Stability and Hopf-bifurcation analysis of an unidirectionally coupled system

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 211, No. 85, 1-17; Staility and Hopf-ifurcation analysis of an unidirectionally coupled system Gang Zhu a,, Junjie Wei a, a. Department of Mathematics, Harin Institute of Technology, Harin, Heilongjiang, 151, P.R. China.. School of Education Science, Harin University, Harin, Heilongjiang, 151, P.R. China. Astract: In this paper, the staility and existence of periodic solutions for an unidirectionally coupled nonlinear system with delays are investigated y comining the linear staility theory and the emedding technique of asymptotically autonomous semiflows, Hopf ifurcation theory and a continuation theorem of coincidence degree theory, respectively. Some numerical simulations are carried out for illustrating the analytical results. Keywords: asymptotic staility; Hopf ifurcation; normal form; coincide degree 1 Introduction It has een shown that coupled nonlinear systems with time delay can exhiit very complex dynamics, such as the appearance of chaotic attractors and chaotic synchronization[1, 2]. Consequently, studies of such systems ecome very important in order to understand their cooperative dynamics. From recent works [3, 4, 5, 6, 7], the Krasovskii-Lyapunov theory [8] is used to discuss the synchronization in the coupled time-delayed system ẋ(t) = ax(t) + f(x(t τ)), ẏ(t) = ay(t) + f(y(t τ)) + k(t)[x(t) y(t)], (1.1) This research is supported y the National Natural Science Foundation of China (No ). Corresponding author. address: weijj@hit.edu.cn EJQTDE, 211 No. 85, p. 1

2 where a and are positive constants, τ > is the time delay, k(t) is the coupling function etween the drive and the response system, f(x) is some nonlinear continuous function. In [7], Senthilkumar et.al. introduced the difference system with the state variale = x(t) y(t) for small values of = [a + k(t)] + f (y(t τ)) τ, τ = (t τ), (1.2) whose coefficients are all time dependent. By use of Krasovskii-Lyapunov theory, they gave the condition a + k(t) > f (y), for t, y R under which the zero solution of Eq.(1.2) is stale, which means that the complete synchronization in the coupled time-delayed systems occurs. The purpose of the present paper is to investigate the coupled system also in view of ifurcation. We choose the coupling function k(t) = k >, which is the coupling strength. Then Eq.(1.1) ecomes ẋ(t) = ax(t) + f(x(t τ)), ẏ(t) = ay(t) + f(y(t τ)) + k[x(t) y(t)], (1.3) and we make the general assumption that f() =, f () = ε. By use of the results of Wei [12] we get the staility of the zero solution and the existence of Hopf ifurcation when the delay varies for the first equation of (1.3). Then y using the center manifold theory and normal form method introduced y Faria and Magalhães[1, 11], we derive an explicit algorithm for determining the direction of the Hopf ifurcation and the staility of the ifurcating periodic solutions. Futhermore we discuss the staility and the existence of periodic solutions of the coupled system (1.3) y using a continuation theorem of coincidence degree theory. The rest of the present paper is organized as follows: in Section 2, we analyze the staility of the zero solution of the first equation of (1.1) including the special and complex cases under which the corresponding characteristic equation has a simple zero root, and discuss the existence of local Hopf ifurcation. In Section 3, we determine the properties of the ifurcating periodic solution. In Section 4, we discuss the staility of the zero solution of the origin coupled system (1.3). Finally, in Section 5 the existence of periodic solutions for system (1.3) are estalished y using a continuation theorem of EJQTDE, 211 No. 85, p. 2

