Restrictions and unfolding of double Hopf bifurcation in functional differential equations

Transcription

1 Restrictions and unfolding of double Hopf bifurcation in functional differential equations Pietro-Luciano Buono Jacques Bélair CRM-837 March Centre de recherches mathématiques, Université de Montréal, CP 618, Succ Centre-ville, Montréal, Québec, H3C 3J7, Canada Département de mathématiques et de statistique, Université de Montréal, CP 618, Succ Centre-ville, Montréal, Québec, H3C 3J7, Canada Institut de génie biomédical, Université de Montréal, CP 618, Succ Centre-ville, Montréal, Québec, H3C 3J7, Canada Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montréal, Québec, Canada

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3 Abstract The normal form of a vector field generated by scalar delay differential equations at nonresonant double Hopf bifurcation points is investigated Using the methods developed by Faria and Magalhães [T Faria and LT Magalhães J Diff Eqs 1 (1995), 181 ] we show that 1) there exists linearly independent unfolding parameters of classes of delay differential equation for a double Hopf point which generically map to linearly independent unfolding parameters of the normal form equations (ordinary differential equations), ) there are generically no restrictions on the possible flows near a double Hopf point for both general and Z -symmetric first order scalar equations with two delays in the nonlinearity, and 3) there always are restrictions on the possible flows near a double Hopf point for first order scalar delay differential equations with one delay in the nonlinearity, and in n th order scalar delay differential equations (n ) with one delay feedback Keywords: Delay differential equations, double Hopf bifurcation, normal forms, center manifold, unfolding

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5 1 Introduction Delay-differential equations share many (but not all) properties with ordinary differential equations This analogy has been made more precise and put on solid theoretical ground as the methods and techniques of geometric dynamical systems theory have been implemented in functional differential equations, see Hale and Verduyn Lunel [11] for numerous references and comments In particular, invariant manifolds for the flow associated with an equation near an equilibrium point have been established, along with their uniqueness and smoothness properties of the manifolds At a bifurcation point, the flow near the equilibrium of the delay-differential equation is essentially governed by the vector field on the centre manifold In this paper, we investigate the flow near double Hopf bifurcation points in scalar first order and n th order scalar delay differential equations by studying the flow on the centre manifold using normal form theory The redeeming feature of centre manifold calculations is the possibility of unfolding degenerate flows in the neighbourhood of invariant sets, in general, and of stationary points in particular In unfolding the flow on the centre manifold, a number of theoretical questions arise The unfolding itself takes place in the framework of ordinary diffential equations, for which most lower codimension cases have been solved [9] For a given class of delay differential equations, it is not a priori obvious, given the circumvoluted reduction procedure involved, that the unfolding of the reduced flow can be obtained from an unfolding of the class of delay differential equations Faria and Magalhães [7] determine parameter families of scalar first order equations leading to reduced flows with appropriate unfolding parameters for several singularities: Hopf, Bogdanov-Takens, and steady-state/hopf We find such parameter families of scalar first order and n th order delay equations for the double Hopf bifurcation, see Theorem 31 and Theorem 41 A natural question concerns the possible restrictions on the flows that can occur on the centre manifold after reduction In this paper, we study this question at double Hopf bifurcation points for the above mentioned classes of delay differential equations This question has been answered in part by Faria and Magalhães [6] They show that any finite jet of an ordinary differential equation can be realized as the centre manifold reduction from a delay-differential equation in R n where n is large enough and the nonlinearity depends on sufficiently many delays Realizability can still be achieved when the number of delays is not sufficient, and this situation is studied by Faria and Magalhães [7] for scalar first order delay differential equations near Hopf, Bogdanov-Takens and steady-state/hopf bifurcation points In particular, realizability holds for the Hopf and Bogdanov- Takens points with one delay and generically for the steady-state/hopf with two delay However, there are strong restrictions on the possible flows near a steady-state/hopf bifurcation point if the nonlinearity depends on a single delay Recently, Redmond et al [13] study the Bogdanov-Takens bifurcation with reflectional symmetry in a scalar first order delay equation with one delay and show that there are no restrictions on the possible phase portraits The determination of possible unfoldings is quite diffferent in a modeling context since it may be leading to different conditions, as pointed out in Hale [1] This becomes particularly significant if our interest lies not so much in assessing all possible behaviours in a class of systems, but rather in trying to determine the range of dynamics accessible in a specific model which depends on a number of parameters The form of the model then becomes a crucial factor in this determination of possible invariant sets, for example For double Hopf bifurcation points, the vector field on the center manifold can be realized by a scalar first order delay differential equation where the number of delays is 4 We study double Hopf bifurcation in scalar first order delay differential equations with one and two delays and in n th order scalar differential equation with delayed feedback We show that, generically, there are no restrictions on the possible flows near a double Hopf bifurcation point for Z -symmetric and general 1

6 scalar first order delay differential equations depending on two delays in the nonlinearity If only one delay is present in the nonlinearity, we compute the normal form to cubic order and show that there always are restrictions on the possible phase portraits See Theorem 31 We study in more detail the equation considered by Bélair and Campbell [1]: they identify, in the Z symmetric scalar equation ẋ(t) = A 1 tanh(x(t τ 1 )) A tanh(x(t τ )) (1) points of double Hopf bifurcation at the boundary of the region of linear stability in the space of the parameters (A 1, A, τ 1, τ ) Using centre manifold calculations, they find restrictions on the possible phase portraits that can appear in the neighbourhood of this singularity We show that these restrictions are due in part to the Z symmetry and to the particular form of (1) We consider equations exhibiting that symmetry in details, recovering and generalizing results from [1] Finally, we study the normal form of the double Hopf bifurcation in n th order scalar delay differential equations A particular example of such equations is the harmonic oscillator with delayed feedback studied by Campbell et al [3] We show that the cubic normal form on the center manifold is given by expressions similar to the cubic normal form for the scalar first order equation with one delay in the nonlinearity Therefore, the same restrictions as for first order equations with one delay apply in this case See Theorem 41 The explicit flow induced by a class of specific functional differential equations on a centre manifold has been made accessible by recent advances in computing power: these calculations have been implemented using symbolic (or analytic) computations, first with Macsyma [14] and more recently with Maple [] In the computation of normal forms of a reduced flow on a centre manifold, Bélair and Campbell [1] used an approach in two steps: they first computed the centre manifold, and then projected the flow from the delay equation on the manifold, then computing the corresponding normal form Faria and Magalhães [5], however, used a different approach, which is the one we employ here: they compute in a single procedure both the centre manifold and the normal form of the flow projected on it Our analysis is the first complete invetigation of the double Hopf bifurcation as it occurs in delay differential equations, and the relationship between unfolded flows on a 4-dimensional centre manifold and the original delay-differential equation: all previous analysis of the restriction question [5, 6, 7, 13] only address unfolding on centre manifolds of dimension three or less The paper is organized as follows Our main results are summarized in Theorem 31 and Theorem 41 The next section is a review of normal form theory for functional differential equations and in particular of the double Hopf bifurcation The proof of Theorem 31 is given in Section 3 The proof of Theorem 41 is given in Section 4 We conclude with a summary and a discussion of our results Some more tedious normal form computations are relegated to Appendix A Normal form for delay differential equations We first recall standard results to fix notation, see [11] Let C = C([ r, ], R n ), L : C R p R n be a continuous linear map and F : R n R p R n a smooth map Consider the retarded functional differential equation: ż(t) = L(µ)z t + F (z t, µ) () where z t C is defined as z t (θ) = z(t + θ) with θ [ r, ] The linear map L(µ) may be expressed in integral form as L(µ)φ = r [dη(θ)]φ(θ),

