Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays

In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.


Introduction
It is known that the van der Pol (VDP) oscillator could model the typical self-excited or self-sustained oscillation. Various coupled van der Pol or van der Pol-Duffing equations have been applied in physics and engineering. Many good results have appeared [1][2][3][4][5][6][7][8][9][10][11][12]. For the following three-dimensional autonomous van der Pol-Duffing type oscillator system: , . ( By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established [1]. Barron has considered the stability of a ring of coupled van der Pol oscillators with non-uniform distribution of the coupling parameter as follows: where 1 , are the coupling parameter corresponding to the ith oscillator. For a modified hybrid van der Pol-Duffing-Rayleigh oscillator for modeling the lateral walking force on a rigid floor: Kumar et al. have studied the stability of the equation (3) by the perturbation and energy balance method [3]. Rompala et al. have considered a system of three van der Pol oscillators [5]. For a ring of four mutually coupled biological systems described by coupled van der Pol oscillators, the stability boundaries and the main dynamical states have been considered on the stability maps by Kadji et al. [6]. A driven van der Pol-like oscillator with a nonlinear friction of higher polynomial order model as follows: The effects of noise correlation on the coherence of a forced van der Pol type birhythmic system has been investigated by Yamapi et al. [7]. It is well known that the time delay is inevitable in many physical and biological phenomena such as manufacturing process, nuclear reactors, rocket motors, mechanical controlling systems, population dynamics, and so on. Naturally the time delay coupled van der Pol equations also have been extensively studied by many researchers [13][14][15][16][17][18][19][20][21][22][23][24]. For example, Li et al. have studied the coupled van der Pol oscillators with two kinds of delays [13]: Zhang and Gu used the theory of normal form and central manifold theorem to discuss the following time delay system [14]: Motivated by the above models, in this paper we consider the following a ring of time delays Duffing-van der Pol-like oscillator with a nonlinear friction of higher polynomial order system: where 0 < : ; , < and 0 < ̃ ≪ 1; %̃ , < , = , < ∈ R for each i=1, 2,..., n, 0 ≤ 2̃ are time delays. Our aim is to investigate the dynamical behavior of n coupled oscillators by means of the generalized Chafee's criterion [25,26].
Obviously, the origin T = 0 (k = 1, 2, ⋯, 2n) is an equilibrium of system (8). The linearization of system (9) at origin is Lemma 1 Assume that the matrix C is a nonsingular matrix, then system (8) (or (9)) has a unique equilibrium.
Proof An equilibrium point * = [ * , * , ⋯ , @ * ] J of system (8) is a constant solution of the following algebraic equation Since * = 0 1 ≤ ≤ , from (11) we get System (12) can be written as a matrix form as the following: where F * = [ * , * , ⋯ , @ * ] J , and According to standard results in linear algebraic, if D is a nonsingular matrix, system (13) has only one solution, namely, the trivial solution. When * = 0 ( 1 ≤ ≤ , matrix D changes to C. The proof is completed. Lemma 2 All solutions of system (8) are uniformly bounded.
Proof Since Re " = 1, 2, ⋯ , 2 < −f < 0 , hence there exists a positive constant g ≥ 1 such that ij k l m i ≤ gj nm . In (15) for ≥ 2 * we have This means that F → 0 as t→ ∞ in system (15). Since 2 ≤ 2 * = 1, 2, ⋯ , , and K F is higher infinitesimal as F → 0, based on the property of delayed differential equation, we know that the trivial solution of system (8) (or (9)) is stable. The proof is completed.
Theorem 2 Assume that system (8) has a unique equilibrium point, for selected parameter values of , , % , and 1 ≤ ≤ . Let the eigenvalues of matrix R= G + I be " 1 ≤ ≤ 2 , the eigenvalues of matrix A be • 1 ≤ ≤ 2 . If there exists at least one eigenvalue • T , ∈^1, 2, ⋯ , 2 } such that Re • T ) > 0, then the unique equilibrium point of system (8) is unstable, implying that system (8) generates an oscillatory solution.
Proof Obviously, if the trivial solution of system (10) is unstable, then the trivial solution of system (9) is also unstable. Therefore, we only need to consider the stability of the trivial solution of system (10). The characteristic equation corresponding to system (10) is det λ ‚ L − L − L j ƒq " = 0 (20) Noting that each characteristic value of matrix B is zero. So we have det λ ‚ L − L − L j ƒq " = ∏ λ − @ Z • = 0 (21) By the assumption, there exits at least one k such that Re (λ † ) = Re • T ) > 0, this means that the trivial solution of system (10) is unstable, implying that the trivial solution of system (8) (or (9)) is unstable. Since all solutions of system (8) (or (9)) are bounded, and system (8) has a unique unstable equilibrium. Based on the generalized Chafee's criterion, this instability of the unique equilibrium will force system (8) then the unique equilibrium point of system (10) is unstable, implying that system (8) generates an oscillatory solution.
Example 1 Consider the case of n =3 in the following:  (26) generates an oscillatory solutions (see Figure 4).
Example 2 Consider the case of n = 4 in the following:

Discussion
When system generates an oscillatory solution, from the Figures we know that time delay affects the oscillatory amplitude and frequency not too much. However, the positive parameter values % , % , % E , % š affect the stability and oscillation of the system. The oscillatory frequency changes too much when different values of % , % , % E , and % š are selected.

Conclusion
In this paper, we have discussed the oscillatory behavior of the solutions on a class of coupled van der Pol-Duffing equations with delays. Based on the generalized Chafee's theory, a simple criterion to guarantee the existence of permanent oscillations, which is easy to check, as compared to predicting the regions of bifurcation have been proposed. In this network, the passive decay rate affects the oscillatory frequency and amplitude. When these time delay systems generate a permanent oscillation, the delays affect oscillatory frequency and amplitude slightly.