Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 3, pp. 531-550, Warsaw 2008 STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH TIME-DELAYED FEEDBACK CONTROL Zhong Hua Liu Xiamen University, Department of Civil Engineering, Xiamen, China e-mail: zhliuzju@yahoo.com Wei Qiu Zhu Zhejiang University, Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Hangzhou, China e-mail: wqzhu@yahoo.com (corresponding author) The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi- integrable Hamiltonian systems with time-delayed feedback control subject to Gaussian white noise excitations is studied. First, the time-delayed fe- edback control forces are approximately expressed in terms of the system state variables without time delay, and the system is converted into anor- dinary quasi-integrable Hamiltonian system. The averaged Itô stochastic differential equations are derived by using the stochastic averagingmethod for quasi-integrableHamiltonian systems.Then, an expression for the avera- ge bifurcation parameter of the averaged system is obtained approximately, and a criterion for determining the stochastic Hopf bifurcation caused by the time-delayed feedback control forces in the original system as the value of the average bifurcation parameter changing is proposed. An example is worked out in detail to illustrate the above criterion and its validity, and to show the effect of the time delay in the feedback control on the stochastic Hopf bifurcation of the system. Key words: stochastic Hopf bifurcation, quasi-integrable Hamiltonian sys- tem, stochastic averaging, time-delayed feedback control 1. Introduction Time delay is usually unavoidable in feedback control systems due to the time spent for measuring and estimating of the system state, calculating and executing of the control forces, etc. This time delay often leads to instability 532 Z.H. Liu, W.Q. Zhu or poor performance of controlled systems. Thus, the issue of handling the time delay has drawnmuch attention of the control community. Systems with time delay under deterministic excitation have been studied by many researchers (Agrawal and Yang, 1997; Atay, 1998; Hu and Wang, 2002; Kuo, 1987; Malek-Zavarei and Jamshidi, 1987; Pu, 1998; Stepan, 1989). The study on those systems under stochastic excitation is very limited. A linearly controlled systemwith deterministic and random time delays excited by Gaussian white noise was treated by Grigoriu (1997) and the stability of such a system was investigated by means of the Lyapunov exponent. The effects of time delay on the controlled linear systems under Gaussian random excitation were studied by Di Paola and Pirrotta (2001) using an approach based on the Taylor expansion of the control force and another approach to find an exact stationary solution. The stochastic averaging method for quasi- integrable Hamiltonian systems with time-delayed feedback control has been proposedby thepresent authors and the effects of the timedelay on the system response and stability were studied in Liu and Zhu (2007, 2008), Zhu and Liu (2007). Hopf bifurcation often occurs in time-delayed deterministic systems and has been studied by using the linear stability analysis method (Hassard et al., 1981; Kuznetstov, 1998), the invariantmanifold reduction and the normal formmethod (Kalmar-Nagy et al., 2001; Xu and Chung, 2003), the averaging method (Stephen and Richard, 2002), the multiple scales method (Das and Chatterjee, 2002). Much work has been done on the effect of noise on the bifurcation (Arnold et al., 1996; Srinamachchivaya, 1990). A procedure was proposed for analysing the stochastic Hopf bifurcation of quasi-nonintegrable Hamiltonian systems (ZhuandHuang, 1999). The studyon stochasticHopf bi- furcation of excitation systemswith time delay is very limited. Longtin (1991) studied the influence of coloured noise on the Hopf bifurcation in a first-order delay differential equation of nonlinear delayed feedback systems. In the present paper, the stochastic Hopf bifurcation of quasi-integrable Hamiltonian systems with time-delayed feedback control is studied. First, the stochastic averaging method for quasi-integrable Hamiltonian systems with time-delayed feedback control is introduced. The time-delayed feedback con- trol forces are expressed in termsof the systemstateswithout timedelay in the average sense. The equations of the systemare reduced to a set of averaged Itô stochastic differential equations by applying the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, the expression for the avera- ge bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation caused by the time-delayed Stochastic Hopf bifurcation of quasi-integrable... 