DOUBLE HOPF BIFURCATION IN DELAYED VAN DER POL DUFFING EQUATION

International Journal of Bifurcation and Chaos, Vol. 3, No. (3) 354 (5 pages) c World Scientific Publishing Company DOI:.4/S8743544 DOUBLE HOPF BIFURCATION IN DELAYED VAN DER POL DUFFING EQUATION YUTING DING,, WEIHUA JIANG and PEI YU, Department of Mathematics, Harbin Institute of Technology, Harbin, 5, P. R. China Department of Applied Mathematics, Western University, London, Ontario, Canada N6A 5B7 pyu@uwo.ca Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. Received October 4, In this paper, we study dynamics in delayed van der Pol Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results. Keywords: Van der Pol Duffing equation; double Hopf bifurcation; normal form; multiple time scales; center manifold reduction.. Introduction Recently, much attention has been focused on the study of high-codimensional bifurcations, since they may exhibit complex dynamical behaviors such as quasi-periodic solutions and chaos. Center manifold theory and normal form theory are usually applied to analyze stability and bifurcation of dynamical systems (e.g. see [Guckenheimer & Holmes, 99; Kuznetsov, 4; Wiggins, 99]). Especially, in delayed differential equations (DDEs), due to time delay, higher-codimension singularities may occur more frequently than that in ordinary differential equations (ODEs). Even for a scalar DDE, it is possible to have bifurcation of limit cycles and even higher-codimension bifurcation phenomena, (e.g. see [Choi & LeBlanc, 6; Hale, 997] and references therein). Several methods have been developed for computing the normal forms of differential equations, for example, multiple time scales (MTS) [Nayfeh, 973, 98] and center manifold reduction (CMR) [Guckenheimer & Holmes, 99; Hale, 997; Hassard et al., 98; Wiggins, 99]. For a dynamical system described by ODEs, multiple time scales method is systematic and can be directly applied to the original nonlinear system, without the application of center manifold theory [Yu, 998, ; Zheng & Wang, ]. In fact, this approach combines the two steps involved in using center manifold theory and normal form theory into one unified step to obtain the normal form and nonlinear transformation simultaneously. Based on multiple time scales, Yu [998,, ] developed Maple programs for computing the normal forms of Hopf bifurcation and other singularities. These programs can be automatically executed by using a computer algebra system for a given ODE system. The basic idea of the center manifold theory is employing successive coordinate transformations to systematically construct a simpler system which has Author for correspondence 354-

Y. Ding et al. Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. less dimension compared to the original system, and thus greatly simplifying the dynamical analysis of the system [Guckenheimer & Holmes, 99; Kuznetsov, 4; Wiggins, 99]. The multiple time scales can also be directly applied to DDEs without the application of center manifold theory [Das & Chatterjee, ; Hu & Wang, 9; Nayfeh, 8]. In contrast, by the center manifold theory, one needs to first change the retarded equations to operator differential equations, and then decompose the solution space of their linearized form into stable and center manifolds. Next, with the adjoint of the operator equations, one computes the center manifold by projecting the whole space to the center manifold, finally calculating the normal form on the center manifold [Faria & Magalhaes, 995; Jiang & Wang, ; Ma et al., 8; Revel et al., ; Wei, 7; Xu & Yu, 4; Yu et al., ; Yuan & Wei, 6]. In the research of nonlinear dynamical systems, the van der Pol Duffing equation is one of the most intensively studied equations. This celebrated equation originally was a model for an electrical circuit with a triode valve, and was extensively studied as a host of a rich class of dynamical behaviors, including relaxation oscillations, quasi-periodicity, elementary bifurcations and chaos [Kuznetsov, 4]. It is well known that the limit cycle oscillations with strong stability property are important in applications, hence, being able to modify their behaviors through feedback is a question of interest. On the other hand, most practical implementations have inherent or feedback delays, the presence of which leads to the appearance of complex phenomena in the autonomous van der Pol Duffing equation, such as Hopf pitchfork bifurcation, double Hopf bifurcation and Bogdanov Takens singularity, etc. [Jiang & Yuan, 7; Jiang & Wei, 8; Wang & Jiang, ; Xu & Chung, 3; Xu et al., 7]. In this paper, we consider the nonresonant double Hopf bifurcation in the following van der Pol equation with delayed feedback: ẍ(t)+ω x(t) [b γx (t)]ẋ(t)+βx 3 (t) = A[x(t τ) x(t)], () where ω, b, γ, β are positive real constants, A is feedback strength, and τ is time delay. Xu ω, = et al. [7] employed the perturbation-incremental scheme (PIS) to investigate the weakly resonant double Hopf bifurcation and dynamics of system (). In this paper, we will study nonresonant double Hopf bifurcation of () by using two methods to compute the normal form of the system, namely, the multiple time scales and a combination of the method of normal forms and the center manifold theorem. A comparison between the normal forms shows that the two methods are equivalent. Furthermore, we carry out the bifurcation analysis and numerical simulations. We will show that there exist a stable fixed point, stable periodic solutions and stable quasi-periodic solutions in the neighborhood of the double Hopf critical point. The rest of the paper is organized as follows. In Sec., we consider the existence of double Hopf bifurcation in the delayed van der Pol system (), and use two methods to derive the normal form associated with double Hopf bifurcation. Then, bifurcation analysis and numerical simulations are presented in Sec. 3. Finally, the conclusion isdrawninsec.4.. Analytical Study In this section, we consider the van der Pol Duffing equation with delayed feedback, described by (), and use the MTS and CMR methods to derive the normal form of the system... System formulation The characteristic equation of the linearized equation of (), evaluated at the trivial equilibrium x =ẋ =,isgivenby: λ bλ + ω + A Ae λτ =. () To find possible periodic solutions, which may bifurcate from a Hopf or double Hopf critical point, let λ =iω (i =,ω >) be a root of (). Substituting the root into () and separating the real and imaginary parts yields { ω + A ω = A cos(ωτ), (3) bω = A sin(ωτ), from which we obtain A +ω b ± (b ω A) 4(ω 4 +Aω ), 354-

