Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 1 (2018), pp. 139 152 Research India Publications http://www.ripublication.com/gjpam.htm Existence of permanent oscillations for a ring of coupled van der Pol oscillators with time delays Chunhua Feng Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL, 36104, USA. Abstract In this paper, a ring of coupled van der Pol oscillators with time delays is investigated. By means of the generalized Chafee s criterion, some sufficient conditions to guarantee the existence of permanent oscillations for the model are obtained. Computer simulations are provided to demonstrate the proposed results. AMS subject classification: 34K11. Keywords: a ring of coupled van der Pol equation, delay, oscillation. 1. Introduction The dynamical behavior of coupled oscillators with or without time delays has been studied by many researchers [1 22]. For example, Barron has investigated the coupled van der Pol oscillators without time delay as follows [5]: x i + a(x 2 i 1)x i = b i(x i 1 2x i + x i+1 ). (1) where 1 i n, b i is the coupling parameter corresponding to the ith oscillator. The stability and synchronization of a large ring of n coupled van der Pol oscillators have been reported. Hirano and Rybicki have considered the following coupled van der Pol equations [6]: u 1 (t) + ε 1(u 2 1 (t) a 1)u 1 (t) + c 11u 1 (t) + c 12 u 2 (t) + c 13 u 3 (t) + +c 1n u n (t) = 0, u 2 (t) + ε 2(u 2 2 (t) a 2)u 2 (t) + c 21u 1 (t) + c 22 u 2 (t) + c 23 u 3 (t) + +c 2n u n (t) = 0, (2) u n (t) + ε n(u 2 n (t) a n)u n (t) + c n1u 1 (t) + c n2 u 2 (t) + c n3 u 3 (t) + +c nn u n (t) = 0.
140 Chunhua Feng where 0 <a i, 0 <ε i 1,c ij R,1 i, j n. By using S 1 degree theory, the authors have investigated the existence of limit cycles for system (2). Recently, Wang and Chen have discussed a ring of coupled van der Pol oscillators with time delay coupling as the following: u 1 (t) (α βu2 1 (t))u 1 (t) + au 1(t) = γu 2 (t τ), u 2 (t) (α βu2 2 (t))u 2 (t) + au 2(t) = γu 3 (t τ), u n 1 (t) (α βu2 n 1 (t))u n 1 (t) + au n 1(t) = γu n (t τ) u n (t) (α βu2 n (t))u n (t) + au n(t) = γu 1 (t τ) (3) where γ is the coupling strength, and τ is the time delay. Chosen γ and τ as the bifurcation parameters, the dynamical behavior arising from the bifurcation is classified [7]. For a two coupled van der Pol oscillators in which the coupling terms have different time delays as follows: { x 1 (t) + ε(x2 1 (t) 1)x 1 (t) + (1 + α)x 1(t) = αy 1 (t τ 2), y 1 (t) + ε(y2 1 (t) 1)y 1 (t) + (1 + α)y 1(t) = αx 1 (t τ 1), (4) Zhang and Gu have studied the existence of the Hopf bifurcation for system (4). The stability and direction of the Hopf bifurcation are also determined by using the normal form theory and the center manifold theorem [19]. Motivated by the above models, in this paper we consider the following general ring of coupled van der Pol system with delays: u u 1 (t) (α 1 β 1 u 2 1 (t))u 1 (t) + a 1u 1 (t) = γ 2 u 2 (t r 2), 2 (t) (α 2 β 2 u 2 2 (t))u 2 (t) + a 2u 2 (t) = γ 3 u 3 (t r 3), u n 1 (t) (α n 1 β n 1 u 2 n 1 (t))u n 1 (t) + a n 1u n 1 (t) = γ n u n (t r n) u n (t) (α n β n u 2 n (t))u n (t) + a nu n (t) = γ 1 u 1 (t r 1) (5) Our goal is to investigate the dynamic behavior of a ring of coupled van der Pol oscillators. Based on the generalized Chafee s criterion: if a time delay system has a unique unstable equilibrium, all solutions of the system are bounded, then this system will generate a nonconstant periodic solution [Theorem 7.4, 24] and [Appendix, 25]. Some existing results in the literature have been extended. 2. Preliminaries For convenience, setting u i (t) = x 2i 1 (t), u i (t) = x 2i(t), r i = τ 2i,α i = ε 2i,β i = b 2i 1,a i = c 2i 1,γ i = k 2i (1 i n), then the coupled system (5) can be rewritten as
