An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks

Transcription

1 An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks J. Xu 1,, K. W. Chung 2 C. L. Chan 2 1 School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 292, P. R. China 2 Department of Mathematics City University of Hong Kong Kowloon, Hong Kong Submissions to SIAM Journal on Applied Dynamical Systems 1 August, 26 Number of Pages: 31 Enclosed: 2nd Revised Manuscript and Response Letter to Reviewers in PDF Format The author to receive correspondence and proofs. Tel: , Fax: xujian@mail.tongji.edu.cn

2 Abstract An efficient method, called the perturbation-incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, delay-induced weak or high-order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay-induced weak resonant double Hopf bifurcation and dynamics in the van der Pol-Duffing and the Stuar-Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively, and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is used as an initial guess for the incremental step which updates the approximate expression iteratively and performs parametric continuation. The analytical predictions on the two systems show that the delayed feedback can, on the one hand, drive a periodic solution into an amplitude death island where the motion vanishes and, on the other hand, create complex dynamics such as quasi-periodic and co-existing motions. The approximate expression of periodic solutions with parameter varying far away from the double Hopf point can be calculated to any desired accuracy by the incremental step. The validity of the results is shown by their consistency with numerical simulations. It is seen that as an analytical tool the PIS is simple but efficient. Key words. delay differential system, double Hopf bifurcation, delayed feedback control, nonlinear dynamics AMS subject classification. 34K6; 37H2; 7K3; 74G1; 93B52 2

3 AN EFFICIENT METHOD FOR STUDYING WEAK RESONANT DOUBLE HOPF BIFURCATION IN NONLINEAR SYSTEMS WITH DELAYED FEEDBACKS JIAN XU, KWOK-WAI CHUNG, AND CHUEN-LIT CHAN Abstract. An efficient method, called the perturbation-incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, delay-induced weak or high-order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay-induced weak resonant double Hopf bifurcation and dynamics in the van der Pol-Duffing and the Stuar-Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively, and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is used as an initial guess for the incremental step which updates the approximate expression iteratively and performs parametric continuation. The analytical predictions on the two systems show that the delayed feedback can, on the one hand, drive a periodic solution into an amplitude death island where the motion vanishes and, on the other hand, create complex dynamics such as quasi-periodic and co-existing motions. The approximate expression of periodic solutions with parameter varying far away from the double Hopf point can be calculated to any desired accuracy by the incremental step. The validity of the results is shown by their consistency with numerical simulations. It is seen that as an analytical tool the PIS is simple but efficient. Key words. delay differential system, double Hopf bifurcation, delayed feedback control, nonlinear dynamics AMS subject classifications. 34K6; 37H2; 7K3; 74G1; 93B52 1. Introduction. Time delay is ubiquitous in many physical systems. In particular, delay differential equations (DDEs) are used to model various phenomena, such as neural [1, 2], ecological [3], biological [4], mechanical [5, 6, 7, 8], controlling chaos [9], secure communication via chaotic synchronization [1, 11] and other natural systems due to finite propagation speeds of signals, finite reaction times and finite processing times [12]. These researches show that the time delay in various systems has not only a quantitative but also qualitative effect on dynamics even for small time delay [13]. Therefore, the investigation of the mechanism of how the delay induces various dynamics of a system becomes important. The qualitative and quantitative theories for DDEs are well developed recently. Many methods and techniques of geometric theory of dynamical systems on ordinary differential equations (ODEs) have been extended to investigate DDEs, such as stability analysis, bifurcation theory, perturbation techniques and so on. In stability analysis, Kalas and Baráková [14] set up a theoretical base for the stability and asymptotic behavior of a two-dimensional delay differential equation. Using a suitable Lyapunov-Krasovskii functional, they transformed a real two-dimensional system to a single equation with complex coefficients and proved the stability and asymp- This work is supported by the strategic research grand No of the City University of Hong Kong and National Nature and Science Foundation of China. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 292, P. R. China (xujian@mail.tongji.edu.cn). Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong (makchung@cityu.edu.hk) (mapeter@cityu.edu.hk). 1

4 2 J. XU, K. W. CHUNG AND C. L. CHAN totic behavior of the system under consideration. Gopalsamy and Leung [15] used perturbation method to study the stability of periodic solutions in delay differential equations. On the aspect of the bifurcation theory of DDEs, the center manifold of a flow is usually adopted to reduce DDEs to finite-dimensional systems. Redmond et. al. [16] studied completely the bifurcations of the trivial equilibrium and computed the normal form coefficients of the reduced vector field on the center manifold. The analysis reveals a Hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two Takens-Bogdanov point in the parameter space. Campbell and LeBlanc [17] used center manifold analysis to investigate 1:2 resonant double Hopf bifurcation in a delay differential equation. It is possible to find period doubling bifurcations near such point for a system without the time delay [18]. Correspondingly, perturbation method is also extended to determine the analytical solutions of DDEs arisen from bifurcations. Instead of center manifold reduction, Das and Chatterjee [19] employed the method of multiple scales (MMS) to obtain analytical solutions close to Hopf bifurcation points for DDEs. In applications, the main steps of the analysis are schemed as follows [2]: a. consider an equilibrium at a critical parameter, b. solve the eigenvalue problem for this equilibrium to find its linear stability, c. localize the critical point in parameter space where bifurcation occurs, d. calculate the eigenvalues and eigenfunctions at the bifurcation point and reduce DDEs on the center manifold, e. compute the appropriate normal form coefficients. In the literature mentioned above, some of them deal with controlled systems with delayed feedbacks. In our previous research [21, 22], it has been found that controlled systems with delayed feedbacks may undergo a double Hopf bifurcation with the time delay and feedback gain varying. However, mechanism has not been explained in details. Such phenomenon has also been observed by Reddy et al [12], Campbell et al [2]. Recently, Buono and Bélair [23] employed the methods developed by Faria and Magalhães [24] to investigate the normal form and universal unfolding of a vector field at non-resonant double Hopf bifurcation points for particular classes of retarded functional differential equations (RFDEs). They represented restrictions on the possible flows on a center manifold for certain singularities. Buono and LeBlanc [25] extended Arnold s theory on universal unfolding of matrices to the case of parameter-dependent linear RFDEs. In studying delay systems governed by DDEs, many authors recommend a center manifold reduction (CMR) as the first step due to the fact that DDEs and RFDEs are infinite dimensional. However, the CMR has its disadvantages. On the one hand, computation of the CMR with normal form is very tedious for a codimension-2 bifurcation. On the other hand, the CMR is invalid for values of the bifurcation parameters far away from the bifurcation point. This constitutes the motivation of the present paper. Our goal is to propose a simple but efficient method, which not only inherits the advantages of the CRM and MMS, but also overcomes the disadvantages of them, to explain mechanism of delay-induced double Hopf bifurcation. To this end, this paper is focused on controlled systems with delayed feedbacks governed by a type of twodimensional delay differential equations and proposes a method called perturbationincremental scheme (PIS). The scheme is described in two steps, namely the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. The perturbation step of the PIS provides not only accurate qualitative prediction and classification but also an analytical expression

