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Research Paper

Stability and Oscillation in coupled Rayleigh-Duffing Oscillators Model with Delays

Chunhua Feng

Department of Mathematics and Computer Science, College of Science, Technology, Engineering and Mathematics, Alabama State University, Montgomery, Alabama, 36104, USA,

*Corresponding author, Chunhua Feng, Email: cfeng@alasu.edu

Received April 18, 2017, revised June 20, 2017, accepted June 21, 2017

Publication Date (Web): June 21, 2017

Abstract

This paper considers a coupled Rayleigh-Duffing oscillators model with delays. Stability and oscillations for the system is investigated. Some sufficient conditions to guarantee the existence of oscillatory solutions are derived. Computer simulations are provided to verify our theoretical results.

Keywords

Coupled Rayleigh-Duffing oscillator model, Delay, Stability, Oscillation

Introduction

It is known that the study of coupled oscillators provides information on emergent properties of the coupled system, such as synchronization, clustering, phase trapping, phase locking, oscillation death, and so on. The dynamic behavior of coupled oscillators has been studied by many authors (Hirano and Rybicki 2003; Guin et al 2017; Giresse and Crepin 2017; Barron 2016; Cveticanin 2014; Das Maharatna 2013; Tang et al 2009; Kuznetsov and Roman 2009; Wang and Chen 2012; Kuznetsov et al 2009; Kuwata et al 2008). In 2003, Hirano and Rybicki considered the following coupled van der Pol equations:

(1)

By using

-degree theory, the authors have investigated the existence of limit cycles of system (1) under some restrictive conditions for parameters and (). The authors also pointed out that the method of

-degree theory should be applied to any n coupled van der Pol equations.

Recently, Guin et al have studied the dynamics of two bilaterally-coupled non-oscillatory Rayleigh-Duffing oscillators:

(2)

Taking direct and anti-diffusive coupling cases into consideration, the authors derived conditions for periodic bifurcation in parameter space analytically and verified them through numerical solution of system equation. Non-oscillatory, periodic, quasi-periodic and chaotic zones of system also were indicated (Guin et al 2017). It is also known that time delay is always ubiquitous in many practical systems such as manufacturing process, population dynamics, controlling systems, network communication system, nuclear reactors, rocket motors, load balancing and instability in parallel computation. Naturally various isolated or coupled van der Pol oscillators, Duffing oscillators, van der Pol-Duffing oscillators with delays have been studied by many researchers in the last decades (Wirkus and Rand 2002; Li et al 2006; Zhang and Gu 2010; Wen et al 2017; Rusinek et al 2014; Zhang et al 2011; Bramburger et al 2014; Jiang and Wei 2008; Maccari 2003; Ma et al 2008; Maccari 2008; Jiang and Yuan 2007). Wirkus and Rand have considered the dynamics of two weakly coupled van der Pol oscillators due to its relevance to coupled laser oscillators, in which the coupling terms have time delays

and as follows:

(3)

In 2006, Li et al studied a mathematical model of coupled van der Pol oscillators with two kinds of time delay as follows:

(4)

By means of averaging method together with truncation of Taylor expansions, the condition necessary for saddle-node and Hopf bifurcations for symmetric modes, namely in-phase and out of-phase modes were determined. Zhang and Gu have discussed the following coupled van der Pol oscillators with time delays:

(5)

The authors found that there exist the stability switches when the delay varies, and the Hopf bifurcation occurs when the delay passes through a sequence of critical values. The stability and direction of the Hopf bifurcation are provided by means of the normal form theory and the center manifold theorem (Zhang and Gu 2010).

Motivated by the above systems, in this paper we extend system (2) to the following coupled Rayleigh-Duffing oscillators with delays:

(6)

The initial condition is

, where By means of the mathematical analysis method, some sufficient conditions to guarantee the existence of limit cycles for the model (6) are obtained.

Computer simulations are provided todemonstrate the correct results.

Preliminaries

It is convenient to write (6) as an equivalent four dimensional first-order system

(7)

Lemma 1 Suppose that for the parameters ai and ri (i=1, 2), the following inequality holds:

(8)

Then system (6) has a unique equilibrium point, it is exactly the zero point.