Double Hopf Bifurcation in Delayed van der Pol Duffing Equation Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. under the assumption: { A + ω >, (A b ) 4ω b >. Further, it follows from (3) that τ (j), = ω, ω, (4) [ ( arccos + ω ) ] ω, +jπ, A for A>, [ ( π arccos + ω ) ] ω, +jπ, A for A<, where j =,,,... Thus, a possible double Hopf bifurcation occurs when two such families of surfaces intersect, with τ c = τ (j) = τ (l),wherej, l =,,,... The equality τ c = τ (j) = τ (l) implies that the linearized system on the trivial equilibrium has two pairs of purely imaginary eigenvalues ±iω and ±iω. Assume ω : ω = k : k, then a possible double Hopf bifurcation with the ratio k : k appears. When k, k Z +, it is called a k : k resonant double Hopf bifurcation; otherwise, it is called a nonresonant double Hopf bifurcation. In this paper, we only consider the nonresonant double Hopf bifurcation, for which ω = ω = k.. Multiple time scales ω A + ω k, (5) k ω A + ω k. We treat the feedback strength A and the delay τ as two bifurcation parameters. From τ c = τ (j) = τ (l), l, j =,,,..., we get the critical value A c. Suppose system () undergoes a double Hopf bifurcation from the trivial equilibrium at the critical point: A = A c, τ = τ c. Further, by the MTS, the solution of () is assumed to take the form: x(t) =ɛx (T,T,T,...)+ɛ x (T,T,T,...) + ɛ 3 x 3 (T,T,T,...)+, (6) where T k = ɛ k t, k =,,,...The derivative with respect to t is now transformed into d dt = + ɛ + ɛ + T T =D + ɛd + ɛ D +, where the differential operator D i = T i, i =,,,... We take perturbations as A = A c + ɛ A ɛ and τ = τ c + ɛτ ɛ in (), where A ɛ is called a detuning parameter [Nayfeh, 8]. To deal with the delayed terms, we expand x j (T τ c ɛτ ɛ,t ɛ(τ c +ɛτ ɛ ),T ɛ (τ c +ɛτ ɛ ),...)atx j (T τ c,t,t ) for j =,, 3,... Then, substituting the solutions with the multiple scales into () and balancing the coefficients of ɛ n (n =,, 3,...) yields a set of ordered linear differential equations. First, for the ɛ -order terms, we have D x +(A c + ω )x bd x A c x τc =, (7) where x = x (T,T,T ) and x τc = x (T τ c,t,t ). Since ±iω and ±iω are the eigenvalues of the linear part of (), the solution of (7) can beexpressedintheformof x (T,T,T ) = G (T,T )sin(ω T )+G (T,T )cos(ω T ) + G 3 (T,T )sin(ω T ) + G 4 (T,T )cos(ω T ). (8) Next, for the ɛ -order terms, we obtain D x +(A c + ω )x bd x A c x τc = D D x + bd x A c (τ ɛ D x τc + τ c D x τc ). (9) Substituting solution (8) into(9) and simplifying, we obtain the following equation: D x +(A c + ω )x bd x A c x τc + P cos(ω T )+P sin(ω T ) + P 3 cos(ω T )+P 4 sin(ω T )=, () where P i (i =,, 3, 4) are given as follows: G P =ω b G + A c τ ɛ ω [G cos(ω τ c )+G sin(ω τ c )] [ G + A c τ c cos(ω τ c ) G ] sin(ω τ c ), 354-3