5 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 3 with high accuracy for both periodic and quasi-periodic motions when bifurcation parameters are closed to the weak resonant point. For those values of bifurcation parameters far away from the resonant point, the analytical expression may not be accurate enough, but it can be considered as an initial guess in the incremental step. Thus this overcomes the disadvantage of the incremental harmonic balance (IHB) method. From the incremental step, the approximate analytical expression is updated iteratively and can reach any desired accuracy. Parametric continuation can also be performed by using the PIS. As for applications of the method, the delay-induced weak resonant double Hopf bifurcation and dynamics in the van der Pol-Duffing and the Stuar-Landau systems with delayed feedback are investigated. The analytical predictions show that the delayed feedback can lead not only to the vanishing of periodic motion in an amplitude death island, but also to more complex dynamics such as quasi-periodic and co-existing motions. The validity of the results is shown by their consistency with numerical simulations. As an analytical tool, the PIS is simple but efficient. The approximate expression obtained from the PIS can achieve any required accuracy which is not possible by the center manifold reduction and other perturbation methods. It should be noted that this work is also an extension of our previous studies [22, 26, 27, 28]. The paper is organized as follows. In section 2, we discuss in details the corresponding linearized systems when two pairs of purely imaginary eigenvalues occur at a critical value of time delay, giving arise to weak resonant double Hopf bifurcations. In section 3, a new method, called the perturbation-incremental scheme, is proposed in two steps to investigate harmonic solutions derived from weak resonant double Hopf bifurcation. As illustrative examples, the van der Pol-Duffing and the Stuar-Landau systems are considered in sections 4 and 5 where the analytical and numerical results are compared. Finally, we close the paper with a summary of our results. 2. Weak Resonant Double Hopf Bifurcation. We consider a type of two first-order delay differential equations with linear delayed feedback and nonlinearities in the following general form (2.1) Ż(t) = C Z(t) + D Z( t τ) + ɛ F ( Z(t), Z(t τ) ) where[ Z(t) = {x(t), ] y(t)} T R[ 2, C and D] are 2 2 real constant matrices such that c11 c C = 12 d11 d and D = 12, F is a nonlinear function in its variable c 21 c 22 d 21 d 22 with F (, ) =, ɛ is a parameter representing strength of nonlinearities, and τ is the time delay. Now we derive some formulae relating to resonant double Hopf bifurcation points. It can be seen from equation (2.1) that Z = is always a equilibrium point or trivial solution of the system. To determine the stability of the trivial solution for τ, we linearize system (2.1) around Z = to obtain the characteristic equation (2.2) det(λ I C D e λ τ ) =, where I is the identity matrix and the perturbation is assumed to have a time dependence proportional to e λ τ. The characteristic equation (2.2) can be rewritten in the form (2.3) λ 2 λ ( c 1 + d 1 e λ τ ) + c d e λ τ + c 2 + det(d) e 2λτ =,

6 4 J. XU, K. W. CHUNG AND C. L. CHAN where (2.4) c 1 = c 11 + c 22 d 1 = d 11 + d 22 c 2 = c 11 c 22 c 12 c 21 c d = c 22 d 11 c 21 d 12 c 12 d 21 + c 11 d 22. The roots of the characteristic equation (2.3) are commonly called the eigenvalues of the equilibrium point of system (2.1). The stability of the trivial equilibrium point will change when the system under consideration has zero or a pair of imaginary eigenvalues. The former occurs if λ = in equation (2.3) or c d + c 2 + det(d) =, which can lead to the static bifurcation of the equilibrium points such that the number of equilibrium points changes when the bifurcation parameters vary. The latter deals with the Hopf bifurcation such that the dynamical behavior of the system changes from a static stable state to a periodic motion or vice versa. The dynamics becomes quite complicated when the system has two pairs of pure imaginary eigenvalues at a critical value of time delay. We will concentrate on such cases. For this, we let c d + c 2 + det(d). Thus, λ = is not a root of the characteristic equation (2.3) in the present paper. Such assumption can be realized in engineering as long as one chooses a suitable feedback controller. It is easy to find explicit expressions for the critical stability boundaries of the following two cases: a. det(d) =, c d + c 2 ; b. det(d), c 11 = c 22, c 12 = c 21, d 11 = d 22, d 12 = d 21 =, c 11 >, c 12 >, d 11. For the case with det(d) = but c d + c 2, substituting λ = a + iω into (2.3), and equating the real and imaginary parts to zero yields (2.5) a 2 ω 2 a c 1 + c 2 e a τ ω sin(τ ω) d 1 + e a τ cos(τ ω) ( c d a d 1 ) =, 2 a ω ω c 1 e a τ ω cos(τ ω) d 1 + e a τ sin(τ ω) ( a d 1 c d ) =. One can derive the explicit expressions for the critical stability boundaries by setting a = in equation (2.5) and obtain (2.6) ω 2 + c 2 + c d cos(τ ω) ω d 1 sin(τ ω) =, ω c 1 c d sin(τ ω) ω d 1 cos(τ ω) =. Eliminating τ from equation (2.6), we have (2.7) ω ± = d 1 2 c c 2 ± (d1 2 c c 2 ) 2 4 (c22 c d2 ) 2 when the following conditions hold: (2.8) c 2 2 c 2 d >, ( d1 2 c c 2 ) 2 > 4 ( c2 2 c d 2 ). Then, two families of surfaces, denoted by τ and τ + in terms of c d and d 1 corresponding to ω and ω + respectively, can be derived from equation (2.5) and be given

