Non-trivial solutions and their stability in a two-degree-of-freedom Mathieu-Dung system

The Mathieu-Duﬃng equation represents a basic form for a parametrically excited system with cubic nonlinearities. In multi- degree-of-freedom systems, other interesting instability conditions take place and are considered in this work. Accordingly, the non-trivial solutions are obtained, especially when the trivial solution is unstable. At resonant frequencies a bifurcation analysis is carried out using the multiple scales method, followed by investigating the eﬀect of an asynchronous parametric excitation. Since various micro- and nano-systems include cubic nonlinearities and subjected to parametric excitation, this work should be of relevant importance.

. In addition, parametric excitation methods showed promising broadband frequency tuning capability for an electrostatically driven nonlinear nano-mechanical string resonator in a strong modal coupling between its elastic modes [15]. These applications in small-scaled systems created, therefore, more interest in this study of parametrically excited nonlinear systems.
Motivated by the mentioned theoretical findings and by the significance of parametric excitation methods in nonlinear applications, the two-degree-of-freedom Mathieu-Duffing equation is discussed in this work in a generic configuration, especially where the broadband destabilization effect is exhibited. Parametrically excited nonlinear two-DoF systems were investigated before in different configurations. In addition to cubic nonlinearities, some systems included either quadratic or coupling nonlinear terms were studied analytically [16,17]. Some works were focused on two-DoF systems for specific applications using numerical [18] or experimental methods [19], where different nonlinear terms were included depending on the application involved. Theoretical works were more interested in systems with bimodal coupled parametric excitation [20,21,22], where the diagonal terms in the parametric excitation matrix were not considered. In these studies, the non-trivial solutions were considered at combination resonances [20,22] using averaging approximation methods and at non-resonant frequencies under asynchronous excitation [21] using the method of normal forms, however, primary resonances were not considered since the diagonal excitation terms did not exist.
In this work, the asynchronous parametric excitation is considered in a two DoF nonlinear system, with a fully populated excitation matrix, providing a generic form of the two-degree-of-freedom Mathieu-Duffing system. Moreover, the non-trivial solutions are discussed at all resonant and non-resonant frequencies, and a more detailed bifurcation analysis using the multiple scales method is carried out. In addition, a special case of a 1:1 internal resonance is also considered since this case appears in systems with degenerate eigenfrequencies as pointed out before in microgyroscopes [10].
To this end the following two DoF nonlinear system is proposed with cubic stiffness and damping nonlinearities having the coefficients γ i and α i , i = 1, 2, respectively, and having natural frequencies ω 1 , ω 2 . Without the given parametric excitation, the two DoF would be rather uncoupled, the coupling is then achieved through the parametric excitation terms, which have the coefficients η 12 and η 21 , where the latter includes a phase shift ζ. In addition, the system includes conventional parametric excitation terms as well with coefficients η ii , i = 1, 2. Thus, there is no forced excitation, that is, the system is purely parametrically excited.

