DELAY-INDUCED BIFURCATIONS IN A NONAUTONOMOUS SYSTEM WITH DELAYED VELOCITY FEEDBACKS

International Journal of Bifurcation and Chaos, Vol. 4, No. 8 (2004) 2777 2798 c World Scientific Publishing Company DELAY-INDUCED BIFURCATIONS IN A NONAUTONOMOUS SYSTEM WITH DELAYED VELOCITY FEEDBACKS JIAN XU and PEI YU Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7 Department of Engineering Mechanics and Technology, Tongji University, Shanghai 200092, China xujian@mail.tongji.edu.cn Received August 2, 2002 Revised June 3, 2003 This paper investigates the bifurcations due to time delay in the feedback control system with excitation. Based on an self-sustained oscillator, the delayed velocity feedback control system is proposed. For the case without excitation, the stability of the trivial equilibrium is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or periodic solutions may occur. For the case with excitation, the main attention is focused on the effect of time delay on the obtained periodic solution when primary resonance occurs in the system under consideration. To this end, the control system is changed to be a functional differential equation. Functional analysis is carried out to obtain the center manifold and then a perturbation approach is used to find periodic solutions in a closed form. Moreover, the unstable regions for the limit cycles are also obtained, predicting the occurrence of some complex behaviors. Numerical simulations are employed to find the routes leading to quasi-periodic motions as the time delay is varied. It has been found that: (i) Time delay can be used to control bifurcations; and (ii) time delay can be applied to generate bifurcations. This indicates that time delay may be used as a switch to control or create complexity for different applications. Keywords: Delayed differential equation; bifurcation; delayed feedback control; quasi-periodic solution.. Introduction The nonautonomous system under consideration, called self-sustained systems with external excitation, is given by ẍ + ω 2 0 x α ẋ + α 3 ẋ 3 = k cos(ωt), () where α α 3 > 0, k > 0. Such a system must have three characteristics in order to generate selfsustained oscillations: a steady source of energy, a part dissipating energy, and a restoring mechanism. It appears very often within several branches of science, such as physics, electronics, biology, chemistry. For example, the friction caused by the relative motion between a mass and moving belt may lead to oscillations, while a solid object emerging in a steady flow may generate various aeroelastic vibrations, etc. These phenomena have been discussed in many books Arnold, 978; Sagdeev et al., 988; Moon, 992; Guckenheimer & Holmes, 993]. Author for correspondence. 2777

2778 J. Xu & P. Yu Self-sustained systems also play a very important role in the study and development of nonlinear science. The earliest mathematical model associated with self-sustained systems was developed by van der Pol and Van der Mark 927] to describe the vacuum tube circuit, which is now called van der Pol equation. The irregular noise, observed from an experiment with an electronic oscillator, was reported by van der Pol but no detailed explanation was given, until Ueda and Akamatsu 98] reinvestigated the system later. They modeled a forced van der Pol type circuit with negative resisting oscillator and found that the irregular noise was indeed a chaotic motion: the Poincaré mapping of computer simulation showed a strange attractor called Ueda attractor. Since then many researchers were attracted to the area and made contributions to the study of the mechanism of complex phenomenon observed from experiments and computer simulations. This gradually formed a foundation in developing the theory, methodology and application of selfsustained oscillation systems. Based on the work of Cartwright et al. 945, 948], Gilles 954], and Holmes 980] obtained various phase portraits and bifurcation results for a type of nonlinear systems with periodic forcing, associated with self-sustained oscillations. Levi 98], on the other hand, used the method of symbol sequence to consider a simplified, piecewise linear model and applied Poincaré map to find the geometrical picture for nonperiodic motions. These results obtained early show that a self-sustained system is truly able to exhibit remarkable complex dynamical behaviors. The self-sustained system often needs to be controlled due to its rich and complex dynamical behaviors. Over the past years, there have been a great deal of research related to various methods to control the complex dynamics for Eq. () Jackson & Grousu, 995]. The method of time delayed feedback control is one of them, which was first given by Pyragas 992]. It is so interesting that many extended investigations have been published recently. A key technique is to derive stability conditions. Nakajima and Ueda 998] proposed a half-period delayed feedback control for some systems with symmetry. An unstable delayed feedback controller was used by Pyragas 200]. Just et al. 997, 998, 999, 2000] employed characteristic function to obtain the stability condition. A general repetitive learning control structure based on the invariant manifold was revealed by Song et al. 2002] to propose a delayed chaos control method for stabilizing unstable periodic orbits. With complex dynamics in other simple delayed autonomous feedbacks systems, Hale and Huang 993], Moiola and Chen 996] contributed to the development of theory and the application of engineering, respectively. Although many remarkable studies have been done as mentioned above, any systematic method for obtaining stability conditions, such as those for the dynamics delayed feedback control for discretetime systems, has not been proposed so far regarding delayed feedback control for continuous-time systems Nakajima, 2002]. On the other hand, a complex dynamic behavior or chaos, which is usually harmful (e.g. causing instability) and should be controlled in most applications, may be useful and has actually been purposely generated for some special applications, such as secure communications Cuomo & Oppenheim, 993; Ushio, 999; Gregory & Rajarshi, 999]. However, few authors use delayed feedback controllers to generate chaos in continuous-time systems. Motivated by these two problems, we intend to believe that applying a time delay to a dynamical system may be one of the best approaches to control or create complex dynamical motions, since the time delay is easy to be controlled and realized in real applications. The time delayed system can be obtained by adding a time delayed feedback to a system. Such a system will be considered in this paper. By adding a time delay (linear or nonlinear) feedback to system (), one can obtain the following delayed velocity feedback control system ẍ + ω 2 0x α ẋ + α 3 ẋ 3 = k cos(ωt) + A(ẋ τ ẋ) + B(ẋ τ ẋ) 3, (2) where x τ = x(t τ) and τ is time delay. The feedback is called negative if A, B < 0, positive if A, B > 0. System (2) is reduced to system () at τ = 0. Although system (2) is obtained through a time delayed feedback control, time delayed systems similar to Eq. (2) (k = 0) have been considered by many researchers, for studying various phenomena appearing in physical systems Wischert et al., 994], biological systems Tass et al., 996], mechanical systems Nayfeh et al., 997], as well as many other natural systems Cushing, 976; Liao & Yu, 998; Cao, 2000; Wulf & Ford, 2000; Yao et al., 200]. This is because time delay is unavoidable due to the finite transmission speed of signals, as well as the limit of system reaction and processing