Permanent oscillations for a ring of coupled van der Pol oscillators... 141 the following equivalent system: x 1 (t) = x 2(t), x 2 (t) = c 1x 1 (t) + ε 2 x 2 (t) + k 4 x 4 (t τ 4 ) b 1 x 2 1 (t)x 2(t), x 3 (t) = x 4(t), x 4 (t) = c 3x 3 (t) + ε 4 x 4 (t) + k 6 x 6 (t τ 6 ) b 3 x 2 3 (t)x 4(t), x 2n 3 (t) = x 2n 2(t), x 2n 2 (t) = c 2n 3x 2n 3 (t) + ε 2n 2 x 2n 2 (t) + k 2n x 2n (t τ 2n ) b 2n 3 x 2 2n 3 (t)x 2n 2(t), x 2n 1 (t) = x 2n(t), x 2n (t) = c 2n 1x 2n 1 (t) + ε 2n x 2n (t) + k 2 x 2 (t τ 2 ) b 2n 1 x 2 2n 1 (t)x 2n(t). (6) Obviously, the origin x i = 0(i = 1, 2,...,2n) is an equilibrium of system (6). The linearization of the coupled system (6) at origin is x 1 (t) = x 2(t), x 2 (t) = c 1x 1 (t) + ε 2 x 2 (t) + k 4 x 4 (t τ 4 ), x 3 (t) = x 4(t), x 4 (t) = c 3x 3 (t) + ε 4 x 4 (t) + k 6 x 6 (t τ 6 ), (7) x 2n 3 (t) = x 2n 2(t), x 2n 2 (t) = c 2n 3x 2n 3 (t) + ε 2n 2 x 2n 2 (t) + k 2n x 2n (t τ 2n ), x 2n 1 (t) = x 2n(t), x 2n (t) = c 2n 1x 2n 1 (t) + ε 2n x 2n (t) + k 2 x 2 (t τ 2 ). The system (6) and (7) can be expressed in the following matrix forms (8) and (9) respectively: X (t) = AX(t) + BX(t τ)+ F(X(t)) (8) where X (t) = AX(t) + BX(t τ) (9) X(t) =[x 1 (t), x 2 (t),...,x 2n (t)] T, X(t τ) =[0,x 2 (t τ 2 ), 0,...,0,x 2n (t τ 2n )] T, F(X(t)) =[0, b 1 x1 2 (t)x 2(t), 0, b 3 x3 2 (t)x 4(t), 0,...,0, b 2n 1 x2n 1 2 (t)x 2n(t)] T. Both A = (a ij ) 2n 2n and B = (b ij ) 2n 2n are 2n 2n matrices as follows: 0 1 0 0 0 0 0 0 0 c 1 ε 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 A = (a ij ) 2n 2n =, 0 0 0 0 0 0 ε 2n 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 c 2n 1 ε 2n
142 Chunhua Feng B = (b ij ) 2n 2n = 0 0 0 0 0 0 0 0 0 0 0 0 k 4 0 0 0 0 0 0 0 0 0 0 0 0 0 k 2n 0 k 2 0 0 0 0 0 0 0. Lemma 2.1. Assume that c 2i 1 = 0(i = 1, 2,...,n), then system (6) has a unique equilibrium point. It is exactly the zero point. Proof. An equilibrium point x =[x 1,x 2,,x 2n ]T of system (6) is a constant solution of the following algebraic equation x 2 = 0, c 1 x 1 + ε 2x 2 + k 4x 4 b 1(x 1 )2 x 2 = 0, x 4 = 0, c 3 x 3 + ε 4x 4 + k 6x 6 b 3(x 3 )2 x 4 = 0, x 2n 2 = 0, c 2n 3 x 2n 3 + ε 2n 2x 2n 2 + k 2nx 2n b 2n 3(x 2n 3 )2 x 2n 2 = 0, x 2n = 0, c 2n 1 x 2n 1 + ε 2nx 2n + k 2x 2 b 2n 1(x 2n 1 )2 x 2n = 0. (10) From system (10), x2i = 0(i = 1, 2,...,n).Since c 2i 1 = 0(i = 1, 2,...,n),therefore we have x2i 1 = 0(i = 1, 2,...,n). This means that system (6) has a unique equilibrium point, namely, the zero point. Lemma 2.2. If parameters b 2i 1 (1 i n) are positive numbers, then all solutions of system (6) are bounded. Proof. To prove the boundedness of the solutions in system (6), we construct a Lyapunov function V(t) = 2n i=1 1 2 x2 i (t). Calculating the upper-right derivative of V(t) through
Permanent oscillations for a ring of coupled van der Pol oscillators... 143 system (6) one can get: V (t) (6) = 2n i=1 x i (t)x i (t) = x 1 x 2 + x 2 [ c 1 x 1 + ε 2 x 2 + k 4 x 4 (t τ 4 ) b 1 x 2 1 x 2]+x 3 x 4 = +x 4 [ c 3 x 3 + ε 4 x 4 + k 6 x 6 (t τ 6 ) b 3 x3 2 x 4]+ +x 2n 3 x 2n 2 +x 2n 2 [ c 2n 3 x 2n 3 + ε 2n 2 x 2n 2 + k 2n x 2n (t τ 2n ) b 2n 3 x2n 3 2 x 2n 2] +x 2n 1 x 2n + x 2n [ c 2n 1 x 2n 1 + ε 2n x 2n + k 2 x 2 (t τ 2 ) b 2n 1 x2n 1 2 x 2n] n n n (1 c 2i 1 )x 2i 1 x 2i + ε 2i x2i 2 + k 2i x 2i (t τ 2i )x 2i 2 i=1 +k 2 x 2 (t τ 2 )x 2n i=1 i=2 n b 2i 1 x2i 1 2 x2 2i. (11) i=1 Noting that as x 2i 1 x 2i (i = 1, 2,...,n)tend to positive or negative infinity, x2i 1 2 x2 2i (i = 1, 2,...,n) are higher order positive infinity than x 2i 1 x 2i. Since 0 < b 2i 1 (i = 1, 2,...,n), therefore, there exists suitably large M > 0 such that V (t) (6) < 0as x i M(i = 1, 2,...,2n). This means that all solutions of system (6) and hence system (5) are bounded. 