7 where η : [ r, ] R n is a function of bounded variation Let L = L(), and rewrite () to exhibit the parameters in the linear map: ż(t) = L z t + [L(µ) L ]z t + F (z t, µ) (3) Let A(µ) be the infinitesimal generator for the flow of the linear system ż = L(µ)z t Let σ(a(µ)) denote the spectrum of A(µ) and Λ µ be the set of eigenvalues of σ(a(µ)) with zero real part The bilinear form (ψ, φ) = ψ()φ() θ r ψ(ξ θ)dη(θ)φ(ξ)dξ (4) is used to decompose C as C = P Q where P is the generalized eigenspace of the eigenvalues in Λ and Q is an infinite dimensional complementary subspace A basis for P is given by Φ Λ = {Φ λ1,, Φ λm } and denote by B be the finite dimensional matrix of the restriction of A to Φ Λ : AΦ Λ = Φ Λ B The set Ψ = col{ψ 1,, Ψ m } is a basis of the dual space P in C with (Ψ, Φ) = I, the identity matrix Faria and Magalhães [5] show that equation (3) can be written as an ordinary differential equation on the Banach space BC of functions from [ r, ] to R n bounded and continuous on [ r, ) with a possible jump discontinuity at Elements of BC are of the form φ+x α where φ C, α R n and X (θ) = for θ [ r, ) and X () = I Let π : BC P be a continuous projection defined by π(φ + X α) = Φ[(Ψ, φ) + Ψ()α] We can write BC = P ker π with the property that Q ker π Decompose z t = Φx t + y t where x t R m and y t ker π D(A) Q 1 where D(A) is the domain of A Equation (3) is equivalent to system ẋ = Bx + Ψ(){[L(µ) L ](Φx + y) + F (Φx + y)} ẏ = A Q 1y + (I π)x {[L(µ) L ](Φx + y) + F (Φx + y)} (5) where A Q 1 : Q 1 ker π is such that A Q 1φ = φ + X [L(φ) φ()] Let F j be the j th Fréchet derivative of F, we take the Taylor expansion of F which transforms (5) to ẋ = Bx + 1 j j! f j 1 (x, y) ẏ = A Q 1y + 1 j j! f j (x, y) (6) where f 1 j (x, y) = Ψ()F j (Φx + y) f j (x, y) = (I π)x F j (Φx + y) Equation () is said to satisfy nonresonance conditions relative to Λ µ if (q, λ) η for all η σ(a ) \ Λ µ, where q is an m-tuple of nonnegative integers, q and λ = (λ 1,, λ m ) For the remainder of the paper, we assume the following hypothesis H1 Card(Λ µ ) < Card(Λ ) for µ small H Equation () satisfies the nonresonance conditions relative to Λ 3

8 Under hypothesis H1 and H, Faria and Magalhães show that system (6) can be put in formal normal form ẋ = Bx + 1 j j! g1 j (x, y) ẏ = A Q 1y + 1 (7) j j! g j (x, y) such that the center manifold is locally given by y = and the equation for the vector field on the center manifold is ẋ = Bx + 1 j! g1 j (x, ) j 1 Double Hopf bifurcation A nonresonant double Hopf bifurcation occurs if the linearization L has a pair of eigenvalues ±iω 1, ±iω with ω 1 /ω Q We can assume that all other eigenvalues have negative real parts This assumption is reasonable since in the cases of interest in this paper, Bélair and Campbell [1] and Campbell et al [3] show that points of double Hopf bifurcation lie at the boundary of the stability region The critical set of eigenvalues is Λ = {ω 1, ω 1, ω, ω } with eigenspace P The restriction of L to P is the matrix B defined above In complex coordinates B is diagonal: B = iω 1 iω 1 iω iω (8) which simplifies the normal form transformations The matrix B generates the torus group T = S 1 S 1 whose action on C is given by (θ 1, θ )(z 1, z ) = (e iθ 1 z 1, e iθ z ) Elphick et al [4] show that a possible normal form commutes with the action of T described above We use this normal form for the double Hopf bifurcation without symmetry and with Z -symmetry The formal normal form is the following, see [8]: ż 1 = p 1 ( z 1, z )z 1 ż = p ( z 1, z )z Truncating the normal form equation to degree three we obtain ż 1 = (iω 1 + c 11 z 1 + c 1 z )z 1 ż = (iω + c 1 z 1 + c z )z (9) where c 11, c, c 1, c 1 are complex numbers Takens [15] shows that nonresonant double Hopf bifurcation is determined to third order if the nondegeneracy conditions Re(c ij ), i = 1, and Re(c 11 )Re(c ) Re(c 1 )Re(c 1 ) are satisfied Let z 1 = r 1 e iρ 1 and z = r e iρ The phase/amplitude equations corresponding to (9) are ṙ 1 = (Re(c 11 )r 1 + Re(c 1 )r )r 1 ṙ = (Re(c 1 )r 1 + Re(c )r )r ρ 1 = ω 1 + Im(c 11 )r 1 + Im(c 1 )r ρ = ω + Im(c 1 )r 1 + Im(c )r (1) 4

9 The possible phase portraits in a neighborhood of a double Hopf point are classified by the dynamics of the planar system given by the amplitude equations (ṙ 1, ṙ ) Let the system depend on parameters (η 1, η ) Then the T action on R C is given by (θ 1, θ )(η 1, η, z 1, z ) = (η 1, η, e iθ 1 z 1, e iθ z ) Then the T -equivariant normal form with parameters is The truncation to quadratic order is ż 1 = p 1 (η 1, η, z 1, z )z 1 ż = p (η 1, η, z 1, z )z ż 1 = iω 1 z 1 + α 1 η 1 z 1 + α η z 1 ż = iω z + β 1 η 1 z + β η z (11) Letting µ 1 = α 1 η 1 + α η and µ = β 1 η 1 + β η, the amplitude equations to cubic order is ṙ 1 = (µ 1 + Re(c 11 )r 1 + Re(c 1 )r )r 1 ṙ = (µ + Re(c 1 )r 1 + Re(c )r )r (1) where µ 1 and µ are unfolding parameters (generically independent) 3 First order scalar delay differential equations We study the restriction on the normal form at a nonresonant double Hopf bifurcation point for the following delay differential equations u = L(u t ) + f(u(t τ 1 ), u(t τ )) (13) u = L(u t ) + f 1 (u(t τ 1 ), u(t τ ) )u(t τ 1 ) + f (u(t τ 1 ), u(t τ ) )u(t τ ) (14) u = L(u t ) + f(u(t τ)) (15) For each equation, L(u t ) = a 1 u(t τ 1 ) + a 1 u(t τ ), and for (13) and(14), f(, ) = Df(, ) = while for (15), f() = Df() = Equation (13) is a general equation depending on two delays Equation (14) is a Z -symmetric equation depending also on two delays, it is a generalization of the system studied by Bélair and Campbell [1] Equation (15) has a nonlinearity depending on only one delay The following result is proved in this section Theorem 31 Suppose that equations (13), (14), or (15) has a nonresonant double Hopf bifurcation point at the origin Then, generically, the two parameter family u = (a 1 + ν 1 )u(t τ 1 ) + (a 1 + ν )u(t τ ) + o(u(t τ 1 ), u(t τ )) (16) is a universal unfolding for the double Hopf bifurcation Moreover, (1) for (13) and (14), generically, there are no restrictions on the possible phase portraits near the double Hopf point, and () for (15) there always are restrictions on the possible phase portraits near the double Hopf bifurcation The proof of the unfolding part is given in Proposition 3 The proof of (1) is given in Proposition 33 and Proposition 35 Finally, the proof of () is given in Proposition 39 5