3 coincidence degree theory, and some numerical simulations are carried out to illustrate the analytic results. We would like to mention that there are several articles focus on Hopf ifurcation and complexity of dynamics in delayed models, y employing the center manifold theorem and normal forms method, see [21, 22, 23] and the references therein. 2 Staility analysis of the uncoupled system Notice that the first equation is uncoupled with the other one, therefore we egin with the investigation of the scalar equation ẋ(t) = ax(t) + f(x(t τ)). (2.1) Clearly the origin is the fixed point of the equation and the linearization of Eq.(2.1) around the origin is given y ẋ(t) = ax(t) + εx(t τ), (2.2) and the characteristic equation associated with Eq.(2.2) is (λ, τ) := λ + a εe λτ =. (2.3) Since Eq.(2.2) and Eq.(2.3) have the same form as that of equations (4) and (5) in [12] except for some constant coefficients, we give the following results without proof. Theorem 2.1. For system (2.1) (i) If ε < a, then all the roots of (2.3) have negative real parts. Furthermore, the zero solution of (2.1) is asymptotically stale for all τ ; (ii) If ε > a, then Eq. (2.3) has at least one positive root, and hence the zero solution of (2.1) is unstale for all τ ; (iii) If ε < a, then there exists τ < τ 1 < < τ j <, such that all the roots of (2.3) have negative real parts when τ [, τ ), and Eq.(2.3) has at least a pair of roots with positive real parts when τ > τ. Furthermore, the zero solution of (2.1) is asymptotically stale when τ [, τ ), and unstale when τ > τ ; (iv) If ε > a, then (2.1) undergoes a Hopf ifurcation at the origin when τ = τ j, j =, 1, 2,, where and τ j = { 1 ω (arccos a + 2jπ), for ε <, ε 1 ω (2π arcsin a + 2jπ), for ε >, j =, 1, 2,, ε ω = 2 ε 2 a 2. EJQTDE, 211 No. 85, p. 3

4 The theorem aove shows that ε = ± a is a critical value of the staility of the zero solution of Eq.(2.1). A mathematical question is whether the zero solution of Eq.(2.1) is stale at this critical situation. The following theorem is to descrie the staility of the zero solution of Eq.(2.1) when ε = ± a. Theorem 2.2. (i) If ε = a, then the zero solution of system (2.1) is asymptotically stale; (ii) If ε = a, then system (2.1) undergoes a fixed point ifurcation; and if f (), the zero solution of system (2.1) is unstale; if f () =, the zero solution of system (2.1) is asymptotically stale when f () <, and unstale when f () >. Proof. (i) When ε = a, the characteristic equation (2.3) ecomes λ + a + ae λτ = (2.4) Oviously, λ = is not a root of Eq.(2.4), and λ = 2a < When τ =. Suppose λ = iω is a root of Eq.(2.4), then we have iω + a + a cosωτ ia sin ωτ =, separating the real and imaginary parts yields a + a cos ωτ =, ω a sin ωτ =, which implies that a 2 + ω 2 = a 2. Furthermore, we get λ =, which is a contradiction since we have known that λ = is not a root of Eq.(2.4). This shows that all roots of Eq.(2.4) have negative real part. (ii) Clearly, λ = is a simple root of the characteristic equation (2.3) with ε = a, since d (, τ) = 1 + aτ >. dλ From theorem 2.1 we know that all roots of characteristic equation (2.3) have negative real parts except λ = when ε = a. In order to study the staility of the zero solution of system (2.1), similar to the method in [13, 14], we employ the center manifold theory and normal form method for FDE introduced y Faria et al. [1, 11]. Let Λ = {} and B =, clearly the non-resonance conditions relative to Λ are satisfied. Therefore there exists a 1-dimensional ODE which governs the dynamics Eq.(2.1) near the origin. EJQTDE, 211 No. 85, p. 4

5 Firstly, we re-scale the time delay y t (t/τ) to normalize the delay so that Eq.(2.1) can e written in the form: ẋ(t) = aτx(t) + τf(x(t 1)). (2.5) Clearly, the phase space for Eq.(2.5) is C := C([ 1, ], R). For ϕ C, define L(ϕ) = aτϕ() + aτϕ( 1). and [ f () F(ϕ) = τ ϕ 2 ( 1) + f ] () ϕ 3 ( 1) + O(ϕ 4 (), ϕ 4 ( 1)). 2! 3! Then Eq. (2.5) can e rewritten in the form: Choosing we otain d dt x(t) = L(x t) + F(x t ). η(θ) = Lϕ = {, θ = 1 or aτ, θ ( 1, ), 1 dη(θ)ϕ(θ). Using the formal adjoint theory for FDEs[8], we decompose C y Λ as C = P Q, where P = spanφ(θ) with Φ(θ) = 1 eing the center space for d dt x(t) = L(x t). Choosing a asis Ψ for the adjoint space P such that < Ψ, Φ >= 1, where <, > is the ilinear form on C C defined y < ψ, ϕ >= ψ()ϕ() θ 1 ξ= ψ(ξ θ)dη(θ)ϕ(θ)dξ. Thus Ψ(s) = (1 + aτ) 1. Taking the enlarged phase space BC = {ϕ : [ 1, ) C, ϕ is continuous on [ 1, ) and lim θ ϕ(θ) exists}, we otain the astract ODE with the form: d dt x(t) = Ax t + X F(x t ). (2.6) EJQTDE, 211 No. 85, p. 5