3. Main Results Fora2n 2n matrix C ={c ij }, the norm will be defined as C = The measure µ(c) = 2n i=1,i =j c ij ) [26]. max 1 j 2n 2n i=1 c ij. I + θc 1 lim, which reduces to µ(c) = max θ 0 + θ (c jj + 1 j 2n Theorem 3.1. Assume that system (6) has a unique equilibrium point and all solutions are bounded for selecting parameter values b 2i 1,c 2i 1,k 2i, and ε 2i (i = 1, 2,...,n). Let α 1,α 2,...,α 2n be 2n characteristic values of matrix A. Assume the following hypotheses: (i) There exists some real number α j such that the following condition holds: α j <(k 2 k 4 k 2n ) 1 n,j {1, 2,...,2n} (12) (ii) Each α i (i = 1, 2,...,2n) is a complex number, there is at least one say α l,l {1, 2,...,2n} has positive real part.
144 Chunhua Feng (iii) Each α i (i = 1, 2,...,2n) is a purely imaginary complex number. Then the unique equilibrium point of system (6) is unstable. system (6) generates a permanent oscillation. Proof. Consider a special case of system (7) as τ 2i = τ (i = 1, 2,...,n), where τ = min{τ 2,τ 4,...,τ 2n }. We have the following matrix form X (t) = AX(t) + BX(t τ ) (13) Let β i (i = 1, 2,...,2n) be the characteristic values of matrix B. Obviously, there is a characteristic root say β j = (k 2 k 4 k 2n ) n 1. From condition (12) we know that βj > 0. The characteristic equation corresponding to system (13) is the following: 2n i=1 (λ α i β i e λτ ) = 0 (14) So, we are led to an investigation of the nature of the roots for some j λ α j β j e λτ = 0 (15) For case (i), the characteristic equation (15) is a transcendental equation, one cannot calculate its roots explicitly. However we claim that equation (15) has a real positive root. Let f (λ) = λ α j β j e λτ. Then f (λ) is a continuous function of λ. Noting that f(0) = α j β j < 0 since α j < β j. On the other hand, f (λ) + as λ +. Therefore, there exists a suitably large positive number L such that L α j β j e Lτ > 0. According to the Intermediate Value Theorem of continuous function, there exists a λ (0,L)such that f(λ ) = 0. In other words, there exists a positive characteristic root of the characteristic equation (15). For case (ii), suppose that the characteristic value λ in (15) is a complex number λ = υ + iω. We show that λ has a positive real part υ. Indeed, based on e iωτ = cos(ωτ ) i sin(ωτ ) from (15) we have υ = Re(α j ) + β j e υ cos(ωτ ) (16) ω = Im(α j ) β j e υ sin(ωτ ) (17) where Re(α j ), Im(α j ) represent the real part and imaginary part of α j respectively. Noting that cos(ωτ ) 1asτ is a suitably small positive number. Since Re(α j )>0, obviously, there is a υ > 0 satisfies equation (16). For case (iii), equation (16) changes to υ = β j e υ cos(ωτ ) (18) Similar to case (ii), the characteristic value λ has a positive real part since β j > 0. Thus, in all cases, there exists a positive characteristic value or there is a positive real