7 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 5 by (2.9) cos( ω τ ) = ω2 c d c 2 c d ω 2 c 1 d 1 c d2 + ω 2 d 1 2, cos( ω + τ + ) = ω2 + c d c 2 c d ω 2 + c 1 d 1 c d2 + ω 2 + d 1 2. It should be noted that ω < ω +. Thus, a possible double Hopf bifurcation point occurs when two such families of surfaces intersect each other where (2.1) τ = τ +. Equation (2.1) not only determines the linearized system around the trivial equilibrium which has two pairs of pure imaginary eigenvalues ±iω and ±iω +, but also gives a relation between ω and ω +. If (2.11) ω : ω + = k 1 : k 2, then a possible double Hopf bifurcation point appears with frequencies in the ratio k 1 : k 2. If k 1, k 2 Z +, k 1 < k 2, k 1 and k 2 1, then such point is called the k 1 : k 2 weak or no low-oder resonant double Hopf bifurcation point. Equations (2.1) and (2.11) form the necessary conditions for the occurrence of a resonant double Hopf bifurcation point. Equation (2.11) yields (2.12) d 1 2 = c c 2 + k k 2 2 k 1 k 2 c22 c d2, if conditions (2.8) are satisfied. Substituting (2.12) into (2.7), one can obtain the frequencies in the simple expressions given by k1 k2 (2.13) ω = c22 c d2, ω + = c22 c d2. k 2 k 1 The other parameters can be determined by equation (2.1) or the following equation arccos ( (c2 c d k 2)+ c 2 2 c 2 d (c d c 1 d 1) k 1 (2.14) = k1 ) c d2 k 2+ c 22 c d2 d 2 1 k 1 k 2 arccos( (c2 c d k 1)+ c 22 c d2 (c d c 1 d 1) k 2 ), c 2 2 c 2 d d 2 1 k 2 c d 2 k 1+ where d 1 is given in equation (2.12). The corresponding value of the time delay at the resonant double Hopf bifurcation point is given by (2.15) τ c = τ = τ + k 1 = k 2 c22 c arccos( (c 2 c d k 1 ) + c22 c d2 (c d c 1 d 1 ) k 2 2 d c d2 k 1 + c 22 c d2 d 2 ). 1 k 2 For the case with det(d) and c 11 = c 22, c 12 = c 21, d 11 = d 22, d 12 = d 21 =, c 11 >, c 12 >, d 11 <, the characteristic equation (2.2) becomes (2.16) λ = c 11 ± ic 12 + d 11 e λτ.

8 6 J. XU, K. W. CHUNG AND C. L. CHAN By substituting λ = a + iω in (2.16), we get (2.17) ω = ω ± = c 12 ± d 2 11 e 2aτ (a c 11 ) 2, a = c 11 ω c21 tan(ωτ), where we consider only one set of curves by choosing ω = c 12 ±. The other set of curves arising due to ω = c 12 ± is implicit in the above since the eigenvalues always occur in complex conjugate pairs. It follows from (2.17) that ω is real only when d 2 11 (a c 11 ) 2 e 2aτ. To obtain the critical boundary, set a =. This gives (2.18) ω = c 12 d 11 2 c 112, ω + = c 12 + d 2 11 c 2 11, and (2.19) τ [j] = 1 ω ( 2jπ cos 1 ( c 11 /d 11 ) ), τ + [j] = 1 ω + ( 2jπ + cos 1 ( c 11 /d 11 ) ), where j = 1, 2,... The necessary conditions for the k 1 : k 2 occurrence of the resonant double Hopf bifurcation can be obtained by setting τ [j] = τ + [j] and ω ω + = k1 k 2. This yields (2.2) c 11 (d 11 ) c = cos(2jπ(k 2 k 1 )/(k 1 + k 2 )), τ 4jk 1 π c =. (k 1 + k 2 )ω Finally, it should be noted that the parameters d 1 and c d cannot be solved in a closed form from equation (2.15) due to the trigonometric function. However, the values can be obtained numerically. Such parameters denoted by ( ) c are called the critical values at the resonant double Hopf bifurcation point with frequencies in the ratio k 1 : k 2. Thus, for given physical parameters τ and D in equation (2.1), one can obtain (2.21) ɛ τ ɛ = τ τ c, ɛ D ɛ = D D c, such that equation (2.1) can be rewritten as (2.22) Ż = C Z + D c Z τc + F (Z, Z τc, Z τc+ɛτ ɛ, ɛ), where Z = Z(t), Z τ = Z(t τ), τ c is given by (2.15) and (2.23) F ( ) = D c [Z τc+ɛτ ɛ Z τc ] + ɛ [D ɛ Z τc+ɛτ ɛ + F (Z, Z τc+ɛτ ɛ )]. Clearly, F = for ɛ = and the resonant double Hopf bifurcation point may occur in the system when ɛ =. 3. Perturbation-Incremental Scheme. Various harmonic solutions with distinct topological structures can occur in a system due to non-resonant and high-order double Hopf bifurcation[32]. When the time delay is absent, the perturbation method [5] can be applied directly to equation (2.22) for small ɛ, and the IHB method [7] to

9 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 7 the system for large ɛ. The key problem of IHB is to find an initial value and in general it is quite difficult. The PI method proposed in [26, 27] can efficiently overcome this disadvantage of the IHB method. However, the above methods must be re-examined and extended to investigate the system with time delay. In this section, we propose a new method, called the perturbation-incremental scheme (PIS) to investigate harmonic solutions derived from weak resonant double Hopf bifurcation of equation (2.1). Our goal is to obtain the harmonic solutions with any desired accuracy and consider the continuation of these solutions when the time delay and feedback gain in (2.1) are taken as bifurcation parameters. The scheme is described in two steps, namely the perturbation step (noted as step one) for bifurcation parameters close to the weak resonant point and the incremental step (noted as step two) for those far way from the bifurcation point Perturbation Step of PIS. It can be seen from the previous section that a double Hopf bifurcation with k 1 :k 2 weak resonance occurs in (2.1) at τ = τ c and D = D c. If τ and D are considered as two bifurcation parameters, then (D c, τ c ) is a double Hopf bifurcation point with weak resonance. In this subsection, we derive the analytic expression of harmonic solutions arising from weak resonant double Hopf bifurcation in the equation (2.1) or (2.22) when τ and D are close to τ c and D c, respectively. For ɛ =, it can be seen from (2.21) and (2.23) that τ = τ c, D = D c and F ( ) =. Thus, the solution of the equation (2.22) may be supposed as (3.1) which results in (3.2) Z (t τ c ) = Z (t) = 2 i=1 2 i=1 [{ aki c ki [{ aki } c ki } { } ] bki cos(k i φ) + sin(k d i φ), ki { } ] bki cos(k i φ k i ωτ c ) + sin(k d i φ k i ωτ c ), ki where τ c is given by (2.15), φ = ωt and ω = ω = ω + is determined by (2.13). k 1 k 2 Substituting (3.1) and (3.2) into the equation (2.22) for ɛ = and using the harmonic balance, one obtains that { } { } bki aki (3.3) M ki = N d ki, ki c ki and (3.4) M ki { aki c ki } { } bki = N ki, d ki where M ki = k i ωi + D c sin(k i ωτ), N ki = C + D c cos(k i ωτ) and I is the { 2 } 2 ãki identity matrix. The equations (3.3) and (3.4) are in fact identical. Let = bki { } N ki aki and note that det(n det(n ki ) b ki ) = det(m ki ). It follows from (3.3) that the ki harmonic solution of the equation (2.22) for ɛ = is given by (3.5) Z (t) = 2 i=1 [Ñki cos(k i φ) + M ] { } ã ki ki sin(k i φ) bki