10 31 The C = P Q decomposition In this section, we write systems (13), (14), (15) as infinite dimensional systems The bases of P and P are respectively where Φ(θ) = (e iω 1θ, e iω 1θ, e iω θ, e iω θ ) Ψ(s) = (ψ 1 ()e iω 1s, ψ ()e iω 1s, ψ 3 ()e iω s, ψ 4 ()e iω s ) t, ψ 1 () = [1 L( θe iω 1θ )] 1 ψ () = ψ 1 () ψ 3 () = [1 L( θe iω θ )] 1 ψ 4 () = ψ 3 () Note that ψ 1 () and ψ 3 () are identical functions of ω 1 and ω respectively Truncate (13), (14) and (15) to cubic order Let F and F 3 be homogeneous polynomials of degree two and three respectively Then the two delay equations are u = L(ν 1, ν )u t + F (u(t τ 1 ), u(t τ )) + F 3 (u(t τ 1 ), u(t τ )), where for equation (14), F The one delay equation is u = L(ν 1, ν )u t + F (u(t τ)) + F 3 (u(t τ)) Let z = (z 1, z 1, z, z ) t and y Q 1 = Q C 1 ([ τ, ], R), then system (6) up to degree three for the three first order equations is ż 1 = iω 1 z 1 + ψ 1 ()([L(ν 1, ν ) L ](Φz + y) + F (Φz + y) + F 3 (Φz + y)) ż 1 = iω 1 z 1 + ψ ()([L(ν 1, ν ) L ](Φz + y) + F (Φz + y) + F 3 (Φz + y)) ż = iω z + ψ 3 ()([L(ν 1, ν ) L ](Φz + y) + F (Φz + y) + F 3 (Φz + y)) (17) ż = iω z + ψ 4 ()([L(ν 1, ν ) L ](Φz + y) + F (Φz + y) + F 3 (Φz + y)) dy dt = A Q 1y + (I π)x (F (Φz + y) + F 3 (Φz + y)) If we remove the dependence on the unfolding parameter, we obtain ż 1 = iω 1 z 1 + ψ 1 ()(F (Φz + y) + F 3 (Φz + y)) ż 1 = iω 1 z 1 + ψ ()F (Φz + y) + F 3 (Φz + y)) ż = iω z + ψ 3 ()(F (Φz + y) + F 3 (Φz + y)) ż = iω z + ψ 4 ()F (Φz + y) + F 3 (Φz + y)) dy dt = A Q 1y + (I π)x (F (Φz + y) + F 3 (Φz + y)) 3 Unfolding of the first order equations The linear equation with unfolding parameters is L(ν 1, ν )u t = (a 1 + ν 1 )u(t τ 1 ) + (a 1 + ν )u(t τ ) (18) Let L = L(, ) The quadratic truncation of (17) in the z 1 and z variables at y = is ż 1 = iω 1 z 1 + ψ 1 ()(ν 1 Φ( τ 1 )z + ν Φ( τ )z + F (Φ( τ 1 )z, Φ( τ )z)) ż = iω 1 z + ψ 3 ()(ν 1 Φ( τ 1 )z + ν Φ( τ )z + F (Φ( τ 1 )z, Φ( τ )z)) (19) 6

11 Equation (11) shows that the normal form to quadratic order is given by ż 1 = iω 1 z 1 + ψ 1 ()(e iω 1τ 1 ν 1 + e iω 1τ ν )z 1 ż = iω z + ψ 3 ()(e iω τ 1 ν 1 + e iω τ ν )z () In polar coordinates z 1 = r 1 e ρ 1 and z = r e ρ, the amplitude equation coming from () is ṙ 1 = (Re(ψ 1 ()e iω 1τ 1 )ν 1 + Re(ψ 1 ()e iω 1τ )ν )r 1 ṙ = (Re(ψ 3 ()e iω τ 1 )ν 1 + Re(ψ 3 ()e iω τ )ν )r Let µ 1 = Re(ψ 1 ()e iω 1τ 1 )ν 1 + Re(ψ 1 ()e iω 1τ )ν µ = Re(ψ 3 ()e iω τ 1 )ν 1 + Re(ψ 3 ()e iω τ )ν Proposition 3 Generically, the independent unfolding parameters (ν 1, ν ) of (16) map to independent unfolding parameters (µ 1, µ ) of the normal form equations Proof: Then Let Q = [ Re(ψ1 ()e iω 1τ 1 ) Re(ψ 1 ()e iω 1τ ) Re(ψ 3 ()e iω τ 1 ) Re(ψ 3 ()e iω τ ) det Q = Re(ψ 1 ())Re(ψ 3 ())(cos(ω 1 τ 1 ) cos(ω τ ) cos(ω 1 τ ) cos(ω τ 1 )) + Re(ψ 1 ())Im(ψ 3 ())(cos(ω 1 τ 1 ) sin(ω τ ) cos(ω 1 τ ) sin(ω τ 1 )) + Im(ψ 1 ())Re(ψ 3 ())(sin(ω 1 τ 1 ) cos(ω τ ) sin(ω 1 τ ) cos(ω τ 1 )) + Im(ψ 1 ())Im(ψ 3 ())(sin(ω 1 τ 1 ) sin(ω τ ) sin(ω 1 τ ) sin(ω τ 1 )) Of course, if τ 1 = τ or ω 1 = ω, then det Q =, but we assume that they are not equal Since det Q is a real analytic function of τ 1 and τ then for an open and dense set of values of (τ 1, τ ), the determinant is nonzero ] 33 Z -symmetric first order scalar equation with two delays Perform the normal form calculations to cubic order where the normal form transformation for the quadratic terms is (z, y) = ( z, ỹ) + σ ( z) Dropping the tilde sign on (z, y), the polynomial of degree three at y = is F 3 (z) = F 3 (Φz) + (d z F (Φz))σ 1(z) + (d y F (Φz))σ (z) (dσ (z))g (z, ), (1) multiplied by the vector (ψ 1 (), ψ 1 (), ψ 3 (), ψ 3 ()) t where g 1 (z, ) since all quadratic terms vanish Dropping the conjugate equations we obtain the system ż 1 = iω 1 z 1 + ψ 1 () F 3 (z) ż = iω z + ψ 3 () F 3 (z) () After normal form transformations of the cubic terms we are left with equation (9) where c 11 = 1 3 ψ 1 () F 3 () c z1 z 1 = 3 ψ 1 () F 3 () 1 z 1 z z c = 1 3 ψ 3 () F 3 () c z z 1 = 3 ψ 3 () F 3 () z z 1 z 1 The genericity result for the Z -symmetric equation is the following (3) 7

12 Proposition 33 Suppose that the Z -symmetric equation (14) has a nonresonant double Hopf bifurcation at Then, generically, there are no restrictions on the values that the coefficients (Re(c 11 ), Re(c 1 ), Re(c ), Re(c 1 )) can take in (1) Proof: Since Z -symmetry forces even degree terms to zero then, F 3 (z) = F 3 (Φ( τ 1 )z, Φ( τ )z) = η(ω 1 )z1 3 + η( ω 1 )z η(ω )z 3 + η( ω )z 3 + ζ(ω 1 )z1z 1 + ζ( ω 1 )z 1z 1 + ζ(ω )zz + ζ( ω )z z + ξ(ω 1, ω )z1z + ξ(ω 1, ω )z1z + ξ( ω 1, ω )z 1z + ξ( ω 1, ω )z 1z + ξ(ω, ω 1 )zz 1 + ξ( ω, ω 1 )z z 1 + ξ( ω, ω 1 )z z 1 + ξ(ω, ω 1 )z 1z + ν(ω 1, ω )z 1 z 1 z + ν(ω 1, ω )z 1 z 1 z + ν(ω, ω 1 )z z z 1 + ν(ω, ω 1 )z z z 1, (4) where η(u) = a 3 e 3iuτ 1 + a 1 e (τ 1+τ )iu + a 1 e (τ 1+τ )iu + a 3 e 3iuτ ζ(u) = 3a 3 e iuτ 1 + a 1 (e iuτ + e ( τ 1+τ )iu ) + a 1 (e ( τ +τ 1 )iu + e iuτ 1 ) + 3a 3 e iuτ ξ(u, v) = 3a 3 e iτ1(u+v) + a 1 e iuτ 1 (e i(vτ 1+uτ ) + e i(uτ 1+vτ ) ) +a 1 e iuτ (e i(vτ 1+uτ ) + e i(uτ 1+vτ ) ) + 3a 3 e iτ (u+v) ν(u, v) = 6a 3 e ivτ 1 + a 1 (e i(vτ 1+uτ 1 uτ ) + e ivτ + e i(vτ 1 uτ 1 +uτ ) ) +a 1 (e i(vτ +uτ 1 uτ ) + e ivτ 1 + e i(vτ +uτ uτ 1 ) ) + 6a 3 e ivτ Using (3) and (4), we compute (c 11, c 1, c, c 1 ) explicitly: c 11 = ψ 1 ()ζ(ω 1 ) c 1 = ψ 1 ()ν(ω, ω 1 ) c = ψ 3 ()ζ(ω ) c 1 = ψ 3 ()ν(ω 1, ω ) We now show that generically (Re(c 11 ), Re(c ), Re(c 1 ), Re(c 1 )) can take arbitrary values Consider (Re(c 11 ), Re(c ), Re(c 1 ), Re(c 1 )) as a linear system in (a 3, a 1, a 1, a 3 ) After tedious computations, one can show that the matrix of coeffcients of (a 3, a 1, a 1, a 3 ) is 3αV 1 3αV 6αV 1 6αV α(v 3 + V 1 c(ω 1 )) α(v 4 + V c(ω )) α(v 3 + V 1 c(ω )) α(v 4 + V c(ω 1 )) α(v 1 + V 3 c(ω 1 )) α(v + V 4 c(ω )) α(v 1 + V 3 c(ω )) α(v + V 4 c(ω 1 )) 3αV 3 3αV 4 6αV 3 6αV 4 (5) where V 1 = cos( β(ω 1 ) + ω 1 τ 1 ), V = cos( β(ω ) + ω τ 1 ), V 3 = cos( β(ω 1 ) + ω 1 τ ), V 4 = cos( β(ω )+ω τ ), c(u) = cos(u(τ 1 τ )), α = ψ 1 (), β(ω 1 ) = arg(ψ 1 ()) and β(ω ) = arg(ψ 3 ()) The determinant of this matrix is 144α 4 (cos(ω 1 (τ 1 τ )) cos(ω (τ 1 τ ))) (V V 3 V 1 V 4 ) Suppose that τ 1 τ and ω 1 (τ 1 τ ) ω (τ 1 τ ) + kπ for all k Z, then the determinant vanishes if and only if V V 3 V 1 V 4 = At a nonresonant double Hopf bifurcation point, Bélair and Campbell [1] show that a 1 cos(ω 1 τ ) = a 1 cos(ω 1 τ 1 ) a 1 cos(ω τ ) = a 1 cos(ω τ 1 ) a 1 sin(ω 1 τ ) = a 1 ω 1 a 1 sin(ω 1 τ 1 ) a 1 sin(ω τ ) = a 1 ω a 1 sin(ω τ 1 ) (6) 8