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 2779 times between synapses. Therefore, it is interesting and important to consider the effect of time delay on the dynamical behavior of natural systems. Xu and Lu 999] proposed an analysis for the delayreduced Hopf bifurcation as k = 0, α = 0 and α 3 = 0. Recently, Reddy et al. 2000] studied the effects of time delayed linear and nonlinear feedbacks on a so-called Stuart Landau system which may be obtained by averaging system (2) and setting k = 0. The system considered by Reddy et al. 2000] showed rich dynamical behavior under linear or nonlinear feedbacks, including several different types of bifurcations and chaos. Other similar studies using different feedbacks can be found in the references provided by Compbell et al. 994, 995a, 995b], which are restricted to the study of limit cycles (periodic solutions). However, for the system with external forcing, i.e. when k 0 in Eq. (2), to our best knowledge, no result has been obtained for the dynamics of such a system with time delayed feedbacks. The main goal of the paper is, based on the model described by Eq. (2), to develop an analytical approach to investigate the effect of time delay. We hope that we can use the time delay to control the system like a switch which either eliminates harmful motions or creates useful motions by varying the values of the time delay. For example, the former can be used to move a system from vibration state to a stable equilibrium and thus increase the equipment life, while the latter may be applied to generate a particular chaotic signal for secure communications Gregory & Rajarshi, 999]. Only the results for velocity feedback are presented in this paper. The position feedback will be discussed in a different paper. In the next section, a linear stability analysis is given for the linearized system of Eq. (2). Section 3 considers Hopf bifurcation solutions and their stabilities. Quasi-periodic solutions derived from the periodic solution bifurcation are studied in Sec. 4, and discussions and conclusions are given in Sec. 5. 2. Stability Analysis of Linearized System In this section, as the first step we analyze the stability of the trivial solution for the linearized system of (2) without excitation, and mainly find the explicit expressions of the critical boundary at which the trivial solution loses stability. This helps the bifurcation analysis in the following sections. The linearized equation of system (2) corresponding to k = 0 can be expressed as ẍ + ω 2 0x α ẋ A(ẋ τ ẋ) = 0. (3) The characteristic equation (3) is given by λ 2 + (A α )λ + ω 2 0 Aλ exp( λτ) = 0, (4) where ω 0 is a real positive constant. To find the explicit expressions of the critical stability boundaries, we will establish one lemma and three theorems, under the following hypothesis: H. The eigenvalues of Eq. (4) are analytical functions of parameters A and τ. We first generalize the result obtained by Bélair et al. 994] to obtain the following lemma. Lemma. Consider the function h(λ, τ) = λ 2 + ω0 2 + λ m j=0 a je τjλ, where a j s are real, τ = (τ, τ 2,..., τ m ) T, and τ j 0, 0 j m. As long as at least one τ j is varied, the number of zeros of the function h(λ, τ) with Re(λ) < 0, counting multiplicities, can be changed only when λ moves to cross the imaginary axis. Proof. Suppose λ = λ(τ) is a zero of equation h(λ, τ) = 0, satisfying < λ < 0. Let R(T ) denote the number of such zeros. Without loss of generality, suppose τ 0 = 0 while the remaining τ j > 0 (j =, 2,...). According to H, λ(τ) is an analytical function of τ, and thus by Rouché s Theorem: there exists an ε > 0, such that for τ ν < ε, λ(τ) and λ(ν) have the same multiplicity. Therefore, if R(T ) has a change, and no zeros appear on the imaginary axis, then such a change can only happen at : i.e. on the open right half complex plane. Thus, there exist a τ and a sequence { τ (i) } such that lim i τ (i) = τ and lim i λ( τ (i) ) =, where stands for complex modulus, and thus h(λ, τ) lim i λ 2 = lim + ω2 0 i λ 2 + a 0 m λ + a j e τjλ λ =, j= due to e τjλ for j m. This contradicts the assumption h(λ, τ) = 0. Based on the above lemma, we can obtain the following theorem. Theorem. The number of eigenvalues of the characteristic equation (4) with negative real parts, counting multiplicities, can be changed only when

2780 J. Xu & P. Yu the eigenvalues become pure imaginary pairs as the time delay τ and the feedback gain A are varied. Proof. It follows the Lemma straightforwardly by noting that Eq. (4) has no trivial roots due to ω0 2 > 0. In order to apply the Lemma and Theorem to Eq. (4), let λ = α+iβ, where α and β are real, be a root of the characteristic equation (4). Without loss of generality, one can assume β > 0 since the roots of Eq. (4) appear as complex conjugate pairs. Then substituting the above λ into Eq. (4), and taking the real and imaginary parts of the resulting equation to equal zero yields Aα + α 2 αα β 2 + ω 2 0 Ae ατ α cos(βτ) + β sin(βτ)] = 0, Aβ + 2αβ α β Ae ατ β cos(βτ) α sin(βτ)] = 0. (5) Now with the Lemma and Theorem, we can derive the explicit expressions for the critical stability boundaries. Setting α = 0 in Eqs. (5) results in ω 2 0 β2 = Aβ sin(βτ), Aβ α β = Aβ cos(βτ). One can eliminate τ from Eqs. (6) to obtain ( ) β 4 ( ) ( ) A α 2 + ω 0 ( α ω 0 ω 0 ω 0 (6) ) ] 2 ( ) β 2 + = 0 (7) ω 0 which, in turn, gives the roots: ( ) 2 β± = + ( A ω 0 ± 2 2 α ω 0 α ω 0 ω 0 4 + α ω 0 ) α ω 0 ( 2 A α ) ω 0 ω 0 ( 2 A α ). (8) ω 0 ω 0 It is easy to see from Eq. (8) that the condition α (2A α ) 0 must be satisfied since β ± are real and positive. This condition actually implies that the critical values of α and A expressed in β and τ, are located in the region bounded by the two straight lines α = 0 and α = 2A (see Fig. ). Fig.. Critical stability boundaries for the trivial solution of Eq. (3) in α A plane (τ fixed): (I) stable region, (II) unstable region. It is noted that Eqs. (6) contain four parameters A, α, β and τ (ω 0 is a constant), but only three of them are independent. When τ = 0, i.e. system (2) is reduced to an ODE (Ordinary Differential Equation) without time delay, and it is thus obvious to see that α = 0 is the critical stability boundary. However, if τ > 0, then the critical stability boundary is changed and we need to find the new boundaries. To achieve this, we establish the following theorem. Theorem 2. For a fixed τ > 0, when β (0, π/τ), the trivial solution of Eq. (2) is asymptotically stable if and only if the values of the parameters (α, A) are located in a region (in the parameter space) bounded by the critical curve, which is solved from Eqs. (6) as α c = S(A c ). Proof. If τ = 0, Eq. (4) has eigenvalues λ = (/2)(α ± α 2 4ω2 0 ), which indicate that the trivial solution of Eq. (3) is stable when α < 0. For any fixed τ > 0, and β (0, π/τ), Theorem can be applied to show that the stability of the trivial solution remains until A reaches a critical value at which λ = iβ, namely, Eq. (4) has a root with zero real part. Further, it follows from the second equation of Eqs. (6) that α c = S(A c ) = 0 when τ = 0. Therefore, α < α c = S(A c ), with the aid of the hypothesis (H), defines the stable region for the trivial solution when τ > 0. The critical stability boundary and stable region for the trivial solution of Eq. (3) established in Theorem 2 are shown in Fig.. The critical