10 8 J. XU, K. W. CHUNG AND C. L. CHAN with Ñk i = N 1 k i det(n ki ), M ki = M 1 k i det(m ki ) for i = 1, 2. Based on the expression in (3.5), we now consider the solution of (2.22) for small ɛ. The harmonic solution of equation (2.22) can be considered to be a perturbation to that of (3.5), given by (3.6) Z(t) = 2 [Ñki cos(k i ωt + ɛσ i t) + M ki sin(k i ωt + ɛσ i t) i=1 ] { a ki (ɛ) b ki (ɛ) where a ki () = ã ki, b ki () = b ki, and σ 1 and σ 2 are{ detuning } parameters. The aki (ɛ) following theorem provides a new method to determine in (3.6). b ki (ɛ) Theorem If W (t) is a periodic solution of the equation }, (3.7) Ẇ (t) = C T W (t) D c T W (t + τ c ), and W (t) = W (t + 2π/ω), then τ c [ Dc T W (t + τ c ) ] T [Z(t) Z(t + 2π/ω)] dt (3.8) [W ()] T [Z(2π/ω) Z()] + 2π/ω [W (t)] T F (Z, Zτc, Z τc+ɛτ ɛ, ɛ) dt =. Proof Multiplying both sides of (2.22) by [W (t)] T and integrating with respect to t from zero to 2π/ω, one has (3.9) 2π/ω [W (t)] T Ż(t)dt = 2π/ω [W (t)] [C T Z(t) + D c Z(t τ c ) + F ] (Z, Z τc, Z τc+ɛτ ɛ, ɛ) dt, where 2π/ω is a period of W (t) in t. The equation (3.9) yields (3.1) ] 2π/ω T [Ẇ (t) + C T W (t) + D T c W (t + τ c ) Z(t) dt + τ c [ Dc T W (t + τ c ) ] T [Z(t) Z(t + 2π/ω)] dt [W ()] T [Z(2π/ω) Z()] + 2π/ω [W (t)] T F (Z, Zτc, Z τc+ɛτ ɛ, ɛ) dt =. The theorem follows from (3.7). { } aki (ɛ) To applied the theorem to determine, one must obtain the expression b ki (ɛ) of W (t) in (3.1). It is easily seen that the periodic solution of the equation (3.7) can been written as 2 [ W (t) = ( Ñk i ) T cos(k i φ) + ( M ki ) T sin(k i φ)] { } p (3.11) ki, q ki i=1 where p ki and q ki are independent constants. Substituting (3.6) and (3.11) into (3.8), neglecting two order terms in power ɛ and noting the independence of p ki and q ki

11 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 9 yield four algebraic equations in a ki (ɛ), b ki (ɛ), σ 1 and σ 2. For ɛ, a ki (ɛ) and b ki (ɛ) are dependent. Therefore, we change this four algebraic equations in polar form by setting (3.12) a ki (ɛ) = b ki (ɛ) = r ki (ɛ) (c 12 sin(θ i ) + d 12 sin(k i ωτ c + θ i )) c 12 (k i ω + d 22 sin(k i ωτ c )) + d 12 (k i ω cos(k i ωτ c ) c 22 sin(k i ωτ c )), r ki (ɛ) (c 22 sin(θ i ) + d 22 sin(k i ωτ c + θ i ) + k i ω cos(θ i )) c 12 (k i ω + d 22 sin(k i ωτ c )) + d 12 (k i ω cos(k i ωτ c ) c 22 sin(k i ωτ c )), where i = 1, 2 and (r k1, r k2, θ 1, θ 2 ) is a polar coordinate system. Using the fact that cos 2 (θ i ) + sin 2 (θ i ) = 1 (i = 1, 2), one can solve r k1 (ɛ), r k2 (ɛ), σ 1 and σ 2 from this four algebraic equations. Thus, when the time delay and feedback gain is very close to the double Hopf bifurcation point (i.e. ɛτ c and ɛd ɛ are very small), the approximate solution in O(ɛ) is expressed as { } rk1 cos((k Z(t) = 1 ω + ɛσ 1 )t + θ 1 ) + r k2 cos((k 2 ω + ɛσ 2 )t + θ 2 ) (3.13), ( ) in terms of (3.6) and (3.12), where ( ) is a complicated expression. Furthermore, when θ 1 and θ 2 are determined from the initial conditions, a ki (ɛ) and b ki (ɛ) in (3.6) can be obtained from (3.12), denoted as a k i (ɛ) and b k i (ɛ). Thus, equation (3.6) becomes (3.14) Z(t) = 2 [Ñki cos(k i ωt + ɛσ i t) + M ki sin(k i ωt + ɛσ i t) i=1 ] { a k i (ɛ) b k i (ɛ) Up to now, an approximate solution is obtained from the perturbation step of the PIS. It should be noted that the solution (3.6) is a perturbation to (3.5) in a ɛ-order magnitude. Besides, the four algebraic equations are obtained in O(ɛ). Therefore, the solution of these four algebraic equations is accurate for small ɛ. However, the case for large ɛ should be investigated further. A large ɛ perturbation yields one of the following cases: a. both ɛτ ɛ and ɛd ɛ are small but the nonlinearity in (2.1) is strong, b. ɛd ɛ is small but ɛτ ɛ is large and the nonlinearity in (2.1) is strong, c. ɛτ ɛ is small but ɛd ɛ is large and the nonlinearity in (2.1) is strong, d. both ɛτ ɛ and ɛd ɛ are large and the nonlinearity in (2.1) is strong. As one will see in sections 4 and 5, the perturbation step of the PIS is still valid for case (a) but invalid for cases (b), (c) and (d). It implies that the approximate solution given in (3.14) is accurate enough to represent the motions near a double Hopf bifurcation point with weak resonance as long as the time delay and feedback gain are close to the point even for very large ɛ. Thus, the second step of the PIS is required for cases (b), (c) and (d) for which the perturbation step is invalid. For these three cases, the values of (τ, D) are far away from (τ c, D c ) Incremental Step of PIS. In this subsection, the incremental step of the PIS is proposed in details for case (b) since we focus on effects of the time delay on equation (2.1) in the present paper. It is an extension of our previous work in [26, 27]. Similar to the formulation in [26, 27], a time transformation is first introduced as (3.15) dφ dt = Φ(φ), Φ(φ + 2π) = Φ(φ), }.