Stability of the trivial solution
As a first step to investigate the dynamics of the system (1), the stability of its trivial solution is discussed. Therefore the system is linearized around the trivial solution to give u ′′ 1 + ω 2 1 u 1 + µ 1 u ′ 1 + η 11 u 1 cos(Ω p t) + η 12 u 2 cos(Ω p t) = 0, u ′′ 2 + ω 2 2 u 2 + µ 2 u ′ 2 + η 21 u 1 cos(Ω p t + ζ) + η 22 u 2 cos(Ω p t) = 0. (2) Since this system is non-autonomous, the eigenvalues cannot be deduced. Therefore, Floquet theory is then used to determine the stability of the trivial solution. To this end, the system is put in a first-order form which could be written in the compact formż = A(t)z.
According to Floquet ansatz, each fundamental matrix Z(t) can be written as where each of Z, P, B is an n × n matrix, P (t) = P (t + T ), and B is constant. This could be verified by translating Z(t) in time with a periodic time T , then (5) gives which means that a translation by a time-period is only a linear transformation by the constant matrix C, called the monodromy matrix. According to Floquet the stability of the trivial solution can be obtained numerically by choosing the initial conditions where I 4×4 is the identity matrix of the size four. The system's fundamental matrix Z(t) is then calculated, which for the given initial conditions constitute the monodromy matrix when t = T = 2π Ω p , where according to (6) By evaluating the eigenvalues of this matrix, we get the Floquet characteristic multipliers µ. Then are the system's Floquet characteristic exponents, and their real parts are found to be Lyapunov characteristic exponents [23]. The same criterion of stability of a fixed point can be extended here for periodic solution of a non-autonomous system. If all Lyapunov exponents are negative, which means that all Floquet multipliers are inside the unit circle of the complex plane, the solution is said to be asymptotically stable. While if any Lyapunov exponent is negative, which means that any Floquet multiplier lies outside the unit circle, the system is said to be unstable. For nonlinear systems, if none of the Floquet multipliers associated with a non-hyperbolic solution lies outside the unit circle, which means that the corresponding Lyapunov exponent is positive, then a nonlinear analysis is necessary to determine the stability [24].
According to this criterion the Lyapunov characteristic exponents Re(λ) are evaluated in the parameter space of parametric excitation frequency and amplitude (Ω p −η). The maximum Re(λ) is calculated, see Fig. 1, and the stability criterion is applied. Plotting only the points in the parameter space where the trivial solution exhibits instability gives the so called stability chart of the system's trivial solution. The stability chart is evaluated in two cases: when all the parametric excitation terms are in phase (synchronous), i.e. ζ = 0, and when they are out of phase (asynchronous), i.e. ζ = −π/2. These two cases are studied and compared through this paper for linear and nonlinear analyses. This study is carried out at the particular phase-shift ζ = −π/2, since it was found to cause total instability [8] and more specifically, it could allow for broadband destabilization and amplification in microsystems [10]. The stability chart for the synchronous excitation case is depicted in Fig. 2, while Fig. 3 shows the asynchronous case, where in each case the blue points represent unstable trivial solutions. Under synchronous excitation the expected Arnold's tongues are found at primary and combination resonant frequencies Ω p = |ω i ± ω j | n , n ∈ N. However, by introducing the phaseshift ζ = −π/2 the instability tongues are merged and a large instability region is formed between both combination frequencies [|ω i − ω j |, ω i + ω j ] resulting in a broadband instability region. This explains the broadband destabilization effect [10]. In order to explain this effect, a horizontal cross-section in the stability chart is taken at a given excitation amplitude showing the maximum Lyapunov exponent max(Re(λ)) as a third axis. This is shown in Fig. 1, where the maximum Lyapunov exponent as a measure of instability is plotted against the excitation frequency Ω p in both cases. It can be observed in the figure that when ζ = 0, resonances occur at the primary resonance frequencies Ω p = 2ω i n and at the summation combination frequency Ω p = ω 1 + ω 2 . However, at the difference combination frequency Ω p = ω 2 − ω 1 an anti-resonance takes place. While in the case of asynchronous parametric excitation ζ = −π/2 resonances occur at all primary and combination frequencies. Moreover, in the latter case a broadband destabilization can be observed between the combination frequencies, where the maximum characteristic Lyapunov exponent is increased in this frequency interval. These findings intrigued the authors to study the dynamics of the system inside the regions of instability under asynchronous excitation and compare it to the conventional synchronous parametric excitation. However, if the trivial solution is destabilized, a linear system would suggest an unbounded response, which does not occur in reality. For these reasons, the system is modeled nonlinearly, and the non-trivial solutions and their stability are studied in the next sections.

Perturbation analysis of the nonlinear system
The method of multiple scales is used to analyze the given problem up to the first order [3,25]. To this end, the linear system of (1) is perturbed to give where all terms but the linear oscillator terms are considered to be small, which is indicated by the perturbation arbitrary parameter ϵ << 1.
One seeks an expansion in the form Inserting (11) and (12) in (1) and separating according to the order of ϵ results in the following: for ϵ 0 , we obtain while for ϵ 1 , the equations read Solving (13) gives where the amplitudes A 1 , A 2 represent the slow time-scale variables, which will exhibit the system's stability in the further calculations, while the exponential expressions represent the fast time-scale periodic solution. Inserting (15) in (14) gives where CC stands for the complex conjugates of the preceding terms in each equation. Since equations (16) have secular terms, which contain e ±iωiT0 , thus, all particular solutions of u 11 and u 12 are unstable, which contradicts being a finer correction in (11). Therefore, the secular terms must vanish. Through enforcing the elimination of secular terms, the slow time-scale dynamics can be then studied. However, these secular terms are found to be dependent on the frequency interval chosen for the solution, whether it is away from resonance frequencies or nearly tuned to them. For this reason, the following sections will represent the different cases according to the resonant conditions. First, we will study the cases of primary parametric resonance at the double of the one of the natural frequencies, that is when Ω p ≃ 2ω i , while the primary parametric resonance have typically the same dynamics. Then we will turn to the interesting case of having 1:1 internal resonance between both DoFs, while one of them being excited parametrically as well. The study of the non-trivial solutions at the summation and difference combination frequencies then follows. Finally and most importantly, the non-trivial stationary solutions are found at non-resonant frequencies which concludes our goal for discussing parametric destabilization and amplification in the broad frequency band as discussed before.