533 feedback control forces in the original system by using the average bifurcation parameter is proposed.An example is given to illustrate the proposed criterion and the results of numerical simulation are obtained to verify the effectiveness of the proposed criterion. The effect of time delay in the control forces on the stochastic Hopf bifurcation is analysed in detail. 2. Stochastic averaging of quasi-integrable Hamiltonian systems with time-delayed feedback control Consider an n-DOF quasi-Hamiltonian system with time-delayed feedback control governed by the following Itô stochastic differential equations dQi = ∂H′ ∂Pi dPi =− [∂H′ ∂Qi +εc′ij ∂H′ ∂Pi +εFi(Qτ,Pτ) ] dt+ √ εσikdBk(t) (2.1) i,j=1,2, . . . ,n k=1,2, . . . ,m where Qi and Pi are generalized displacements and momenta, respectively, Q= [Q1,Q2, . . . ,Qn] ⊤,P= [P1,P2, . . . ,Pn] ⊤;H′ =H′(Q,P) is twice differen- tiableHamiltonian; ε is a small positive parameter; εc′ij = εc ′ ij(Q,P) represent the coefficients of quasi linear damping; Bk(t) are standardWiener processes and √ εσik represent amplitudes of stochastic excitations; εFi(Qτ,Pτ) with Qτ = Q(t− τ) and Pτ = P(t− τ) denote the time-delayed feedback control forces, τ is the time delay, and εFi(Qτ,Pτ)= 0 when t∈ [0,τ]. Assume that theHamiltonian H′ associatedwith system (2.1) is separable and of the form H′ = n∑ i=1 H′i(qi,pi) H ′ i = 1 2 p2i +G(qi) (2.2) where G(qi)­ 0 is symmetric with respect to qi = 0, and with minimum at qi =0, i.e., theHamiltonian systemwithHamiltonian H ′ is integrable andhas a family of periodic solutions around the origin.When ε is small, the solution to Eq. (2.1) is of the form (Huang et al., 2000; Zhu et al., 2003) Qi(t)=AicosΦi(t) Pi(t)=−Ai dΘi dt sinΦi(t) (2.3) Φi(t)=Θi(t)+Γi(t) 534 Z.H. Liu, W.Q. Zhu where cosΦ(t) and sinΦ(t) are called generalized harmonic functions. For quasi-integrable Hamiltonian systems, Ai(t) and Γi(t) are slowly varying pro- cesses and the averaged value of the instantaneous frequency dΘi/dt is equal to ωi(Ai). For a small delay time τ, wehave the following approximate expres- sions for time-delayed state variables Qi(t− τ)=Ai(t−τ)cosΦi(t− τ)≈Ai(t)cos[ωi(t− τ)+Γi(t)] = =Qi(t)cosωiτ− Pi ωi sinωiτ (2.4) Pi(t− τ)=−Ai(t− τ) dΘi(t− τ) dt sinΦi(t− τ)≈ ≈−Ai(t)ωi sin[ωi(t− τ)+Γi(t)] =Picosωiτ+Qi(t)ωi sinωiτ Thus, the time-delayed feedback control forces εFi(Qτ,Pτ) can be approxi- mately expressed in terms of system state variables without the time delay. Note that the numerical results in the present paper and inLiu andZhu (2007, 2008), Zhu and Liu (2007) show that Eqs. (2.4) holds even for larger τ. The terms εF(Qτ,Pτ) in Eqs. (2.1) can be split into two parts: one has the effect of modifying the conservative forces and the other modify- ing the damping forces. The first part can be combined with −∂H′/∂Qi to form overall effective conservative forces −∂H/∂Qi with a new Hamiltonian H =H(Q,P;τ) andwith ∂H/∂Pi = ∂H ′/∂Pi.The secondpartmaybecombi- nedwith −εc′ij∂H′/∂Pj to constitute effective damping forces −εmij∂H/∂Pi with mij =mij(Q,P;τ).With these accomplished,Eqs. (2.1) canbe rewritten as dQi = ∂H ∂Pi dt dPi =− (∂H ∂Qi +εmij ∂H ∂Pj ) dt+ √ εσikdBk(t) (2.5) i,j=1,2, . . . ,n k=1,2, . . . ,m where H =H(Q,P;τ),mij =mij(Q,P;τ). Eqs. (2.5) is the Itô equations for quasi-integrable Hamiltonian systems without time delay. Assume that theHamiltonian systemwithHamiltonianH is still integrable and nonresonant. That is, the Hamiltonian system has n independent first integrals H1,H2, . . . ,Hn, which are in involution. The term ”in involution” implies that the Poisson bracket of any two of H1,H2, . . . ,Hn vanishes. In principle, n pairs of action-angle variables Ii, θi can be introduced for an Stochastic Hopf bifurcation of quasi-integrable... 