Stability and Hopf-bifurcation analysis of an unidirectionally coupled system ∗

In this paper, the stability and existence of periodic solutions for an unidirectionally coupled nonlinear system with delays are investigated by combining the linear stability theory and the embedding technique of asymptotically autonomous semiflows, Hopf bifurcation theory and a contin- uation theorem of coincidence degree theory, respectively. Some numerical simulations are carried out for illustrating the analytical results.


Introduction
It has been shown that coupled nonlinear systems with time delay can exhibit very complex dynamics, such as the appearance of chaotic attractors and chaotic synchronization [1,2].Consequently, studies of such systems become 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) + bf (x(t − τ )), ẏ(t) = −ay(t) + bf (y(t − τ )) + k(t)[x(t) − y(t)], ( where a and b are positive constants, τ > 0 is the time delay, k(t) is the coupling function between 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 variable ∆ = x(t) − y(t) for small values of ∆ whose coefficients are all time dependent.By use of Krasovskii-Lyapunov theory, they gave the condition a + k(t) > |bf ′ (y)|, for t ≥ 0, y ∈ R under which the zero solution of Eq.(1.2) is stable, which means that the complete synchronization in the coupled time-delayed systems occurs.
By use of the results of Wei [12] we get the stability of the zero solution and the existence of Hopf bifurcation when the delay varies for the first equation of (1.3).Then by using the center manifold theory and normal form method introduced by Faria and Magalhães [10,11], we derive an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.Futhermore we discuss the stability and the existence of periodic solutions of the coupled system (1.3) by using a continuation theorem of coincidence degree theory.The rest of the present paper is organized as follows: in Section 2, we analyze the stability 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 bifurcation.In Section 3, we determine the properties of the bifurcating periodic solution.In Section 4, we discuss the stability 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 established by using a continuation theorem of EJQTDE, 2011 No. 85, p. 2 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 bifurcation and complexity of dynamics in delayed models, by employing the center manifold theorem and normal forms method, see [21,22,23] and the references therein.

Stability analysis of the uncoupled system
Notice that the first equation is uncoupled with the other one, therefore we begin with the investigation of the scalar equation ẋ(t) = −ax(t) + bf (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 by ẋ and the characteristic equation associated with Eq.(2.2) is 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 b , then all the roots of (2.3) have negative real parts.Furthermore, the zero solution of (2.1) is asymptotically stable for all τ ≥ 0; (ii) If ε > a b , then Eq. (2.3) has at least one positive root, and hence the zero solution of (2.1) is unstable for all τ ≥ 0; (iii) If ε < − a b , then there exists , such that all the roots of (2.3) have negative real parts when τ ∈ [0, τ 0 ), and Eq.(2.3) has at least a pair of roots with positive real parts when τ > τ 0 .Furthermore, the zero solution of (2.1) is asymptotically stable when τ ∈ [0, τ 0 ), and unstable when τ > τ 0 ; (iv) If |ε| > a b , then (2.1) undergoes a Hopf bifurcation at the origin when τ = τ j , j = 0, 1, 2, • • • , where EJQTDE, 2011 No. 85, p. 3 The theorem above shows that ε = ± a b is a critical value of the stability of the zero solution of Eq.(2.1).A mathematical question is whether the zero solution of Eq.(2.1) is stable at this critical situation.
From theorem 2.1 we know that all roots of characteristic equation (2.3) have negative real parts except λ = 0 when ε = a b .In order to study the stability 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 by Faria et al. [10,11].
Let Λ = {0} and B = 0, 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, 2011 No. 85, p. 4 Firstly, we re-scale the time delay by t → (t/τ ) to normalize the delay so that Eq.(2.1) can be written in the form: (2.5) Clearly, the phase space for Eq.(2.5) is and Then Eq. (2.5) can be rewritten in the form: Choosing we obtain Using the formal adjoint theory for FDEs [8], we decompose C by Λ as C = P ⊕ Q, where P = spanΦ(θ) with Φ(θ) = 1 being the center space for Choosing a basis Ψ for the adjoint space P * such that < Ψ, Φ >= 1, where Taking the enlarged phase space we obtain the abstract ODE with the form: Here for any ϕ ∈ C, and X 0 is given by The definition of the continuous projection allows us to decompose the enlarged space by Λ as BC = C ⊕ Kerπ.Since π commutes with A in C 1 , and using the decomposition the abstract ODE (2.6) is therefore decomposed as the system Since therefore the local invariant manifold of system (2.1) tangent to P at the origin satisfying y(θ) = 0 and the flow on this manifold is given by the following 1-dimensional ODE: Since a, b and τ are all positive, the zero solution of ODE (2.8) is unstable when f ′′ (0) = 0; and if f ′′ (0) = 0, the zero solution of ODE (2.8) is asymptotically stable when f ′′′ (0) < 0, and unstable when f ′′′ (0) > 0, and so is the zero solution of system (2.1).The proof is completed.

