4. Periodic γ-Hamiltonian systems

Once the linear Hamiltonian systems become periodic, i.e., the matrix H tð Þ of the system in Eq. (18) possesses a periodically time-varying H tðÞ¼ H tð Þ þ Ω , the underlying monodromy matrix becomes μ-symplectic and γ-Hamiltonian.

Definition 28 Any linear periodic system that can be written as

$$
\dot{\mathbf{x}} = J[H(\mathbf{t}) + \mathbf{y}I] \mathbf{x} \tag{33}
$$

with H tðÞ¼ H tð Þ þ Ω will be named linear periodic γ-Hamiltonian system, where <sup>x</sup><sup>∈</sup> <sup>R</sup><sup>2</sup><sup>n</sup> and <sup>H</sup><sup>T</sup>ðÞ¼ <sup>t</sup> H tð Þ are a 2<sup>n</sup> � <sup>2</sup><sup>n</sup> matrix and <sup>γ</sup> <sup>≥</sup> 0.

Remark 29 According to Lemma 18, the state transition matrix Φð Þ t; t<sup>0</sup> of Eq. (33) is μ-symplectic, in particular, the state transition matrix evaluated over one period Ω.

Corollary 30 The monodromy matrix <sup>M</sup> <sup>¼</sup> <sup>e</sup><sup>R</sup><sup>Ω</sup> and the matrix R of the periodic system in Eq. (33) are μ-symplectic and γ-Hamiltonian matrices, respectively, with <sup>μ</sup> <sup>¼</sup> <sup>e</sup>�2γ<sup>Ω</sup>.

Proof 31 From the definition of a <sup>μ</sup>-symplectic matrix MTJM <sup>¼</sup> eR<sup>Ω</sup> � �<sup>T</sup> J e<sup>R</sup><sup>Ω</sup> � � <sup>¼</sup> <sup>μ</sup>J, we obtain e<sup>R</sup>T<sup>Ω</sup> <sup>¼</sup> <sup>μ</sup>Je�R<sup>Ω</sup><sup>J</sup> �<sup>1</sup> <sup>¼</sup> <sup>μ</sup>J I2<sup>n</sup> � <sup>R</sup><sup>Ω</sup> <sup>þ</sup> RRΩ<sup>2</sup> <sup>2</sup> � RRRΩ<sup>3</sup> <sup>3</sup>! <sup>þ</sup> … <sup>þ</sup> <sup>R</sup>kΩ<sup>4</sup> <sup>k</sup>! þ … n o<sup>J</sup> �1 <sup>¼</sup> <sup>μ</sup>e�JRJ�<sup>1</sup> <sup>Ω</sup> <sup>¼</sup> <sup>e</sup>�2γ<sup>Ω</sup>e�JRJ�1<sup>Ω</sup> thus e<sup>R</sup>T<sup>Ω</sup> <sup>¼</sup> <sup>e</sup>�2γΩ�JRJ�<sup>1</sup> <sup>Ω</sup> ) <sup>R</sup>TJ <sup>þ</sup> JR ¼ �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

$$z(t) = P(t)\mathbf{x}(t)\tag{34}$$

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

$$
\dot{z}(t) = Rz(t). \tag{35}
$$

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 ¼ 1, the characteristic polynomial of the monodromy matrix M associated with the system in Eq. (33) becomes PMð Þ¼ <sup>λ</sup> <sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>a</sup><sup>λ</sup> <sup>þ</sup> <sup>μ</sup> with <sup>a</sup> ¼ �trð Þ <sup>M</sup> . 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 1ð Þ ; 0 or ð Þ �1; 0 (see Figure 2). Note that the term �a is equal to the transformation in Eq. (13):

$$\delta = \lambda + \frac{\mu}{\lambda} = \text{tr}(\mathcal{M}) = -a$$

Theorem 32 For n ¼ 1, the system in Eq. (33) is asymptotically stable if and only if the inequality

$$|a| < (1+\mu)$$

is satisfied.

Proof 33 Since the multipliers only leave the unit circle on the points λ ¼ 1 or λ ¼ �1, the stability boundaries are given by

$$P\_M(\mathbf{1}) = \left(\mathbf{1}\right)^2 + a(\mathbf{1}) + \mu = a + (\mu + \mathbf{1})$$

$$P\_M(-\mathbf{1}) = \left(-\mathbf{1}\right)^2 + a(-\mathbf{1}) + \mu = -a + (\mu + \mathbf{1})$$

This means that a þ ð Þ μ þ 1 . 0 and �a þ ð Þ μ þ 1 . 0 must be fulfilled; thus, j j a , ð Þ 1 þ μ .