12 1 J. XU, K. W. CHUNG AND C. L. CHAN where φ is the new time. Thus, Φ(φ) can be approximately expanded in a truncated Fourier s series about φ as (3.16) m Φ(φ) = (p j cos j φ + q j sin j φ). j= In the φ domain, (2.1) or (2.22) is rewritten as (3.17) ΦZ = C Z + D Z τ + ɛ F ( Z, Z τ ), where prime denotes differentiation with respect to φ, D = D c + ɛ D ɛ, τ = τ c + ɛτ ɛ. If φ 1 corresponds to t τ, it follows from (3.15) that (3.18) dt = dφ Φ(φ) = dφ 1 Φ(φ 1 ) = Φ(φ) dφ 1 dφ = Φ(φ 1). We note that φ 1 φ is a periodic function in φ with period 2π. Similarly, φ 1 φ can also be expanded in a truncated Fourier s series about φ as (3.19) m φ 1 = φ + (r j cos jφ + s j sin jφ). j= The integration constant of (3.18) provides information about the delay τ. Since φ 1 is the new time corresponding to t τ, it follows from (3.18) that (3.2) t = τ = dt 1 = t τ φ φ 1 φ φ 1 dθ Φ(θ). dθ Φ(θ) To consider the continuation with the delay τ as the bifurcation parameter, ɛ and D are kept fixed such that D is close to D c. If (3.17) possesses a periodic solution at τ = τ = τ c + ɛτ ɛ { and the } expression {(3.14) provides } a sufficiently accurate a representation, where either k1 (ɛ) a b = or k2 (ɛ) k 1 (ɛ) b =, then a periodic solution k 2 (ɛ) at τ = τ + τ can be expressed in a truncated Fourier series as (3.21) m Z = (a j cos j φ + b j sin j φ), j= where a j, b j R 2, m k 2 > k 1. Correspondingly, one has (3.22) m Z τ = (a j cos jφ 1 + b j sin jφ 1 ) j=

13 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 11 For τ =, one can easily obtain that { } a Ñ ki (ɛ) ki a j = b j = k k i (ɛ) i, j k i (3.23) { } a M ki (ɛ) ki b j = b j = k k i (ɛ) i, j k i, where i = 1, 2. The coefficients of Φ(φ) in (3.16) are given by { } a ω + ɛσ 1 /k 1 for k2 (ɛ) b =, k 2 (ɛ) (3.24) p = { } a ω + ɛσ 2 /k 2 for k1 (ɛ) b =, k 1 (ɛ) and p j =, q j = for all j >. From (3.19) and (3.2), one has also (3.25) r = p τ, and r j =, s j = for all j >. A small increment of τ from the initial value of τ to τ + τ yields small increments of the following quantities (3.26) Z Z + Z, Z τ Z τ + Z τ, Φ Φ + Φ, and φ 1 φ 1 + φ 1. Substituting (3.26) into (3.17) and (3.18), and expanding in Taylor s series about an initial solution, one can obtain linearized incremental equations by ignoring all the non-linear terms of small increments as below Z Φ(φ) + Φ(φ) Z C Z D Z τ (3.27) ɛ ( F (Z,Z τ ) Z Z F (Z,Zτ ) Z τ ) Z τ = C Z + D Z τ + ɛ F ( Z, Z τ ) Φ(φ)Z, (3.28) φ 1 Φ(φ) + Φ(φ) φ 1 Φ(φ 1 ) Φ (φ 1 ) φ 1 = Φ(φ 1 ) Φ(φ)φ 1, where the subscript represents the evaluation of the relevant quantities corresponding to the initial solution. From (3.16), (3.19), (3.21) and (3.22), one has (3.29) (3.3) Φ(φ) = m j= ( p j cos jφ + q j sin jφ), Φ (φ) = m j=1 j( q j cos jφ p j sin jφ), φ 1 = m j= ( r j cos jφ + s j sin jφ), φ 1 = m j=1 j( s j cos jφ r j sin jφ).

14 12 J. XU, K. W. CHUNG AND C. L. CHAN (3.31) Z = m j= ( a j cos jφ + b j sin jφ), Z = m j=1 j( b j cos jφ a j sin jφ), and (3.32) Z τ = m ( a j cos jφ 1 + b j sin jφ 1 ) + Z τ φ 1. φ 1 j= In addition, a small increment of τ to τ + τ also yields the linearized incremental equation of (3.2) as (3.33) φ φ 1 Φ(θ) Φ 2 (θ) dθ + φ φ 1 Φ(φ 1 ) = dθ φ 1 Φ(θ) τ τ, which implies, for φ =, (3.34) ξ Φ(θ) Φ 2 (θ) dθ + φ 1() dθ = Φ(α) ξ Φ(θ) τ τ. where ξ = φ 1 (). Substituting (3.29)-(3.32) into (3.27) and using the harmonic balance method, one obtains the linearized equation (3.27) in terms of the increments a j, b j, p j, q j, r j and s j as m j= (Ψ 1,j a j + Ψ 2,j b j + Ψ 3,j p j + Ψ 4,j q j + Ψ 5,j r j + Ψ 6,j s j ) (3.35) = Λ 1, where Ψ 1,j = jφ(φ) sin jφ C cos jφ D cos jφ 1 ɛ Ψ 2,j = jφ(φ) cos jφ C sin jφ D sin jφ 1 ɛ Ψ 3,j = Z cos jφ, Ψ 4,j = Z sin jφ, Ψ 5,j = D Zτ φ 1 cos jφ F Z τ Ψ 6,j = D Zτ φ 1 sin jφ F Z τ Z τ φ 1 Z τ φ 1 cos jφ, sin jφ, Λ 1 = C Z + D Z τ + ɛ F ( Z, Z τ ) Φ(φ)Z, (3.36) ( ( F Z F Z ) cos jφ + F Z τ cos jφ 1, ) sin jφ + F Z τ sin jφ 1,

15 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 13 Similarly, from (3.28) and (3.34), we obtain, respectively, (3.37) m [Ψ 7,j p j + Ψ 8,j q j + Ψ 9,j r j + Ψ 1,j s j ] = Λ 2, j= and (3.38) m [Ψ 11,j p j + Ψ 12,j q j + Ψ 13,j r j ] = Λ 3, j= where Ψ 7,j = φ 1 cos jφ cos jφ 1, Ψ 8,j = φ 1 sin jφ sin jφ 1, Ψ 9,j = jφ(φ) sin jφ Φ (φ 1 ) cos jφ, Ψ 1,j = jφ(φ) cos jφ Φ (φ 1 ) sin jφ, (3.39) Ψ 11,j = ξ Ψ 12,j = ξ cos jθ Φ 2 (θ) dθ, sin jθ Φ 2 (θ) dθ, Ψ 13,j = 1 Φ(ξ), Λ 2 = Φ(φ 1 ) Φ(φ)φ 1, Λ 3 = ξ dθ Φ(θ) τ τ. Since Ψ i,j (1 i 13, 1 j m) and Λ k (1 k 3) are periodic functions in φ, they can be expressed in Fourier series the coefficients of which can easily be obtained by the method of Fast Fourier Transform (FFT). Let a ij, b ij R (1 i 2, j m) be the i-th element in a j and b j, respectively. By comparing the coefficients of 2(2m + 1) + 1 harmonic terms of (3.35), 2m + 1 of (3.37) and (3.38), a system of linear equations is thus obtained with unknowns a ij, b ij, p j, q j, r j and s j in the form 2 m i=1 j= (A k,ij a ij + B k,ij b ij )+ (3.4) m j= (P k,j p j + Q k,j q j + R k,j r j + S k,j s j ) = T k, where k = 1, 2,..., 3(2m + 1) + 2 and T k are the residue terms. The values of a j, b j, p j, q j, r j and s j are updated by adding the original values and the corresponding incremental values. The iteration process continues until T k for all k (in practice, T k is less than a desired degree of accuracy). The entire incremental process proceeds by adding the τ increment to the converged value of τ, using the previous solution as the initial approximation until a new converged solution is obtained. We note that