Permanent oscillations for a ring of coupled van der Pol oscillators... 145 part of the complex characteristic value of the system (13), implying that the trivial solution of system (13) is unstable based on the basic theorem of the differential equation. Both in case (ii) and case (iii), the instability of trivial solution is dependent with the time delays. Noting that in a time delay system, as the value of delay increases the instability of the trivial solution still maintains [23]. So for any values of time delays τ 2i ( τ,i = 1, 2,...,n),the trivial solution of system (9) (or system (7)) is unstable. Recall that in system (6), the nonlinear terms b 2i 1 x2i 1 2 (t)x 2i(t)(i = 1, 2,...,n) are higher order infinitesimal when x 2i 1 and x 2i (i = 1, 2,...,n) tend to zero. So the instability of the unique equilibrium of system (7) implies that system (6) has a unique unstable equilibrium. This instability of the unique equilibrium combined with the boundedness of the solutions will force system (6) to generate a permanent oscillation. Theorem 3.2. Assume that system (6) has a unique equilibrium point and all solutions are bounded for selecting parameter values b 2i 1,c 2i 1,k 2i, and ε 2i (i = 1, 2,...,n). If the following condition holds ( B )eτ exp( τ µ(a) ) >1 (19) where τ = min{τ 1,τ 2,...,τ 2n }. Then the unique equilibrium point of system (7) is unstable. System (6) generates a permanent oscillation. Proof. To prove the instability of the equilibrium point of system (7), it only needs to prove that the trivial solution of system (7) is unstable. We still consider system (13). Noting that when X(t) > 0, d X(t) = dx(t), and d X(t) = dx(t) as X(t) < 0. dt dt dt dt From (13) we have Let z(t) = 2n i=1 d X(t) dt x i (t). Then we have µ(a) X(t) + B X(t τ ) (20) dz(t) µ(a)z(t)+ B z(t τ ) (21) dt Specifically dy(t) = µ(a)y(t)+ B y(t τ ) (22) dt If the trivial solution of system (22) is stable, then the characteristic equation associated with (22) given by λ = µ(a)+ B e λτ (23) will have a real negative root say λ 0, and we have from (23) λ 0 B e λ 0 τ µ(a) (24)
146 Chunhua Feng Using the formula e x ex for x>0 one can get 1 B e λ 0 τ µ(a) + λ 0 = B τ e µ(a) τ e ( µ(a) + λ0 )τ ( µ(a) + λ 0 )τ ( B eτ )e τ µ(a) (25) The last inequality contradicts equation (19). Hence, our claim regarding the instability of the trivial solution of system (22) is valid. According to the comparison theorem of differential equation we have z(t) y(t). Noting that the trivial solution is unstable in system (22), this implies that the trivial solution of (11) (thus system (13)) is unstable. Similar to Theorem 3.1, we claim that the trivial solution of system (6) is unstable. Since all solutions of system (6) are bounded, this instability of the unique equilibrium point together with the boundedness of the solutions lead system (6) to generate a permanent oscillation. 4. Simulation results Example 4.1. First we consider four coupled van der Pol equation: x 1 (t) = x 2(t), x 2 (t) = c 1x 1 (t) + ε 2 x 2 (t) + k 4 x 4 (t τ 4 ) b 1 x 2 1 (t)x 2(t), x 3 (t) = x 4(t), x 4 (t) = c 3x 3 (t) + ε 4 x 4 (t) + k 6 x 6 (t τ 6 ) b 3 x 2 3 (t)x 4(t), x 5 (t) = x 6(t), x 6 (t) = c 5x 5 (t) + ε 6 x 6 (t) + k 8 x 8 (t τ 8 ) b 5 x 2 5 (t)x 6(t), x 7 (t) = x 8(t), x 8 (t) = c 7x 7 (t) + ε 8 x 8 (t) + k 2 x 2 (t τ 2 ) b 7 x 2 7 (t)x 8(t). (26) We fixed ε 2 = 0.0045,ε 4 = 0.0075,ε 6 = 0.0065,ε 8 = 0.0085, and first select time delays as τ 2 = 0.5,τ 4 = 0.8,τ 6 = 1.2,τ 8 = 1.5. The parameter values are k 2 = 1.55,k 4 = 1.65,k 6 = 1.25,k 8 = 1.35,b 1 = 0.85,b 3 = 1.45,b 5 = 0.65,b 7 = 1.15,c 1 = 1.35,c 3 = 1.85,c 5 = 1.45,c 7 = 1.15. The eigenvalues of matrix A are 0.0022 ± 1.1619i, 0.0037 ± 1.3601i, 0.0032 ± 1.2042i, 0.0042 ± 1.0724i. All the eigenvalues have positive real parts. It is easy to check the conditions of Theorem 3.1 are satisfied. System (26) generates a a permanent oscillation (see Fig. 1). In order to see the effect of time delays, we increase the time delays as τ 2 = 1.5,τ 4 = 1.8,τ 6 = 2.2,τ 8 = 2.5. The dynamical behavior still holds (see Fig.2).