#### 4.2 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

$$P\_M(\lambda) = \lambda^4 + a\lambda^3 + b\lambda^2 + a\mu\lambda + \mu^2 \tag{36}$$

where <sup>a</sup> ¼ �trð Þ <sup>M</sup> and 2<sup>b</sup> <sup>¼</sup> ð Þ trð Þ <sup>M</sup> <sup>2</sup> � tr <sup>M</sup><sup>2</sup> . 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

$$
\delta = \lambda + \frac{\mu}{\lambda} \tag{37}
$$

to obtain the auxiliary polynomial

$$Q\_M(\delta) = \delta^2 + a\delta + b - 2\mu. \tag{38}$$

The symmetry of the eigenvalues yield

$$a = -\text{tr}(\mathcal{M}) = \lambda\_1 + \frac{\mu}{\lambda\_1} + \lambda\_3 + \frac{\mu}{\lambda\_3} = \delta\_1 + \delta\_2.1$$

The transition boundaries are characterized by having at least one eigenvalue at j j λ ¼ 1. The simplest cases are if λ ¼ 1 (δ ¼ 1 þ μ) or λ ¼ �1 (δ ¼ �1 � μ). These points overlay if a real-valued multiplier leaves the unit circle at the point 1ð Þ ; 0 or ð Þ 0; �1 (see the cases c, d, e, f, or g in Figure 2). Substituting these two values into Eq. (36) gives

$$b = -a(\mathbf{1} + \mu) - \left(\mathbf{1} + \mu^2\right) \tag{39}$$

and

$$b = a(\mathbf{1} + \boldsymbol{\mu}) - (\mathbf{1} + \boldsymbol{\mu}^2). \tag{40}$$

Considering the case λ∈ C, we search the transition boundary line when two complex multipliers leave the unit circle at points different to 1ð Þ ; 0 and 0ð Þ ; �1 (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 <sup>μ</sup> <sup>p</sup> . Here, <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>x</sup> <sup>þ</sup> iy, <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>μ</sup> λ1 , <sup>λ</sup><sup>3</sup> <sup>¼</sup> <sup>x</sup> � iy, and <sup>λ</sup><sup>4</sup> <sup>¼</sup> <sup>μ</sup> λ3 . At j j <sup>λ</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>p</sup> <sup>¼</sup> <sup>1</sup> � �, it follows that

$$
\lambda\_1 = \infty + i\mathfrak{y}, \quad \lambda\_2 = \mu(\infty - i\mathfrak{y}), \quad \lambda\_3 = \mathfrak{y} - i\mathfrak{y}, \quad \lambda\_4 = \mu(\infty + i\mathfrak{y}).
$$

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

$$\begin{aligned} \delta\_1 &= \lambda\_1 + \frac{\mu}{\lambda\_1} = \varkappa(\mathbf{1} + \mu) + i\wp(\mathbf{1} - \mu) \\ \delta\_2 &= \lambda\_3 + \frac{\mu}{\lambda\_3} = \varkappa(\mathbf{1} + \mu) - i\wp(\mathbf{1} - \mu) \end{aligned}$$

Adding δ<sup>1</sup> and δ<sup>2</sup> gives

$$
\delta\_1 + \delta\_2 = 2\mathfrak{x}(\mathbf{1} + \mu). \tag{41}
$$

From Eq. (38) we obtain

$$\delta\_{1,2} = \frac{-a}{2} \pm \frac{\sqrt{a^2 + 8\mu - 4b}}{2}. \tag{42}$$

Note that for δ<sup>1</sup> and δ<sup>2</sup> to become complex, the inequality

$$4b \ge a^2 + 8\mu$$

must be fulfilled. Adding δ<sup>1</sup> and δ2, one obtains

$$
\delta\_1 + \delta\_2 = -\mathfrak{a} \tag{43}
$$

Equating Eqs. (41) and (43) yields

$$2\mathfrak{x}(1+\mu) = -\mathfrak{a}.\tag{44}$$

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

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

$$
\lambda^2 - \lambda \delta + \mu = 0
$$

and

$$
\lambda = \frac{\delta \pm \sqrt{\delta^2 - 4\mu}}{2}. \tag{45}
$$

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

$$\lambda\_1 = \frac{1}{4}\left(-a + \sqrt{w - 4\mu - 2b + a^2}\right) + \frac{i}{4}\left(\sqrt{w + 4\mu + 2b - a^2} + \sqrt{-(a^2 + 8\mu) + 4b}\right)$$