Proof Noting that system (7) is an equivalent version of (6). Therefore, we only consider the uniqueness of the equilibrium point of system (7). An equilibrium point

of system (7) is a constant solution of the following algebraic equation:

(9)

from (9) we have

Since

(10)

(10) can be written as a matrix form

(11)

The determinant of matrix is . According to the linear algebraic knowledge, if the determinant of matrix A is not equal to zero, then A is a nonsingular matrix. System (10) has a unique trivial solution. In the case x1*=0, x3*=0, the determinant of matrix A reduces

to Therefore, condition (8) holds, there exists a unique trivial solution of system (7). In other words, system (7) has a unique equilibrium point.

Lemma 2 Suppose that for the parameters bi and di we have bi > 0, di > 0 (i=1, 2), then all solutions of system (7) are bounded.

Proof To show the boundedness of the solutions of system (7), we construct a Lyapunov function Calculating the derivative of V (t) through system (7) we have

(12)

(i=1, 2, 3, 4) tend to positive infinity , , and are higher order infinity than , , and

noting that as

as xi(t) > N (i=1, 2, 3, 4). This means that all solutions of system (7), and hence system (6) are bounded. In this paper for a vector

Since bi > 0, di > 0 (i=1, 2), hence there exists suitably large N > 0 such that

, and a four by four matrix B=(bij), the norms is defined by: the measure of the matrix B is defined by

, which for the chosen norms reduces to ] (Gopalsamy 1992).

Main Results

The linearized system of (7) is the following:

(13)

System (13) can be written as a matrix form:

(14)

where

both P and Q are four by four matrices: ,

.

Theorem 1 Suppose that

< 0, and the norm of matrix Q satisfies

(15)

Then the trivial solution of system (14) is stable, implying that the trivial solution of system (7) is stable.

Proof Construct a Lyapunov functional as the follows:

(16)

Noting that (j=1, 2, 3, 4), = as xi(t) > 0, and |xi(t)|=-xi(t) as xi(t) < 0. From < 0, implying that c1 <0, c2 < 0. Therefore, we have

(17)

+ (18)

(19)

+ (20)

Therefore, we get

+

(21)

Since

<0, and ||Q|| < - , we have . This means that the trivial solution of system (14) is stable. Noting that (i=1, 2, 3, 4) is a higher order infinitesimal as xi(t) tend to zero. Therefore, the stability of trivial solution of system (14) implies that the trivial solution of system (7) is also stable.

Theorem 2 Suppose that for selecting parameters, system (7) has a unique equilibrium point and all solutions are bounded. If the following characteristic equation of the matrix P:

(22)

has a positive eigenvalue

or there is a positive real part of a complex eigenvalue (Re ). Then the trivial solution of linearized system (14) is unstable, implying that there exists a limit cycle of system (7), namely, a periodic solution.

in system (14) as follows:

Proof To avoid unnecessary complexity, we consider the case that

(23)

and . Since equation (22) has a positive eigenvalue or there is a positive real part of a complex eigenvalue, without loss of generality, we assume that

Assume that the eigenvalues of the equation (22) are

or , where . Let Q1, Q2, Q3, and Q4 be four eigenvalues of the matrix Q. The characteristic equation corresponding to system (23) is the following:

(24)

where I is a four by four identity matrix. Since the eigenvalues are

and Qi (i=1, 2, 3, 4) of matrices P and Q, respectively, from (24) we have

(25)

Thus, we are led to an investigation of the nature of the roots of equations:

(26)

Specifically, we consider equation:

(27)

Equation (27) is a transcendental equation. Generally speaking we cannot find all solutions of this equation. However, we prove that there exists a positive characteristic root of equation (27) under the assumptive conditions. Indeed, let

then is a continuous function of . Obviously, since . Noting that.