13 Hence V V 3 V 1 V 4 simplifies to a real analytic function of τ 1 a 1 a 1 ((sin(ω τ 1 )ω 1 sin(ω 1 τ 1 )ω ) sin(β(ω 1 )) sin(β(ω )) + ω 1 cos(ω τ 1 ) sin(β(ω 1 )) cos(β(ω )) ω cos(ω 1 τ 1 ) cos(β(ω 1 )) sin(β(ω ))) Since the zeroes of nonzero analytic functions are isolated, then for an open and dense set of values of τ 1, we have that V V 3 V 1 V 4 Hence, generically, there are no restrictions on the cubic coefficients of the normal form Bélair and Campbell [1] compute the normal form at a double Hopf bifurcation to cubic order for the delay differential equation ẋ(t) = A 1 tanh(x(t T 1 )) A tanh(x(t T )) (7) Equation (7) is Z -symmetric with a 1 = and a 1 = They show that there are relations between the coefficients of the cubic monomials of the normal form Therefore, not all possible phase portraits in a neighborhood of the origin in parameter space are realized near the double Hopf bifurcation point We recover their result Corollary 34 Suppose that F 3 (x, y) = a 3 x a 3 y 3 3 Then where Re(c 11 ) = Re(c 1 ) = Re(c 11 ) and Re(c 1 ) = Re(c ) 3Re(ψ 1 ()) a 1 cos(ω 1 τ 1 )(a 3 a 1 a 3 a 1 ) + 3Im(ψ 1()) a 1 [(a 3 a 1 a 3 a 1 ) sin(ω 1 τ 1 ) + a 3 a 1 ω 1 ] and Re(c ) = 3Re(ψ 3 ()) a 1 cos(ω τ 1 )(a 3 a 1 a 3 a 1 ) + 3Im(ψ 3()) a 1 [(a 3 a 1 a 3 a 1 ) sin(ω τ 1 ) + a 3 a 1 ω ] Moreover, if Re(c 11 ) and Re(c ), then the double Hopf bifurcation is determined to third order Proof: Set a 1 = a 1 = in (Re(c 11 ), Re(c ), Re(c 1 ), Re(c 1 )) to obtain the result Then use conditions (6) Now, Re(c 11 )Re(c ) Re(c 1 )Re(c 1 ) = 3Re(c 11 )Re(c ) Thus the nondegeneracy conditions for the vector field to be determined to third order are satisfied if Re(c 11 ) and Re(c ) We now discuss the possible restrictions on the phase portraits near the nonresonant double Hopf bifurcation point We rewrite system (1) as in Guckenheimer and Holmes [9] ṙ 1 = r 1 (µ 1 + r 1 + br ) ṙ = r (µ + cr 1 + dr ), (8) where d = Re(c )/ Re(c ) = ±1, c = Re(c 1 )/ Re(c 11 ) and b = Re(c 1 )/ Re(c ) In Table 1, we reproduce Table 75 of [9] which shows the twelve unfolding cases for the nonresonant double Hopf bifurcation Corollary 34 implies that sgn d = sgn c Table 1 shows that the unfoldings II, IVa, IVb, V, VIIa, and VIIb are not possible in this case 9

14 Table 1: The twelve unfolding cases of (8) Case Ia Ib II III IVa IVb V VIa VIb VIIa VIIb VIII d b c d bc + (+) (+) + ( ) + + ( ) 34 First order scalar equation with two delays For the general scalar delay equation, the calculation of the cubic normal form requires lengthy calculations and the size of the expressions for the coefficients of the cubic terms become quickly unmanageable Instead, we use Proposition 33 to obtain a similar result for general scalar equations Proposition 35 Suppose that the scalar delay differential equation (13) has a nonresonant double Hopf bifurcation at Then, generically, there are no restrictions on the values that the coefficients (Re(c 11 ), Re(c 1 ), Re(c ), Re(c 1 )) can take in (1) Proof: Recall that the cubic polynomial in the normal form is given by multiplying by Ψ() the following expression: F 3 (z) = F 3 (Φz) + (d z F (Φz))σ 1(z) + (d y F (Φz))σ (z) (9) The coefficients (c 11, c 1, c, c 1 ) are functions of the coefficients of (a, a 11, a ) of F and (a 3, a 1, a 1, a 3 ) of F 3 Let T be matrix (5), Re(c 11 ) a 3 Re(c 1 ) C = Re(c ) and C a 1 3 = Re(c 1 ) From (3) and (9), we see that the coefficients of the cubic terms can be written as a 1 a 3 C = T C 3 + R(a, a 11, a ), (3) where R(a, a 11, a ) is a vector in R 4 Hence, for any C R 4 and coefficients (a, a 11, a ), by Proposition 33, generically, we can find C 3 such that equation (3) is satisfied 35 First order scalar equation with one delay The quadratic and cubic nonlinearities are F (u) = a u and F 3 (u) = a 3 u In Faria and Magalhães [5], it is shown that the homogeneous polynomials g i (x, y) of (7) are given by g i (x, y) = f i (x, y) [D x U j (x)bx A Q 1(U j (x))] 1