6 Here for any ϕ C, and X is given y Aϕ = ϕ(θ) + X (L ϕ ϕ()), X = { I, θ =, θ [ 1, ). The definition of the continuous projection π : BC P, π(ϕ + X α) = Φ[< Ψ, ϕ > +Ψ()α] allows us to decompose the enlarged space y Λ as BC = C Kerπ. Since π commutes with A in C 1, and using the decomposition x t = Φx(t) + y, x(t) C, y = y(θ) Q 1, the astract ODE (2.6) is therefore decomposed as the system ẋ = Bx + Ψ()F(Φx + y), ẏ = A Q 1 + (I π)x F(Φx + y). (2.7) Since Ψ()F(Φx+y) = τ 1 + aτ [ f () 2! (x + y( 1)) 2 + f () (x + y( 1)) ]+O(4), 3 3! therefore the local invariant manifold of system (2.1) tangent to P at the origin satisfying y(θ) = and the flow on this manifold is given y the following 1-dimensional ODE: ẋ(t) = τ 1 + aτ [ f () 2! x 2 + f ] () x 3 + O(4). (2.8) 3! Since a, and τ are all positive, the zero solution of ODE (2.8) is unstale when f () ; and if f () =, the zero solution of ODE (2.8) is asymptotically stale when f () <, and unstale when f () >, and so is the zero solution of system (2.1). The proof is completed. 3 Hopf ifurcation analysis In section 2, we otain the conditions under which Eq. (2.1) undergoes a Hopf ifurcation at some critical values of τ. In this section we shall study EJQTDE, 211 No. 85, p. 6

7 the direction and staility of the ifurcating periodic solutions. The method we use here is ased on the normal form method and center manifold theory introduced y Faria et al. [1, 11]. For convenience, let τ = τ j +µ, µ R, then µ = is the ifurcation value of Eq. (2.1). Defining L µ (ϕ) = a(τ j + µ)ϕ() + ε(τ j + µ)ϕ( 1), [ f () F µ (ϕ) = (τ j + µ) ϕ 2 ( 1) + f ] () ϕ 3 ( 1) + O(ϕ 4 (), ϕ 4 ( 1)). 2! 3! Then we can rewrite Eq.(2.1) in the following form: ẋ(t) = L µ (x t ) + F µ (x t ). (3.1) Using the normal form method introduced y Farai directly, we get and L (ϕ(θ)) = aτ j ϕ() + ετ j ϕ( 1), L (θe iω θ ) = ετ j e iω = (aτ j + iτ j ω ), L (1) = aτ j + ετ j, L (e 2iω θ ) = aτ j + ετ j e 2iω = aτ j + (aτ j + iτ j ω ) 2 ετ j, F(x 1 e iω θ + x 2 e iω θ + x x 4 e 2iω θ, ) = τ j 2! f ()(x 1 e iω + x 2 e iω + x 3 + x 4 e 2iω ) 2 + τ j 3! f ()(x 1 e iω + x 2 e iω + x 3 + x 4 e 2iω ) 3 + O(4) = B (2,,,) x B (1,1,,) x 1 x 2 + B (1,,1,) x 1 x 3 + B (,1,,1) x 2 x 4 + B (2,1,,) x 2 1 x 2 +. (3.2) Comparing the coefficients, we have B (2,,,) = τ j 2 f ()e 2iω = 1 2τ j ε 2f ()(aτ j + iτ j ω ) 2, B (1,1,,) =τ j f (), B (1,,1,) =τ j f ()e iω = 1 ε f ()(aτ j + iτ j ω ), (3.3) B (,1,,1) =τ j f ()e iω = 1 ε f ()(aτ j + iτ j ω ), B (2,1,,) = τ j 2 f ()e iω = 1 2ε f ()(aτ j + iτ j ω ). EJQTDE, 211 No. 85, p. 7