535 integrable Hamiltonian system of n-DOF. Nonresonance means that the n frequencies ωi = dθi/dt do not satisfy the following resonant relation kuiωi =0(ε) (2.6) where kui are integers. Introduce transformations Hεr =Hr(Q,P,ε) r=1,2, . . . ,n (2.7) The Itô stochastic differential equations for Hεr are obtained from Eqs. (2.5) by using the Itô differential rule as follows dHεr = ε ( −mij ∂H ∂Pj ∂Hεr ∂Pi + 1 2 σikσjk ∂2Hεr ∂Pi∂Pj ) dt+ √ εσik ∂Hεr ∂Pi dBk(t) (2.8) r,i,j =1,2, . . . ,n k=1,2, . . . ,m where Pi are replaced by H ε r in terms of Eq. (2.7). It is seen from Eqs. (2.5) and (2.7) that Qi are rapidly varying processes while H ε r are slowly vary- ing processes. According to the Khasminskii theorem (Khasminskii, 1967), H ε = [Hε1,H ε 2, . . . ,H ε n] ⊤ convergesweakly to an n-dimensional vector diffusion process H= [H1,H2, . . . ,Hn] ⊤ in a time interval O(ε−1) as ε→ 0. For each bounded and continuous real-valued function f(H), the expression ”Hεr co- nverges weakly to Hr” means ∫ f(H)dPε(H) → ∫ f(H)dP(H) as ε → 0, where Pε(H) and P(H) are, respectively, the joint probability distributions of Hε and H. The error between the solutions of the original and averaged systems is of the order ε. The Itô stochastic differential equations for this n-dimensional vector dif- fusion process can be obtained by applying the time averaging to Eq. (2.8). The result is dHr = ar(H)dt+σrk(H)dBk(t) r=1,2, . . . ,n k=1,2, . . . ,m (2.9) where H= [H1,H2, . . . ,Hn] ⊤;Bk(t) are independent unitWiener processes ar(H)= ε 〈 −mij ∂H ∂Pj ∂Hr ∂Pi + 1 2 σikσjk ∂2Hr ∂Pi∂Pj 〉 t brs(H)=σrk(H)σsk(H)= ε 〈 σikσjk ∂Hr ∂Pi ∂Hs ∂Pj 〉 t (2.10) 〈[·]〉t = lim T→∞ 1 T t0+T∫ T [·]dt 536 Z.H. Liu, W.Q. Zhu Note that Hr are kept constant in performing the time averaging. The time averaging in Eqs. (2.10)may be replaced by space averaging. For example, suppose that the Hamiltonian is separable and equal to the sum of n independent first integers, i.e. H(q,p)= n∑ r=1 Hr(qr,pr) (2.11) and for each Hr there is a periodic orbit with the period Tr. Then, the ave- raged drift and diffusion coefficients in Eqs. (2.10) become ar(H)= ε T ∮ ( −mij ∂H ∂Pj ∂Hr ∂Pi + 1 2 σikσjk ∂2Hr ∂Pi∂Pj ) n∏ u=1 (∂Hu ∂Pu )−1 dqu (2.12) brs(H)= ε T ∮ ( σikσjk ∂Hr ∂Pi ∂Hr ∂Pj ) n∏ u=1 (∂Hu ∂Pu )−1 dqu where ∮ [·] ∏n u=1(· · ·)dqu represents an n-fold loop integral and T =T(H)= n∏ u=1 Tu = ∮ n∏ u=1 (∂Hu ∂Pu )−1 dqu (2.13) Note that averaged Eq. (2.9) ismuch simpler than original Eqs. (2.5). The dimension of the former equation is only a half of that of the later equation. Averaged equation (2.9) contains only slowly varying process, while Eqs. (2.5) contains both rapidly and slowly varyingprocesses. Furthermore, the averaged equation can be used to study the long-term behaviour of the system, such as stability, stationary response and first-passage failure, since the convergen- ce of Hεr to the diffusion process holds even for t → ∞ (Blankenship and Papanicolaou, 1978; Kushner, 1984). 3. Determination of Hopf bifurcation by the average bifurcation parameter It is well known that for a Duffing-van der Pol oscillator under parametric excitation of Gaussian white noise, stochastic Hopf bifurcation consists of a dynamical bifurcation (D-bifurcation) and a phenomenological bifurcation (P-bifurcation) (Arnold et al., 1996). Before the D-bifurcation, the trivial so- lution is asymptotically stable with probability one and the stationary joint Stochastic Hopf bifurcation of quasi-integrable... 