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 278 boundaries for τ = 0., 0.3, 0.4 and 0.6 are shown by thin solid, dash, dot dash and thick solid lines, respectively. It is seen that each critical boundary divides the (α, A) parameter plane into two parts: (I) and (II), corresponding to the stable and unstable regions. It should be noted from Fig. that the topological structure of the critical boundary for a larger τ is different from that for a smaller τ. Therefore, it is important to study how the variation of the time delay τ affects the dynamical behavior of the system. This may be achieved by solving τ from Eqs. (6) in terms of the other parameters. By noting ω 2 0 β2 + < 0 and ω 2 0 β2 > 0, we may obtain τ + j, A, α, ω 0 ] = β + β + ( 2jπ + cos ( ) α )] A ( (2j + 2)π cos ( ) α )] A if A < 0, if A > 0 ; (9) and τ j, A, α, ω 0 ] = ( (2j + 2)π cos ( ) β ( 2jπ + cos ( ) β α )] A α )] A if A < 0, if A > 0. (0) The following theorem establishes the stability conditions with respect to the time delay τ. Theorem 3. The trivial solution of Eq. (3) is asymptotically stable when α, A and τ satisfy one of the following conditions: (i) α < min{0, 2A}, τ > 0; (ii) A < α /2 < 0, 0 < τ < τ + 0, A, α, ω 0 ], τ j, A, α, ω 0 ] < τ < τ + j +, A, α, ω 0 ]; (iii) A > α /2 > 0, τ j, A, α, ω 0 ] < τ < τ + j +, A, α, ω 0 ]. Here j = 0,, 2,.... τ + j, A, α, ω 0 ] and τ j, A, α, ω 0 ] are given by Eqs. (9) and (0), respectively. Proof (i) In fact, α < min{0, 2A} is equivalent to α (2A α ) < 0. It can be seen from Eq. (8) that under this condition Eq. (4) has only real eigenvalues. For τ = 0, α < min{0, 2A} yields a stable trivial solution. Then by Theorem one can conclude that the stability of the trivial solution for τ > 0 may remain until Eq. (4) has at least one zero eigenvalue. This is not possible as discussed above. (ii) For A < α /2 < 0 and τ > 0, the stability of the trivial solution remains until τ reaches its first critical value τ + 0, A, α, ω 0 ] at which the real part of the eigenvalue becomes zero. If the stability of the trivial solution is changed, this critical curve should be the one on which dα/dτ > 0. From Eq. (4) one may find and dλ dτ = Aλ 2 A(λτ ) + e λτ (A α + 2λ) dα dτ = G Aβ 2 (A α ) cos βτ α=0 () A 2β sin βτ], (2) where G = ie iβτ (iα + 2β) + A(e iβτ + iβτ) 2 is real positive. Hence, at τ = τ + j, A, α, ω 0 ](j = 0,, 2,...), dα dτ = G β+(a 2 α )A cos(β + τ) α=0 A 2 2Aβ + sin(β + τ)] = G β 2 + (A α ) 2 A 2 2(ω 2 0 β2 + )] = G β 2 + α (α 2A) 2(ω 2 0 β2 + )] > 0, since ω 2 0 β2 + < 0. (3) (iii) Similar to (ii), one can prove this part. The results stated in Theorem 3 are depicted in Fig. 2 where ω 0 = 0. The two cases for α = and α = are shown in Figs. 2(a) and 2(b), respectively. The stable region is indicated by (I) while the unstable region by (II). Solid and dash lines denote the critical boundaries for τ + and τ, respectively.

2782 J. Xu & P. Yu (a) (a) (b) Fig. 2. Critical stability boundaries for the trivial solution of Eq. (3) in τ A plane (α fixed): (a) α =, (b) α = +. (I) stable region, (II) unstable region; solid line for τ + and dashed line for τ. (b) Fig. 3. Phase portrait of system (2) for ω 0 = 0, α 3 = 0.3, τ = 0., B = and k = 0: (a) stable focus when (α, A) = (0.,.0) region (I), (b) stable limit cycle when (α, A) = (.0,.0) region (II). From the proof of Theorem 3 and Fig. 2 we have observed: (i) Hopf bifurcations may only occur on the critical stability boundaries. (ii) It is possible to have a double Hopf bifurcation (associated with two pairs of purely imaginary eigenvalues), as shown in Fig. 2(b) denoted by points P, P 2 and P 3. This paper focuses on the analysis of periodic solutions, which take place from codimension-one Hopf bifurcations, as well as the dynamical behavior of the system when the solutions lose stability. The double Hopf bifurcation will be discussed in detail in another paper. 3. Hopf Bifurcation Solutions and Their Stability In the previous section we have analyzed the stability of the trivial solutions. It has been shown that when the parameters are varied to cross the critical boundary given in Eqs. (6) due to the time delay τ, the trivial solution loses its stability, leading to Hopf bifurcation (periodic) solutions. This is demonstrated in Fig. 3, where the parameter value used for the stable focus see Fig. 3(a)] is chosen from region (I), while that used for the stable limit cycle given in Fig. 3(b) is taken from region (II). It is interesting to note from Fig. 3(a) that the time delay does change the stability of the trivial