Y. Ding et al. Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. P = A c τ ɛ ω [G sin(ω τ c ) G cos(ω τ c )] ω G b G + A c τ c [ G cos(ω τ c )+ G sin(ω τ c ) ], G 3 P 3 =ω b G 4 + A c τ ɛ ω [G 3 cos(ω τ c )+G 4 sin(ω τ c )] [ G4 + A c τ c cos(ω τ c ) G ] 3 sin(ω τ c ), P 4 = A c τ ɛ ω [G 3 sin(ω τ c ) G 4 cos(ω τ c )] ω G 4 b G 3 + A c τ c [ G3 cos(ω τ c )+ G 4 sin(ω τ c ) ]. To avoid occurrence of secular terms in the solution of (), the coefficients of cos(ω i T ), sin(ω i T )(i =, ) in () must be set to zero, i.e. P i =(i =,, 3, 4). Therefore, G, G, G 3 and G 4 are solved from the four linear equations in terms of G, G, G 3 and G 4. Then, Eq. () is reduced to D x +(A c + ω )x bd x A c x τc =, () where x = x (T,T,T )andx τc = x (T τ c, T,T ), and thus, the particular solution of () is x (T,T,T )=. () Further, for the ɛ 3 -order terms, we similarly obtain D x 3 +(A c + ω )x 3 bd x 3 A c x 3τc = D D x D D x D x A ɛ x γx D x βx 3 + A ɛx τc A c (τ ɛ D x τc + τ c D x τc + τ c D x τc + τ ɛ D x τc ) + b(d x +D x )+ A c (τ ɛd x τc + τ c D x τ c +τ ɛ τ c D D x τc ), (3) where x 3 = x 3 (T,T,T ) and x 3τc = x 3 (T τ c,t,t ). Substituting the solutions (8) and() into (3) and letting the coefficients of the terms which may generate secular terms in the solution equal to zero, yields the derivatives G T, G T, G 3 T and G 4 T expressed in terms of G, G, G 3 and G 4. The above procedure can in principle continue indefinitely (to any high order). Finally, the equations for Ġ, Ġ, Ġ 3 and Ġ4 are given by {Ġ = ɛd G + ɛ D G +, (4) Ġ = ɛd G + ɛ D G +, {Ġ3 = ɛd G 3 + ɛ D G 3 +, (5) Ġ 4 = ɛd G 4 + ɛ D G 4 +. Let G = R sin(θ (t)), G = R cos(θ (t)), G 3 = R sin(θ (t)) and G 4 = R cos(θ (t)). Substituting these expressions into (4) and(5), and truncating the equations at O(ɛ 3 ) yields the following normal form in polar coordinates: R = A cω [a cos(ω τ c )+b sin(ω τ c )] (a + b )τ R τ ɛ + [b cos(ω τ c ) b a sin(ω τ c )] c a + R A ɛ b + Q τ ɛ R 3βb + ω γa (a + b ) R R 3βb + ω γa 4(a + b ) R 3, R = A cω [a cos(ω τ c )+b sin(ω τ c )] (a + b )τ R τ ɛ + [b cos(ω τ c ) b a sin(ω τ c )] c a + R A ɛ b + Q τ ɛ R 3βb + ω γa (a + b ) R R 3βb + ω γa 4(a + b ) R 3, Θ = δ + H τ ɛ T + 3a β b ω γ 4(a + b ) R + 3a β b ω γ (a + b ) R, Θ = δ + H τ ɛ T + 3a β b ω γ (a + b ) R + 3a β b ω γ 4(a + b ) R, (6) 354-4

where a i =ω i A c τ c sin(ω i τ c ), b i = A c τ c cos(ω i τ c ) b, Double Hopf Bifurcation in Delayed van der Pol Duffing Equation Q i = ω i(a i A c b iτ c +a i ω i b i + A c b3 i τ c + ω i b4 i + a4 i ω i a ia c ω ib i τ c a i A c τ c ω ib i a3 i A c τ c ω i )sin(ω iτ c ) τ c (a i + b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )) + A cτ c (a i b i ω ib i a i ω i + a 3 i + τ cω i b 3 i + a i τ cω i b i )cos(ω i τ c ) A c ω i τ c (a i + b i ) τ c (a i + b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )) A cω i (a i ω i a 3 i ω ib i a ib i a i τ cω i b i τ c ω i b 3 i )sin(ω iτ c ) τ c (a i + b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )) + ω i A c(b 3 i a ib i ω i + a i b i a 3 i τ cω i a i b i ω iτ c )cos(ω i τ c ) τ c (a i +, b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )) Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. δ i = [a i(ω i + ω + A c A c cos(ω i τ c )) b i A c sin(ω i τ c )]τ ε (a i + b i )τ c [a i(cos(ω i τ c ) ) + b i sin(ω i τ c )]A ε a i +, b i H i = [ω3 i (a i + b i ) A cτ c ω i (a 3 i τ cω i b 3 i a i b i +a i ω i b i + a i τ c ω i b i )] cos(ω iτ c ) τ c (a i + b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )) + A cτ c ω i (a i + b i ) ω i A cτ c (ω i b i τ cω i b 3 i a i ω i + a 3 i + a ib i + a i τ cω i b i )sin(ω i τ c ) τ c (a i + b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )) A cω i (a 3 i ω i a i ω i + ω i b i + τ cω i b3 i + a iω i b i + a i τ cω i b i)cos(ω i τ c ) τ c (a i + b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )) A cω i (ω i b 3 i a iω i b i + a i ω ib i a 3 i τ cω i a iτ c ω i b i )sin(ω iτ c ) τ c (a i +, i =,. b i ) (A c τ c cos(ω i τ c )b ω i sin(ω i τ c )).3. Center manifold reduction In this section, we compute the normal form near the double Hopf bifurcation critical point (A c,τ c ) using the CMR method. First, let ẋ = y. Then, system () can be rewritten as dx dt = y(t), dy dt = ω x(t)+[b γx (t)]y(t) βx 3 (7) (t) + A[x(t τ) x(t)]. Rescale the time by t (t/τ) to normalize the delay so that system (7) becomes dx d t = τy( t), dy d t = bτy( t) ω τx( t)+aτ[x( t ) x( t)] γτx ( t)y( t) βτx 3 ( t). (8) The trivial equilibrium of (8) isx = y =.Atthe critical point (A, τ) =(A c,τ c ), we choose τ c N, θ =, η(θ) =, θ (, ), τ c N, θ =, with ( ) N = ω A, c b ( ) N =. A c Then, the linearized equation of (8) at the trivial equilibrium is dx( t) = L X t d t, where L φ = dη(θ)φ(θ),φ C=C([, ], R ), and the bilinear form on C C ( stands for adjoint) is 354-5

