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.
- Hamiltonian systems
- periodic systems
- Mathieu equation
- parametric excitation
- parametric resonance
- symplectic maps
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 . 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 , and a numerical analysis may require considerable computation time. In this contribution, an extension of the theory developed in  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 . 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 -Hamiltonian and -symplectic matrices. With the last definitions, we prove that the state transition matrix of any -Hamiltonian system is -symplectic. The relevance of the symplectic matrices or symplectic maps lies on their symmetry which allows simplifying many computations and analysis . 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 is called symplectic if it satisfies
and is the identity matrix.
Note that for the following relations hold: , , , and . The determinant of a symplectic matrix is 1 (), and and are symplectic matrices themselves. If and are of the same dimensions and symplectic, then is also symplectic because . Finally and importantly, the inverse of a symplectic matrix always exists and is also symplectic:
The set of the symplectic matrices of dimension forms a group. The corresponding characteristic polynomial of a symplectic matrix
is a reciprocal polynomial:
This is equivalent to stating that the coefficients of satisfy the relation or rewriting as a matrix product
Since is real, if is an eigenvalue of , then so are , , and , where the bar indicates the complex conjugate. Equivalently, the eigenvalues of a symplectic matrix are reciprocal pairs. This property is called reflexivity . 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 , the following transformation is proposed in :
where . This transforms the characteristic polynomial of degree to an auxiliary polynomial of degree , while keeping all pertinent information of the original polynomial .
2.2 Hamiltonian matrices
Definition 2 The matrix ( ) is said to be Hamiltonian if and only if
Let be the characteristic polynomial of , then is an even polynomial, and it only has even powers. Thus, the eigenvalues of are symmetric with respect to the imaginary axis, i.e., if is an eigenvalue of , then is an eigenvalue, too. Furthermore, if the matrix is real, and 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 -symplectic matrices
The next definitions and properties attempt to generalize the classical definitions above.
Definition 3 is called -symplectic matrix if
is satisfied for .
Lemma 4 The determinant of a -symplectic matrix is .
To see the proof of the last lemma, see Appendix A. If is a -symplectic matrix, is a -symplectic matrix, and the set of -symplectic matrix matrices does not form a group.
Lemma 5 The characteristic polynomial of a -symplectic matrix satisfies
Corollary 7 The eigenvalues of a -symplectic matrix satisfy the symmetry
The product of each pair of eigenvalues contributes with to , and there are of these pairs; therefore, . If all eigenvalues have the same magnitude, i.e., , then . From this we find that , independent of . This may be interpreted as a “symmetry” with respect to a circle of radius . Since is real if is an eigenvalue of , then ,, and are also eigenvalues of . Moreover, the eigenvalues are symmetric with respect to the -circle: if there is an eigenvalue inside of the -circle, then there must be another eigenvalue outside (see Figure 1a for a visualization).
Remark 8 Due to Eq. (9) , the characteristic polynomial of the -symplectic matrix satisfies the following relations:
rewritten as a product of matrices yields
Remark 9 By applying the transformation
the characteristic polynomial of degree , associated to a -symplectic matrix, is reduced to an auxiliary polynomial of degree . For instance,
Note that the property of the characteristic polynomial of a -symplectic matrix in Eq. (9) reduces to Eq. (4) at . Then Eq. (12) represents the “symmetry” of the characteristic polynomial for all . Although the definition of -symplectic matrices appears in , 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 -Hamiltonian matrices.
2.4 -Hamiltonian matrices
Definition 10 A matrix ( ) is called -Hamiltonian matrix if for some ,
Lemma 11 is -Hamiltonian if and only if is Hamiltonian.
Proof 12 If is -Hamiltonian, then which can be rewritten as . Hence, is Hamiltonian.
Lemma 13 If is -Hamiltonian and if , then .
Proof 14 Recall that if , then . Then if , then , since is Hamiltonian and consequently which is equivalent to .