16 14 J. XU, K. W. CHUNG AND C. L. CHAN the value of m may be changed during the continuation so as to ensure sufficient accuracy of the solution. The stability of a periodic solution can be determined by the Floquet method [34, 35]. Let ζ R 2 be a small perturbation from a periodic solution of (2.1). Then, (3.41) dζ dϕ = 1 Φ [A(ϕ, ϕ 1)ζ + B(ϕ, ϕ 1 )ζ τ ] + O(ζ 2, ζ τ 2 ), F (Z,Zτ ) F (Z,Zτ ) where A(ϕ, ϕ 1 ) = C + ɛ Z and B(ϕ, ϕ 1 ) = D + ɛ Z τ. The entities of A and B are all periodic functions of ϕ with period 2π, which can be determined by using the incremental procedure. The time delay interval I 1 = [ τ, ] corresponds to I 2 = [α, ] in the ϕ domain. Discrete points in I 2 are selected for the computation of Floquet multipliers. From the incremental procedure, the Fourier coefficients of ϕ 1 in (3.3) are obtained. Assume that ϕ = β when ϕ 1 = and let I 3 = [, β]. For each ϕ I 3, there corresponds a unique ϕ 1 I 2. We choose a mesh size h = and discrete points β N 1 ϕ (i) = ih ( i N 1) in I 3, which correspond to ϕ (i) 1 = ϕ 1 (ϕ (i) ) in I 2. Let ζ(ϕ (i) 1 ) be the (i + 1)-th unit vector in R 2. By applying numerical integration to (3.1), we obtain the monodromy matrix M as (3.42) M = [ζ(ϕ () 1 + 2π), ζ(ϕ (1) 1 + 2π),, ζ(ϕ (N 1) 1 + 2π)]. The eigenvalues of M are used to determine the stability of the periodic solution. One of the eigenvalues or Floquet multipliers of M must be unity which provides a check for the accuracy of the calculation. If all the other eigenvalues are inside the unit circle, the periodic solution under consideration is stable; otherwise, it is unstable. Although the incremental step described above is for case (b), it can be extended in a similar way to cases (c) and (d). In this section, we have proposed the two steps of the PIS in details. As for applications, two examples will be investigated in the next two sections to illustrate the validity of the PIS. 4. Weak Resonant Double Hopf Bifurcation for van der Pol-Duffing System with Delayed Feedback. First, we consider the van der Pol-Duffing oscillator with linear delayed position feedback governed by (4.1) ü (α ɛ γ u 2 ) u + ω 2 u + ɛ β u 3 = A(u τ u) where α, γ, β are positive constants, A <, τ time delay, ɛ small perturbation parameter and u τ = u(t τ). The system (4.1) can be expressed in the form of (2.1), where (4.2) which imply, from (2.4) (4.3) c 11 =, c 12 = 1, c 21 = ω 2 A, c 22 = α, d 11 =, d 12 =, d 21 = A, d 22 =, c 1 = α, c 2 = A + ω 2, d 1 =, c d = A. Substituting (4.2) into (2.7)-(2.9), one obtains that if (4.4) 2 A + ω 2 >, ( 2 A + α 2 ) 2 4 α 2 ω 2 >,

17 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 15 Table 4.1 Some critical values at possible weak resonant double Hopf bifurcations for the system (4.1). j ω : ω + α A c ω τ c 3: : : : : : : : : : then (4.5) α ω ± = 2 A 2 + ω 2 ± ( ) 2 A α2 α 2 2 ω 2, (4.6) cos(ω ± τ) = 1 + ω 2 2 ω ±, sin(ω ± τ) = α A A ω ±. For A < and α, it follows from (4.6) that (4.7) τ ± [j] = 1 [ ( 2jπ cos ( 1) 1 + ω2 ω 2 )] ±, ω ± A where j = 1, 2, 3,, and ω ± are determined by (4.5). With aids of (2.12) and (2.15), the necessary conditions in terms of the critical values A c and τ c for the occurrence of resonant double Hopf points with frequencies in the ratio k 1 : k 2 are given by (4.8) α 2 2 (A c + ω 2 ) + (k k 2 2 )ω k 1 k 2 2Ac + ω 2 =, (4.9) τ c = τ + = τ, and the corresponding frequencies are (4.1) ω = k1 k 2 ω 2Ac + ω 2, ω + = k2 k 1 ω 2Ac + ω 2, where ω is a constant. For α, A c cannot be solved from (4.9) in a closed form, but they can be easily solved numerically for a fixed ω. Some values of the possible weak resonant double Hopf bifurcation are shown in Table (4.1) for ω = 1. In Fig. 1, we plot the diagrams for the case with α = in Table (4.1). The grey color regions show the stable trivial solutions for system (4.1), i.e. amplitude death regions. The intersection points located on the amplitude death regions are two double Hopf bifurcation points with 3:5 resonance. To obtain the neighboring solutions derived from such double Hopf

18 16 J. XU, K. W. CHUNG AND C. L. CHAN bifurcation, we let A = A c + ɛa ɛ and τ = τ c + ɛτ ɛ for a given α, where ɛa ɛ and ɛτ ɛ are very small. Thus, system (4.1) can be rewritten as (4.11) Ż(t) = C Z(t) + D c Z( t τ c ) + F (Z(t), Z(t τ c ), Z(t τ c ɛτ ɛ ), ɛ), where (4.12) Z(t) = { F = { u(t) v(t) } [, C = 1 (A c + ω 2 ) α ] [, D c = A c A c (u τc+ɛτ ɛ u τc ) + ɛ ( A ɛ (u τc+ɛτ ɛ u) u 2 (βu + γv) ) }, ], where u τ = u(t τ) τ 6 3 : A FIG. 1 High-order resonant double Hopf bifurcation diagram with frequencies in the ratio ω : ω + = k 1 : k 2 for α = and ω = 1., where solid line represents τ +, dashing τ, grey region amplitude death. As an example, we consider the case shown in Fig. 1 with ω = 1, α =.17468, where a 3:5 weak resonant double Hopf bifurcation occurs in system (4.1) at A c = and τ c = (cf. Lemma 8.15 in [36]). It follows from (3.11), (3.12) and (3.13) that (4.13) W (t) = { } w1 (t), w 2 (t)