Y. Ding et al. Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. θ ψ(s),φ(θ) = ψ()φ() ψ(ξ θ)dη(θ)φ(ξ)dξ, ξ= in which φ C, ψ C. Then, the phase space C is decomposed by Λ = {±iω, ±iω } as C = P Q, where Q = {ϕ C:(ψ, ϕ) =, for all ψ P },andthebasesforp and its adjoint P are given respectively by ( ) e iω τ cθ e iω τ cθ e iω τ cθ e iω τ cθ Φ(θ) =, iω e iω τ cθ iω e iω τ cθ iω e iω τ cθ iω e iω τ cθ and h (b iω )e iω τ cs h e iω τ cs h Ψ(s) = (b +iω )e iω τ cs h e iω τ cs h (b iω )e iω τ cs h e iω τ cs, h (b +iω )e iω τ cs h e iω τ cs where h j =(b iω j A c τ c e iω jτ c ),j =,. We also use the same bifurcation parameters given by A = A c + A ε and τ = τ c + τ ε in (8), where A ε and τ ε are perturbation parameters, and denote ε =(A ε,τ ε ). Then (8) can be written as dx( t) = L(ε)X t + F (X t,ε), (9) d t where ( L(ε)X t = (τ c + τ ε )y t () ) b(τ c + τ ε )y t () ω (τ c + τ ε )x t () + (A, c + A ε )(τ c + τ ε )[x t ( ) x t ()] and ( ) F (X t,ε)= γ(τ c + τ ε )x t ()y t () β(τ. c + τ ε )x () 3 t We now consider the enlarged phase space BC of functions from [, ] to R, which are continuous on [, ) with a possible jump discontinuity at zero. This space can be identified as C R. Thus, its elements can be written in the form ϕ = ϕ + X c,whereϕ C, c R and X is a matrix-valued function, defined by X (θ) =for θ [, ) and X () = I. In the BC,(9) becomes an abstract ODE, du d t = Au + X F (u, ε), () where u C, and A is defined by A : C BC, and Au = du d t + X [ L u du() d t F (u, ε) =[L(ε) L ]u + F (u, ε). By the continuous projection π : BC P, π(φ + X c)=φ[(ψ,φ)+ψ()c], we can decompose the enlarged phase space by Λ = {±iω τ c, ±iω τ c } ], as BC = P Ker π, where Ker π = {φ + X c : π(φ + X c)=}, denoting the Kernel under the projection π. Letη =(η, η,η, η ) T, v t Q := Q C Ker π,anda Q the restriction of A as an operator from Q to the Banach space Ker π. Further, denote u t = Φη + v t. Then, Eq. () is decomposed as dη d t = Bη +Ψ() F (Φη + v t,ε), dv t d t = A Q v t +(I π)x F (Φη + v t,ε), () where B =diag{iω, iω, iω, iω }. Next, let M denote the operator defined in V 6 (C4 Ker π ), with M : V 6 (C4 ) V 6 (C4 ), (M p)(η, ε) =D ηp(η, ε)bη Bp(η, ε), where V 6 (C4 ) represents the linear space of the second-order homogeneous polynomials in six 354-6