(>0) such that According to the well known Intermediate Value Theorem, there exists a positive value of say such that

Therefore, there exists a suitably large

. In other words, equation (27) has a positive characteristic root. So the trivial solution of system (23) is unstable. According to the basic property of delayed equation: the increase of time delay, the instability of the trivial solution still maintained. Therefore, the trivial solution of system (14) is unstable for any

(i=1, 2, 3, 4). The instability of the trivial solution of system (14) implies that the trivial solution of system (7) is unstable. Since system (7) has a unique equilibrium point and all solutions are bounded, the instability of the trivial solution of system (7) will force this system to generate a limit cycle, namely, a periodic solution (Chafee 1971).

and in equation (27), we prove that the characteristic root of equation (27) has a positive real part. Let then separating the real and imaginary parts, we have

If

(28)

Obviously, there exists a

satisfies equation (28). Thus, the trivial solution of system (23) is unstable, implying that system (7) generates a limit cycle, namely, a periodic solution.

Theorem 3 Suppose that for selecting parameters, system (7) has a unique equilibrium point and all solutions are bounded. Assume that the characteristic values of the matrix P are four complex numbers, then the trivial solution of linearized system (7) is unstable, implying that system (7) has a limit cycle, namely, a periodic solution.

Proof We still consider system (23). The characteristic equation of system (23) is the following:

(29)

Since the characteristic values of the matrix P all are complex numbers, this means that every characteristic root of system (29) is complex number. Apparently, and Thus the trivial solution of system (23) is unstable, implying that the trivial solution of system (7) is unstable. Therefore, system (7) generates a limit cycle, namely, a periodic solution.

Simulation Results

This simulation is performed by means of system (7). The parameters are selected as the follows:

Table 1. Parameter values

Figure 1. Convergence of the solutions, a1 = - 0.38, a2 = - 0.35, c1 = - 1.66, c2 = - 1.75.

Figure 2. Convergence of the solutions, a1 = 0.05, a2 = 0.15, c1 = - 2.55, c2 = - 2.65

Figure 3. Oscillatory behavior of the solutions, delays: [3.5, 3.6, 3.8, 3.5].

Figure 4. Oscillatory behavior of the solutions, delays: [7.5, 7.6, 7.8, 7.5].

In Figure 1, the parameters a1 and a2, c1and c2 all are negative values. We see that ||Q|| = 0.20, and

= -0.35. The conditions of Theorem 1 are satisfied. Based on Theorem 1, the trivial solution is convergent. In Figure 2 we select both a1 = 0.05 and a2 = 0.15 are positive values. Obviously,

= 0.15 and the conditions of Theorem 1 are not satisfied. However, the trivial solution is still convergent. This means that Theorem 1 is only a sufficient condition. In Figure 3, we select both c1 and c2 are positive values. Based on the parameter values of Table 1, the characteristic roots of equation (22) are 2.2720, 2.2176, -0.0220, -0.0676, and ||Q|| = 1.3.

Figure 5. Time delays affect the convergence of the solutions.

Obviously, 2.2176 > 1.3 = ||Q|| . The conditions of Theorem 2 are satisfied. When time delays are selected as 3.5, 3.6, 3.8, and 3.5, system (7) has a periodic solution. In Figure 4, the parameter values of di, ki (i=1, 2) are different from in Figure 3, and we increase the time delays as 7.5, 7.6, 7.8, and 7.5, we see that the oscillatory solution is still maintained, only oscillatory frequency and amplitude are slightly changed. This means that the parameter values of ci and c2 affect the stability of the solutions. In order to see the effect of time delays, in Figure 5 we select the parameter values almost the same as in Figure 2, one can see that the solution is convergent as time delays are 2.7, 2.8, 3.3, and 3.5, respectively. However, when we increase the time delays as 3.5, 3.8, 3.6, and 3.5, respectively, an oscillatory solution is appeared.

Conclusion

By means of mathematical analysis method, this paper discussed a coupled Rayleigh-Duffing oscillators model with delays. Some theorems are given to guarantee the stability and oscillation of the solutions. Computer simulation indicated that our criteria are only sufficient conditions. Figure 5 indicated that time delay will induce oscillation. However, what are the critical values of time delays between convergence and oscillation of the solutions is still an open problem.