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 2783 solution. For the case shown in this figure, the trivial solution should be unstable if the time delay is not present. However, it becomes stable with a small time delay. This suggests that the time delay is very important for analyzing the dynamical behavior of a system such as stability, and we therefore need to consider the effect of time delay on Hopf bifurcation solutions, as shown in Fig. 3(b). 3.. Reduction on center manifold For a system described by ordinary differential equations (with or without external forcing), perturbation techniques may be applied to find closed form solutions for periodic motions Yu, 998]. Usually, for such an equation, periodic solutions and their stabilities can be analyzed with the aid of center manifold theory and normal form theory. In fact, the perturbation technique presented in the references Yu, 998] combines the two theories to directly obtain a normal form on the center manifold. For the system (2) considered in this paper, the perturbation approaches applied to ordinary differential equation are no longer applicable since here it is a time delayed differential equation. Functional analysis needs to be employed to obtain the center manifold of system (2) in order to find the closed form of the stability conditions for the periodic solution. To achieve this, letting α = α c + µ ε, A = A c + µ 2 ε, and rescaling x εx, k ε 3/2 k (which implies that the excitation is soft, i.e. k > 0) { F (t, u 2t ) = in Eq. (2) yields or ẍ α c ẋ + A c (ẋ ẋ τ ) + ω 2 0 x = εµ ẋ + µ 2 (ẋ τ ẋ) α 3 ẋ 3 + k cos(ωt) + B(ẋ τ ẋ) 3 ], (4) ẍ + (A c α c )ẋ + ω 2 0 x A cẋ τ = εf(x, ẋ, ẋ τ, t), (5) where f(x, ẋ, ẋ τ, t) = µ ẋ + µ 2 (ẋ τ ẋ) α 3 ẋ 3 + k cos(ωt) + B(ẋ τ ẋ) 3, α c and A c are the critical values, satisfying Eqs. (6). Consequently, Eq. (4) may be rewritten as u = u 2, u 2 = ω 2 0 u + (α c A c )u 2 + A c u 2τ + ε(µ µ 2 )u 2 + µ 2 u 2τ ] + εk cos Ωt α 3 u 3 2 + B(u 2τ u 2 ) 3 ], (6) where u = x, u 2τ = u 2 (t τ). To transform Eqs. (6) into a functional differential equation, let C = C( τ, 0], R 2 ), and then for any φ C, define φ = sup τ θ 0 φ(θ) and u t (θ) = u(t + θ), τ θ 0. Thus, (6) becomes u = L(0)u t + εl(µ, µ 2 )u t + εf (t, u 2t ), (7) where u = (u, u 2 ) T, u 2τ = u 2 (t τ) = 0 τ δ(θ + τ)u 2 (t+θ)dθ = 0 τ δ(θ+τ)u 2tdθ, and F : R + C R 2, given by 0 k cos Ωt α 3 u 3 2t (0) + Bu 2t( τ) u 2t (0)] 3 }. (8) L(0) : C R 2 is a linear operator for the critical case, expressed by L(0)φ = where 0 τ dη(s)]φ(s), (9) dη(s) ] 0 δ(s) = ω0 2δ(s) (α ds. c A c )δ(s) + A c δ(s + τ) (20) Similarly, L(µ, µ 2 )φ = 0 τ dη(s, µ, µ 2 )]φ(s) (2) with dη(s, µ, µ 2 )= 0 0 0 (µ µ 2 )δ(s) + µ 2 δ(s + τ) Further, for φ C( τ, 0], R 2 ), we define dφ(θ) for θ τ, 0), D(0)φ = dθ L(0)φ for θ = 0, D(µ, µ 2 )φ = { 0 for θ τ, 0), L(µ, µ 2 )φ for θ = 0, ] ds. (22) (23) (24)

2784 J. Xu & P. Yu and where F (t, φ) { = 0 for θ τ, 0), Qφ = F (t, φ) for θ = 0, 0 k cos Ωt α 3 φ 3 2 (0) + Bφ 2( τ) φ 2 (0)] 3 (25) }. (26) Finally Eq. (7) can be rewritten as a functional differential equation: u t = D(0)u t + εd(µ, µ 2 )u t + εqu t. (27) From the discussion given in the previous section, we know that the characteristic equation (4) has a single pair of purely imaginary eigenvalues Λ = ±iβ c for ε = 0, β c (0, π/τ), and a fixed τ 0. Therefore, C can be split into two subspaces as C = P Λ Q Λ, where P Λ is a two-dimensional space spanned by the eigenvectors of the operator D(0) associated with the eigenvalues Λ, and Q Λ is the complementary space of P Λ. Furthermore, for ψ(ξ) C (0, τ], R 2 ), where C is the dual space of C, define dψ(ξ) for 0 < ξ τ, D dξ (0)ψ = 0 dη T (s)]ψ( s) for ξ = 0, τ (28) and then for φ C and ψ C, we can define the Ψ(ξ) = 2 l 2 + m 2 bilinear form ψ, φ = ψ T (0)φ(0) 0 τ θ 0 ψ T (ξ θ)dη(θ)]φ(ξ)dξ. (29) Hence, D (0) and D(0) are adjoint operators. Now suppose q(θ) and q (θ) are the eigenvectors of D(0) and D (0), respectively, corresponding to the eigenvalues iβ c and iβ c, i.e. D(0)q(θ) = iβ c q(θ), D (0)q (ξ) = iβ c q (ξ). (30) By a direct calculation, we obtain { } q(θ) = e iβcθ, (3) iβ c and { } iβc q + A c sin(β c τ)] (ξ) = N e iβcξ, (32) where N = /(l im), l = τβ c A c sin(β c τ), m = 2β c + A c sin(β c τ) + τβ c cos(β c τ)]. Thus, it is easy to show that q, q = and q, q = 0. Further, it follows from Eqs. (30) (32) that the real bases for P Λ and its dual space are given by Φ(θ) = (ϕ, ϕ 2 ) = ( 2 Re(q(θ)), 2 Im(q(θ))) and Ψ(ξ) = (ψ, ψ 2 ) = ( 2 Re(q (ξ)), 2 Im(q (ξ))), respectively, where the factor 2 is required for the normalization. Then, the basic solution matrices can be written as Φ(θ) = ] cos βc θ sin β c θ 2 (33) β c sin β c θ β c cos β c θ and ω 2 0 β c (m cos(β c ξ) + l sin(β c ξ) ω 2 0 ( l cos(β c ξ) + m sin(β c ξ) β c. (34) l cos(β c ξ) m sin(β c ξ) m cos(β c ξ) + l sin(β c ξ) Next, by defining v (v, v 2 ) T = Ψ, u t (which actually represents the local coordinate system on the two-dimensional center manifold, induced by the basis Ψ), one can then, with the aid of Eqs. (33) and (34), decompose u t into two parts to obtain u t = u P Λ t + u Q Λ t = Φ Ψ, u t + u Q Λ t = Φv + u Q Λ t, (35) which implies that the projection of u t on the center manifold is Φv. Now substituting Eq. (35) into Eq. (27) and then applying the bilinear operator (29) with Ψ given by Eq. (34) to the resulting equation yields Ψ, (Φ v + u Q A t ) = Ψ, D(0) + εd(µ, µ 2 ) + εq](φv + u Q A t )