Permanent oscillations for a ring of coupled van der Pol oscillators... 147
148 Chunhua Feng Figure 3: Continued.
Permanent oscillations for a ring of coupled van der Pol oscillators... 149 Figure 4: Continued.
150 Chunhua Feng Example 4.2. We then consider seven coupled van der Pol equation: x 1 (t) = x 2(t), x 2 (t) = c 1x 1 (t) + ε 2 x 2 (t) + k 4 x 4 (t τ 4 ) b 1 x1 2 (t)x 2(t), x 3 (t) = x 4(t), x 4 (t) = c 3x 3 (t) + ε 4 x 4 (t) + k 6 x 6 (t τ 6 ) b 3 x3 2 (t)x 4(t), x 5 (t) = x 6(t), x 6 (t) = c 5x 5 (t) + ε 6 x 6 (t) + k 8 x 8 (t τ 8 ) b 5 x5 2 (t)x 6(t), x 7 (t) = x 8(t), x 8 (t) = c 7x 7 (t) + ε 8 x 8 (t) + k 10 x 10 (t τ 10 ) b 7 x7 2 (t)x 8(t), x 9 (t) = x 10(t), x 10 (t) = c 9x 9 (t) + ε 10 x 10 (t) + k 12 x 12 (t τ 12 ) b 9 x9 2 (t)x 10(t) x 11 (t) = x 12(t), x 12 (t) = c 11x 11 (t) + ε 12 x 12 (t) + k 14 x 14 (t τ 14 ) b 11 x11 2 (t)x 12(t), x 13 (t) = x 14(t), x 14 (t) = c 13x 13 (t) + ε 14 x 14 (t) + k 2 x 2 (t τ 2 ) b 13 x13 2 (t)x 14(t). (27) We fixed ε 2 = 0.0015,ε 4 = 0.0025,ε 6 = 0.0035,ε 8 = 0.0024,ε 10 = 0.0018,ε 12 = 0.0016,ε 14 = 0.0024 and first select time delays as 0.5, 0.4, 0.3, 0.5, 0.6, 0.2, 0.35. The parameter values are k 2 = 1.55,k 4 = 1.65,k 6 = 0.98,k 8 = 1.35,k 10 = 1.45,k 12 = 1.55,k 14 = 1.65; b 1 = 0.95,b 3 = 1.25,b 5 = 1.65,b 7 = 1.15,b 9 = 1.25,b 11 = 1.05,b 13 = 1.15,c 1 = 1.45,c 3 = 1.65,c 5 = 1.45,c 7 = 1.15,c 9 = 1.35,c 11 = 1.25,c 13 = 1.15. Corresponding to system (27), µ(a) = 1.65, B = 1.65, and τ = 0.2. Then B eτ exp( τ µ(a) ) = 6.4486 > 1. The condition (19) is satisfied. System (27) has a unique unstable equilibrium point and thus generates a permanent oscillation (see Fig. 3). Again we increase the value of c i as c 1 = 2.45,c 3 = 2.65,c 5 = 2.45,c 7 = 2.15,c 9 = 2.35,c 11 = 2.25,c 13 = 2.15. Condition B eτ exp( τ µ(a) ) = 5.2798 > 1 still holds. There is also a permanent oscillation (Fig. 4). Increase the values of c i such that the oscillatory frequency is a slightly change. 5. Conclusion In this paper, we have discussed the dynamical behavior of n coupled van der Pol equations with time delays. The existence of a limit circle which is easy to check, as compared to the general bifurcating method. Some simulations are provided to indicate the effectness of the criterion. Time delays only affect the oscillatory frequency when there exists a limit circle of the system. Remark 5.1. Theorem 3.2 is only a sufficient condition. Condition (19) can not hold when τ is suitably large (exp( τ µ(a) ) 0 when τ ). However, based on the instability of the solution in a delayed differential equation, when we select a τ such that the condition (19) holds, then for any delays larger than τ, the instability of the solution is still maintained. In other words, the permanent oscillation still holds.
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