Double Hopf Bifurcation in Delayed van der Pol Duffing Equation Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. variables (η, η,η, η,ε) with coefficients in C 4. Then, it is easy to verify that one may choose the decomposition V 6 (C4 )=Im(M ) Im(M )c with complementary space Im(M )c spanned by the elements A ε η e, τ ε η e, A ε η e, τ ε η e, A ε η e 3, τ ε η e 3, A ε η e 4, τ ε η e 4,wheree i (i =,, 3, 4) are unit vectors. Consequently, the normal form of (9) onthe center manifold associated with the equilibrium (, ) near A ε =,τ ε = has the form dη d t = Bη + g (η,,ε)+h.o.t., where g is the function giving the quadratic terms in (η, ε) for v t =, and is determined by g (η,,ε) = Proj (Im(M f ))c (η,,ε), where f (η,,ε) is the function giving the quadratic terms in (η, ε) for v t = defined by the first equation of (). Thus, the normal form, truncated at the quadratic order terms, is given by dη d t =iω τ c η h A ε τ c (e iω τ c )η + h [ω + ω A c(e iω τ c )]τ ε η, dη d t =iω τ c η h A ε τ c (e iω τ c )η + h [ω + ω A c(e iω τ c )]τ ε η, () where h j =(b iω j A c τ c e iω jτ c ) (j =, ). To find the normal form up to third order, similarly, let M 3 denote the operator defined in V 4 3 (C4 Ker π ), with M 3 : V 4 3(C 4 ) V 4 3(C 4 ), (M 3 p)(η, ε) =D ηp(η, ε)bη Bp(η, ε), Therefore, the normal form up to third-order terms is given by dη d t = Bη +! g (η,,ε)+ 3! g 3 (η,,ε)+h.o.t., (3) where 3! g 3(η,, ) = 3! (I P I,3)f 3(η,, ), and f 3 (η,, ) is the function giving the cubic terms in (η, ε, v t )forε =,andv t = is defined by the first equation of (). Finally, the normal form on the center manifold arising from () becomes dη d t =iω τ c η h A ε τ c (e iω τ c )η + h [ω + ω A c(e iω τ c )]τ ε η + h τ c (iω γ +3β)η η +h τ c (iω γ +3β)η η η, dη d t =iω τ c η h A ε τ c (e iω τ c )η + h [ω + ω A c(e iω τ c )]τ ε η + h τ c (iω γ +3β)η η +h τ c (iω γ +3β)η η η, (4) where h j =(b iω j A c τ c e iω jτ c ), j =,. With the polar coordinates: η = R e iθ, η = R e iθ, combining with (3), we finally obtain the amplitude and phase equations of (4) on the center manifold as dr = µ R + P R 3 dt + P R R, where V 4 3 (C4 ) denotes the linear space of the thirdorder homogeneous polynomials in four variables (η, η,η, η ) with coefficients in C 4. Then, one may choose the decomposition V 4 3 (C4 )=Im(M 3 ) Im(M 3 )c with complementary space Im(M 3 )c spanned by the elements η η e, η η η e, η η e, η η η e, η η e 3, η η η e 3, η η e 4, η η η e 4,where e i (i =,, 3, 4) are unit vectors. where dr dt dθ dt dθ dt = µ R + P R R + P R 3, = ω + δ + Q R +Q R, = ω + δ +Q R + Q R, (5) µ i = [a i cos(ω i τ c )+b i sin(ω i τ c )]A c ω i τ ε (a i + b i )τ c + [b i cos(ω i τ c ) b i a i sin(ω i τ c )]A ε a i +, b i 354-7