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 2785 which, with the aid of Eq. (35), in turn results in Ψ, Φ v = Ψ, D(0)Φ v + ε Ψ, D(µ, µ 2 )Φ v + ε Ψ, Q(Φv + u Q A t ) I v = D Λ v + ε Ψ, L(µ, µ 2 )Φ v + ε Ψ, F (t, Φv + u Q A t ) ] 0 βc v = v + εd ε v + εn ε (v), (36) β c 0 where the order O() linear part D Λ v is now in the Jordan canonical form, as expected. The D ε is the coefficients of the O(ε) linear term, while N ε (v) represents the nonlinear terms contributed from the original system to the center manifold, given by D ε = N ε (v) = 2β c (l 2 + m 2 ) lµ2 sin(β c τ) l(µ µ 2 ) + lµ 2 cos(β c τ) mµ 2 sin(β c τ) m(µ µ 2 ) + mµ 2 cos(β c τ) { } 2 l l 2 + m 2 (k cos(ωt) 2 2 α 3 βc 3 v3 2 + 2 2Bβc 3 sin(β cτ)v + (cos(β c τ) )v 2 ] 3 ). m ], (37) It should be noted from Eqs. (37) that N ε (v) only takes the leading (cubic) order terms from the nonlinear function Ψ, F (t, Φv + u Q A t ) and thus the part of u Q A t does not affect N ε (v). In the following, based on Eq. (36), we shall discuss the bifurcation solutions and their stabilities. 3.2. Frequency response for a primary resonance When ε = 0, the solution of Eq. (36) can be expressed as v = (r cos(β c t + θ), r sin(β c t + θ)) T, (38) where r and θ are constants while β c is the vibration frequency corresponding to the critical point defined by α = α c, A = A c. When ε 0, since excitation is weak soft in order of ε (see Eq. (4)), we need to consider the primary resonance, and thus let Ω = β c + εσ, (39) where σ is a detuning parameter with σ = O(). We want to find the periodic solutions of the system with small ε, which can be considered as perturbing the linear system v = D Λ v. Therefore, we may assume that the solution of Eq. (36) can be still expressed in the form (38), but now r and θ are assumed functions of t rather than constants. Then substituting Eqs. (38) and (39) into Eq. (36) and applying averaging theorem Guckenheimer & Holmes, 993], one can transform the variable v into new variables r and γ as r = 2(l 2 + m 2 ) {2mβ cr(µ µ 2 ) 3mβ 3 c r 3 (α 3 + 3B) + 2 kl cos(γ) + m sin(γ)] + 2mrβ c (6Br 2 β 2 c + µ 2) cos(β c τ) + 2lβ c r(3bβ 2 c r2 + µ 2 ) sin(β c τ) 3Br 3 β 3 c r3 m cos(2β c τ) + l sin(2β c τ)]}, γ = σ + 2(l 2 + m 2 )r { 2lβ cr(µ µ 2 ) + 3lβc 3 r 3 (α 3 + 3B) 2 km cos(γ) l sin(γ)] + 2lrβc (6Br 2 βc 2 + µ 2) cos(β c τ) (40) + 2mβ c r3bβ 2 c r 2 + µ 2 ] sin(β c τ) + 3Bβ 3 c r 3 l cos(2β c τ) m sin(2β c τ)]}, where r = r(t), γ = γ(t) = σεt θ(t), ( ) = d( )/d(εt).

2786 J. Xu & P. Yu To find the solution for a steady-state motion, setting r = γ = 0 in Eqs. (40) yields 2 kl cos(γ) + m sin(γ)] = 2mβc r(µ µ 2 ) + 3mβ 3 c r 3 (α 3 + 3B) 2mrβ c (6Br 2 β 2 c µ 2 ) cos(β c τ) 2lβ c r(3bβ 2 c r 2 + µ 2 ) sin(β c τ) + 3Br 3 β 3 c r3 m cos(2β c τ) + l sin(2β c τ)], 2 kl sin(γ) m cos(γ)] = 2rσ(l 2 + m 2 ) + 2lβ c r(µ µ 2 ) 3lβ 3 c r3 (α 3 + 3B) (4) 2lrβ c (6Br 2 β 2 c + µ 2) cos(β c τ) 2mβ c r3bβ 2 c r2 + µ 2 ] sin(β c τ) 3Bβ 3 c r 3 l cos(2β c τ) m sin(2β c τ)]. It then follows from Eqs. (4) that the steady-state motion is periodic with a frequency which equals that of the excitation, and the first-order approximation of the periodic motion can be written as x(t) = u (t) = u t (0) = 2 v + O(ε) = 2 r cos(ωt γ) + O(ε), (42) where r and γ are the solutions of Eqs. (4), representing the nontrivial fixed points of the averaged system (40). It can be seen from Eq. (42) that the response of Eq. (36) or Eq. (2) is synchronized with the external excitation since both Ω and γ are constants. This shows a fundamentally different phenomenon compared with the case without excitation (i.e. when k = 0): when k = 0, system (2) can exhibit multiple states Reddy et al., 2000; Nayfeh et al., 997], while for the case k 0, the system cannot have multiple states, as discussed above. Eliminating γ from Eqs. (4) results in the bifurcation equation with parameters σ, µ 2, τ and B, given by g st = 0, (43) where g st = 2k 2 + 80B 2 r 6 βc 6 + 54Br 6 α 3 βc 6 + 9r 6 α 2 3βc 6 36Br 4 βc 4 µ 2r 4 α 3 βc 4 µ + 4r 2 βc 2 µ 2 + 72Br 4 βc 4 µ 2 + 2r 4 α 3 βc 4 µ 2 8r 2 βc 2 µ µ 2 + 8r 2 βc 2 µ2 2 + 36Blr4 βc 3 σ + 2lr4 α 3 βc 3 σ 8lr 2 β c µ σ + 8lr 2 β c µ 2 σ + 4l 2 r 2 σ 2 + 4m 2 r 2 σ 2 2r 2 β c 35B 2 r 4 βc 5 + 2µ 2 (3r 2 α 3 βc 3 2β c µ + 2β c µ 2 + 2lσ) + 2Br 2 βc 2 (3r2 α 3 βc 3 2β cµ + 4β c µ 2 + 2lσ)] cos(β c τ) + 6Br 4 βc 3 (8Br 2 βc 3 + 3r 2 α 3 βc 3 2β c µ + 4β c µ 2 + 2lσ) cos(2β c τ) 8B 2 r 6 βc 6 cos(3β c τ) + 24Bmr 4 βc 3 σ sin(β c τ) + 8mr 2 β c µ 2 σ sin(β c τ) 2Bmr 4 βc 3 σ sin(2β cτ). (44) When τ = 0, Eq. (43) becomes 2k 2 + 4r 2 (µ 2 + 4σ2 )ω 2 0 2r4 α 3 µ ω 4 0 + 9r 6 α 2 3ω 6 0 = 0, (45) which agrees with the result obtained by Nayfeh and Mook 979] for an oscillating system without time delay. 3.3. Bifurcations from periodic motions First it is noted from Eqs. (42) (44) that the nonzero roots of the algebraic equation (43) are the nontrivial fixed points of Eqs. (40) and actually represent the periodic solutions of system (2).