15 where Uj is the nonlinear part of the normal form transformation and f i denote the terms of degree i obtained after normal form computations to degree i 1 Thus, because of assumption H1 and H the polynomial Uj is determined by solving D x U j (x)bx A Q 1(U j (x)) = f i (x, ) (31) Note that f (x, ) = f (x, ) In our case, let (σ 1(z), σ (z)) be the nonlinear part of the normal form transformation for quadratic polynomials where σ (z)(θ) = q = h q1,q,q 3,q 4 (θ)z q 1 1 z q 1 z q 3 z q 4, with q = q 1 + q + q 3 + q 4 and h q1,q,q 3,q 4 (θ) Q 1 Then (31) becomes [ ] [ ] σ ω 1 z 1 σ σ z 1 ω z σ z σ z 1 z 1 z z (z) = ΦΨ()a (Φ( τ)z) (3) with boundary conditions σ (z)() L(σ (z)) = a (Φ( τ)z) A rough expression for the normal form transformation of the quadratic polynomial of the ẏ equation is given here Proposition 36 σ(z)(θ) = a (P 1 (θ, ω 1, ω )z1 + P 1 (θ, ω 1, ω )z 1 + P 1 (θ, ω, ω 1 )z + P 1 (θ, ω, ω 1 )z + P (θ, ω 1, ω )z 1 z 1 + P (θ, ω, ω 1 )z z + Q 1 (θ, ω 1, ω )z 1 z + Q 1 (θ, ω 1, ω )z 1 z + Q (θ, ω 1, ω )z 1 z + Q (θ, ω 1, ω )z 1 z ) where P 1, P, Q 1 and Q are smooth functions of θ, ω 1 and ω The proof of Proposition 36 is found in Appendix A coefficients of the normal form We now give expressions for the cubic Proposition 37 The coefficients of the cubic terms in the normal form are given below: Re(c 11 ) = 3a 3 Re(ψ 1 ()e iω1τ ) + a [ ω 1 1 Re(ψ 1()e iω1τ )Im(ψ 1 ()e iω1τ ) 4ω 1 Re(ψ 1()e iω1τ )Im(ψ 3 ()e iωτ ) + (4ω1 ω) 1 (ω 1 Re(ψ 3 ()e iωτ )Im(ψ 1 ()e iω1τ ) + ω Im(ψ 3 ()e iωτ )Re(ψ 1 ()e iω1τ )) ] + a Re[ψ 1 ()(e iω1τ P ( τ, ω 1, ω ) + e iω1τ P 1 ( τ, ω 1, ω ))] [ Re(c 1 ) = 6a 3 Re(ψ 1 ()e iω1τ ) + 4a ω1 1 (Re(ψ 1()e iω1τ )Im(ψ 1 ()e iωτ ) + Im(ψ 1 ()e iω 1τ )Re(ψ 3 ()e iω τ )) + ω 1 Im(ψ 3()e iω τ )Re(ψ 1 ()e iω 1τ ) + (ω 1 4ω ) 1 (ω 1 Re(ψ 3 ()e iω τ )Im(ψ 1 ()e iω 1τ ) ω Re(ψ 1 ()e iω 1τ )Im(ψ 3 ()e iω τ )) ] + a Re[ψ 1 ()(e iω 1τ P ( τ, ω, ω 1 ) + e iω τ Q ( τ, ω 1, ω ) + e iω τ Q 1 ( τ, ω 1, ω ))] Letting c 11 = c 11 (ω 1, ω ) and c 1 = c 1 (ω 1, ω ) then Re(c ) = Re(c 11 (ω, ω 1 )) and Re(c 1 ) = Re(c 1 (ω, ω 1 )) 11

16 Proof: Recall first that ψ 1 () = ɛ(ω 1 ) and ψ 3 () = ɛ(ω ) for some function ɛ The quadratic and cubic polynomials are given below: F (Φ( τ)z) = a (e iω1τ z1 + e iω1τ z 1 + e iωτ z + e iωτ z + z 1 z 1 + e iτ(ω 1+ω ) z 1 z + e iτ(ω 1 ω ) z 1 z + e iτ(ω 1 ω ) z 1 z + e iτ(ω 1+ω ) z 1 z + z z ), F 3 (Φ( τ)z) = a 3 (e 3iω1τ z e iω1τ z1z 1 + 3e iτ(ω +ω 1 ) z1z + 3e iτ(ω ω 1 ) z z1 + 3e iω1τ z 1 z 1 + 6e iωτ z 1 z 1 z + 6e iωτ z z 1 z 1 + 3e iτ(ω +ω 1 ) z 1 z + 6e iω1τ z z 1 z + 3e iτ(ω +ω 1 ) z z 1 + e 3iω1τ z e iτ(ω ω 1 ) z 1z + 3e iτ(ω +ω 1 ) z z 1 + 3e iτ(ω ω 1 ) z 1 z + 6e iω 1τ z z 1 z + 3e iτ(ω +ω 1 ) z z 1 + e 3iω τ z 3 + 3e iω τ z z + 3e iω τ z z + e 3iω τ z 3 ) We perform the computations for the system in complex coordinates and then take the appropriate real parts Equation (1) gives the cubic terms after normal form transformation of the quadratic terms The part of the coefficients c ij (i, j = 1, ) coming from F 3 (Φ( τ)z) + (d z F (Φ( τ)z))σ 1(z) are found using the result of Knobloch [1] on the computation of the cubic normal form for ODEs The remaining part of the coefficients is computed from d y (F (Φ( τ)z + y)) y= σ = a Φ( τ)zσ (z) (33) Thus, [ ( a 11 = 3ψ 1 ()a 3 e iω1τ + a ψ 1 ()e iω 1τ 1 iim(ψ 1 ()e iω1τ ) ) iω 1 3 ψ 1()e iω 1τ 4 iim(ψ 3 ()e iτω i ( ) ω1 Re(ψ iω 4ω1 ω 3 ()e iτω ) + ω iim(ψ 3 ()e iτω ) )] + a ψ 1 ()[e iω1τ h 1,1,, ( τ) + e iω1τ h,,, ( τ)] [ a 1 = 6ψ 1 ()a 3 e iω1τ + 4a ψ 1 ()e iω 1τ ω1 1 (Im(ψ 1 ()e iω1τ ) ire(ψ 3 ()e iωτ )) + ω 1 Im(ψ 3 ()e iωτ ) + ψ 1()e iω 1τ i(ω 1 ω ) + ψ 1()e iω1τ i(ω 1 + ω ) ] i (ω ω1 4ω 1 Re(ψ 3 ()e iωτ ) + ω iim(ψ 3 ()e iωτ )) + a ψ 1 ()[e iω1τ h,,1,1 ( τ) + e iωτ h 1,,,1 ( τ) + e iωτ h 1,,1, ( τ)] It is a straightforward computation using formulae (11a) and (11b) of Knobloch [1], (33) and Proposition 36 to verify that c = c 11 (ω, ω 1 ) and c 1 = c 1 (ω, ω 1 ) Taking the real parts yields the result Corollary 38 If a = then Re(c 1 ) = Re(c 11 ) = 6a 3 Re(ψ 1 ()e iω 1τ ) and Re(c 1 ) = Re(c ) = 6a 3 Re(ψ 3 ()e iω τ ) As in Corollary 34, the double Hopf bifurcation is determined to third order if Re(c 11 ) and Re(c ) 1

17 If a =, since Re(c 1 ) = Re(c 11 ) and Re(c 1 ) = Re(c ), the restrictions on the possible phase portraits near a double Hopf point are similar to the restrictions stated after Corollary 34 Now, letting a, a priori many more unfolding cases are possible since sgn d and sgn c need not be equal anymore However, we now show that there always are restrictions on the possible flows near the double Hopf point for fixed values of ω 1, ω and τ Before, we state the result, we perform some transformations on the expressions for the cubic coefficients From Proposition 37 the coefficients in the normal form can be written as Re(c 11 ) = p 1 a 3 + p a, Re(c 1 ) = q 1 a 3 + q a Re(c 1 ) = r 1 a 3 + r a, Re(c ) = s 1 a 3 + s a, where p 1, p, q 1, q, r 1, r, s 1, s are constants since the calculation is made for ω 1, ω and τ fixed Now, if the determinant of M = [ p 1 s 1 p s ] is nonzero, we can write Re(c 1 ) = γ 1 Re(c 11 ) + γ Re(c ) and Re(c ) = δ 1 Re(c 11 ) + δ Re(c ), where (γ 1, γ ) t = M 1 (q 1, q ) t and (δ 1, δ ) t = M 1 (r 1, r ) t Hence, b = Re(c 1) Re(c ) = γ Re(c 11 ) 1 Re(c ) ± γ and c = Re(c 1) Re(c 11 ) = ±δ Re(c ) 1 + δ Re(c 11 ) (34) We now state the result Note that the proof of the proposition also gives a method to determine which restrictions occurs for a particular system Proposition 39 Assume the nondegeneracy condition det M is satisfied Then there always are restrictions on the possible flows of system (15) near a nonresonant double Hopf bifurcation point Proof: We need to show that for all values of a and a 3, there are some combinations of signs of b, c and d which are prohibited The equations Re(c 11 ) = p 1 a 3 +a a = and Re(c ) = s 1 a 3 +s a = define two parabolae passing through (, ) in (a, a 3 ) space Then there is always at least one case of signs of Re(c 11 ) and Re(c ) which cannot occur simultaneously for any value of (a, a 3 ) Let p /p 1 s /s 1, then if a, a 3 are chosen so that d = sgn (Re(c )) = 1, this forces Re(c 11 ) < Similarly, let s /s 1 p /p 1, if a, a 3 are chosen so that d = +1, this forces Re(c 11 ) > Fix s /s 1 p /p 1 and d = +1, then Re(c 11 ) > and let = Re(c 11 )/Re(c ) Then b = γ 1 + γ and c = δ 1 + δ / So b and c are defined for values of > only and are monotone functions of on (, ) Thus, b and c vanish for at most one value of each Therefore, there are at most three intervals where b and c have constant signs Hence, there is always a choice of signs of b and c which is restricted A similar argument holds when p /p 1 s /s 1 and d = 1 This proves the result 4 n th order scalar equation with delayed feedback, n Consider now the n th order delay differential equation (n ) u (n) + β 1 u (n 1) + + β n u = f(u(t τ)) (35) where f() =, β j (j = 1,, n) are constants and τ is the time delay This equation generalizes the harmonic oscillator with delayed feedback ü + β 1 u + β u = f(u(t τ)) (36) 13