8 By Faria s normal form theory, the flow of Eq.(3.1) on the center manifold of the origin is given in polar coordinates (ρ, ξ) y equation { ρ = µα ()ρ + Kρ 3 + O(µ 2 ρ + (ρ, µ) 4 ), where [ K = Re 1 1 L (θe iωθ ) ξ = ω + O( (ρ, µ) ), ( B (2,1,,) B (1,1,,)B (1,,1,) + B (2,,,)B (,1,,1) L (1) 2iω L (e 2iωθ ) )]. (3.4) Sustituting (3.2) and (3.3) into (3.4), and since we have that α () >, we otain the following theorem. Theorem 3.1. If K < (respectively, K > ), there exists a unique nontrivial periodic orit in the neighorhood of ρ = for µ > (respectively, µ < ) and no nontrivial periodic orits for µ < (respectively, µ > ); and the ifurcating periodic solution on the center manifold is oritally asymptotically stale (respectively, unstale). Particularly, the staility of the ifurcating periodic solutions of (2.1) and that on the center manifold are coincident at the first critical value τ. Corollary 3.2. If f () =, then the nontrivial periodic orit exists in the neighorhood of ρ = for µ > ( µ < ) and the ifurcating periodic solutions on the center manifold are oritally asymptotically stale ( unstale) when εf () < (> ). In fact, from (3.3) it follows that B (1,1,,) = B (1,,1,) = B (2,,,) = B (,1,,1) =, and B (2,1,,) = 1 2ε f ()(aτ j + iτ j ω ). Sustituting the coefficients aove into (3.4), it follows that K = f () 2ε aτ j + a 2 τj 2 + τj 2 ω 2. (1 + aτ j ) 2 + τj 2ω2 And hence from (3.4) and theorem 3.1, the conclusion is reached. EJQTDE, 211 No. 85, p. 8

9 4 Staility analysis of the coupled system So far, we have investigated the staility and Hopf ifurcation for the first equation of system (1.3). In the following we will focus on the staility of the zero solution of the coupled system. The theory we use here is from[15, 16], and its notions are discussed elow. Lemma 4.1. ([16]) Let e e a locally asymptotically stale equilirium of Θ and W s (e) = {x X : Θ(t, x) e, t } its asin of attraction(or stale set). Then every pre-compact Φ orit whose ω Φ limit set intersects W s (e) converges to e. To make use of the lemma aove, similar to [17], we introduce some notations first. Define C := C([ τ, ], R), and d(x, y) = max x(θ) y(θ), θ [ τ,] for any x, y C. Suppose Ω is an open suset of C, F C(Ω, R), and F is Lipschitz in each compact set in Ω. Consider the following equation ẋ(t) = F(x t ) + G(t), t σ, x σ = φ, (4.1) with (σ, φ) R C. E.q.(4.1) is a 1-dimensional non-autonomous retarded differential equation and has a unique solution through any given initial function. Let x(s, φ)(t) e the unique solution through (σ, φ) and well-defined for all t [σ τ, σ +α]. Here we assume further that the maximal existence interval of the solutions is [σ τ, ). From [8], we know that x(s, φ)(t) is continuous in σ, φ, t for σ R, φ C and t [σ τ, ). Consider a mapping Φ : C C, which is defined as Φ(t, s, φ) = x t (s, φ) C, with x t (s, φ)(θ) = x(x, φ)(t + θ). It can e verified that Φ is continuous non-autonomous semiflow on C y the existence and uniqueness of solutions. Further we consider the corresponding autonomous equation ẋ(t) = F(x t ), x = ψ, (4.2) for ψ C. Under the same assumptions, let y(ψ)(t) e the unique solution through (, ψ), and define Θ : [, ) C C as Θ(t, ψ) = y(ψ), with y t (ψ)(θ) = y(ψ)(t + θ). Similarly we can verify that Θ is a continuous autonomous semiflow. Before stating the main theorem, we give the following property of Φ and Ψ defined aove. EJQTDE, 211 No. 85, p. 9