Primary parametric resonance
For this case the parametric excitation frequency is detuned around the primary resonance frequency, then where σ p = O(1) is the detuning parameter. Inserting (17) in (16) and equating secular terms to zero yields Since these equations contain complex variables A 1 , A 2 , a transformation in polar coordinates is suggested as follows, Substituting (19) in (18) and separating real and imaginary terms gives where θ 1 = σ p T 1 − 2ϕ 1 and θ 2 = ϕ 2 . Stationary solutions of the system (1) is obtained by calculating the fixed points of (20), that is, D 1 a i = 0 and D 1 θ i = 0, i = 1, 2. Thus, the fixed points are Obviously, the steady state amplitude of the second oscillator a 2 vanishes, while that of the first one a 1 is given by the resonance equation Thereby, either we have a trivial solution The frequency-response curve corresponding to the non-trivial solution (23) is presented in Fig. 4 for the parameter values ω 1 = 1, µ 1 = 0, γ 1 = 0.07, α 1 = 0.03 and η 11 = 0.2. A nonlinear resonance behavior is exhibited, and a stable limit cycle is born after a bifurcation at σ p = −0.1. Although the system incorporates Duffing-type nonlinearities, the non-trivial solution here differs substantially from that of a forced Duffing oscillator with regard of the type of excitation. At the point (σ p = 0.1) another bifurcation occurs, where a smaller unstable limit cycle appears in addition instead of the polar one used in the analysis. This is depicted as having a saddle trivial fixed point in the first interval and a stable focus. While in the second interval, the stable focus replaced by a saddle and another stable focus, whilst the trivial solution turns to be stable again. Thus in the second interval two stable solutions exist, at the trivial fixed point and a non-trivial one.
A further insight could be drawn from this analysis by setting the detuning parameter σ p = 0 and at the same time canceling the linear damping, i.e. µ 1 = 0. In this case the non-trivial solution (23) reduces to which will be called Γ for further analysis. As we see in this expression the amplitude of the oscillation is determined mainly by the parametric excitation amplitude η 11 and the nonlinear terms γ 1 , α 1 . Here it is obviously clear that in the case of the absence of any of them we will be left with only the trivial solution. Although it might seem to contradict the fact that the unforced Duffing oscillator has non-trivial solutions even without any parametric excitation, but the main difference here is that the linear damping was set to zero [26]. The Γ value is considered here to represent a non-trivial solution neither with a perturbing linear damping nor with the detuning of the excitation frequency. The effect of perturbing this solution through varying both of them will be studied next.
Substituting the value of Γ back in (22) while keeping the the detuning parameter σ p = 0 but allowing the linear damping µ 1 to vary gives which is clearly a quadratic equation in a 1 . Using combined parameters Ξ 1 , Ξ 2 for the terms between brackets and rewriting it gives which has a solution of For relatively high excitation amplitudes with respect to system parameters, we find the combined parameters Ξ 1 , Ξ 2 and Γ to have values smaller than one. In this case Ξ 2 will dominate the solution. And in order to have a real-valued a 1 , i.e. positive-valued a 2 1 , Ξ 2 must then be negative. Thus a real valued a 1 can only take place if positive square root solution solution was selected and if Ξ 2 < 0. Reading the Ξ 2 term from (28) and applying this condition which could be confirmed by numerically solving (28) and plotting the solution in Fig. 6(a). However, if the excitation amplitudes were relatively small, giving the combined parameters values larger than one, we find Ξ 1 to be dominating. In this case, a real-valued a 1 is only possible for µ 1 < 0, as shown in the same figure. The plotted solution in Fig. 6 corresponds to the solution of the main resonance equation at σ p = 0 in Fig. 4, where only a stable solution exists. However if another value of σ p was chosen which includes an unstable solution as well, this should give another dimension to the problem. Thereby, by returning back to (22) it can be written in which gives the admissible amplitude values shown in Fig.7 by varying the linear damping µ 1 again. The figure shows isolated stable and unstable stationary solutions. This is particularly interesting, since the variation of the linear damping could cause an abrupt increase or decrease in the amplitude of the response at a bifurcation point. This high sensitivity of the response at the bifurcation point could be of a significant importance for systems, where high sensitivity is pursued using a bifurcation control scheme.