537 probabilitydensity of thedisplacement andvelocity is theDiracdelta function. TheD-bifurcation occurswhen the largest Lyapunov exponent vanishes.After theD-bifurcation andbeforeP-bifurcation, the trivial solution is unstable and the stationary joint probability density of the displacement andvelocity is uni- model with the peak at the origin. After the P-bifurcation, the trivial solution is still unstable, while the stationary joint probability density of the displace- ment and velocity becomes crater-like. The interval between theD-bifurcation and P-bifurcation is called the bifurcation interval. The analysis of stochastic Hopf bifurcation can be greatly simplified by using the stochastic averaging method for quasi Hamiltonian systems. By using the relationship between the one-demensional diffusion process and its boundaries, Zhu and Huang (1999) proposed a criterion for determining sto- chastic Hopf bifurcation (bothD-bifurcation and P-bifurcation) in quasi non- integrable Hamiltonian systems using the diffusion exponent, draft exponent and a character value. In the following, we will generalize this criterion to quasi-integrable Hamiltonian systems with time-delayed feedback control. Based on Eq. (2.11), we introduce the following new variable αr = Hr H r=1,2, . . . ,n (3.1) Note that ∑n r=1αr = 1, so only n − 1 variables for αr in Eq. (3.1) are independent. In the following, we take the first n − 1 variables for a ′ = [α1,α2, . . . ,αn−1] as independent variables with αn replaced by αn = = 1− ∑n−1 r=1 αr. The Itô equations for H and αr can be obtained from Eq. (2.9) by using the Itô differential rule as follows dH =Q(a′,H;τ)dt+Σk(a ′,H;τ)dBk(t) dαr =mr(a ′,H;τ)dt+ σ̃rk(a ′,H;τ)dBk(t) (3.2) r=1,2, . . . ,n−1 k=1,2, . . . ,m where Q(a′,H;τ)= n∑ r=1 ar(a ′,H;τ) Σk(a ′,H;τ)= n∑ r=1 σrk(a ′,H;τ) 538 Z.H. Liu, W.Q. Zhu mr(a ′,H;τ)=−αr n∑ s=1 1 H as(a ′,H;τ)+ −1 2 n∑ s=1 m∑ k=1 1 H2 σrk(a ′,H;τ)σsk(a ′,H;τ)+ (3.3) + 1 2 αr n∑ s,s′=1 m∑ k=1 1 H2 σsk(a ′,H;τ)σs′k(a ′,H;τ)+ 1 H ar(a ′,H;τ) σ̃rk(a ′,H;τ)= 1 H σrk(a ′,H;τ)−αr n∑ s=1 1 H σsk(a ′,H;τ) For the one-dimensional diffusion process H(t) governed byEq. (3.2)1, the boundary H →∞must be either an entrance or a repulsively natural in order that the trivial solution H =0 is stable inprobability or H(t) has a stationary probability density, i.e., the boundary H →∞must be either an entrance or repulsively natural during the first and second bifurcation. In the following, we will focus our attention on the qualitative change in sample behaviour of H(t) near the boundary H =0 during the first and second bifurcation. For the one-dimensional diffusionprocess reduced fromhigher-dimensional systems undergoing parametric excitations by using the stochastic averaging, the boundaries H = 0 and H → ∞ are often singular and the sample be- haviour of the process near the boundaries are characterised by the diffusion exponent, the drift exponent and the character value (Lin andCai, 1995). For a singular left boundary of the first kind, i.e., Σk(a,0;τ) = 0, the diffusion exponent αl, the drift exponent βl and the character value cl are defined as follows b′(a′,H;τ)= (Σk(a ′,H;τ)) 2 =O(Hαl) αl > 0 as H → 0 Q(a′,H;τ) =O(Hβl) βl > 0 as H → 0 cl(a ′;τ)= lim H→0+ 2Q(a′,H;τ)Hαl−βl b′(a′,H;τ) (3.4) where O(·) denotes the order of magnitude of (·). For a singular right boun- dary of the second kind, i.e., m(∞)→∞, the diffusion exponent αr, the drift exponent βr and the character value cr are defined as follows b′(a′,H;τ) = (Σk(a ′,H;τ)) 2 =O(Hαr) αr > 0 as H →∞ Q(a′,H;τ)=O(Hβr) βr > 0 as H →∞ cr(a ′;τ) = lim H→+∞ 2Q(a′,H;τ)Hαr−βr b′(a′,H;τ) (3.5) Stochastic Hopf bifurcation of quasi-integrable... 539 The criteria for classification of the singular boundary based on values of the diffusion exponent, the drift exponent and the character value can be found in Tables in Lin and Cai (1995). ConsideringEqs. (3.4), one can obtain the following asymptotic expression for the stationary probability density of H(t) p(H;τ) =O { H−αl exp [ cl H∫ 0 x(βl−αl) dx ]} as H → 0 (3.6) Two cases can be identified. Case 1. βl−αl =−1. In this case p(H;τ)=O(Hv) as H → 0 (3.7) with v(a′;τ)= cl(a ′;τ)−αl (3.8) Particularly, when βl = 1 and αl = 2, the diffusion and drift coefficients in Eq. (3.2)2 are linear. Introduce an average bifurcation parameter v(τ) defined by v(τ)= ∫ Ω v(a′;τ)p(a′;τ) da′ Ω= { a ′ ∣∣∣ n∑ r=1 αi =1, 0¬αi ¬ 1 } (3.9) where p(a′;τ) is the stationary solution to the Fokker-Plank-Kolmogorov (FPK) equation associated with the Itô differential equations in Eq. (3.2)2. Equation (3.7) is non-integrable and the probability density p(H;τ) is the delta function if v(τ) < −1. When −1 < v(τ) < 0, Eq. (3.7) is integrable and a stationary probability density p(H;τ) exists with a peak at H =0. If v(τ) > 0, then Eq. (3.7) is integrable and p(H;τ) exists with a peak away from H =0. Thus, the first bifurcation (D-bifurcation) occurs at v(τ) =−1 and the second bifurcation (P-bifurcation) at v(τ)= 0provided that the right boundary H →∞ is an entrance or repulsivelynatural. It is interesting tonote that the condition for thefirst bifurcationhere is consistentwith that obtained from the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution. Case 2. βl−αl 6=−1. In this case p(H)=O { H−αl exp [ cl 1+βl−αl H(βl−αl+1) ]} as H → 0 (3.10) 540 Z.H. Liu, W.Q. Zhu which cannot be expressed in the form of Eq. (3.7). It can be shown that it is impossible for p(H;τ) to have apeak at or near H =0when it exists. In other words, in this case, although the first bifurcation may occur, it is impossible for the second bifurcation to occur. The D-bifurcation and P-bifurcation of H(t) implies a stochastic Hopf bifurcation of original system (2.1). In other words, the stochastic Hopf bifur- cation of a quasi-integrable Hamiltonian system with time-delayed feedback control can be determined by examining the sample behaviour of the one- dimensional averaged diffusion process at its boundaries H =0 and H →∞. Thefirstbifurcationoccurswhen v(τ)=−1, andthesecondbifurcationoccurs when v(τ) = 0 while the right boundary is either an entrance or repulsively natural. 4. Example To illustrate the above criterion for stochastic Hopf bifurcation, consider two coupledRayleigh oscillators with time-delayed feedback control subject to pa- rametric excitations of Gaussian white noise. The equations of motion of the system are of the form Ẍ1+ ( −β′10+β11Ẋ21 +β12Ẋ22 ) Ẋ1+ω ′ 1 2 X1 = =−η1Ẋ1τ +f11Ẋ1W1(t)+f12Ẋ2W2(t) (4.1) Ẍ2+ ( −β′20+β21Ẋ21 +β22Ẋ22 ) Ẋ2+ω ′ 2 2 X2 = =−η2Ẋ2τ +f21Ẋ1W1(t)+f22Ẋ2W2(t) where Xi are generalized coordinates; β ′ i0 and βij (i,j = 1,2) are damping coefficients; ω′i are natural frequencies of the two linear oscillators; Wk(t) (k = 1,2) are independent Gaussian white noises with intensities 2Dkk; −ηiẊiτ represent the time-delayed feedback control forces. Here we study the effects of time delay in the feedback control forces on the stochastic Hopf bifurcation of system (4.1). Following Eqs. (2.4), the time-delayed feedback control forces can be expressed in terms of system state variables without time delay as follows −ηiẊiτ =−ηiẊicosω′iτ−ηiω′iXi sinω′iτ i=1,2 (4.2) Stochastic Hopf bifurcation of quasi-integrable... 541 On the right hand side of Eq. (4.2), the first terms are dissipative, while the second terms are conservative. They can be combined, respectively, with the damping terms and conservative terms of Eq. (4.1) to constitute effective damping terms and effective conservative terms. Let X1 = Q1, X2 = Q2, Ẋ1 = P1, Ẋ2 = P2. By applying the stochastic averaging method for quasi-integrable Hamiltonian systems to modified Eq. (4.1), the following averaged Itô equations can be obtained in the nonresonant case dH1 = [ (β10+2f 2 11D11)H1− 3 2 β11H 2 1 −β12H1H2+f212D22H2 ] dt+ +σ11dB1(t)+σ12dB2(t) (4.3) dH2 = [ (β20+2f 2 22D22)H2− 3 2 β22H 2 2 −β21H1H2+f221D11H1 ] dt+ +σ21dB1(t)+σ22dB2(t) with Hi = 1 2 (P2i +ω 2 iQ 2 i) ω 2 i =ω ′ i 2 +ηiω ′ i sinω ′ iτ βi0 =β ′ i0+ηicosomega ′ iτ b11 =σ1jσ1j =3D11f 2 11H 2 1 +2f 2 12D22H1H2 (4.4) b22 =σ2jσ2j =3D22f 2 22H 2 2 +2f 2 21D11H1H2 b12 = b21 =σ1jσ2j =0 The Itô differential equations associated with H =H1+H2 and α1 =H1/H can be obtained by using the Itô differential rule as follows dH =(Q1H+Q2H 2)dt+Σ1HdB1(t) (4.5) dα1 =m1dt+ σ̃1dB1(t) where Q1 =(β10+2f 2 11D11+f 2 21D11)α1+(β20+2f 2 22D22+f 2 12D22)(1−α1) Q2 =− 3 2 β11α 2 1− 3 2 β22(1−α1)2− (β12+β21)α1(1−α1) Σ21 =3f 2 11D11α 2 1+3f 2 22D22(1−α1)2+2(f221D11+f212D22)α1(1−α1) m1 = (1 2 −α1 ) ϕ(α1)+2α1(1−α1)(λ1−λ2) (4.6) 542 Z.H. Liu, W.Q. Zhu σ̃21 =α1(1−α1)ϕ(α1) ϕ(α1)= aα21+ bα1+ c c=G12 a=G12+G21−G11−G22 b=G11+G22−2G12 G11 =3f211D11 λ1 = 1 2 β10+ 1 4 f211D11 λ2 = 1 2 β20+ 1 