10 Lemma 4.2. [17] If G(t), t, then Φ is the asymptotically autonomous -with limit-semiflow Θ. Theorem 4.3. If a ε < a, then the zero solution of the coupled system (1.3) is asymptotically stale. Proof. Consider the second equation of (1.3) as follows ẏ(t) = (a + k)y(t) + f(y(t τ)) + kx(t) = F(y t ) + G(t) (4.3) and the corresponding autonomous equation ẏ(t) = (a + k)y(t) + f(y(t τ)) = F(y t ) (4.4) From theorem 2.1,we know that the zero solution of (2.1) is asymptotically stale, which implies that G(t) as t. Therefore Φ defined y (4.3) is asymptotically autonomous with limit-semiflow Θ defined y (4.4), where lemma 4.2 is used. Next we egin to investigate the asymptotic ehavior of the autonomous system (4.4). Similar to theorem 2.1, it can e proved that ε < a+k. the zero solution of (4.4) is asymptotically stale when a+k Furthermore, let e = e the stale equilirium of Θ on C and then the intersection of C and e s asin of attraction is nonempty. On the other hand, it is known that every Φ-orit is pre-compact y Ascoli-Arzela theorem. Therefore, lemma 4.1 implies that the zero solution of Eq.(4.3) is asymptotically stale. Notice that [ a, a ) [ a+k, a+k ), this completes the proof. 5 Existence of periodic solutions in the coupled system Results in section 3 show that under certain conditions, there exist nonconstant periodic solutions to (2.1) due to Hopf ifurcation when τ lies in some neighorhood of each ifurcation value. In the following we assume that these conditions ensuring the appearance of Hopf ifurcation are met. For convenience, we denote y D the region where τ lies and ifurcating periodic solutions for (2.1) exist. Our purpose is to otain sufficient conditions for the existence of the periodic solutions to the original coupled system ẋ(t) = ax(t) + f(x(t τ)), ẏ(t) = ay(t) + f(y(t τ)) + k[x(t) y(t)], (5.1) EJQTDE, 211 No. 85, p. 1

11 y employing the coincide degree theory from Gaines and Mawhin[18]. To make use of the continuation theorem of coincidence degree theory, similar to [17, 19, 2], we need to introduce following notations. Let X, Y e real Banach spaces, L : DomL X Y e a Fredholm mapping of index zero, and let P : X X, Q : Y Y e continuous projectors such that ImP = KerL, KerQ = ImL and X = KerL KerP, Y = ImL ImQ. Denote y L P the restriction of L to DomL KerP. Denote y K P : ImL KerP DomL the inverse of L P, and denote y J : ImQ KerL an isomorphism of ImQ onto KerL. For convenience, we also cite elow the continuation theorem. Lemma 5.1. [18] Let Ω X e an open ounded set and let N : X Y e a continuous operator which is L compact on Ω(i.e.,QN : Ω Y and K P (I Q)N : Ω Y are compact). Assume (i) for each λ (, 1), x Ω DomL, Lx λnx; (ii) for each x Ω KerL, QNx, and deg{jqn, Ω KerL, }. Then Lx = Nx has at least one solution on Ω Doml. To use the continuation theorem of coincidence degree theory, we take X = Y = {y(t) C(R, R) : y(t + ω) = y(t)}. With the maximal norm, X and Y are Banach spaces. Set L : DomL X, Ly = ẏ, where DomL = {y(t) C 1 (R, R)}. Define two projectors P and Q as Py = Qy = 1 ω ω y(t)dt, y X. Clearly, KerL = R, ImL = {y X : ω y(t)dt = } is closed in X and dimkerl = ImL = 1. Hence, L is a Fredholm mapping of index. Furthermore, through an easy computation, we find that the inverse K P of L P has the form K P : ImL DomL KerP, K P (y) = t y(s)ds 1 ω ω u y(s)dsdu, t [, ω]. We are now in a position to state and prove our main result. Theorem 5.2. If ε > a, then system (1.3) has at least one periodic solution for τ D. EJQTDE, 211 No. 85, p. 11