Y. Ding et al. δ i = [a i(ω i + ω + A c A c cos(ω i τ c )) b i A c sin(ω i τ c )]τ ε (a i + b i )τ c P ii = 3βb i + ω i γa i 4(a i + b i ), P =P, P =P, Q i = 3a iβ b i ω i γ 4(a i + b i ), [a i(cos(ω i τ c ) ) + b i sin(ω i τ c )]A ε a i +, b i Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. with a i =ω i A c τ c sin(ω i τ c ), b i = A c τ c cos(ω i τ c ) b, i =,. Remark. In the MTS method, we assume that the solution is in a real form, while in the CMR method, we consider the solution to be in a complex form. In fact, the solution to Eq. (7) can be expressed in either a complex form or a real form, that is, x = K e iω T + K e iω T + K e iω T + K e iω T = G (T,T )sin(ω T ) + G (T,T )cos(ω T ) + G 3 (T,T )sin(ω T ) + G 4 (T,T )cos(ω T ). These two different form solutions become identical under the polar coordinate transformations: K = R e iθ, K = R e iθ and G = R sin(θ ), G = R cos(θ ), G 3 = R sin(θ ), G 4 = R cos(θ ). Therefore, in the CMR method, we use the polar coordinates: η = R e iθ, η = R e iθ in order to keep consistence..4. Comparison of the MTS and CMR methods Equation (6) is the normal form derived by using the MTS method, and Eq. (5) is the normal form derived by using the CMR method. Comparing the two normal forms, we have found that Eq. (6) has four more terms Q i τ ɛr i, H i τ ɛ (i =, ) than Eq. (5) does, which come from the Taylor expansions of x j (T τ c ɛτ ɛ,t ɛ(τ c + ɛτ ɛ ),T ɛ (τ c + ɛτ ɛ )) at x j (T τ c,t,t )forj =,, 3,...In fact, in the CMR method, we treat x(t τ) as a linear term of t, and thus no higher-order terms appear, while in the MTS method, we expand x(t τ) in T,T,... as x j (T τ c ɛτ ɛ,t ɛ(τ c + ɛτ ɛ ),T ɛ (τ c + ɛτ ɛ )) at x j (T τ c,t,t ) up to infinite orders, which results in (ɛτ ɛ ) (ɛτ ɛ) 4 + (ɛτ ɛ) 6 (ɛτ ɛ)! 4! 6! 8! =cos(ɛτ ɛ ), + ɛτ ɛ + (ɛτ ɛ) 3 (ɛτ ɛ) 5 3! 5! = sin(ɛτ ɛ ), + (ɛτ ɛ) 7 7! + but we only take terms up to ɛ -order in the MTS method. That is why Eq. (6) has additional Q i τ ɛ R i and H i τ ɛ (i =, ) terms. Or simply, in the MTS method, if we do not expand x(t τ) int,t,... (which would be the same as the CMR method), then these two normal forms are identical. Nevertheless, the small difference has very little effect on the dynamical analysis. 3. Bifurcation Analysis and Numerical Simulation In this section, we first give a bifurcation analysis based on the normal form (5),andthenpresent some numerical simulation results. 3.. Bifurcation analysis For the normal form (5),accordingtothesigns of P P, there exist two different cases, i.e. simple case (with no periodic solutions) and difficult case (with periodic solutions) [Guckenheimer & Holmes, 99]. Here, we are interested in the difficult case, i.e. when P P <. Without loss of generality, we assume P > andp <. Let r j = P jj R j, (j =, ). Then, we have the following planar system in terms of r and r : {ṙ = r (µ + r κr ), (6) ṙ = r (µ + χr r ), where κ = P P, χ = P Note that M =(, ) is always an equilibrium of (6). The two semi-trivial equilibria are given in terms of perturbation parameters as M =( µ, ) and M =(,µ ), which bifurcate from the origin on the bifurcation lines L : µ =andl : µ =, respectively. There may also exist a nontrivial equilibrium M 3 = ( µ κµ κχ, µ χµ κχ ). For this equilibrium to exist, it needs κχ. The nontrivial equilibrium M 3 collides with a semi-trivial one on P. 354-8

Double Hopf Bifurcation in Delayed van der Pol Duffing Equation Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. the bifurcation line T : µ κµ = or T : µ χµ =.If( χ)µ +( κ)µ <, the fixed point M 3 is a sink, otherwise M 3 is a source. Therefore, we further need to consider the bifurcation line T 3 :( χ)µ +( κ)µ =. In order to give a more clear bifurcation picture, choose A c =.359, τ c =6.7583, b =.774, β =.3, γ =and ω ω = 5. Then, the characteristic equation () has two pairs of purely imaginary eigenvalues Λ = {±iω, ±iω }, and all the other eigenvalues have negative real part. Assume that system () undergoes a double Hopf bifurcation from the equilibrium (, ). By a simple calculation, we obtain ω =.8, τ () =.557, τ () =6.7583, Re (λ ) >, (7) ω =.85, τ () =.9433, τ () =6.7583, Re (λ ) <, and µ =.437τ ε +.34A ε, µ =.356τ ε.63a ε, P =.54, P =.449, P =.4338, P =.69, κ =.8587 and χ = 4.6585. Therefore, the critical bifurcation lines become: L : A ε =.3653τ ε, L : A ε =.59τ ε, T : A ε =.756τ ε, T : A ε =.336τ ε, T 3 : A ε =.376τ ε, as shown in the bifurcation diagram (see Fig. ). Since there does not exist unstable manifold containing the equilibrium, according to the center manifold theory, the solutions on the center manifold determine the asymptotic behavior of the solutions of the original system (). Therefore, if Eq. (6) has one or two asymptotically stable (unstable) semi-trivial equilibria M and M,then() has one or two asymptotically stable (unstable) periodic solutions in the neighborhood of the trivial equilibrium. If Eq. (6) has an asymptotically stable (unstable) equilibrium M 3,then()has an asymptotically stable (unstable) quasi-periodic solution in the neighborhood of (, ). So, we shall call the periodic solution the source (respectively, saddle, sink) periodic solution of () ifthesemitrivial equilibrium of (6) is a source (respectively, saddle, sink), and call the quasi-periodic solution the source (respectively, saddle, sink) quasi-periodic solution of () when the nontrivial equilibrium of (6) is a source (respectively, saddle, sink). For the bifurcation behaviors of the original system () in the neighborhood of the trivial equilibrium, the above critical bifurcation boundaries divide the (A ε,τ ε ) parameter plane into seven regions (see Fig. ). We explain the bifurcations in the clockwise direction, starting from B and ending at B.First,inregionB, there is only one trivial equilibrium which is a saddle. When the parameters are varied across the line L from region B (a) (b) Fig.. (a) Critical bifurcation lines in the (τ ε,a ε) parameter space near (τ c,a c); (b) the corresponding phase portraits in the (r,r )plane. 354-9