4 f222D22 G22 =3f 2 22D22 G12 =2f 2 12D22 G21 =2f 2 21D11 For H(t) governed by Eqs. (4.5) at H →∞, the diffusion exponent αr =2, the drift exponent βr =2. If βij > 0 (i,j=1,2), then Q2 < 0, the boundary H →∞ is an entrance. At the boundary H =0, the diffusion exponent, the draft exponent and the character value are αl =2 βl =1 cl = 2Q1 Σ21 = cl(α1;τ) (4.7) v(α1;τ)= cl(α1;τ)−2 The stationary solution p(α1;τ) to theFPKequation associatedwith Itô Eqs. (4.6) is p(α1;τ)= C ϕ(α1) F(α1) (4.8) where F(α1)=    exp (4(λ1−λ2)√ ∆ ln ∣∣∣ 2aα1+ b− √ ∆ 2aα1+ b+ √ ∆ ∣∣∣ ) for ∆> 0 exp (8(λ1−λ2)√ −∆ arctan ∣∣∣ 2aα1+ b√ −∆ ∣∣∣ ) for ∆< 0 exp (8(λ1−λ2) 2aα1+ b ) for ∆=0 (4.9) C = 4(λ1−λ2) F(1)−F(0) ∆= b2−4ac The average bifurcation parameter v(τ) can be obtained as follows v(τ)= 1∫ 0 v(α1;τ)p(α1;τ) dα1 (4.10) The stochastic Hopf bifurcation of system (4.1) can be determined by using the average bifurcation parameter v(τ). If v(τ)<−1, the probability density Stochastic Hopf bifurcation of quasi-integrable... 543 p(H;τ) is the delta function and the system is stable; if −1< v(τ)< 0, the probability density p(H;τ) exists with a peak at H =0. If v(τ)> 0, the pro- bability density p(H;τ) exists with a peak away from H =0. Thus, the first bifurcation (D-bifurcation) occurs at v(τ) = −1 and the second bifurcation (P-bifurcation) occurs at v(τ)= 0. Some numerical results for the stochastic Hopf bifurcation of system (4.1) caused by the time-delayed feedback control are shown in Figs.1-5. The sta- bility of system (4.1) is shown in parameter plane (β′10,β ′ 20) in terms of the location of the origin O(0,0) in the plane relative to D-bifurcation and P-bifurcation curves. Fig. 1. Results for τ =0. (a) D-bifurcation and P-bifurcation curves and point O(0,0) in plane (β′ 10 ,β′ 20 ). (b) Stationary probability density p(H) at point O(0,0). (c) Stationary probability density p(H1,H2) at point O(0,0). (d) Stationary probability density p(q1,p1) of the first oscillator at point O(0,0). (e) Stationary probability density p(q2,p2) of the second oscillator at point O(0,0). The parameters are: β11 =β12 =β21 =β22 =0.005, ω ′ 1 =1.0, ω′ 2 =1.414, 2D1 =0.01, 2D2 =0.01, η1 = η2 =0.02, f11 = f12 = f21 = f22 =1 544 Z.H. Liu, W.Q. Zhu The result for τ = 0 is shown in Fig.1. It is seen that without the time delay, system (4.1) is stable and the stationary probability densities p(H), p(H1,H2), p(q1,p1) and p(q2,p2) are all Dirac delta functions. Fig. 2. Results for τ =1.0. (a) D-bifurcation and P-bifurcation curves and point O(0,0) in plane (β′ 10 ,β′ 20 ). (b) Stationary probability density p(H) at point O(0,0). (c) Stationary probability density p(H1,H2) at point O(0,0). (d) Stationary probability density p(q1,p1) of the first oscillator at point O(0,0). (e) Stationary probability density p(q2,p2) of the second oscillator at point O(0,0). The parameters are the same as those in Fig.1 The result for τ = 1.0 is shown in Fig.2. It is seen that in this case system (4.1) is unstable and the time delay τ is in the bifurcation interval. All the stationary probability densities are normalizable functions with a peak at the origin. It implies that the D-bifurcation occurs in system (4.1) with τD value between 0 and 1. This inference is verified by the value τD = 0.9107 determined by v(τD)=−1. Stochastic Hopf bifurcation of quasi-integrable... 545 Fig. 3. Results for τ =1.5. (a) D-bifurcation and P-bifurcation curves and point O(0,0) in plane (β′ 10 ,β′ 20 ). (b) Stationary probability density p(H) at point O(0,0). (c) Stationary probability density p(H1,H2) at point O(0,0). (d) Stationary probability density p(q1,p1) of the first oscillator at point O(0,0). (e) Stationary probability density p(q2,p2) of the second oscillator at point O(0,0). The parameters are the same as those in Fig.1 The result for τ = 1.5 is shown in Fig.3. It is seen that system (4.1) is unstable andpostD-bifurcation andP-bifurcation.The stationary probability densities p(H) and p(H1,H2) are normalizablewith their peaks away fromthe origin and stationary probability density p(q2,p2) is crater-like. It implies that the P-bifurcation occurs in the second oscillator of system (4.1) of τP value between 1.0 and 1.5. This inference is verified by τP =1.1803 determined by v(τP)= 0. 