Internal resonance under parametric excitation
The case of 1:1 internal resonance stands to be relevant for systems involving degenerate or similar eigenvalues ω 1 ≃ ω 2 . This happens to be the case for structures with axis-symmetric geometry. A motivating example is the micro-ring gyroscope [10], where it was shown that the degeneracy of eigenvalues or even the nearness to one another leads to a large broadband destabilization effect between the difference and summation combination frequencies. In the nonlinear case, however, an additional phenomenon takes place, which is the internal resonance, provided that one eigenfrequency is in the vicinity of the other. In this case we introduce an additional detuning parameter σ in where while the previous one (17) remains effective. Returning to (16) and this time inserting both (17) and (33) gives the solvability conditions In order to investigate the exchange of energy when only one DoF is parametrically excited, we put η 22 = 0, and the phase shift is firstly not taken into consideration, i.e. ζ = 0. As previously done, we insert (19) in (34) then separate real and imaginary parts to yield where θ 1 = ϕ 1 + ϕ 2 + (σ in − σ p )T 1 and θ 2 = −2ϕ 1 + σ p T 1 . The steady state solutions are then sought when D 1 a 1 = 0, D 1 a 2 = 0, with D 1 θ 1 = 0 and D 1 θ 2 = 0, or D 1 ϕ 1 = σ p /2 and D 1 which when substituted in (35) and solved for a 1 and a 2 give the two resonance equations 9(γ 2 2 + α 2 2 ω 6 2 )a 6 2 + 24ω 2 (−γ 2 σ p + 2γ 2 σ in + α 2 µ 2 ω 3 2 )a 4 2 + 16ω 2 2 (µ 2 2 + (σ p − 2σ in ) 2 )a 2 2 − 4η 2 21 a 2 1 = 0,  The resonance curves show the typical "M" shape due to the internal resonance, moreover, the hardening nonlinear stiffness causes all curves to bend to the right. In addition, multiple stationary points could be calculated by detuning the excitation frequency. This results in a complex phase space containing various fixed points at a given excitation frequency. However, when the external parametric excitation η 11 is turned off, the internal resonance's typical behavior vanishes (see Fig. 9), and the frequency-response curves show only stable and unstable limit cycles similar to the case of primary parametric resonance discussed before. Nevertheless, a distinction should be made here between this case and the case of parametric resonance by observing the solvability conditions in both cases, i.e. in (18) and (34). In the case of primary resonance we have only one excitation source, which is η 11 . The energy, however, could not be transferred to the second degree of freedom, due to the absence of internal resonance. In the case discussed here we do have three excitation sources, η 11 , η 12 and η 21 . If the first one is turned off, the other two remain effective, causing a transfer of energy at the primary resonance frequency. These coupling excitation terms will show up again to be influencing the system's behavior under combination resonance, as will be discussed afterwards. In other words, while having zero η 11 and η 22 , we have a cross parametric resonance through η 12 and η 21 . This leads to the response depicted in Fig. 9.

Summation parametric resonances
We apply the same analysis as before for the case of combination parametric resonances, where the parametric excitation frequency is in the neighborhood of the summation or the difference frequencies, that is Ω p ≃ |ω 1 ± ω 2 |. Beginning with the summation resonance we introduce the detuning parameter σ s , where then as before the resonance condition (38) is then inserted in (16), to give the solvability conditions Putting the amplitudes A 1 (T 1 ), A 2 (T 1 ) in polar form according to (19), substituting in (39) and separating real and imaginary terms gives where θ 1 = σ s T 1 − ϕ 1 − ϕ 2 and θ 2 = ϕ 2 . Stationary solutions are then obtained by calculating the fixed points of (40), that is when D 1 a i = 0 and D 1 θ i = 0, for i = 1, 2. However, in this case the effect of the asynchronicity of the parametric excitation, i.e. the presence of the phase shift ζ, heavily influences the non-trivial solutions. Therefore, we present the stationary solutions in two different cases, synchronous ζ = 0 and asynchronous ζ = −π/2. This latter particular phase-shift is chosen according to our analysis of the trivial solution discussed before. If ζ = 0, then the fixed points of equations (40) could be determined from while under asynchronous excitation, ζ = −π/2, the equations read From these equations it can be observed that only the η 21 excitation terms are changed. Furthermore, although the modulation equations (40) were in four variables, only three of them influence the vector fields excluding θ 2 , which lead to three algebraic equations in three variables to determine the fixed points.