18 studied by Campbell et al [3] In this section, we prove the following unfolding result for equation (35) Theorem 41 Suppose that (35) has a nonresonant double Hopf bifurcation point at the origin Then, generically, the two parameter family of delay differential equations u (n) + β 1 u (n 1) + + (β n + ν 1 )u = (a 1 + ν )u(t τ) + o(u(t τ)) (37) provides a universal unfolding for the double Hopf bifurcation However, generically, there always are restrictions on the possible flows of (35) near a double Hopf bifurcation point The proof of Theorem 41 is given by Lemma 4 and Proposition The C = P Q decomposition Truncate f to degree three in its Taylor expansion f(u(t τ)) = a 1 u(t τ) + a u (t τ) + a 3 u 3 (t τ) and rewrite (35) as a system of n first order delay differential equations u = v 1 v 1 = v v n 1 = β 1 v n 1 β n u + a 1 u(t τ) + a u (t τ) + a 3 u 3 (t τ) (38) So, L(u t, v t ) = v 1 v n β 1 v n 1 β n v 1 + a 1 u(t τ), F (u t) = a u (t τ) + a 3 u 3 (t τ) At a double Hopf point, the basis of P is given by the columns of Φ = [Φ 1,, Φ n ] t where Φ j = ((iω 1 ) j 1 e iω 1θ, ( iω 1 ) j 1 e iω 1θ, (iω ) j 1 e iω θ, ( iω ) j 1 e iω θ ) The basis of the adjoint problem is given by the rows of Ψ = [Ψ 1,, Ψ n ] with Ψ j = (Ψ 1 j, Ψ j, Ψ 3 j, Ψ 4 j) t where Ψ = (Φ t, Φ) 1 Φ t and (, ) is the bilinear form (4) Let (u, v 1,, v n 1 ) t = Φz + y where y = (y 1,, y n ) t Q C 1 ([ τ, ], R n ) We rewrite (38) as an infinite dimensional system Note that F is only function of u = Φ 1 ( τ)z + y 1, thus ż = Bz + Ψ()F (Φ 1 ( τ)z + y 1 ) ẏ = A Q 1y + (I π)x F (Φ 1 ( τ)z + y 1 ) (39) where B is (8) Now, F (Φ 1 ( τ)z + y 1 ) = a (Φ 1 ( τ)z + y 1 ) + a 3 (Φ 1 ( τ)z + y 1 ) 3 14

19 Hence (39) becomes ż = Bz + Ψ n ()(a (Φ 1 ( τ)z + y 1 ) + a 3 (Φ 1 ( τ)z + y 1 ) 3 ) ẏ = A Q 1y + (I π)x a (Φ 1 ( τ)z + y 1 ) + a 3 (Φ 1 ( τ)z + y 1 ) 3 ) (4) 4 Unfolding of the n th order equation We choose the following unfolding for the n th order delay differential equation L(ν 1, ν )(u t, v 1,, v n 1 ) = v 1 v n β 1 v n 1 (β 1 ν 1 )v 1 + (a 1 + ν )u(t τ) Thus, (L(ν 1, ν ) L )Φz = ν 1 (ω 1 (z 1 z 1 ) + ω (z z )) + ν Φ 1 ( τ)z The quadratic terms computed from (5) are given by Ψ()[L(ν 1, ν ) L ]Φz = Ψ n ()(ν 1 (ω 1 (z 1 z 1 ) + ω (z z )) + ν Φ 1 ( τ)z) The normal form to degree two is given by equation (11) ż 1 = iω 1 z 1 + (Ψ 1 n()ω 1 )ν 1 z 1 + (Ψ 1 n()e iω 1τ )ω z 1 ż = iω z + (Ψ 3 n()ω )ν 1 z + (Ψ 3 n()e iω τ )ω z and after transformation to polar coordinates the radial part becomes ṙ 1 = (ω 1 Re(Ψ 1 n())ν 1 + Re(Ψ 1 n()e iω 1τ )ν )r 1 ṙ = (ω Re(Ψ 3 n())ν 1 + Re(Ψ 3 n()e iω τ )ν )r Lemma 4 Generically, the independent unfolding parameters (ν 1, ν ) of (37) map to independent unfolding parameters (µ 1, µ ) of the normal form equations Proof: As in Proposition 3, it is easy to check that if ω 1 ω, the determinant of [ ] ω 1 Re(Ψ 1 n()) Re(Ψ 1 n()e iω1τ ) ω Re(Ψ 3 n()) Re(Ψ 3 n()e iωτ ) is nonzero for an open and dense set of values of τ In particular, note that it is necessary to have a parameter as coefficient of the u(t τ) term while the other unfolding parameter can be chosen in front of any other term 15

20 43 Normal form of the n th order scalar equation In this section, we discuss the normal form of the n th order scalar delay differential equation (35) We proceed with normal form transformations of (4) Consider the normal form transformation for quadratic terms (z, y) = ( z, ỹ) + (S ( z), T ( z)) (41) where T (z) = [T 1 (z),, T n (z)] t After this transformation the ż equation becomes { [ ż = Bz + Ψ n () a 3 (Φ 1 ( τ)z) 3 Φ1 ( τ)z + a S z (z) 1 + Φ 1( τ)z S 1 z (z) (4) 1 + Φ 1( τ)z S z (z) 3 + Φ ] } 1( τ)z S 4 z (z) + a Φ 1 ( τ)(z)t 1 (z) (43) This equation is very similar to the ż equation of the scalar first order equation (15) in normal form to cubic order Hence, modulo the computation of T 1 (z), the cubic coefficients c ij are given by Proposition 37 We now prove (4) The cubic terms after normal form transformation (41) are given by F 3 (z) = F 3 (Φz) + (d z F (Φ 1 ( τ)z + y 1 )) y1 =S (z) + (d y F (Φ 1 ( τ)z + y 1 )) y1 =T (z) Now, F 3 (Φ 1 ( τ)z) + (d y F (Φ 1 ( τ)z + y 1 )) y1 =T (z) =, a 3 (Φ 1 ( τ)z) 3 + a Φ 1 ( τ)(z)t 1 (z) d z F (Φ 1 ( τ)z)s (z) = a Φ 1 ( τ)z z 1 Φ 1 ( τ)z Φ 1 ( τ)z z 1 z Φ 1 ( τ)z z Thus ż is given by (4) where only T 1 (z) enters in the cubic terms after normal form transformation of the quadratic terms of the dy/dt equation Computation of T In the case of the n th order equation (35), equation (31) for the quadratic terms is D z T (z)bz A Q 1(T (z)) = (I π)x = f (z) (44) a (Φ 1 ( τ)z) Recall that A Q 1y = ẏ +X (L(y) ẏ()) and (I π)x = X ΦΨ() Thus, (44) reduces to solving for T (z) the system D z T (z)bz T (z) = ΦΨ() f (z) (45) with boundary conditions T (z)() + L(T (z)) = f (z) (46) 16 S 1 S S 3 S 4