Remark 15 If in the last lemma all the eigenvalues of the Hamiltonian matrix have zero real parts, then the real parts of the eigenvalues of the -Hamiltonian matrix are identical to . Thus, the eigenvalues of the -Hamiltonian matrix are symmetric with respect to the vertical line 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 -Hamiltonian matrices is symmetric with respect to the real axis and a vertical line at . Then the eigenvalues of a real -Hamiltonian matrix are placed: (i) in quadruples symmetrically with respect the real axis and the line , (ii) pairs on the line and symmetric with the real axis, and (iii) real pairs symmetric with the line . All cases are shown in Figure 1b .
By the last lemma, the characteristic polynomial of the -Hamiltonian satisfies
Thus, depends only on coefficients. For instance, for ,
. Equating the coefficients leads to , , and finally to
Similarly, the polynomials for the lowest values of read
Furthermore, by applying the transformation
the polynomial can be written as an auxiliary polynomial which only has even coefficients, namely,
3. Linear -Hamiltonian systems
Definition 16 If there is a differentiable function called Hamiltonian function (energy) ,, which satisfies
then it is called a Hamiltonian system. If is a quadratic function with respect to and , then the system is a linear Hamiltonian system.
It is easy to prove that if does not depend on , is a first integral. However, this is no longer true in the time-periodic case. In the time-periodic case, even for , the integration of the equations is not possible. Any linear Hamiltonian system can be written as
where is a symmetric matrix (Hermitian in the complex case). Herein, the variables used in the definition satisfy . Therefore, the dimension of real Hamiltonian systems is always even. Finally, note that the product 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  for more details).
If is -Hamiltonian matrix, or equivalently, is a Hamiltonian matrix for some ; then it follows from Eq. (16) that
for some matrix . From the last equation , we obtain
Any -Hamiltonian matrix may be written as in Eq. (17), which motivates the next definition.
Definition 17 Any linear system that can be written as
with , , and is called a linear -Hamiltonian system.
Lemma 18 The state transition matrix of a linear -Hamiltonian system in Eq. (18) is -symplectic with .
Proof 19 Let be be the state transition of Eq. (17) , and then
Differentiating the product gives
Since , we get 1
Therefore, is -symplectic.
Lemma 20 Consider the transformation
Proof 21 From the definition , thus , and from Eq.(20)
where , but ; therefore .
3.1 Mechanical, linear -Hamiltonian system
Consider any mechanical system described by the equation
where , , and the constant matrices and such that and . Then there always exists a linear transformation such that
(e.g., see ). Therefore, applying the transformation yields
where . Eq. (22) can be rewritten as a first-order system by introducing the state vector :
where . Let
be an orthogonal matrix satisfying , and also , one can introduce the transformation , and Eq. (23) gives
Since and , the matrix
is also symmetric . Therefore, Eq. (25) can be cast into the -Hamiltonian linear system form if is approximated as . In the special case , is given exactly given by .
3.2 Periodic linear systems
where , , and are the fundamental periods.
Theorem 22 (Floquet) The state transition matrix of the system in Eq. (26) may be factorized as
In addition is a periodic matrix of the same period , and is in general a complex constant matrix [ 18 ].
Definition 23 We define the monodromy matrix associated to the Eq. (26) as
The monodromy matrix may be defined as , but we use only the spectrum of the monodromy matrix, . From.
, because and are -periodic. This last relation shows that and are similar matrices and possess the same spectrum. Moreover, if in the Floquet theorem, then based on and ; we have
Definition 24 The eigenvalues of the monodromy matrix are called characteristic multipliers or multipliers. The numbers , not unique, defined as , are called characteristic exponents or Floquet exponents.
Corollary 25 (Lyapunov-Floquet Transformation) If we define the change of coordinates
where 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 coefficients2 (). However, the matrix is not always real (e.g., see [10, 20]). In the present discussion, we only use its spectrum .
For analyzing as , we assume that the initial conditions are given at . Then for any , may be expressed as , where and . Applying the well-known properties of the state transition matrix, the solution can be written as
Analyzing the last expression, the terms and are bounded; the following three cases can be distinguished:
the solution is bounded is bounded, and if and , is a simple root of the minimal polynomial of .
or and is a multiple root of the minimum polynomial of .
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.
Corollary 27 The system in Eq. (26) is:
Asymptotically stable .
