Partial periodic oscillation : an interesting phenomenon for a system of three coupled unbalanced damped Duffing oscillators with delays

In this paper, a system of two coupled damped Duffing resonators driven by a van der Pol oscillator with delays is studied. Some sufficient conditions to ensure the periodic and partial periodic oscillations for the system are established. Computer simulation is given to demonstrate our result.


Introduction
The dynamics of isolated and coupled Duffing oscillators or Duffing-van der Pol oscillators with or without time delays is an important topic of research in different fields of science and engineering.For example, the magneto-elastic mechanical systems [5], large amplitude oscillation of centrifugal governor systems [30], nonlinear vibration of beams and plates [14], electroencephalogram signals model [3], micro-electro-mechanical systems resonators [11], fluid flow and gas flow induced vibration [23], a weak signal detection method [35], are modeled by the nonlinear Duffing equations or Duffing-van der Pol equations.Many researchers have studied various Duffing systems [4,9,12,16,17,22,29,37].Recently, the study of nonlinear dynamics of micro-electro-mechanical systems (MEMS) and nano-electro-mechanical systems (NEMS) has grown rapidly over the last decades.It is known that the fundamental study of coupled nonlinear oscillators is very important in understanding the emergent behavior of complex dynamical systems in MEMS or NEMS.Analysis of simple cases as the building blocks in MEMS or NEMS can gain insight into larger complicated systems.In 2009, Karabalin et al. have discussed a system of two coupled nonlinear nano-electro-mechanical resonators using a structure of doubly clamped beams with a shared mechanical ledge.The authors modeled the behavior of the two strongly interacting nonlinear resonators by a coupled equations of motion for the beam as follows [8]: x 1 (t) + γ 1 x 1 (t) + ω 2  1 x 1 (t) + α 1 x 3 1 (t) + D(x 1 (t) − x 2 (t)) = g D1 (t), x 2 (t) + γ 2 x 2 (t) + ω 2  2 x 2 (t) + α 2 x 3 2 (t) + D(x 2 (t) − x 1 (t)) = g D2 (t). (1.1) By using the standard methods of secular perturbation theory, the complex nonlinear behavior of the system has been demonstrated.Nonlinear response of one oscillation can be modified by driven the second oscillation.Complicated frequency-sweep response curves are found when both oscillations are driven into their strongly nonlinear range.The nonlinear behavior of coupled equations can be understood, controlled, and exploited.In order to understand the emergent behavior of complex dynamical systems and develop novel NEMS devices, Wei et al. [19,26] have investigated the dynamics of a periodically driven Duffing resonator coupled to a van der Pol oscillator using the standard two time scales approach.The motion of the coupled dynamical system is described by the following equations: Due to the difference of order of magnitude about the coupling stiffness and other parameters, however, it is not easy to investigate the effects of the coupling stiffness on the steady state response using the multiple time scales analysis method.Therefore, Leung et al. have discussed the following damped Duffing resonator driven by a van der Pol oscillator [13]: By solving nonlinear algebraic equations, highly accurate bifurcation frequencies for various parameters are provided.The effects of the nonlinear damping, coupling stiffness on the angular frequency and amplitude of steady state response are studied.The obtained results were in good agreement with respect to the numerical integration solutions.Rand and Wong have considered a system of four coupled phase-only oscillators.The qualitative dynamics is depended upon a parameter representing coupling strength.This work has been used to MEMS artificial intelligence decision-making devices [18].It is known that time delay is ubiquitous in many physical systems, for example due to finite switching speeds of amplifiers in electronic circuits, finite signal propagation times in networks and circuits, and so on.Recently, many researchers have studied the dynamical behavior of various isolated and coupled time delay systems [6,20,27,28,33,34,36].Zhang et al. have investigated three coupled van der Pol oscillators with delay as follows [34]: (1.4) By using a symmetric Hopf bifurcation theory, the Hopf bifurcations at zero point appear as the delay increases and the existence of multiple periodic solutions are also established.
Luo and Huang have studied a discontinuous dynamics of a non-linear, self-excited, frictioninduced, periodically forced oscillator [15].Tchakui and Woafo have discussed the dynamics of three unidirectionally coupled autonomous Duffing oscillators and application to inchworm piezoelectric motors [24].Verichev et al. have investigated the dynamics of a "flexiblerotor/limited-power-excitation-source" system [25].The vibration of the mass unbalance of the rotating component in a power plant has been studied by Kim et al. [10].For dynamic analysis of a system of Van der Pol-Duffing oscillators with delay coupled, Zang et al. have investigated the existence of Hopf bifurcation and the bifurcation periodic solution [31].Rusinek et al. have discussed the dynamics of a time delayed Duffing oscillator [21].Motivated by the above research work, in this paper we shall extend Leung's system to the following model: (1.5) where x i = x i (t) represents coordinate, ε i , Ω i (i = 1, 2, 3), k j (j = 1, 2) are the damping coefficient, linear frequency and nonlinear stiffness of the Duffing resonator respectively.p i , q i (i = 1, 2, 3) are the coupling linear stiffness between the three resonators.It is well known that the Duffing oscillator is a nonlinear second order differential equation.The equation describes the motion of a damped oscillator with a complex potential than in simple harmonic motion.The Duffing oscillator is an example of a dynamical system that exhibits chaotic behavior.In system (1.5), the first two Duffing oscillators are coupled and driven by a van der Pol oscillator, in which the system appeared a partial oscillation under some restrictive conditions.It is an interesting phenomenon.By means of mathematical analysis method, some sufficient conditions to ensure the periodic and partial periodic oscillations of system (1.5) were obtained.Numerical simulation is provided to support our result.It should be emphasized that if the constants ε i , Ω i , p i , q i , τi (i = 1, 2, 3), k j (j = 1, 2) are different values, then the method of Hopf bifurcation is very hard to deal with system (1.5).This is due to the complexity of finding the bifurcating parameter.