Y. Ding et al..4...4 4 6 8.4.. Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. Fig...4 4 6 8 Simulated solution of system () for(a ε,τ ε)=(.5,.5), showing a stable fixed point. to B, the trivial equilibrium becomes a sink, and an unstable periodic solution O (saddle) appears from the trivial solution due to a Hopf bifurcation. Similarly, when the parameters are changed from region B to B 3, another periodic solution O (sink) occurs from the trivial solution due to a Hopf bifurcation while the trivial equilibrium becomes a saddle. In region B 4, a quasi-periodic solution (sink) occurs from the periodic solution O due to a Neimark Sacker bifurcation, and O is changed to a saddle from a sink. Further, the quasi-periodic solution (sink) becomes a source when the parameters are varied across line T 3 from region B 4 to B 5, and when the parameters are further changed from region B 5 to B 6 crossing the line T,thequasiperiodic solution collides with the periodic solution O and then disappears, and O becomes a source. When the parameters are further varied across line L from region B 6 to B 7, the periodic solution O collides with the trivial solution and then disappears, and the trivial solution becomes a source from a saddle. Finally, when the parameters are varied across line L from region B 7 to B,theperiodic solution O (saddle) collides with the trivial solution and then disappears, and the trivial solution becomes a saddle from a source. 3.. Numerical simulation To demonstrate the analytic results obtained in Sec. 3., here we present some numerical simulation results. We choose three groups of perturbation parameter values: (A ε,τ ε ) = (.5,.5), (.,.) and (.5,.5), belonging to the regions B, B 3 and B 4, corresponding to a stablefixedpointshowninfig., a stable periodic solution as depicted in Fig. 3, and a stable quasiperiodic solution, see Fig. 4 respectively. It is clear that the numerical simulations agree with the analytical predictions. 4. Conclusion and Discussion In this paper, we have discussed double Hopf bifurcation in delayed van der Pol Duffing equation. We derived the normal form of double Hopf bifurcation by using multiple time scales and center manifold reduction methods. A comparison between the two methods shows that the two normal forms are identical if the higher-order terms obtained in the MTS method, due to the expansion in the delayed variable with respect to time scales, are ignored. Moreover, bifurcation analysis near the double Hopf critical point is given, showing that the system may exhibit a stable fixed point, periodic solutions, and quasi-periodic solutions in the neighborhood of the critical point. Numerical simulations are given to verify the analytical predictions. The normal form method and bifurcation analysis presented in this paper are for local dynamical behaviors. But, surprisingly, we have found that even for parameter values not chosen in the 354-

Double Hopf Bifurcation in Delayed van der Pol Duffing Equation.... 45 46 47 48 49 5..5..5 Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only..... 45 46 47 48 49 5 (a).5..5..... Fig. 3. Simulated solution of system () for(a ε,τ ε)=(.,.): (a) the time history; (b) the phase portrait, showing a stable periodic solution..... 7 75 8..5..5 (b).... 7 75 8 (a).5..5..... (b) Fig. 4. Simulated solution of system () for(a ε,τ ε)=(.5,.5): (a) the time history; (b) the phase portrait, showing a stable quasi-periodic solution. 354-

Y. Ding et al. 4 4 5 6 4 Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. 5 5 5 (a) 4 6 4 4 Fig. 5. Simulated solution of system () for(a ε,τ ε)=( 3, 6): (a) the time history; (b) the phase portrait, showing a stable periodic solution. 7 75 8 4 3 (b) 4 4 7 75 8 (a) 3 4 (b) Fig. 6. Simulated solution of system () for(a ε,τ ε)=( 3, 6): (a) the time history; (b) the phase portrait, showing a stable periodic solution. 354-

Double Hopf Bifurcation in Delayed van der Pol Duffing Equation 7 75 8 5 6 4 Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. 5 7 75 8 (a) 4 6 Fig. 7. Simulated solution of system () for(a ε,τ ε)=(3, 6): (a) the time history; (b) the phase portrait, showing a stable quasi-periodic solution. 3 7 75 8 85 9 8 6 4 (b) 5 5 5 7 75 8 85 9 (a) 4 (b) Fig. 8. Simulated solution of system () with a nonlinear delayed feedback: (a) the time history; (b) the phase portrait, showing a chaotic attractor. 354-3