Stable , if are simple roots of the minimum polynomial of .
Unstable or which is a multiple root of minimum polynomial of .
4. Periodic -Hamiltonian systems
Once the linear Hamiltonian systems become periodic, i.e., the matrix of the system in Eq. (18) possesses a periodically time-varying , the underlying monodromy matrix becomes -symplectic and -Hamiltonian.
Definition 28 Any linear periodic system that can be written as
with will be named linear periodic -Hamiltonian system, where and are a matrix and .
Remark 29 According to Lemma 18, the state transition matrix of Eq. (33) is -symplectic, in particular, the state transition matrix evaluated over one period .
Corollary 30 The monodromy matrix and the matrix of the periodic system in Eq. (33) are -symplectic and -Hamiltonian matrices, respectively, with .
Proof 31 From the definition of a -symplectic matrix , we obtain thus .
This corollary states the main relation in our analysis. The symmetry of the -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 -Hamiltonian system can be reduced to a linear time-invariant -Hamiltonian system
The next two subsections are based on  and are adapted for characteristic polynomials of -symplectic matrices.
4.1 Stability of a system with one degree of freedom
For , the characteristic polynomial of the monodromy matrix associated with the system in Eq. (33) becomes with . According to the Lemma 18, is -symplectic. Then, there are two multipliers symmetric to the circle of radius and the real axis. Therefore, the multipliers only can leave the unit circle at the coordinates or (see Figure 2 ). Note that the term is equal to the transformation in Eq. (13):
Theorem 32 For , 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 or , the stability boundaries are given by
This means that and must be fulfilled; thus, .
4.2 Stability of a system with two degrees of freedom
For , the characteristic polynomial of the monodromy matrix associated with the system in Eq. (33) reads
where and . There are four multipliers, and due to the symmetry with respect to the -circle, they can be categorized in the position configurations depicted in Figure 2 .
Respecting that the characteristic polynomial is associated with a -symplectic matrix, we can use the transformation
to obtain the auxiliary polynomial
The symmetry of the eigenvalues yield
The transition boundaries are characterized by having at least one eigenvalue at . The simplest cases are if () or (). These points overlay if a real-valued multiplier leaves the unit circle at the point or (see the cases c, d, e, f, or g in Figure 2 ). Substituting these two values into Eq. (36) gives
Considering the case , we search the transition boundary line when two complex multipliers leave the unit circle at points different to and (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 . Here, , , , and . At , it follows that
Hence, the transformation in Eq. (13) follows:
Adding and gives
From Eq. (38) we obtain
Note that for and to become complex, the inequality
must be fulfilled. Adding and , one obtains
The real part of the eigenvalues results from Eq. (37)
with the abbreviation . Consequently, the real part of is
and substituting into Eq. (44) results in
which can be solved for to obtain the transition boundary curve
Theorem 34 The Eq. (33) when is asymptotically stable if and only if the inequalities are fulfilled:
From this analysis, the multipliers position in relation to the unit circle and -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
Consider two coupled and damped Mathieu equations of the following form:
Following the procedure presented in Section 3.1, the system in Eq. (52) can be cast into the -Hamiltonian form in Eq. (33) if and , i.e., the coefficient matrices and are symmetric. In this case, the coupled Mathieu equations present all the properties of the periodic -Hamiltonian system defined in Eq. (33) for and . 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 , , , . 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 -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 -circle, while the other two are real with . 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 , with , and parametric combination resonances of summation type occur at [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 and . 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 .
The definitions of -symplectic and -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 -Hamiltonian system is -symplectic. The symmetry properties of the eigenvalues of -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.
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 , Rim proposed an elementary proof that real symplectic matrices have determinant one; following the same procedure, we prove that the symplectic matrices have determinant . From the definition , therefore it is necessary to prove that is false. Considering the matrix since and , the matrix has real and greater than eigenvalues:
Now from the definition and rewriting ,
denotes the subblocks of as follows:
with ; thus if and ; then
We rewrite (54) with the unitary transformation :
since are real, the complex conjugation commute with the determinant, and then
and from Eq. (53)
then and ; therefore .
- 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 , and in general a numerical solution needs to be calculated.