12 Proof. Consider the second equation of (5.1) ẏ(t) = (a + k)y(t) + f(y(t τ)) + kx(t). (5.2) Then x(t) is a ω-periodic solution of (2.1) when τ D. To complete the proof, it suffices to show that (5.2) has an ω-periodic solution. Define N : X Y as Ny = (a + k)y(t) + f(y(t τ)) + kx(t). Notice that QN : X X takes the form QN(y) = 1 ω ω [ (a + k)y(t) + f(y(t τ)) + kx(t)]dt. By some computation, we can show that K P (I Q)N : X Y takes the form K P (I Q)N(y) = 1 ω t ω u [ (a + k)y(s) + f(y(s τ)) + kx(s)]ds [ (a + k)y(s) + f(y(s τ)) + kx(s)]dsdu +( 1 2 t ω ω ) [ (a + k)y(t) + f(y(t τ)) + kx(t)]dt. The integration form of the terms of oth QN and K P (I Q)N imply that they are continuously differentiale with respect to t and that they map ounded continuous functions to ounded continuous functions. By the Ascoli-Arzela theorem, we see that QN(Ω), K P (I Q)N(Ω) are relatively compact for any open ounded set Ω X. Therefore, N is L compact on Ω for any open ounded set Ω X. Corresponding to the operator equation Ly = λn y, λ (, 1), we have ẏ(t) = λ[ (a + k)y(t) + f(y(t τ)) + kx(t)]. (5.3) Suppose that y(t) X is a solution of (5.3) for some λ (, 1). On one hand, we choose t M [, ω] and t m [, ω] such that y(t M ) = max t [,ω] y(t), y(t m) = min t [,ω] y(t). Then it is clear that ẏ(t M ) = ẏ(t m ) =. From this and (5.3), we otain y(t) y(t m ) = 1 a + k [f(y(t m τ)) + kx(t m )], y(t) y(t M ) = 1 a + k [f(y(t M τ)) + kx(t M )], EJQTDE, 211 No. 85, p. 12

13 which implies that 1 a + k (M + k x ) y(t) 1 a + k (M + k x ), where M > satisfies f(u) M, u X since f is continuous. On the other hand, suppose y KerL satisfies QNy =, and notice that KerL = R, which means that y(t) = y is a constant, we otain QNy = 1 ω This implies that y = 1 a + k ω [ (a + k)y + f(y) + kx(t)]dt = (a + k)y + f(y) + 1 ω ω kx(t)dt =. [ f(y) + 1 ω ] kx(t)dt 1 ω a + k (M + k x ). Taking A = 1 a+k (M + k x + 1) and Ω = {y(t) X : y < A}, then it is clear that Ω satisfies condition (i) and the first part of condition (ii) in Lemma 5.1. Furthermore, take J = I : ImQ KerL, x x and y a straightforward computation, we see that deg[jqn, KerL Ω, ]. The conclusion now follows from Lemma (5.1). This completes the proof. 6 An example Taking f(x) = sin x as an example, then Eq.(2.1) has the form: ẋ(t) = ax(t) sin(x(t τ)). (6.1) Oviously x = is the fixed point of (6.1), and f() =, ε := f () = 1, f () =, f () = 1. Basing on the discussion aove, we get the following conclusions directly. Theorem 6.1. For Eq. (6.1) (i) If < < a, then the zero solution of (6.1) is asymptotically stale for all τ ; EJQTDE, 211 No. 85, p. 13

14 (ii) If > a, then there exists τ < τ 1 < < τ j < such that the zero solution of (6.1) is stale when τ [, τ ), and unstale for τ > τ, and Eq.(6.1) undergoes a Hopf ifurcation at the origin when τ = τ j, j =, 1, 2,, where τ j is defined as τ j = 1 ω ( arccos( a ) + 2jπ ), j =, 1, 2. and ω = 2 a 2. And the ifurcating periodic solution exists for τ > τ j, and the ifurcating periodic solutions on the center manifold are stale. Particularly, the ifurcating periodic solutions from the first ifurcation vale τ are asymptotically stale. If we take a = 1, = 3, k = 1, then (5.1) ecomes ẋ(t) = x(t) 3 sin(x(t τ)), ẏ(t) = y(t) 3 sin(y(t τ)) + [x(t) y(t)]. (6.2) Clearly, f() = f () =, ε := f () = 1, f () = 1, ω = 2 a 2 = 2.828, τ = Then y the discussion aove, the zero solution of (6.2) is asymptotically stale when τ [,.6756), and has periodic solutions in the right neighorhood of τ j. Acknowledgment The authors wish to express their special gratitude to the reviewers for their helpful comments and suggestions given for this paper. EJQTDE, 211 No. 85, p. 14

15 Figure 1: Numerical simulations for system (6.2) with τ =.5 [,.6756) shows that the origin is asymptotically stale Figure 2: Wave plot for (6.2) with τ =.7 > τ shows that the ifurcating periodic solutions are stale. EJQTDE, 211 No. 85, p. 15

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