Y. Ding et al. Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only...5.5..5..5.7.8.9...3.4.5 Fig. 9. The simulated Poincaré map of the chaotic attractor shown in Fig. 8(b) with Poincaré sectionx(t τ) =. neighborhood of the critical point, most simulation results are periodic or quasi-periodic solutions (see Figs. 5 7). This seemingly noncomplex dynamics is perhaps due to system () only containing time delay in a (feedback) linear term. If we introduce time delay in nonlinear terms, we may obtain more complicated dynamical behavior, such as chaos. Indeed, for example, when we change the linear delay feedback A[x(t τ) x(t)] to a nonlinear delay feedback Ax(t τ)[x(t τ) x(t)], and select A =5,τ =4,ω =,b =.774,β =.3 and γ =, we obtain chaotic motion, as shown in Figs. 8 and 9. Acknowledgments The work was supported by the National Natural Science Foundation of China (NSFC) and the National Science and Engineering Research Council of Canada (NSERC), the Heilongjiang Provincial Natural Science Foundation (No. A86), and the Program of Excellent Team in HIT. The first author also acknowledges the financial support received from the China Scholarship Council for visiting Western University. References Choi, Y. & LeBlanc, V. G. [6] Toroidal normal forms for bifurcations in retarded functional differential equations I: Multiple Hopf and transcritical/multiple Hopf interaction, J. Diff. Eqns. 7, 66 3. Das, S. L. & Chatterjee, A. [] Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlin. Dyn. 3, 33 335. Faria, T. & Magalhães, L. T. [995] Normal form for retarded functional differential equations and applications to Bogdanov Takens singularity, J. Diff. Eqns., 4. Guckenheimer, J. & Holmes, P. [99] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd edition (Springer, NY). Hale, J. [997] Theory of Functional Differential Equations (Springer, NY). Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. [98] Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge). Hu, H. & Wang, Z. [9] Singular perturbation methods for nonlinear dynamic systems with time delays, Chaos Solit. Fract. 4, 3 7. Jiang, W. & Yuan, Y. [7] Bogdanov Takens singularity in van der Pol s oscillator with delayed feedback, Physica D 7, 49 6. Jiang, W. & Wei, J. [8] Bifurcation analysis in van der Pol s oscillator with delayed feedback, J. Comput. Appl. Math. 3, 64 65. Jiang, W. & Wang, H. [] Hopf-transcritical bifurcation in retarded functional differential equations, Nonlin. Anal. 73, 366 364. Kuznetsov, Y. A. [4] Elements of Applied Bifurcation Theory, 3rd edition (Springer, NY). Ma, S., Lu, Q. & Feng, Z. [8] Double Hopf bifurcation for van der Pol Duffing oscillator with parametric delay feedback control, J. Math. Anal. Appl. 338, 993 7. Nayfeh, A. H. [973] Perturbation Methods (Wiley- Interscience, NY). Nayfeh, A. H. [98] Introduction to Perturbation Techniques (Wiley-Interscience, NY). Nayfeh, A. H. [8] Order reduction of retarded nonlinear systems The method of multiple scales versus center-manifold reduction, Nonlin. Dyn. 5, 483 5. Revel, G., Alonso, D. M. & Moiola, J. L. [] Interactions between oscillatory modes near :3 resonant Hopf Hopf bifurcation, Chaos, 436. Wang, H. & Jiang, W. [] Hopf-pitchfork bifurcation in van der Pol s oscillator with nonlinear delayed feedback, J. Math. Anal. Appl. 368, 9 8. Wei, J. [7] Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 483 498. Wiggins, S. [99] Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, NY). Xu, J. & Chung, K. W. [3] Effects of time delayed position feedback on a van der Pol Duffing oscillator, Physica D 8, 7 39. 354-4

Int. J. Bifurcation Chaos 3.3. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 3/5/3. For personal use only. Double Hopf Bifurcation in Delayed van der Pol Duffing Equation Xu, J. & Yu, P. [4] Delay-induced bifurcations in nonautonomous system with delayed velocity feedbacks, Int. J. Bifurcation and Chaos 4, 777 798. Xu, J., Chung, K. W. & Chan, C. L. [7] An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedback, SIAM J. Appl. Dyn. Syst. 6, 9 6. Yu, P. [998] Computation of normal forms via a perturbation technique, J. Sound Vibr., 9 38. Yu, P. [] Symbolic computation of normal forms for resonant double Hopf bifurcations using a perturbation technique, J. Sound Vibr. 47, 65 63. Yu, P. [] Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales, Nonlin. Dyn. 7, 9 53. Yu, P., Yuan, Y. & Xu, J. [] Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback, Commun. Nonlin. Sci. Numer. Simul. 7, 69 9. Yuan, Y. & Wei, J. [6] Multiple bifurcation analysis in a neural network model with delays, Int. J. Bifurcation and Chaos 6, 93 93. Zheng, Y. & Wang, Z. [] Stability and Hopf bifurcation of a class of TCP/AQM networks, Nonlin. Anal. Real World Appl., 55 559. 354-5