Stability analysis
In order to determine the stability of the obtained non-trivial solutions, the solution is perturbed using a 1 (T 1 ) = a 10 + ∆a 1 (T 1 ), a 2 (T 1 ) = a 20 + ∆a 2 (T 1 ), θ 1 (T 1 ) = θ 10 + ∆θ 1 (T 1 ), compactly written z(T 1 ) = z 0 + ∆z(T 1 ), which is then inserted in the modulation equations (40) to give where J = ∂f (∆z,z0) ∂∆z | ∆z=0 is the Jacobian matrix evaluated at the fixed point, N LT represents the nonlinear terms. After eliminating the nonlinear terms, the linearized system presents an eigensystem with the eigenvectors being tangent to the system's nonlinear manifolds. The stability of the fixed point of the nonlinear system can be deduced from the eigenvalues of the linearized system, as long as the fixed point is hyperbolic according to Hartman-Grobman theorem [27]. In this view, if all the eigenvalues at the investigated fixed point have negative real parts, the fixed point and the corresponding solution is considered asymptotically stable. While the existence of a single positive eigenvalue implies the instability of the solution. However, if the largest eigenvalue is strictly zero, then the stability of the solution cannot be determined by a linear analysis [24].
In this synchronous excitation case the stationary solutions are deduced through solving the equations (41), then the stability of each fixed point is determined as previously discussed and the result is then plotted in Fig. 10, where the blue and red points represent stable and unstable limit cycles respectively. The parametric resonance curve is shown to be similar to primary resonances (see Fig. 4), where the amplitude of the second degree of freedom a 2 is lower than the first one a 1 .
In the other case, when ζ = −π/2, the fixed points of (40) are determined through solving (42). The resonance curves in this case is then depicted in Fig. 11. In this figure it can be observed that all the non-trivial stationary solutions, or limit cycles, are found to be stable for the give detuning interval. However, the resonance curves of both degrees of freedom have different profiles. Furthermore, detuning the excitation frequency have opposite effects on the amplitudes of the limit cycles of both degrees of freedom; by positive detuning a 2 exhibits higher amplitude than a 1 , while by negative detuning the opposite occurs.

Difference parametric resonance
The perturbation analysis is carried out for the difference parametric resonance case, where the parametric excitation frequency is in the neighborhood of the difference frequency or Ω p ≃ |ω 2 − ω 1 |. In this case the parametric excitation frequency is detuned through where σ d is the detuning parameter, and ω 2 is assumed to be larger than ω 1 without loss of generality. By inserting this condition again into (16) the solvability conditions become Before indulging into the procedure of finding the non-trivial solutions, we can observe the differences in the solvability conditions between summation and difference resonances. By comparing (47) and (39), we can obviously see that all terms are the same in both equations except for the coupling term in each equation, which causes the vast difference in the end results.
Following the same procedure as before, the modulation equations are found to be where in this case we put θ 1 = (ϕ 2 − ϕ 1 ) − T 1 σ d and θ 2 = ϕ 2 . The modulation equations are then separated into real and imaginary parts, and solved to obtain the stationary solutions when D 1 θ 1 = 0, D 1 θ 2 = 0, D 1 a 1 = 0 and D 1 a 2 = 0. The resulting equations are again given in two cases. Thus for the synchronous excitation case ζ = 0, they yield while if the parametric excitation is asynchronous, i.e. ζ = −π/2, this gives By solving the equations in the synchronous excitation case, an interesting result is observed: non-trivial solutions do not exist. This comes inline with our stability analysis of the trivial solution, where no resonance was found at the difference combination frequency, instead an anti-resonance could be detected. According to this observation, a correspondence between the stability of the trivial solution and the existence of the non-trivial ones can be proposed.
By solving the equations (50) the non-trivial stationary solutions are calculated, their stability is investigated and plotted in Fig. 12. The resonance curves are shown to be significantly similar and mirrored around the zero detuning parameter when compared to those calculated around the summation resonance. This result should be related to the difference in the solvability conditions (39) and (47) where the excitation terms in each case are found to be the complex conjugate of the corresponding ones in the other case.