with the abbreviation w ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �4a<sup>2</sup>μ þ ð Þ b þ 2μ 2 q . Consequently, the real part of λ is

$$\infty = \frac{1}{4} \left( -a + \sqrt{w - 4\mu - 2b + a^2} \right),$$

and substituting into Eq. (44) results in

$$\frac{1}{2}\left(-a+\sqrt{w-4\mu-2b+a^2}\right)(1+\mu)=-a$$

which can be solved for b to obtain the transition boundary curve

$$b = \frac{\mu^4 + 2\mu^3 + 2\mu^2 + 2\mu + a^2\mu + 1}{\left(1 + \mu\right)^2}. \tag{46}$$

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

$$b = \frac{1}{\mu} \left( \mu^4 + \mu^3 + 2\mu^2 + \mu + 1 \right) \tag{47}$$

$$b = \mu^2 + 4\mu + \mathbf{1} \tag{48}$$

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

Theorem 34 The Eq. (33) when n ¼ 2 is asymptotically stable if and only if the inequalities are fulfilled:

$$b \ge -a(\mathbf{1} + \mu) - \left(\mathbf{1} + \mu^2\right),\tag{49}$$

$$b \ge a(\mathbf{1} + \boldsymbol{\mu}) - (\mathbf{1} + \boldsymbol{\mu}^2),\tag{50}$$

$$b \le \frac{\mu^4 + 2\mu^3 + 2\mu^2 + a^2\mu + 2\mu + \mathbf{1}}{\left(1 + \mu\right)^2}. \tag{51}$$

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.

#### Figure 3.

Multiplier map in the case of n ¼ 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 ¼ 2. The dashed circle depicts the μ-circle.

## 5. Coupled Mathieu equations

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

$$
\begin{bmatrix} \ddot{\boldsymbol{z}}\_1 \\ \ddot{\boldsymbol{z}}\_2 \end{bmatrix} + \begin{bmatrix} \Theta\_{11} & \Theta\_{12} \\ \Theta\_{21} & \Theta\_{22} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{z}}\_1 \\ \dot{\boldsymbol{z}}\_2 \end{bmatrix} + \left( \begin{bmatrix} \boldsymbol{a}\_1^2 & \mathbf{0} \\ \mathbf{0} & \boldsymbol{a}\_2^2 \end{bmatrix} + \beta \begin{bmatrix} \mathbf{Q}\_{11} & \mathbf{Q}\_{12} \\ \mathbf{Q}\_{21} & \mathbf{Q}\_{22} \end{bmatrix} \begin{bmatrix} \boldsymbol{z}\_1 \\ \boldsymbol{z}\_2 \end{bmatrix} \cos\left(\boldsymbol{\nu}t\right) \right) \begin{bmatrix} \mathbf{z}\_1 \\ \mathbf{z}\_2 \end{bmatrix} = \mathbf{0}.\tag{52}
$$

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 Θ<sup>12</sup> ¼ Θ<sup>21</sup> and Q<sup>12</sup> ¼ Q21, i.e., the coefficient matrices Θ and Q are symmetric. In this case, the coupled Mathieu equations present all the properties of the periodic γ-Hamiltonian system defined in Eq. (33) for n ¼ 2 and Ω ¼ 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 ω2 <sup>1</sup> <sup>¼</sup> 8, <sup>ω</sup><sup>2</sup> <sup>2</sup> ¼ 26, Q<sup>11</sup> ¼ Q<sup>22</sup> ¼ 2, Q<sup>12</sup> ¼ Q<sup>21</sup> ¼ �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 j j 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.

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

#### 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<sup>12</sup> ¼ Θ<sup>21</sup> ¼ 0 and Θ<sup>11</sup> ¼ Θ<sup>22</sup> ¼ 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<sup>12</sup> ¼ Θ<sup>21</sup> ¼ 0 and Θ<sup>11</sup> ¼ Θ<sup>22</sup> ¼ 0:3.

Additionally, parametric primary resonances occur at parametric excitation frequencies ν ¼ 2ωi=k, with k∈ℕþ, and parametric combination resonances of summation type occur at ν ¼ ð Þ ω<sup>1</sup> þ ω<sup>2</sup> =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 <sup>a</sup> ¼ �trð Þ <sup>M</sup> and 2<sup>b</sup> <sup>¼</sup> ð Þ trð Þ <sup>M</sup> <sup>2</sup> � tr <sup>M</sup><sup>2</sup> . 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.