19 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 17 with w 1 (t) =.17468p k1 cos( t).17468p k2 cos( t) q k1 cos( t) q k2 cos( t) (4.14) p k1 sin( t).12345q k1 sin( t) p k2 sin( t).17574q k2 sin( t), w 2 (t) = p k1 cos( t) + p k2 cos( t) q k1 sin( t) q k2 sin( t), (4.15) a k1 = r k1 sin(θ 1 ), b k1 = r k1 cos(θ 1 ) r k1 sin(θ 1 ), a k2 = 1.247r k2 sin(θ 2 ), b k2 = r k2 cos(θ 2 ) r k2 sin(θ 2 ), and u(t) = r k1 cos(θ 1 + t ( ɛσ 1 )) + r k2 cos(θ 2 + t ( ɛσ 2 )) (4.16) v(t) = r k1 sin(θ 1 + t ( ɛσ 1 )) r k2 sin(θ 2 + t ( ɛσ 2 )) Substituting (4.12), (4.13), (4.14) and (4.16) into (3.8), noting p 1, p 2, q 1 and q 2 being independent and cos 2 (θ i ) + sin 2 (θ i ) = 1 (i = 1, 2) yield r k1 ɛ ( A ɛ r k1 2 β r k2 2 β σ τ ɛ ) =, r k1 ɛ ( ) A ɛ r 2 k1 γ r 2 k2 γ σ τ ɛ =, (4.17) r k2 ɛ ( ) A ɛ r 2 k1 β r 2 k2 β 6.865σ τ ɛ =, r k2 ɛ ( A ɛ r k1 2 γ r k2 2 γ σ τ ɛ ) =. It can be seen from (4.16) that r k1, r k2, σ 1 and σ 2 determine the feature of motions of the system (4.11) when the double Hopf point at (A c, τ c ) is perturbed by A ɛ and τ ɛ for given values of ɛ, β and γ. Such motion can be amplitude death (r k1 = and r k2 = ), periodic (r k1 = and r k2 or r k1 and r k2 = ) and quasi-periodic (r k1 and r k2 ). Therefore, it is necessary to classify the solutions of the algebraic equation (4.17) in the neighborhood of the double Hopf point (A c, τ c ). To this end, one has firstly to distinguish so called simple and difficult double Hopf cases (see, for example, section 8.6 in [36]). In the simple case, the truncated cubic amplitude system (4.17) has no periodic orbits and addition of any 4th- and 5th-order terms does not change its bifurcation diagram. In the difficult case, one has to consider the truncated 5th-order amplitude system to stabilize the Hopf bifurcation there. This Hopf bifurcation implies the existence of 3D invariant tori in the full 4D system on the central manifold. Both cases can be seen when one takes β = γ and β γ =, respectively, as shown in Figs. 2(a) and 2(b). We do not discuss the simple case shown in Fig. 2(a) here as a similar case will be discussed at length in the next section.

20 18 J. XU, K. W. CHUNG AND C. L. CHAN As for the difficult case with γ =, solving the equation (4.17) yields that (rk1, rk2 ) = (rk1, rk2 ) = (, ) is always a root and that up to three other roots (in the positive quadrant) can appear, as follows: q ² ² (rk1, rk2 ) = (rk1 1, rk2 ) = ( A².66777τ, ) for A² τ <, β q β ² ² ) for A² τ <, (rk1, rk2 ) = (rk1, rk2 1 ) = (, A² τ β β (4.18) q q A².19497τ² ² (rk1, rk2 ) = (r12, r22 ) = ( A².46195τ, ), β β for A².46195τ² β >, A² τ² β <, σ1 = (A² τ² ), σ2 = (A² τ² ). The stability of the solution (4.16) determined by (4.18) can be easily analyzed with aids of (4.17). Thus, the parameter plane (A, τ ) in the neighborhood of (Ac, τc ) is divided into seven regions (I)-(VII) bounded by (4.18), as shown in Fig.2(b), which is very similar to that produced by Guckenheimer and Holmes (cf Fig in [32]). τ τ A (a) simple case: γ = 1, β = A (b) difficult case: γ =, β = 4 FIG. 2 Classification and bifurcation sets of the solution for system (4.11) due to 3:5 resonant double Hopf bifurcation where solid lines,dashing, and dot-dashing represent boundaries and amplitude death region is in grey for (a) the simple case: γ = 1, β = and (b) difficult case: γ =, β = 4. In Fig.2(b), there are a stable trivial solution (, ) and an unstable periodic solution (, rk2 1 ) in region (I) which is an amplitude death region. With (A, τ ) changing to region (II), the trivial solution loses its stability and no local solution appears. Two unstable solutions at (,) and (rk1 1, ) exist in region (III). When (A, τ ) enters into region (IV), there are three unstable solutions given by (,), (rk1 1, ) and (, rk2 1 ). The stable non-trivial solutions of system (4.17) occur in regions (V) and (VI), determined by (rk1 2, rk2 2 ) and (rk1 1, ), respectively. The other unstable solutions in region (V) are at (, ), (rk1 1, ) and (, rk2 1 ), and those in region (VI) at (, ), (, rk2 1 ). It is easily seen from (4.16) that the stable solution in region (V) is quasi-periodic as σ1 /σ2 is not a rational number. Especially, the Hopf bifurcation of the nontrivial equilibrium at (rk1, rk2 ) = (r12, r22 ) occurs in the cubic amplitude system (4.17). This leads to the boundary between regions (VII) and (VIII), and a 3D invariant tori in

21 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 19 the full 4D system on the central manifold occurs in region (VIII). Thus, one has to consider the truncated 5th-order amplitude system to observe the cycle generated by the Hopf bifurcation. Following section 8.6 in [36], one can obtain the border between regions (VIII) and (V), given by (4.19) τ = A 211.4A 2, A A c, along which the cycle coexists with three saddles at (r k1, r k2), (r k11, r k2) and (r k1, r k21). Thus, the cycle disappears via a heteroclinic bifurcation when the parameter is varied from region (VIII) to (V). These results are also sketched in Fig. 2(b). Now, the numerical simulation is employed to examine the validity and accuracy of step one of the PIS (or perturbation step of the PIS) for the difficult case as shown in Fig. 2(b). To this end, the following two cases are considered. The values of the delay and gain are chosen to be, firstly, close to and, secondly, far away from the double Hopf bifurcation point at (A c, τ c ) in Fig. 2(b). The Runge-Kutta scheme is adopted to produce the numerical results, where β = 4, γ =, and the other parameters are the same as those in Fig.1. The gain A is kept fixed for the two cases as the effect of the time delay on the system under consideration is the prime concern..5 u max.14 (a) u max (b) ɛ ɛ.5 u max (c) u max 1 (d) ɛ ɛ FIG. 3 A comparison among the approximate solution (4.16) (solid), solution from step two of the PIS (thick blue) and the numerical simulation (crossing symbols) in Max(u) vs. ɛ for the periodic solution of system (4.1) with (A, τ) located in region (VI) in Fig.2: (a) A =.3348, τ = 5.852, (b) A =.33, τ = 5.7, (c) A =.31, τ = 5.4, and (d) A =.33, τ = 5.2. First, we consider the case of A kept fixed and τ varying in region (VI). Fig. 3 show a comparison between the approximate solution (4.16) represented in solid line and the numerical simulation from system (4.1) represented in crossing symbols for (a) A =.3348, τ = 5.852, (b) A =.33, τ = 5.7, (c) A =.31, τ = 5.4, and (d) A =.33, τ = 5.2. The values of (A, τ) in Figs. 3(a) and (b) are close