References

Barron MA (2016) Stability of a ring of coupled van der Pol oscillators with non-uniform distribution of the coupling parameter, J. Appl. Res. Tech., 14, 62-66

Bramburger J, Dionne B and LeBlanc VG (2014) Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback, Nonlinear Dynamics, 78, 2959-2973

Chafee N (1971) A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl., 35, 312-348

Chun R. Zhang, Bao. D. Zheng and Lian C.Wang (2011) Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay, Appl. Math. Comput., 217, 7155-7166

Cveticanin L (2014) Coupled Van der Pol oscillator with non-integer order connection, Commun Nonlinear Sci Numer Simulat, 19, 3362-3373

Das S and Maharatna K (2013) Fractional dynamical model for the generation of ECG like signals from filtered coupled Van-der Pol oscillators, Comput. Meth. Prog. Biomedi., 112, 490-507

Giresse TA and Crepin KT (2017) Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative, Chaos, Solitons & Fractals, 98, 88-100

Gopalsamy GK (1992) Stability and oscillations in delay different equations of population dynamics, pp. 210-258, Kluwer Academic, Dordrecht

Guin AK, Dandapathak M, Sarkar S and Sarkar BC (2017) Birth of oscillation in coupled non-oscillatory Rayleigh-Duffing oscillators, Commun Nonlinear Sci Numer Simulat, 42, 420-436

Hirano N and Rybicki S (2003) Existence of limit cycles for coupled van der Pol equations, J. Diff. Eqs., 195, 194-209

Kuwata N, Hoshi Y and Nohara BT (2008) Analysis of coupled van der Pol oscillators and implementation to a myriapod robot, IFAC Proceedings, 41, 767-772

Kuznetsov AP and Roman JP (2009) Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol-Duffing oscillators. Broadband synchronization, Phys. D, 238, 1499-1506

Kuznetsov AP, Stankevich NV and Turukina LV (2009) Coupled van der Pol-Duffing oscillators: Phase dynamics and structure of synchronization tongues, Phys. D, 238, 120-1215

Jia S. Tang, Feng Han, Han Xiao and Xiao Wu (2009) Amplitude control of a limit cycle in a coupled van der Pol system, Nonlinear Analysis: TMA, 71, 2491-2496

Jian M. Zhang and Xin S. Gu (2010) Stability and bifurcation analysis in the delay-coupled van der Pol oscillators, Appl. Math. Model., 34, 2291-2299

Jiang WH and Wei JJ (2008) Bifurcation analysis in van der Pol oscillators with delaed feedback, J. Comput. Appl. Math., 213, 604-615

Jiang WH and Yuan Y (2007) Bogdanov-takens singularity in van der Pol’s oscillator with delayed feedback, Phys. D, 227, 149-161

Li XY, Ji JC and Hansen CH (2006) Dynamics of two delay coupled van der Pol oscillators, Mech. Res. Commun., 33, 614-6

Maccari A (2003) Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback, Int. J. Non-Linear Mechanics, 38, 123-131

Maccari A (2008) Vibration amplitude control for a van der Pol-Duffing oscillator with time delay, J. Sound Vib., 317, 20-29

Rusinek R, Weremczuk A, Kecik K and Warminski J (2014) Dynamics of a time delayed Duffing oscillator, Int. J. Non-Linear Mech., 65, 98-106

Su Q. Ma, Qi S. Lu and Zhao S. Feng (2008) Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control, J. Math. Anal. Appl., 338, 993-1007

Wang X and Chen FQ (2012) Global dynamics of two coupled parametrically excited van der Pol oscillators, Math. Comput. Simul., 82, 1551-1571

Wen SF, Shen YJ, Yang SP and Wang J (2017) Dynamical response of Mathieu-Duffing oscillator with fractional-order delayed feedback, Chaos, Solitans & Flactals, 94, 54-62

Wirkus S and Rand R (2002) The dynamics of two coupled van der Pol oscillators with delay coupling, Nonlinear Dynamics, 30, 205-221

Citation:

Chunhua Feng (2017) Stability and oscillation in coupled Rayleigh-Duffing oscillators model with delays, Frontiers in Science, Technology, Engineering and Mathematics, Volume 1, Issue 1, 27-37