546 Z.H. Liu, W.Q. Zhu Fig. 4. Results for τ =2.0. (a) D-bifurcation and P-bifurcation curves and point O(0,0) in plane (β′ 10 ,β′ 20 ). (b) Stationary probability density p(H) at point O(0,0). (c) Stationary probability density p(H1,H2) at point O(0,0). (d) Stationary probability density p(q1,p1) of the first oscillator at point O(0,0). (e) Stationary probability density p(q2,p2) of the second oscillator at point O(0,0). The parameters are the same as those in Fig.1 The result for τ = 2.0 is shown in Fig.4. The difference between Fig.4 and Fig.3 is that in this case both stationary probability densities p(q1,p1) and p(q2,p2) are crater-like, which means both oscillators of system (4.1) are post P-bifurcation. Unfortunately, this second P-bifurcation of system (4.1) can not be predicted by using the criterion proposed in the present method. The result for τ =3.0 is shown inFig.5. System(4.1) is also unstable andpost D-bifurcation andP-bifurcation.The stationary probability densities p(q1,p1) and p(q2,p2) are typical crater-like herein. Stochastic Hopf bifurcation of quasi-integrable... 547 Fig. 5. Results for τ =3.0. (a) D-bifurcation and P-bifurcation curves and point O(0,0) in plane (β′ 10 ,β′ 20 ). (b) Stationary probability density p(H) at point O(0,0). (c) Stationary probability density p(H1,H2) at point O(0,0). (d) Stationary probability density p(q1,p1) of the first oscillator at point O(0,0). (e) Stationary probability density p(q2,p2) of the second oscillator at point O(0,0). The parameters are the same as those in Fig.1 5. Conclusions In the present paper, a criterion for determining of the stochastic Hopf bi- furcation of quasi-integrable Hamiltonian systems with time-delayed feedback control has beenproposedbased on the stochastic averagingmethod for quasi- integrableHamiltonian systems.The time-delayed feedback control forces have been approximately expressed in terms of the system state variables without time delay. The expression for the average bifurcation parameter of the avera- 548 Z.H. Liu, W.Q. Zhu ged system has been derived. The stochastic Hopf bifurcation caused by the time-delayed feedback control forces in the original system has been examined by using the average bifurcation parameter. The effect of time delay in feed- back control on the stochastic Hopf bifurcation has been analysed in detail. The results show that the time delay in the feedback control forcesmay result in a stochastic Hopf bifurcation in coupled Rayleigh oscillators. Acknowledgements The work reported in this paper has been supported by the National Natural Science Foundation of China under grants No. 10332030 and 10772159 and the Re- search Fund for the Doctoral Program of Higher Education of China under Grant No. 20060335125. References 1. Agrawal A.K., Yang J.N., 1997, Effect of fixed time delay on stability and performance of actively controlled civil engineering structures,Earthquake En- gineering and Structural Dynamics, 26, 1169-1185 2. Arnold L., Sri Namachchivaya N., Schenk-Hoppe K.R., 1996, Toward and understanding of stochastic Hopf bifurcation: a case study, International Journal of Bifurcation and Chaos, 6, 1947-1975 3. Atay F.M., 1998, Van der Pol’s oscillator under delayed feedback, Journal of Sound and Vibration, 218, 333-339 4. BlankenshipG., PapanicolaouG.C., 1978, Stability and control of stocha- stic systemswith wide-band disturbances, SIAM Journal of Applied Mathema- tics, 34, 437-476 5. Das S.L., Chatterjee A., 2002, Multiple scales without center manifold reductions for delay differential equations near hopf bifurcations, Nonlinear Dynamics, 30, 323-335 6. Di Paola M., Pirrotta A., 2001, Time delay induced effects on control of linear systems under random excitation, Probabilistic Engineering Mechanics, 16, 43-51 7. Grigoriu M., 1997, Control of time delay linear systemswithGaussianwhite noise,Probabilistic Engineering Mechanics, 12, 89-96 8. Hassard B., Kazarinoff N., Wan Y.H., 1981,Theory and Applications of Hopf Bifurcation, Cambridge University Press, London Stochastic Hopf bifurcation of quasi-integrable... 