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 2787 (a) (b) (c) Fig. 4. Comparison of the results between the computer simulation based on Eq. (2) and the analytical solutions derived from Eq. (4) for different values of time delay: (a) τ = 0.02, (b) τ = 0.3, (c) τ = 0.6, (d) τ =.0. Solid lines for analytical solutions, dashed line with triangles for numerical results. (d) Therefore, studying the bifurcations from the periodic solution of system (2) is equivalent to considering the bifurcations from the fixed points of system (40). General frequency amplitude response can be obtained from Eq. (43). Since this paper focuses on the study of dynamical behavior of system (2), in particular, the effect of the time delay on the periodic solutions excited by the external forcing, the critical point should be located on the boundary between the cases with delay and without delay. Thus, the critical point is characterized by τ c = 0 which in turn results in see Eqs. (6)] α c = A c = 0, and therefore, l = 0, m = 2β c. Hence, the first-order periodic solution can be found from Eq. (42) where r and γ are determined by Eqs. (4). Several sampled analytical (stable) solutions corresponding to different values of the time delay τ have been solved using the above analytical formulas and are shown in Fig. 4, where numerical simulation (stable) solutions based on the original system (2) are also presented. The four chosen values of τ are 0.02, 0.3, 0.6 and.0, corresponding to Figs. 4(a) 4(d), respectively. It is seen that the system can exhibit different amplitude (large and small) periodic solutions. The figure clearly shows that the numerical simulation results agree very well with the analytical predictions. In particular, for small amplitude motions (when τ = 0.3 and τ =.0) the numerical solutions are almost identical to the analytical results. This implies that the analytical approach presented in this paper can provide very accurate predictions for periodic motions, no matter τ is small or large. Now based on the analytical formulas obtained above, consider the nonhyperbolic fixed points of

2788 J. Xu & P. Yu (a) (b) (c) (d) (e) Fig. 5. Nonhyperbolic fixed points for (a) τ = 0, (b) τ = 0.079, (c) τ = 0.57, (d) τ = 0.47, (e) τ = 0.549 and (f) τ = 0.628. (f)

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 2789 system (40) which exist only in the case when the following equation has at least one real root, given by where g st r = 0, (46) g st r = 2rβ2 c {540B 2 r 4 β 4 c + 62Br 4 α 3 β 4 c + 27r 4 α 2 3β 4 c 72Br 2 β 2 c µ 24r 2 α 3 β 2 c µ + 4µ 2 + 44Br2 βc 2 µ 2 + 24r 2 α 3 βc 2 µ 2 8µ µ 2 + 8µ 2 2 + 6σ2 2405B 2 r 4 βc 4 + 4µ 2 (3r 2 α 3 βc 2 µ + µ 2 ) + 2Br 2 βc 2 (9r 2 α 3 βc 2 4µ + 8µ 2 )] cos(β c τ) + 6Br 2 βc 2 (54Br 2 βc 2 + 9r 2 α 3 βc 2 4µ + 8µ 2 ) cos(2β c τ) 54B 2 r 4 βc 4 cos(3β c τ) + 96Br 2 βc 2 σ sin(β cτ) + 6µ 2 σ sin(β c τ) 48Br 2 βc 2 σ sin(2β cτ)}, (47) in which l = 0 and m = 2β c have been used. Then the nontrivial steady state motions (r 0) may undergo bifurcation at some critical parameter values. Such a bifurcation is called nontrivial bifurcation. It is well-known that the bifurcation points are determined by Eqs. (43) and (46), in which g st and g st / r can be considered as functions of σ and r with a number of parameters. To demonstrate the effect of time delay on bifurcation points, we eliminate σ from Eq. (47) by using Eq. (43), and then plot the derivative on the (r, ( g st / r)/500) plane. Therefore, the intersection points of curves with the r-axis are the bifurcation points. Since we are mainly interested in the dynamic behavior of the system with respect to the time delay, we choose τ as a varying parameter and fix other parameters, and the results are shown in Fig. 5, where k = /2, µ =, α 3 = /3, β c = 0, µ 2 = and B = 0.5. It can be seen from Fig. 5(a) that when τ = 0, there exist two bifurcation points because the system has a large amplitude excitation. This is a well-known result: large amplitude excitation can cause instability and bifurcation Guckenheimer & Holmes, 993]. With τ slightly increasing to 0.079, the nontrivial bifurcation still holds, as shown in Fig. 5(b). When τ further increases to 0.57 see Fig. 5(c) and 5(d)], the intersection points disappear and therefore there does not exist nontrivial bifurcation points. One can imagine that there may exist one bifurcation point for a value of τ between 0.079 and 0.57. In this degenerate case the curves are tangent to the r-axis. However, when τ continues to increase to, say, larger than 0.54, the bifurcation points are resumed. This is depicted in Figs. 5(e) and 5(f). The above discussions indicate that the analytical results can be used to predict (a) (b) Fig. 6. Frequency response curves for the values of the time delay as shown in Fig. 4: (a) τ = 0 solid line, τ = 0.079 dash line, τ = 0.57 dotted line, (b) τ = 0.47 solid line, τ = 0.549 dash line, τ = 0.628 dotted line. the number of nontrivial bifurcation points occurring in the frequency-response curves, and to show the important influence of time delay on the existence of bifurcation points.

