^3 - 4\eta a\_4 \right) s^3 + a\_2 s^2 + \left( -96\eta^5 + 8\eta^3 a\_4 - 2\eta a\_2 \right) + a\_0 \\ n &= 4: \\ P\_A(s) &= s^8 - 8\eta s^7 + a\_6 s^6 + \left( 112\eta^3 - 6\eta a\_6 \right) s^5 + a\_4 s^4 + \left( -896\eta^5 + 40\eta^3 a\_6 - 4\eta a\_4 \right) s^3 \\ &+ a\_2 s^2 + \left( 2176\eta^7 - 96\eta^5 a\_6 + 8\eta^3 a\_4 - 2\eta a\_2 \right) + a\_0. \end{aligned}$$

Coupled Mathieu Equations: γ-Hamiltonian and μ-Symplectic DOI: http://dx.doi.org/10.5772/intechopen.88635

Figure 2. All configurations of multiplier positions with respect to the unit (in solid line) and μ-circle (dashed line).

Furthermore, by applying the transformation

$$
\phi = \mathfrak{s} - \mathfrak{y},
\tag{15}
$$

the polynomial PAð Þs can be written as an auxiliary polynomial QAð Þ ϕ which only has even coefficients, namely,

$$P\_A(s) = s^{2n} + a\_{2n-1}s^{2n-1} + a\_{2n-2}s^{2n-2} + \dots + a\_2s^2 + a\_1s + a\_0$$

$$Q\_A(\phi) = \phi^{2n} + q\_{2n-2}\phi^{2n-2} + \dots + q\_2\phi^2 + q\_0$$

For instance,

$$\begin{aligned} n &= 1: \\ Q\_A(\phi) &= \phi^2 + a\_0 - \chi^2 \\ n &= 2: \\ Q\_A(\phi) &= \phi^4 + \left(a\_2 - 6\chi^2\right)\phi^2 + 5\chi^4 - a\_2\chi^2 + a\_0 \end{aligned}$$

$$\begin{aligned} n &= 3: \\ Q\_A(\phi) &= \phi^6 + \left(a\_4 - 15\gamma^2\right)\phi^4 + \left(a\_2 - 6a\_4\gamma^2 + 75\gamma^4\right)\phi^2 - 61\gamma^6 + 5a\_4\gamma^4 - a\_2\gamma^2 + a\_0 \\ n &= 4: \\ Q\_A(\phi) &= \phi^8 + \left(a\_6 - 28\gamma^2\right)\phi^6 + \left(a\_4 - 15a\_6\gamma^2 + 350\gamma^4\right)\phi^4 \\ &+ \left(a\_2 - 6a\_4\gamma^2 + 75a\_6\gamma^4 - 1708\gamma^6\right)\phi^2 + 1385\gamma^8 - 61a\_6\gamma^6 + 5a\_4\gamma^4 - a\_2\gamma^2 + a\_0. \end{aligned}$$