549 9. HuH.Y.,WangZ.H., 2002,Dynamics of ControlledMechanical Systemswith Delayed Feedback, Springer-Verlag, Berlin 10. Huang Z.L., Zhu W.Q., Suzuki Y., 2000, Stochastic averaging of strongly nonlinear oscillators under combined harmonic and white noise excitations, Journal of Sound and Vibration, 238, 233-256 11. Kalmar-Nagy T., Stepan G., Moon F.C., 2001, Subcritical Hopf bifurca- tion in the delay equationmodel for machine tool vibrations,Nonlinear Dyna- mics, 26, 121-142 12. Khasminskii R.Z., 1967, Sufficient and necessary conditions of almost sure asymptotic stability of a linear stochastic system, Theory of Probability and Applications, 11, 390-405 13. Kuo B.C., 1987,Automatic Control Systems, Prentice-Hall, EnglewoodCliffs, NJ 14. KushnerH.J., 1984,Approximation andWeakConvergenceMethods for Ran- domProcesses, withApplications to Stochastic SystemsTheory, TheMITPress, Cambridge,MA 15. Kuznestov Y.A., 1998, Elements of Applied Bifurcation Theory, Springer, NewYork 16. Lin Y.K., Cai G.Q., 1995,Probabilistic Structural Dynamics: Advanced The- ory and Applications, McGraw-Hill, NewYork 17. Liu Z.H., Zhu W.Q., 2007, Stochastic averaging of quasi-integrableHamilto- nian systems with delayed feedback control, Journal of Sound and Vibration, 299, 178-195 18. Liu Z.H., Zhu W.Q., 2008, Asymptotic Lyapunov stability with probability one of quasi integrable Hamiltonian systems with delayed feedback control, Automatica, in print 19. Longtin A., 1991, Noise-induced transitions at a Hopf bifurcation in a first- order delay-differential equation,Physical Review A, 44, 4801-4813 20. Malek-Zavarei M., Jamshidi M., 1987, Time-Delay Systems: Analysis, Optimization and Applications, North-Holland, NewYork 21. Pu J.P., 1998, Time delay compensation in active control of structures,ASCE Journal of Engineering Mechanics, 124, 1018-1028 22. Sri Namachchivaya N., 1990, Stochastic bifurcation, Appl. Math. Comput., 38, 101-159 23. Stepan G., 1989, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific and Technical, Essex 24. Stephen W., Richard R., 2002, The dynamics of two coupled van der Pol oscillators with delay coupling,Nonlinear Dynamics, 30, 205-221 550 Z.H. Liu, W.Q. Zhu 25. Xu J., ChungK.W., 2003, Effects of time delayed position feedback on a van der Pol-Duffing oscillator,Physica D, 180, 17-39 26. Zhu W.Q., Huang Z.L., 1999, Stochastic Hopf bifurcation of quasi- nonintegrable Hamiltonian systems, International Journal of Non-linear Me- chanics, 34, 437-447 27. ZhuW.Q.,HuangZ.L.,DengM.L., 2003,Optimal bounded control of first- passage failure of quasi-integrableHamiltonian systemswithwide-band random excitation,Nonlinear Dynamics, 33, 189-207 28. ZhuW.Q., Liu Z.H., 2007,Response of quasi-integrableHamiltonian systems with delayed feedback bang-bang control,Nonlinear Dynamics, 49, 31-47 Stochastyczna bifurkacja Hopfa w quasi-całkowalnych układach Hamiltonowskich sterowanych w pętli sprzężenia zwrotnego z opóźnieniem Streszczenie Wpracyzajęto się problememstochastycznejbifurkacjiHopfaquasi-całkowalnych układówHamiltonowskich owielu stopniach swobodypoddanychwymuszeniubiałym szumem z układem sterowania opartym na pętli sprzężenia zwrotnego z opóźnie- niem. Najpierw znaleziono przybliżone wyrażenia na siły sterujące w funkcji zmien- nych stanu układu bez opóźnienia, a następnie przetransformowano go postaci quasi- całkowalnej, Hamiltonowskiej. Wyprowadzono stochastyczne równania różniczkowe Itô zapomocąmetodyuśrednianiaukładówquasi-całkowalnych.Znalezionoprzybliżo- ną postaćwyrażenia na parametr bifurkacyjny uśrednionego układu i zaproponowano kryterium stwierdzające obecność stochastycznej bifurkacji Hopfa wywołanej siłami sterującymi z opóźnieniemna podstawiewartości zmiany tego parametru.Opracowa- no szczegółowo przykład do ilustracji działania tego kryterium i zakresu jego stoso- walności oraz do prezentacji wpływu opóźnienia w pętli sterownia na stochastyczną bifurkację Hopfa badanego układu. Manuscript received January 9, 2008; accepted for print May 16, 2008