Preliminaries
Let ) where u i = u i (t) (i = 1, 2, . . ., 6).The matrix form of system (2.1) is as follows: where , where l i = −p i − q i (i = 1, 2, 3).Obviously, the linearized system of (2.2) is the following: If the unique equilibrium point of system (2.1) is unstable, then system (2.1) generates a limit cycle.In other words, there exists a periodic oscillatory solution of system (2.1).
Lemma 2.4.For selected parameter values Ω i , p i , q i (i = 1, 2, 3), if M is a nonsingular matrix, then system (2.1) has a unique equilibrium point.where Proof.
) is a constant solution of the following algebraic equation (2.5) We first consider the homogeneous system associated with system (2.5) as follows: Since M is a non-singular matrix, the determinant of the coefficient matrix of system (2.6) does not equal to zero.According to the algebraic basic theorem, system (2.6) implies that In other words, system (2.6) has a unique zero point.We see that the third equation of system (2.5) is the same as the third equation of system (2.6).Note that g(u 3 both are monotone functions, and only g(0) = h(0) = 0.This implies that u * = (0, 0, 0, 0, 0, 0) T is the unique equilibrium point of system (2.1).The proof is completed.
For a matrix D = (d ij ) 6×6 , we adopt the matrix norm D = max 1≤j≤6 ∑ 6 i=1 |d ij |, and the matrix measure , then all solutions of system (2.1) are bounded.
Proof.Note that time delay affects the stability of the solutions, it does not affect the boundedness of the solutions.To avoid unnecessary complexity, consider a special case of system (2.1) as τ 1 = τ 3 = τ 5 = τ * , in the following: The matrix form of system (2.8) is the following: 1+τ * > 0, and ) be any solution of (2.7).Consider a Lyapunov functional t > τ * .(2.12) Calculating the upper right derivative D + V of V along the solution of (2.9), we derive that From the definition of V and D + V < 0, this implies the boundedness of the solutions of system (2.9) [7].