21 The n components of (45) are given by [ ] [ ] T j ω 1 z 1 T j T j z 1 ω z T 1 z z 1 z 1 z z where j runs from 1 to n In particular, [ T 1 ω 1 z 1 T ] [ 1 T 1 z 1 ω z T ] 1 z z 1 z 1 z z T 1 (z) = Φ j Ψ n ()a Φ 1 (z), (47) T 1 (z) = Φ 1 Ψ n ()a Φ 1 (z) The only difference lies in solving the boundary conditions for T 1 (z) which involves the knowledge of T j (z) for j =, n We know already that the c ij coefficients in the case of the n th order equation are identical to the c ij coefficients of Proposition 37 up to the T 1 term Consider now the boundary conditions: T 1 (z)() T (z)() = (48) T n 1 (z)() T n (z)() = T n (z)() + β 1 T n 1 (z)() + + β n T 1 (z)() a 1 T 1 (z)( τ) = a (Φ 1 ( τ)z) Since (35) has constant coefficients, equation (48) factors into subsystems ḣ 1 (q1,q,q3,q4) () h (q1,q,q3,q4) = ḣ n 1 (q1,q,q3,q4) () hn (q1,q,q3,q4) = ḣ n (q1,q,q3,q4) () + + β nh 1 (q1,q,q3,q4) () a 1h (q1,q,q3,q4) ( τ) = a ξ (q1,q,q3,q4) (49) where ξ (q1,q,q3,q4) is the coefficient of z with power (q1, q, q3, q4) in (Φ 1 ( τ)z) Proposition 43 The polynomial T 1 (z) found by solving (47) and (49) is of the same form as σ in Proposition 36 Proof: See Lemma A1 and Lemma A6 in the appendix Therefore the following result follows Proposition 44 The coefficients of the cubic terms of the normal form of (35) are given by Proposition 37 where the polynomials P 1, P, Q 1 and Q depend on the boundary condition (48) Proof: The proof follows from equation (4) and Proposition 43 Proposition 44 implies that Proposition 39 applies directly to n th order scalar delay equations 5 Discussion We have presented an analysis of the relationship between projected flows associated with ordinary differential equations on centre manifolds and the delay-differential equation from which they originate, in the case of a double Hopf bifurcation We have seen that the universal unfolding of the vector field around the singular point may or may not have restrictions, moreover restrictions are 17

22 also influenced by the modeling context in which the delay equation arises As pointed out in [1], there is a difference between unfolding such a singularity in general, and unfolding in the context of modeling using a specific class of delay-differential equations Indeed, the restrictions introduced by the specific structure of the model put conditions on the possible range of parameters allowed in the unfolding The ensuing range of invariant sets is thus limited by the framework in which the model is developed This shifts some of the burden of the analysis from the purely mathematical considerations to the derivation of the model itself It thus becomes paramount to have a properly derived systems of functional differential equations to adequately translate the biological or mechanical systems under study Our analysis is the first one addressing the double Hopf bifurcations Previous investigations [5, 6, 7, 13] have considered simpler bifurcations, all leading to centre manifolds of dimension three or less We have made use of symmetric bifurcation techniques, explaining in general terms the intriguing simplifying relation, discovered in [1], relating the two cubic terms in the scalar first order equation with two delays The role of the symmetry of the hyperbolic tangent in that analysis becomes transparent with the calculations presented here We have only studied, albeit in some details, scalar equations of arbitrary order The only caveat is the necessity for a double Hopf bifurcation point to exist, which is impossible in the case of a first order equation with a single delay The same formal analysis can be extended to systems of functional differential equations Our preliminary calculations indicate a fundamental increase in algebraic difficulties, not all of which can be overcome by the use of symbolic manipulation software, such as MAPLE It is hard to predict how much of our analysis can thus be extended to large scale systems What is clear, though, is the benefit from this investigation for the purposes of modeling biological systems using delay differential equations, and the insight provided into the possible behaviours around singular stationary solutions of the delay equations A Proof of Proposition 36 and Proposition 43 To prove Proposition 36 and Proposition 43 we need to solve equations for σ and T We begin by writing equations (3) and (47) in a suitable form for easy integrating The integration is done in the lemmae that follow and the boundary conditions are used to determine the integrating constants We write the defining condition equations for σ and Tj for all j: [ ] [ ] σ ω 1 z 1 σ σ z 1 ω z σ [ z σ z 1 z 1 z z (x) + X σ () L(σ(x)) ] (5) = a [X ΦΨ()](Φ 1 ( τ)z) (51) where Φ stands for Φ in (3) and it stands for Φ j in (47) Similarly Ψ() stands for Ψ() in (3) and for Ψ n () in (47) Recall that Φ 1 = (e iω1θ, e iω1θ, e iωθ, e iωθ ) Equation (5) is split in two linear differential equations: ] ] and ω 1 [ σ z 1 z 1 σ z 1 z 1 ω [ σ z z σ z z σ () L(σ (x)) = a (Φ( τ)z) σ (x) = a ΦΨ()(Φ 1 ( τ)z), (5) Let ḣ be differentiation with respect to θ Equation (5) can be written in matrix form ḣ = Ah + f, (53) 18

23 where f = a ΦΨ()(Φ 1 ( τ)z), and Now h = (h,,,, h,,,, h,,,, h,,,, h 1,1,,, h,,1,1, h 1,,1,, h,1,,1, h,1,1,, h 1,,,1 ), ω 1 ω 1 ω ω ω 1 ω 1 A = ω ω ω 1 ω ω ω 1 ω 1 ω ω ω 1 ΦΨ() = Re(ψ 1 ()e iω 1θ ) Re(ψ 3 ()e iω θ ), and set Re(ψ 1 ()) = ξ(ω 1 ), Re(ψ 3 ()) = ξ(ω ), Im(ψ 1 ()) = ζ(ω 1 ) and Im(ψ 3 ()) = ζ(ω ) for some ξ and ζ Let H(a) = ξ(a) cos(aθ) ζ(a) sin(aθ), then ΦΨ() = H(ω 1 ) + H(ω ) Since ξ(a) = ξ( a) and ζ( a) = ζ(a) then H is an even function and so is ΦΨ() Since H(θ, ω 1 ) = H(θ, ω 1 ), then ḣ,,, = ω 1 h 1,1,, (H(θ, ω 1 ) + H(θ, ω ))a e iω 1τ ḣ,,, = ( ω 1 )h 1,1,, (H(θ, ω 1 ) + H(θ, ω ))a e iω 1τ Therefore, h,,, (θ, ω 1 ) = h,,, (θ, ω 1 ) The same relationship holds between h,,, and h,,, but with ω 1 replaced by ω The system then reduces to two four dimensional systems where and the matrices are ḣ1 = A 1 h 1 + f 1 and ḣ = A h + f (54) h 1 = (h,,,, h,,,, h 1,1,,, h,,1,1 ), h = (h 1,,1,, h,1,,1, h,1,1,, h 1,,,1 ), f 1 = a ΦΨ()(e iω 1τ, e iτω,, ) t f = a ΦΨ()(e iτ(ω 1+ω ), e iτ(ω 1+ω ), e iτ(ω 1 ω ), e iτ(ω 1 ω ) ) t A 1 = The boundary conditions are ω 1 ω 1 ω ω ω 1, A = ω ω 1 ω 1 ω ω ω ω 1 ḣ 1 () L(h 1 ) = (a e iω 1τ, a e iτω, a, a ) t ḣ () L(h ) = (a e iτ(ω 1+ω ), a e iτ(ω 1+ω ), a e iτ(ω 1 ω ), a e iτ(ω 1 ω ) ) t (55) The following lemma gives h 19