2790 J. Xu & P. Yu Equation (43) is a quadratic algebraic equation of σ, and thus the closed form solution of the frequency-response relation given in terms of σ can be determined by Eq. (43). The frequency-response curves (bifurcation diagrams) with the values of time delay used in Fig. 5 are shown in Fig. 6, where the solid, dashed and dotted lines correspond to the values of τ = 0, 0.079 and 0.57 in Fig. 6(a), and τ = 0.47, 0.549 and 0.628 in Fig. 6(b). These results agree with the analytical predications shown in Fig. 5, i.e. for τ = 0, 0.079, 0.549 and 0.628, there exists nontrivial bifurcations, while for τ = 0.57 and 0.47, there is no bifurcation. The results obtained in Figs. 5 and 6 suggest that the time delay plays a very important role in the analysis of the dynamic behavior of the system, and may be used as a control switch at the bifurcation point. With this switch, one may not only control instability (i.e. transform unstable motions to stable ones) but also may generate complex dynamical motions such as chaos. 3.4. Stability analysis of the periodic solutions As discussed in the previous subsection, system (2) can exhibit, depending upon the time delay τ and the detuning parameter σ (with other parameters fixed), one, two or three periodic solutions, which are described by the first-order approximation (42). To determine the stability of these periodic solutions, we assume that the states in the neighborhood of the nontrivial fixed points are given in the form r = r 0 + r, γ = γ 0 + γ, (48) where r 0 and γ 0 are the nontrivial fixed points of Eqs. (40). Now substituting Eqs. (48) into Eqs. (40), with the aid of Eq. (43), and only keeping linear terms with respect to r and γ yields the following differential equation: ( r) = 2 µ 2 µ 2( cos β c τ) 9 ( ) ] 4 α 3βc 2r2 0 8Br2 0 β2 c sin βc τ 4 r 2 + k 2 cos(γ 0 ) γ, 2β c ] ( γ) = 3Bβc 2 k r 0 sin(β c τ)( cos(β c τ)) 2 2β c r0 2 cos(γ 0 ) r k 2 2β c r 0 sin(γ 0 ) γ, where prime denotes the differentiation with respect to εt. By assuming the solution of Eqs. (49) in the form of r = ( r) 0 exp(ελt) and γ = ( γ) 0 exp(ελt) and then substituting them into Eqs. (49) results in 2 µ 2 µ 2( cos β c τ) 9 ( ) βcτ 4 4 α 3βc 2 r0 2 8Br2 0 β2 c sin Λ] ( r) 0 2 + k 2 cos(γ 0 )( γ) 0 = 0, 2β c ] 3Bβc 2r k 0 sin(β c τ)( cos(β c τ)) 2 2β c r0 2 cos(γ 0 ) ( r) 0 ] k 2 sin(γ 0 ) + Λ ( γ) 0 = 0. 2β c r 0 Note that one may use Eqs. (4) to eliminate cos γ 0 and sin γ 0 from Eqs. (50). Then, a nontrivial solution of Eqs. (50) exists if and only if the determinant of its coefficient matrix equals zero, i.e. (49) (50) Λ 2 pλ + q = 0, (5)

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 279 where p = µ µ 2 + µ 2 cos(β c τ) 3β c r 2 3B + α 3 4B cos(β c τ) + B cos(2β c τ)], q = 4 (µ2 2µ µ 2 + 2µ 2 2) + σ 2 + 2 (µ µ 2 )µ 2 cos(β c τ) + 3 2 β2 c r2 3Bµ α 3 µ + 6Bµ 2 + α 3 µ 2 + 4Bµ cos(β c τ) 8Bµ 2 cos(β c τ) α 3 µ 2 cos(β c τ) Bµ cos(2β c τ) + 2Bµ 2 cos(2β c τ)] + 27 6 β4 c r 4 20B 2 + 6Bα 3 + α 2 3 30B 2 cos(β c τ) 8Bα 3 cos(β c τ) + 2B 2 cos(2β c τ) + 2Bα 3 cos(2β c τ) 2B 2 cos(3β c τ)] + σ{µ 2 sin(β c τ) + 3Bβ 2 c r 2 2 sin(β c τ) sin(2β c τ)]}. (a) (b) Fig. 7. Stability regions of the periodic response for primary resonance: (a) τ = 0, (b) τ = 0.5.

2792 J. Xu & P. Yu Fig. 8. Nontrivial points and their stability of Eqs. (40) versus τ for primary resonances, where σ = 0: solid line stable node, thicker solid line stable focus, dashed line saddle node, dotted dash line unstable focus, dotted line unstable node. When q < 0, the roots of Eqs. (50) are real, having opposite signs, hence the predicted periodic solutions (i.e. nontrivial fixed points) correspond to saddle points and therefore are unstable. The saddle points are located in the interior region bounded by the curve q = 0. It follows from Eqs. (49) that the curve q = 0 is the locus of vertical tangents, which forms a part of the critical boundary of the nontrivial fixed points between stable and unstable solutions. This is called saddle-node bifurcation. Since the discriminant of Eq. (5) is p 2 4q, the nontrivial fixed points associated with p 2 4q < 0 are focal points, while those associated with p 2 4q > 0 are nodal points. These points are stable (unstable) when p < 0 (p > 0). Therefore, the nontrivial fixed points (or the periodic solutions) are stable if and only if either σ or τ satisfies the conditions p < 0 and q > 0. The stability classification discussed above is shown in Fig. 7(a) (τ = 0) and Fig. 7(b) (τ = 0.5), where the critical stability boundaries divide the plane into stable and unstable regions. Other parameter values used in the figure are given as follows: µ = µ 2 = B =, α 3 = 0.3, k = 0.7, and β c = 0. Comparing Figs. 7(a) and 7(b) indicates that time delay can cause a drastic change in the response. As an example, we present a detailed result for the case when σ = 0 to demonstrate the effect of time delay on the stability of the nontrivial points. This is shown in Fig. 8, where the parameter values are the same as that used in Fig. 7. First note that g st given by Eq. (44) is a periodic function of τ with period 2π/β c, implying that the dynamic behavior shown in Fig. 8 appears periodically. It can be observed from Fig. 8 that the same dynamic behavior which occurs at the point a 2 is repeated at the point a 0, and a 7 due to symmetry. The β c used in Fig. 8 is 0, and thus the period in τ should be 2π/β c 0.628, which is exactly the difference in τ between a 2 and a 0. Similar situations happen at points a, a 9 and a 8, and a 3, a, as well as a 4 (a 5 ), a 2 (a 3 ), and so on. It can be seen from Fig. 8 that for small time delay, there exist three types of nontrivial fixed points belonging to the upper, middle and lower branches, denoted by the solid, dotted and dash lines, respectively. The solid (thin or thick) lines represent stable fixed points and others unstable ones. The fixed points located on the upper branch is stable while the fixed points on the other two branches are unstable. Further note that as the time delay τ is increasing, a point on the lower branch becomes an unstable focus from an unstable node when it crosses the point a at which τ = 0.05. A point on the upper branch is a stable node while a point on the middle branch is a saddle node. It should be