22 2 J. XU, K. W. CHUNG AND C. L. CHAN to (A c, τ c ) while those of Figs. 3(c) and (d) far away. In Figs. 3(a) and (b), the analytical prediction is in good agreement with the numerical result. This implies that the periodic solutions obtained from the presented method are accurate even for large ɛ as long so (A, τ) are near to the double Hopf bifurcation point at (A c, τ c ). With (A, τ) drifting away from (A c, τ c ), the accuracy decrease as shown in Figs. 3(b) and (d). This can be seen clearly from Figs. 4(a) and (b), where τ is decreased from τ c while A is kept fixed. 3.5 u max (a) u max (b) τ τ FIG. 4 A comparison between the approximate solution (4.16) (solid), solution from step two of the PIS (thick blue) and the numerical simulation (crossing symbols) in Max(u) vs. τ for the periodic solution of the system (4.1) when (A, τ) is located in region (VI) of Fig.2, where (a) ɛ =.1, (b) ɛ = 1. and A = Similar conclusion can be obtained for (A, τ) in Region (V). The time history of the quasi-periodic solution of system (4.1) is illustrated in Figs. 5(a) and (b) with small and large ɛ, where the approximate solution (4.16) is represented in solid line and the numerical simulation crossing symbols. It follows from Fig. 5 where good agreement is observed between the analytical prediction and the numerical simulation, that step one of the PIS can provide an analytical expression with high accuracy in the neighborhood of the double Hopf bifurcation point even for the quasi-periodic motions. However, with (A, τ) drifting away from the bifurcation point, the method becomes invalid as shown in Fig. 6. u (a) t u (b) t FIG. 5 A comparison between the approximate solution given in (4.16) (solid) and the numerical simulation (crossing symbols) in time history for the quasi-periodic solution of system (4.1), where (a) ɛ =.1, (b) ɛ = 1, and A=-.34,τ=5.85 are chosen in region (V) of Fig.2. and close to (A c, τ c).

23 AN EFFICIENT METHOD FOR DELAYED SYSTEMS 21.4 u t FIG. 6 A comparison between the approximate solution (4.16) (solid) and the numerical simulation (line with diamond symbols) in time history for the quasi-periodic solution of system (4.1) when (A, τ) is in region (V) but far away from (A c, τ c), where ɛ = 1, A =.5, τ=4..2 (a).6.4 (b) v.1 v u u FIG. 7 A comparison between the approximate solution (4.16) (solid), solution (thick blue) from step two of PIS and the numerical simulation (crossing symbols) in phase plane for the periodic solution of system (4.1), where (a) τ = 5.85, (b) τ = 3; ɛ=1 and A =.3365 throughout. From Figs. 3 to 6, one can see that step one of the PIS not only provides a fairly good prediction qualitatively but also an analytical expression with high accuracy for both periodic and quasi-periodic motions when parameters are chosen to be close to the weak resonant point. However, the analytical expression is not accurate enough quantitatively when the bifurcation parameters drift away from the bifurcation point. The approximate expression is now considered as an initial guess for step two of the PIS which traces the periodic solutions for bifurcation parameters far away from (A c, τ c ). To this end, we choose ɛ = 1 and (τ, A) = (5.85,.3365) which is close to (A c, τ c ) as the starting point. It follows from Fig. 2(b) that this point is located in region (VI) at which the stable periodic solution is given by (4.2) { Z(t) = r k11 cos(( ɛσ 1 ) t + θ 1 ) r k11 sin(( ɛσ 1 ) t + θ 1 ) where r k11 and σ 1 are determined by (4.18), τ ɛ = τ τ c, A ɛ = A A c and θ 1 is determined by initial values. Thus, θ 1 = by setting v() =. The solution (4.2) is },

24 22 J. XU, K. W. CHUNG AND C. L. CHAN Table 4.2 Coefficients of u, v, Φ(φ) and φ 1 for solution derived from step two of the PIS in Fig. 7(b). m = 8, A =.38, ɛ = 1., τ = 3. i a 1i b 1i a 2i b 2i p i q i r i s i in good agreement with the numerical one as depicted in Fig. 7(a) but such agreement disappears for τ = 3 as shown in Fig. 7(b). If the solution (4.2) is considered as an initial guess of step two, then the initial coefficients in the incremental solution given by (3.21), (3.16) and (3.19) can be expressed as (cf. (3.23), (3.24) and (3.25)) (4.21) a =, a 1 = { } { rk11, b 1 = a j = b j = for j = 2,..., m, r k11 }, (4.22) p = σ 1 3, p j = q j = for j = 1,...m, and (4.23) r = p τ = 5.85 ( σ 1 3 ), r j = s j = for j = 1,...m. With the incremental procedure from τ = 5.85 down to τ = n τ = 3 (n Z +, τ 1) and step two of the PIS in terms of (3.26)-(3.42), a converged periodic solution at τ = 3 can easily be obtained after a few iterations as shown in Fig. 7(b). It follows from Fig. 7(b) that although the approximate solution derived from the Theorem is far away from the numerical solution, an accurate solution can be obtained thought the incremental process of the PIS. The updated solution by step two of the PIS is listed in Table 4.2. Similarly, we can use step two to consider the other cases of Figs. 3 and 4. However, the incremental step of the PIS has not been presented here for the continuation of quasi-periodic solutions shown in Figs. 5 and 6. This will be investigated in future. Finally, the numerical simulation verifies that there is a stable trivial solution in region (I) but not any stable solution in regions (II), (III), (IV) and (VII). 5. Stuar-Landau System with Time Delay. As the second example, we consider the Stuar-Landau System with a limit cycle oscillator governed by (5.1) ẋ = α x ω y (x 2 + y 2 )x, ẏ = ω x + α y (x 2 + y 2 ) y, where ω is the frequency of the oscillator, α a real positive constant. It is easily seen that system (5.1) has a stable limit cycle of amplitude α with frequency ω.