24 Lemma A1 The solutions to equations (54) are ( ) ( h 1 (θ) = e θa 1 K 1 + e θa 1 θ ) e sa 1 f 1, h (θ) = e θa K + e θa θ e sa f where K 1 = a (Ã(ω 1), Ã(ω ), B(ω 1 ), B(ω )) t, K = a ( χ 1 (ω 1, ω ), χ 1 ( ω 1, ω ), χ (ω 1, ω ), χ ( ω 1, ω )) t, θ e sa 1 f 1 ds = a (A(θ, ω 1, ω ), A(θ, ω, ω 1 ), B(θ, ω 1, ω ), B(θ, ω, ω 1 )) t θ e sa f ds = a (χ 1 (θ, ω 1, ω ), χ 1 (θ, ω 1, ω ), χ (θ, ω 1, ω ), χ (θ, ω 1, ω )) t, cos( 1 ω 1 θ) sin( ω1 θ) exp(θa 1 ) = cos( 1 ω θ) sin( ω θ) sin( ω 1 θ) cos( ω 1 θ) and exp(θa ) = sin( ω θ) cos( ω θ) cos(ω 1 θ) cos(ω θ) sin(ω 1 θ) sin(ω θ) sin(ω 1 θ) cos(ω θ) cos(ω 1 θ) sin(ω θ) sin(ω 1 θ) sin(ω θ) cos(ω 1 θ) cos(ω θ) cos(ω 1 θ) sin(ω θ) sin(ω 1 θ) cos(ω θ) sin(ω 1 θ) cos(ω θ) cos(ω 1 θ) sin(ω θ) cos(ω 1 θ) cos(ω θ) sin(ω 1 θ) sin(ω θ) cos(ω 1 θ) sin(ω θ) sin(ω 1 θ) cos(ω θ) sin(ω 1 θ) sin(ω θ) cos(ω 1 θ) cos(ω θ) Proof: The proof follows from the following Lemma A, Lemma A3 and Lemma A5 Lemma A θ e sa 1 f 1 ds = a (A(θ, ω 1, ω ), A(θ, ω, ω 1 ), B(θ, ω 1, ω ), B(θ, ω, ω 1 )) t θ e sa f ds = a (χ 1 (θ, ω 1, ω ), χ 1 (θ, ω 1, ω ), χ (θ, ω 1, ω ), χ (θ, ω 1, ω )) t Proof: We consider first the integral θ e sa 1 f 1 ds = a θ e sa 1 ( ΦΨ())(e iω 1τ, e iω τ,, ) t ds which separates into four integrals where ΦΨ() = H(s, ω 1 ) + H(s, ω ) Each integral is of the form θ (H(s, ω 1 ) + H(s, ω ))J (s, a)ds, for some function J where a = ω 1 or a = ω Hence it is easy to see that if I(ω 1, ω ) = θ (H(s, ω 1 ) + H(s, ω ))J (s, ω 1 )ds

25 then It is easy to check that I(ω, ω 1 ) = θ (H(s, ω 1 ) + H(s, ω ))J (s, ω )ds thus e sa (e iτ(ω 1+ω ), e iτ(ω 1+ω ), e iτ(ω 1 ω ), e iτ(ω 1 ω ) ) t = (δ 1 (s, ω 1, ω ), δ 1 (s, ω 1, ω ), δ (s, ω 1, ω ), δ (s, ω 1, ω )) t, θ e sa f ds = θ (H(s, ω 1) + H(s, ω ))(δ 1 (s, ω 1, ω ), δ 1 (s, ω 1, ω ), δ (s, ω 1, ω ), δ (s, ω 1, ω )) t ds Since H(s, a) is even in a, then H(ω 1, ω )δ i (s, ω 1, ω ) = H( ω 1, ω )δ i (s, ω 1, ω ) = H(a, b)δ i (s, a, b) for i = 1,, where a = ω 1 and b = ω and the result follows Lemma A3 The sets of matrices A 1 and A respectively of the form a b x y z w c d M 1 = b a and M y x w z = z w x y d c w z y x, where x ±y and z ±w for all nonzero matrices, are fields Moreover, M 1 (α(ω 1, ω ), α(ω, ω 1 ), β(ω 1, ω ), β(ω, ω 1 )) t = (α 1 (ω 1, ω ), α 1 (ω, ω 1 ), β 1 (ω 1, ω ), β 1 (ω, ω 1 )) t (56) Proof: The determinants are det(m 1 ) = (a + b )(c + d ) and det(m ) = ((z w) + (x + y) )((y x) + (z + w) ) which vanish only for the zero matrix Commutativity and property (56) are verified by a simple computation Remark A4 Note that in Lemma A1, exp(θa 1 ) is an element of A 1 and exp(θa ) is in A since A 1 A 1 and A A Moreover, if M = A m for any integer m or M = exp(θa ), then an easy calculation shows that M (χ(ω 1, ω ), χ( ω 1, ω ), ξ(ω 1, ω ), ξ( ω 1, ω )) t = (χ 1 (ω 1, ω ), χ 1 (, ω 1, ω ), ξ 1 (ω 1, ω ), ξ 1 ( ω 1, ω )) t (57) Proof of Proposition 36 From Lemma A1 and Lemma A3 we see that the multiplication and addition in the expressions for h yields the desired result Lemma A5 The constants K 1 and K found using the boundary conditions with L(, ) coming from (18) have the form K 1 = a (Ã(ω 1), Ã(ω ), B(ω 1 ), B(ω )) K = a ( χ 1 (ω 1, ω ), χ 1 ( ω 1, ω ), χ (ω 1, ω ), χ ( ω 1, ω )) 1

26 Proof: Writing the boundary equation using the solutions h 1 (θ) computed before we obtain (A 1 a 1 e τ 1A 1 a 1 e τ A 1 )K 1 = f 1 () + a 1 e τ 1A 1 τ1 e sa 1 f 1 (s)ds + a 1 e τ A 1 τ e sa 1 f 1 (s)ds + (e iω 1τ, e iω τ,, ) t = (α(ω 1, ω ), α(ω, ω 1 ), β(ω 1, ω ), β(ω, ω 1 )) t, where the last equality is easily shown using Lemma A By Lemma A3, (A 1 a 1 e τ 1A 1 a 1 e τ A 1 ) 1 is of the form M 1 and the result follows The vector K is computed in the same way using Remark A4 Lemma A6 The constants K 1 and K found using the boundary conditions (48) have the same form as in Lemma A5 Proof: Let T j (z) = q = h j q 1,q,q 3,q 4 (θ)z q 1 1 z q 1 z q 3 z q 4, where j = 1,, n, q = q 1 + q + q 3 + q 4 and h j q 1,q,q 3,q 4 (θ) Q 1 For j = 1,, n, let h j 1 = (h j,,,, h j,,,, h j 1,1,,, h j,,1,1) and h j = (h j 1,,1,, h j,1,,1, h j,1,1,, h j 1,,,1) Then using the solutions of equation (54) for h j i, we replace in the boundary conditions (48) By Lemma A1, h j 1 = e θa 1 (K j 1 + f1()) j where the superscripts of K and f are indices setting K1 = K 1 and f1() j has the same form as in Lemma A Thus for h j 1 we obtain A 1 K 1 + f 1 () + K 1 1 = A 1 K1 n + f1 n 1 () + K1 n 1 = A 1 K1 n 1 + f1 n 1 () + β 1 K1 n 1 + β n K 1 + a 1 e τa 1 K 1 = a (e iω1τ, e iωτ,, ) t ) Solve K j 1 in terms of K 1 : K j 1 = ( 1) j A j 1K 1 +( 1) j (A j 1 1 f 1 () A j 1 f 1 1 ()+ ±f j 1 1 ()) Replacing in the last equation and putting on the right hand side all the terms which do not contain K 1 we obtain ( 1) n (A n 1 + β 1 A n β n I + a 1 e τa 1 )K 1 = a (g 1 (ω 1, ω ), g 1 (ω, ω 1 ), g (ω 1, ω ), g (ω, ω 1 )) t Using Lemma A3 in the preceding equation yields the result The other vectors h j i the same way and yield similar results are handled in Acknowledgements The research presented here was supported by the Natural Sciences and Engineering Research Council (NSERC, Canada) [Postdoctoral Fellowship to PLB, Research Grant to JB], the Fonds pour la Formation de Chercheurs et l Aide à la recherche (FCAR, Québec) [Team Grant to JB] and the Comité d étude et d administration de la recherche (CEDAR, Université de Montréal) [Centre Grant to the CRM]