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 2793 (a) Fig. 9. Poincaré maps for system (2) showing quasi-periodic motions for small time delay with τ (a 2, a 3 ): (a) τ = 0.0295, (b) τ = 0.05. (b) pointed out that each point on the three branches represent a stable or unstable limit cycle for the original system (2). Therefore, the periodic solutions associated with the upper branch points lose stability at the saddle node bifurcation point a 2 where τ = 0.0285. Thus, time delay increasing from zero system (2) may first exhibit large stable amplitude periodic motion, corresponding to the upper branch, and then lose stability at a 2. However the system returns to a small stable amplitude periodic motion at the point a 3 (τ = 0.08), corresponding to a stable focus for Eqs. (40). This focus remains until point a 4 at which it changes to a node but keeps its stability. 4. Delay-Reduced Quasi-Periodic Solutions The analytical study as mentioned above indicates that no stable periodic solutions exist for τ (a 2, a 3 ), τ (a 6, a 7 ), τ (a 0, a ) and so on, as shown in Fig. 8. Such phenomenon appears periodically in time delay τ. There exists an unstable branch in such intervals (associated with unstable focus) which is connected with a stable branch (stable focus), for example, at point a 3. This implies that a Hopf bifurcation may occur at points a 3, a 6 and so on, which is a secondary Hopf bifurcation at these points for the original system (2). Thus it raises a question: if a stable periodic solution loses stability at these points, when τ is varied to cross these points from right to left, where does the limit cycle go? There are two possibilities: either the amplitude of the periodic solution goes to infinity or it becomes a more complicated bounded motion. To this end, we will here apply a numerical approach to pursue such complex dynamical behavior and to demonstrate validity of the analytical results with delay varying. Based on the information given in Fig. 8 about the bifurcation property of nontrivial solutions, we will apply a sixth-order Runge Kutta approach (RK6) to the original system (2) in order to explore the possible dynamical motions for τ (a 2, a 3 ), τ (a 6, a 7 ), τ (a 0, a ) and so on. In all numerical simulations, the initial values for the equation are taken as x(t) = 0 for t = 0 and ẋ(t) = 0 for τ < t 0. Poincaré mapping is used and the Poincaré plane is taken from the three-dimensional (x(t), x(t τ), ẋ(t τ)) space as x(t τ) = 0, ẋ(t τ) > 0. The numerical results are shown in Figs. 9 and 0, which show the trend of the change of solutions and bifurcations to quasi-periodic solutions with respect to time delay τ. Figure 9 is for small values of τ with τ (a 2, a 3 ), while Fig. 0 demonstrates the behavior for large values of τ with τ (a 6, a 7 ). The characteristics of the motion when τ = 0.0295 which are larger than the value of τ at the point a 2 (see Fig. 8) are shown in Fig. 9(a). As predicted by the analytical study, the periodic solution of the system loses its stability and becomes a quasiperiodic motion: the Poincaré map shows a closed curve. With τ increasing further, the quasi-periodic

2794 J. Xu & P. Yu (a) Fig. 0. Poincaré maps for system (2) showing quasi-periodic motions for large time delay with τ (a 6, a 7 ): (a) τ = 0.53, (b) τ = 0.58. (b) motion remains as shown in Fig. 9(b). When τ is varied to cross the point a 3 (see Fig. 8), a small stable periodic solution appears. This implies that the point a 3 is a Hopf bifurcation point, called a secondary Hopf bifurcation. This trend may be of importance and interesting to applications in chaos control and secure communications due to its simplicity. Similarly, a route from a periodic solution to a quasi-periodic motion and vice verse may also occur for larger time delay, for example, in the interval τ (a 6, a 7 ). This is depicted in Fig. 0, where τ (0.520, 0.599). In addition, the numerical results shown in Figs. 9 and 0 represent the validity of the analytical prediction: unstable periodic solutions may occur periodically in time delay τ. One can clearly see that the period is 0.628 sec. This periodicity is particularly important since it gives much large intervals for time delay in order to control or create complex motions. 5. Discussions and Conclusions Based on a nonautonomous model with time delayed feedback, we have developed an analytical approach to consider the effect of time delay on self-sustained oscillations. The feedback studied in this paper involve time delay in linear and nonlinear (cubic) terms. The analytical results have been used to predict the effect of time delay on the bifurcations as well as on the critical stability boundaries of periodic solutions. It is also predicted that quasiperiodic motions may occur in unstable regions for limit cycles. The results have been shown in various bifurcation diagrams with respect to the system parameters, detuning σ or time delay τ. Although the chaotic motions have not been revealed for the system under consideration, the methodology provided by the present paper suggests: (i) Time delay can be used to control bifurcations and chaos. By comparing to OGY method Ott et al., 990] and the computer simulation performed by Pyragas 992], the theoretical method and results may be considered as an additional approach for chaos control. (ii) Time delay can be used to generate bifurcations and chaos. This may have potential applications in some special applications since a time delayed system has infinite dimension and thus may generate higher dimensional chaos. Moreover, control of time delay is easier to be realized in practice than other control methods. The main attention of this paper is concentrated on the effect of time delay on the dynamical behavior of the system. The effect of feedback gains are not particularly discussed in this paper since many researchers have studied such effects Reddy et al., 2000]. It may be noted in our obtained results (see the figures) that the feedback coefficients A and B are positive. That is, we only considered positive feedback. The frequency curves for the negative feedback can also be derived from Eq. (43). For the linear feedback (i.e. B = 0), we plot Figs. (a) (negative feedback) and (b) (positive feedback) to show a comparison between the negative and

Delay-Induced Bifurcations in a Nonautonomous System with Delayed Velocity Feedbacks 2795 (a) (b) Fig.. Effect of linear time delay feedback on frequency response for primary resonances: (a) negative feedback, µ 2 = 0. solid line, µ 2 = 2 dash line, µ 2 = 5 dotted dash line, (b) positive feedback, µ 2 = 0.3 solid line, µ 2 = 2 dash line, µ 2 = 5 dotted dash line. positive feedback. Figure (a) show that since the topological structures of the frequency curves corresponding to the negative feedback remains the same as the gains are varied, bifurcations cannot occur from the nontrivial fixed points. For positive feedback, however, topological changes can be observed from the frequency-response curves, as shown in Fig. (b), when the feedback gains are changing. This indicates that the number of nontrivial fixed points can be changed with the variation of positive feedback gains, which causes the system to jump (bifurcation) from one state to another. This suggests that the negative feedback is not very interesting but the positive feedback is a necessary condition to allow the system to have chaotic motions. Similar results can be obtained for the nonlinear feedback (i.e. A = 0) as shown in Figs. 2(a) and 2(b). For Eq. (2), problems concerning the multiple solutions or bifurcations can be reduced to studying

2796 J. Xu & P. Yu (a) (b) Fig. 2. Effect of nonlinear time delay feedback on frequency response for primary resonances: (a) negative feedback, B = 0.5 solid line, B = 2 dash line, B = 5 dotted dash line, (b) positive feedback, B = 0.5 solid line, B = 2 dash line, B = 5 dotted dash line. how the solutions r of Eq. (43) vary with the parameters τ and σ when the other parameters are fixed. It is seen that the time delay affects the topological structure of the bifurcations. One is led naturally to consider such questions as: are there other new possible bifurcation structures for Eqs. (43) and (46)? The bifurcation structure of Eq. (43) with Z 2 - symmetry may be investigated in terms of the singularity theory Golubisky & Schaeffer, 985] and the generation of the isola structure (see Figs. 6, and 2). It follows from the theory that two problems must be considered Chen & Xu, 996, 997]. The first problem is called recognition, that is, given a new equation h = 0, when is Eq. (43) equivalent to it? Such a step can simplify Eq. (43) since pathological functions h are chosen in some simple forms. The second problem is to classify the qualitatively different bifurcations occurring in Eq. (43). This is