Periodic and partial periodic oscillations
Note that k 1 , k 2 and ε 3 are constants, u 3 1 , u 3 3 and u 2 5 are high order infinitesimal as u 1 , u 3 and u 5 tend toward to zero respectively.So, the unique equilibrium point which is exactly the zero point of system (2.1) and system (2.3), have the same stability or instability.The oscillatory behavior of the solution of system (2.3) implied that the solution of system (2.1) is also oscillatory.Assume that ε i > 0 (i = 1, 2, 3) and all solutions of system (2.1) are bounded.We first point out that the component u 6 of the trivial solution of system (2.3) is unstable.Consider the subsystem constructing by the fifth and sixth equations of system (2.3) as follows (u 1 = u 3 = 0): Obviously, u 6 (t) = 0 (t 1 < t < t 2 ).This means that the component u 6 (t) of the trivial solution of system (2.3) is unstable.It is easy to see that u 5 (t) is also unstable since u 5 (t) = u 6 (t).Therefore, for unbalanced damped Duffing oscillators model (2.3), if the components u 1 , u 2 , u 3 , and u 4 of the trivial solution are globally asymptotically stable, then the system generates a partial periodic oscillation.In order to discuss the asymptotic stability of components u 1 , u 2 , u 3 , and u 4 , we investigate the subsystem constructed by the first four equations of system (2.3) (u 5 (t) = 0): For convenience, we make the change of variables as y [32].We can then rewrite system (3.5) as the following equivalent system where τ = τ 1 +τ 3 2 .The matrix form of (3.6) is as follows: where Theorem 3.1.Suppose that there exists a unique equilibrium point and all solutions of system (2.1) are bounded.Let ε i > 0 (i = 1, 2), γ 1 , γ 2 , γ 3 , γ 4 be the eigenvalues of matrix A 1 + B 1 , which has a negative real part, namely, Re γ i < 0 (i = 1, 2, 3, 4).We set γ = min{| Re Then system (2.1) has a partial periodic oscillation.
Proof.Since ε i > 0 (i = 1, 2), and all solutions of system (2.1) are bounded, according to the above analysis, the component u 6 and u 5 are unstable.Therefore, we only need to show that the components u i or y i (i = 1, . . ., 4) are stable.Then system (2.1) generates a partial periodic oscillation.Consider system (3.7) for t ≥ τ we have leading to and hence ) where Y τ = sup t∈[− τ, τ] Y(t) .From (3.8), there exists a positive constant α (α < γ) such that for arbitrary t > τ the following inequality holds: From (3.11) and (3.12), we get Inequality (3.13) implies the global asymptotic stability of the equilibrium point of system (3.8).This suggests that the equilibrium point of system (3.6) is globally asymptotically stable.So system (2.1) has a partial periodic oscillation.
In the following let A T and A −1 be the transpose and the inverse of a square matrix A respectively.A > 0 (< 0) will be denoted a positive (negative) definite matrix A. Theorem 3.2.Suppose that there is a unique equilibrium point and all solutions of system (2.1) are bounded.In addition, there is a positive definite four by four matrix P and a positive diagonal four by four matrix Q such that A T then system (2.1) has a partial periodic oscillation.
Proof.Similar to Theorem 3.1, we only need to prove that the components y i (i = 1, . . ., 4) of the trivial solution of system (3.7) are stable.For system (3.7),we will employ the following positive definite Lyapunov functional The upper right derivative of W(t) along the trajectories of the system (3.7) is obtained as follows: This means that the trivial solution of system (3.7) is asymptotically stable, and implies that system (2.1) for arbitrary τ 1 and τ 3 has a partial periodic oscillation.
Theorem 3.3.Suppose that system (2.1) has a unique equilibrium point and all solutions of system (2.1) are bounded.If A 1 + B 1 > 0, then system (2.1) generates a periodic oscillation.
Proof.According to Lemma 2.3 we only need to show that the equilibrium point of subsystem (3.7) is unstable since the components u 5 and u 6 of the equilibrium point of system (2.3) are unstable.The characteristic equation associated with system (3.7) is given by: (3.17) Note that (3.17) is a transcendental equation and λ may be a complex number.We prove that there exists a positive eigenvalue of (3.17) under the condition When λ is sufficiently large, say λ = λ * > 0, e −λ * τ is sufficiently small, and f (λ * ) = λ * − A 1 − B 1 e −λ * τ > 0, then there exists λ = λ, λ ∈ (0, λ * ) such that f ( λ) = 0.This means that there is a positive eigenvalue of the characteristic equation (3.17) for any time delay τ, implying that the equilibrium point of system (2.1) is unstable for arbitrary time delays τ 1 and τ 3 , and system (2.1) generates a periodic oscillation.
Since α j1 > 0, one can select a suitable delay τ * such that β j1 cos(ωτ * ) > −α j1 .Therefore, f (0) = −α j1 − β j1 cos(ωτ * ) < 0. Noting that e −στ * → 0 as στ * → +∞, obviously, there exists a suitably large σ(> 0) such that f ( σ) = σ − α j1 − β j1 e − στ * cos(ωτ * ) > 0. By the continuity of f (σ), there exists a positive σ * ∈ (0, σ) such that f (σ * ) = 0. Thus, there is an eigenvalue of the characteristic equation associated with system (3.20) which has a positive real part.This means that the trivial solution of system (3.20) is unstable, implying that the trivial solution of system (2.3) for τ 1 = τ 3 = τ 5 = τ * is unstable.For a time delay system, if the trivial solution is unstable, then the instability of the trivial solution will be maintained as time delay increases.So for any delays the trivial solution of system (2.3) is also unstable.This implies that system (2.1) generates a periodic oscillation.We select a suitable delay such that the system has an oscillatory solution.This oscillation is said to induce by time delay.The proof is completed.

Conclusion
This paper discussed a system of two coupled damped Duffing oscillators driven by a van der Pol oscillator with delays.Some sufficient conditions to ensure the periodic and partial periodic oscillations for the system are established.Interestingly, this partial periodic oscillation is induced by unbalanced damped oscillators.When periodic and partial periodic oscillations occur, delays only affect the oscillation frequency.The study of micro-electro-mechanical phenomena and nano-electro-mechanical phenomena often requires experimental methods that can accurately control and manipulate the interaction between micro-and nano-objects.Our results are helpful for developing of novel MEMS or NEMS devices, which can precisely control of these nanoscale interactions, provide an ideal platform for interacting with the microand nano-world.