Non-resonant limit cycles
As illustrated in section 2, the uniqueness of the phase-shifted parametric excitation lies in the destabilization of the trivial solution in a broad band of excitation frequencies. This means, when the trivial solution turns unstable, we should look for a non-trivial solution. In this case, as previously noted, the non-trivial solutions represent limit cycles due to the parametric excitation. In order to find these solutions, the system (10) is restudied without specifying a resonance condition. However, in order to capture the required phenomenon the approximation up to the second order is then required. In this case we rewrite (11) and (12) to be and Inserting (51) and (52) in the perturbed differential equations (10) gives an additional couple of equations for the second order ϵ 2 , which are (53b) While the excitation frequency Ω p is chosen to be away from all resonance frequencies, the solvability conditions of (16) in the non-resonant case then read which show no influence of the parametric excitation. However, as pointed out in section 2, the trivial solution shows instability in a broad band of frequencies, which suggests the presence of secular terms at non-resonant frequencies that depend on parametric excitation terms. Moreover, in a nonlinear system this could lead to the presence of non-trivial solutions or limit cycles. For this reason, the analysis is then extended to the second order approximation. Up to this point it was not required to obtain the first order correction terms u 11 , u 12 , since we were only interested in the amplitudes of the basic solution u 01 , u 02 . In this current case the first order correction terms are needed to solve (53), noting that the solutions should be functions of three time scales here T 0 , T 1 , T 2 , which also applies to (15) and (16). By eliminating the secular terms and solving (16), this yields A 2 e iω2T0 + CC. Furthermore, (15) can be rewritten using (19) to yield which shows the main frequency of the oscillations of each degree of freedom to be ω 1 and ω 2 modified by the slow varying ϕ 1 and ϕ 2 respectively. To summarize all the non-trivial solutions obtained: the analysis carried out at the non-resonant frequencies represents a global solution with respect to the excitation frequency, whereas the solutions obtained in the previous sections at resonant frequencies correspond to local solutions. In combining the global and local pictures of the system response under asynchronous excitation (ζ = −π/2), see figures 4, 11, 12 and 13, we obtain a full representation of the dynamics of the system (1) at all excitation frequencies.
Furthermore and most importantly, stable non-trivial solutions are proved to exist at non-resonant frequencies, where the trivial solution exhibit instability. This result affirms our hypothesis, which is the broadband parametric amplification acquired through this method of excitation as a result of a broadband destabilization of the trivial solution discussed in section 2.

Conclusion
Nonlinear time-periodic systems exhibit several types of instability which occur due to different reasons. Resonances and transfer of energy between coupled degrees of freedom contribute to this destabilization, moreover, the addition of a phase-shift between the coupling parametric excitation terms add other non-trivial stationary solutions. Through the variation of the linear damping coefficient at the primary parametric resonance, the non-trivial solutions seemed to be limited in domain, but more interestingly it could cause isolated stationary solutions as well when the excitation frequency is detuned in a region of multiple stationary points. In addition, the case of internal resonance shows an influence of each excitation term. Even when the intrinsic excitation terms (diagonal terms in the parametric excitation matrix) are turned off, the coupling excitation terms could cause a cross excitation in both degrees of freedom. At the summation combination frequency, the non-trivial solutions were obtained for both synchronous and asynchronous excitations. In the former case the resonance curves are shown to be similar to those occurring at the primary resonance frequency, however in the latter case only stable limit cycles could be found for the given parameter values. Furthermore, it could be shown that non-trivial solutions appear at the difference frequency under asynchronicity, which is not the case when the phase-shift vanishes. This is found to be in accordance with the stability analysis of the trivial solution, since at the difference frequency an anti-resonance occurs causing the stabilization of the trivial solution. Finally, limit cycles are found at non-resonant frequencies under asynchronous parametric excitation. This last result supports the aim of this work to find a broadband parametric amplification through the destabilization of the trivial solution.
In summary, the system's dynamics are investigated at all resonant and non-resonant frequencies, thus giving a global picture for the system's response. Through the analysis, the influence of the asynchronous parametric excitation on the stability of the trivial and non-trivial solutions is highlighted, which corresponds directly to parametric amplification. Therefore, these findings provide a better understanding of nonlinear dynamics of parametrically excited M-DoF systems which are particularly important for micro and nano-systems, such as micro and nanomechanical resonators. Through these conclusions, the proposed method of asynchronous parametric excitation is shown to be promising for the sake of increasing the amplification and thereby sensitivity of these systems in the sensors industry.