Hopf bifurcation analysis
In section 2, we obtain the conditions under which Eq. (2.1) undergoes a Hopf bifurcation at some critical values of τ .In this section we shall study EJQTDE, 2011 No. 85, p. 6 the direction and stability of the bifurcating periodic solutions.The method we use here is based on the normal form method and center manifold theory introduced by Faria et al. [10,11].
In fact, from (3.3) it follows that Substituting the coefficients above into (3.4), it follows that And hence from (3.4) and theorem 3.1, the conclusion is reached.
EJQTDE, 2011 No. 85, p. 8 So far, we have investigated the stability and Hopf bifurcation for the first equation of system (1.3).In the following we will focus on the stability of the zero solution of the coupled system.The theory we use here is from [15,16], and its notions are discussed below.
To make use of the lemma above, similar to [17], we introduce some notations first.
Further we consider the corresponding autonomous equation for ψ ∈ C.Under the same assumptions, let y(ψ)(t) be the unique solution through (0, ψ), and define Θ : [0, ∞) × 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 above.From theorem 2.1,we know that the zero solution of (2.1) is asymptotically stable, which implies that G(t) → 0 as t → ∞.Therefore Φ defined by (4.3) is asymptotically autonomous with limit-semiflow Θ defined by (4.4), where lemma 4.2 is used.Next we begin to investigate the asymptotic behavior of the autonomous system (4.4).Similar to theorem 2.1, it can be proved that the zero solution of (4.4) is asymptotically stable when − a+k b ≤ ε < a+k b .Furthermore, let e = 0 be the stable equilibrium of Θ on C and then the intersection of C and e's basin of attraction is nonempty.On the other hand, it is known that every Φ-orbit is pre-compact by Ascoli-Arzela theorem.Therefore, lemma 4.1 implies that the zero solution of Eq.(4.3) is asymptotically stable.Notice that [− a b , a b ) ⊂ [− a+k b , a+k b ), 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 bifurcation when τ lies in some neighborhood of each bifurcation value.In the following we assume that these conditions ensuring the appearance of Hopf bifurcation are met.For convenience, we denote by D the region where τ lies and bifurcating periodic solutions for (2.1) exist.Our purpose is to obtain sufficient conditions for the existence of the periodic solutions to the original coupled system ẋ EJQTDE, 2011 No. 85, p. 10 by employing the coincide degree theory from Gaines and Mawhin [18].
To make use of the continuation theorem of coincidence degree theory, similar to [17,19,20], we need to introduce following notations.
Let X, Y be real Banach spaces, L : DomL ⊂ X → Y be a Fredholm mapping of index zero, and let P : X → X, Q : Y → Y be continuous projectors such that ImP = KerL, KerQ = ImL and X = KerL⊕KerP, Y = ImL ⊕ ImQ.Denote by L P the restriction of L to DomL ∩ KerP .Denote by K P : ImL → KerP ∩DomL the inverse of L P , and denote by J : ImQ → KerL an isomorphism of ImQ onto KerL.
For convenience, we also cite below the continuation theorem.Clearly, KerL = R, ImL = {y ∈ X : ω 0 y(t)dt = 0} is closed in X and dimKerL = ImL = 1.Hence, L is a Fredholm mapping of index 0. Furthermore, through an easy computation, we find that the inverse K P of L P has the form We are now in a position to state and prove our main result.