University of GroningenGeneric Hopf-Neimark-Sacker bifurcations in feed-forward systemsBroer, Henk W.; Vegter, GertPublished in:NonlinearityDOI:10.1088/0951-7715/21/7/010IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2008Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Broer, H. W., & Vegter, G. (2008). Generic Hopf-Neimark-Sacker bifurcations in feed-forward systems.Nonlinearity, 21(7), 1547-1578. https://doi.org/10.1088/0951-7715/21/7/010CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 11-10-2022IOP PUBLISHING NONLINEARITYNonlinearity 21 (2008) 1547–1578 doi:10.1088/0951-7715/21/7/010Generic Hopf–Neı̆mark–Sacker bifurcations infeed-forward systemsHenk W Broer and Gert VegterInstitute of Mathematics and Computing Science, University of Groningen, PO Box 800,9700 AV Groningen, The NetherlandsE-mail: H.W.Broer@math.rug.nl and G.Vegter@math.rug.nlReceived 29 October 2007, in final form 24 April 2008Published 10 June 2008Online at stacks.iop.org/Non/21/1547Recommended by A ChencinerAbstractWe show that generic Hopf–Neı̆mark–Sacker bifurcations occur in the dynamicsof a large class of feed-forward coupled cell networks. To this end wepresent a framework for studying such bifurcations in parametrized familiesof perturbed forced oscillators near weak resonance points. Our approach isbased on fine-tuning existing normal form techniques. We then apply thisframework to show that certain cells in the feed-forward networks exhibit thisforced oscillator dynamics and, hence, undergo generic Hopf–Neı̆mark–Sackerbifurcations. These bifurcations correspond to the occurrence of resonancetongues in parameter space with a ‘standard’ geometry.Mathematics Subject Classification: 37G15, 34C25, 34C27(Some figures in this article are in colour only in the electronic version)1. IntroductionIn the general theory of dynamical systems an important problem has been, and to some extentstill is, how one system of equations of motion can describe both very orderly (e.g. laminar)and extremely complicated (say, turbulent) behaviour. In the 1940s and 1950s Hopf, Landauand Lifschitz [25,26,28,29] partially answered this question. A slightly different answer wasproposed by Ruelle and Takens in the 1970s [30, 31]. Both approaches were reconciled andunified in [3] for further information also see [12, 16]. The idea is that the dynamical systemat hand depends on parameters and that by repeated Hopf bifurcation the system graduallyacquires ‘more frequencies’ and thus starts living on ever higher dimensional tori, which thenoccur as ‘centre manifolds’ in a possibly even much higher dimensional state space. The0951-7715/08/071547+32$30.00 © 2008 IOP Publishing Ltd and London Mathematical Society Printed in the UK 15471548 H W Broer and G VegterFigure 1. Three-cell linear feed-forward network.dynamics on these tori can be either periodic, quasi-periodic or, in the case of flows fromdimension three on, also chaotic.One further question is whether this ‘generic’ scenario can also take place in certainsubclasses of dynamical systems, such as coupled cell networks. This paper partly deals withthis problem in the case of feed-forward systems. In particular we shall consider two stages ofthe scenario, concerning the Hopf bifurcation and the Hopf–Neı̆mark–Sacker bifurcation. Theformer of these deals with the bifurcation from stationary to periodic dynamics in the secondcell, and the asymptotics of this Hopf bifurcation has turned out to be different from thegeneral case [21]. The latter deals with the third cell and concerns the Hopf–Neı̆mark–Sackerbifurcation from a periodic orbit to an invariant two-torus. Here we show that the transition is‘generic’ in the sense that the dynamics displays the characteristic alternation of quasi-periodicand phase-locked states, where the latter occurs in resonance tongues in parameter space.Coupled cell systems. A coupled cell system is a network of dynamical systems, or ‘cells’,coupled together. These networks are represented by a directed graph with nodes representingcells and edges representing couplings between these cells. See, e.g. Golubitsky et al [23]. Weconsider the three-cell feed-forward network shown in figure 1, where the first cell is coupledexternally to itself. The network represents the following system of ordinary differentialequations (ODEs):ẋ1 = F(x1, x1),ẋ2 = F(x2, x1),ẋ3 = F(x3, x2),(1)with xj ∈ R2. In our case, each cell satisfies the same dynamic law; only different choices ofinitial conditions may lead to different kinds of dynamics for each cell.Synchrony breaking and quasiperiodicity in feed-forward networks. Golubitsky et al [21]describe synchrony-breaking Hopf bifurcations in a restricted class of feed-forward networks,with the first cell in equilibrium and the periods of the second and third cell being equal.Curiously, the amplitude of the periodic state in the second cell grows as λ1/2, whereas theamplitude of the third cell grows as λ1/6. Here λ is the ‘Hopf parameter’. Elmhirst andGolubitsky [19] show that this phenomenon occurs generically within this class of systems.The λ1/6 growth rate is due to resonance; the third cell is forced periodically at a frequencynear the Hopf frequency. McCullen et al [17] use this phenomenon of resonance near theHopf bifurcation to demonstrate experimentally that the feed-forward network can be used asan efficient frequency filter/amplifier.In [21] it is shown analytically that a generic Hopf bifurcation can occur from a periodicsolution leading to a stable solution. Numerical examples show that the network dynamicsmay exhibit very different states in different cells. In a particular example, the dynamicscorresponds to the first cell being in equilibrium, the second cell exhibiting time periodicbehaviour, whereas the dynamics of the third cell is (conditionally) quasi-periodic.Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1549Figure 2. Resonance tongues in the Arnold-family [14]. The tongue boundaries correspond tosaddle-node bifurcations. The interior of each tongue also contains a period-doubling bifurcationline.Main results and contribution of the paper. It was generally believed that the dynamicphenomena in feed-forward networks are much more limited than those found in genericdynamical systems, but that, instead, the results of [21] describe generic behaviour of suchnetworks. We refute this belief and show that such systems may exhibit fully generic Hopf–Neı̆mark–Sacker bifurcations, with its characteristic alteration of quasi-periodic and phase-locked states [1, 5, 6, 12, 16, 27]. In particular, we show how this class of dynamic laws maygive rise to time evolutions that are equilibria in cell 1, periodic in cell 2 and exhibit the genericHopf–Neı̆mark–Sacker phenomenon in cell 3. See also figures 2 and 4.The detailed analysis of the dynamics of the third cell requires a fine-tuned version of awell-known normal form procedure. We show that a rather large class of families of perturbedforced oscillators, which contains the dynamics of our network cells as a special subclass, canbe brought into a simple polynomial normal form near a p : q resonance. As can be concludedfrom this normal form, Hopf–Neı̆mark–Sacker bifurcations occur generically in such families.Our analysis of cell dynamics requires detailed information on higher order terms of thenormal form. Although the normal form procedure is well known and has been applied invarious contexts [15, 27, 32, 33], the standard procedure does not provide this information.Therefore, in appendices A and B we extend the standard procedure considerably. Althoughnot the primary topic of this paper, this part of our work can be applied in many settingsinvolving forced oscillators and is, therefore, of independent interest.Related work on resonance tongues. Heuristically speaking resonance occurs in theinteraction of various oscillatory parts of a system, when the corresponding frequencies arerationally related. In our context the periodic dynamics of the second cell forces the dynamics ofthe third cell, where a ‘secondary’ Hopf bifurcation takes place, generating a second oscillatory1550 H W Broer and G Vegtermotion. We study the cases where these two motions are in certain resonances. As said earlier,we show that in the present coupled cell context a generic Hopf–Neı̆mark–Sacker bifurcationoccurs. It turns out that the subharmonic periodic motion, that with regard to its rotationnumber fits with the p : q resonance at hand, occurs when the parameter ranges over a tongue-shaped region in parameter space. Along the boundaries of the tongue, the stable and unstablebranches of the subharmonic motion annihilate each other in a saddle-node bifurcation. Thetip of the tongue corresponds to a more degenerate bifurcation.This kind of scenario occurs in many other situations also. An illustration of this is theArnol’d family of circle maps [1], given by x → x + 2πα + β sin x. Here in the (α, β) planetongues appear with their tips in (α, β) = (pq, 0) and stretching out into the regions β = 0.See figure 2. The order of tangency of the tongue boundaries at the tip, as q → ∞, has thesame asymptotics as in the Hopf–Neı̆mark–Sacker case. Also compare with [16,18] and [5,6].Also in the worlds of Hamiltonian, reversible or equivariant systems resonance widelyoccurs. In all cases the resonant dynamics in parameter space is organized by the bifurcationset, which often contains tongues. As another example we refer to the parametrically forcedpendulum, both in linear and nonlinear versions, where in the former case we deal with Hill’sequation. For general reference, also including cases with quasi-periodic forcing, we referto [2, 7, 9, 10, 11, 13, 15].Overview of the paper. In section 2 we introduce the class of feed-forward networks ofcoupled cells, in which each cell has the same dynamics. For a large class of cell dynamicswe analyse the time evolution of a linear chain of coupled cells, where each cell, except forthe first one, is driven by the dynamics of its predecessor in the chain. We show that the thirdcell and its successors may exhibit forced oscillator dynamics, giving rise to generic Hopf–Neı̆mark–Sacker bifurcations near weak resonance points. Section 3 presents a specific classof such linear networks in which such bifurcations occur. The resonance tongues in parameterspace, a characteristic feature of such bifurcations, are analysed and turn out to exhibit the‘standard’ geometry.The technical details of the normal form techniques for the class of networks studied in thispaper are presented in the appendices. Appendix A shows how to bring the linear part of thecell dynamics near the Hopf–Neı̆mark–Sacker bifurcation point into the Jordan normal form.This step has to precede the application of the normal form procedure, which is presented inappendix B. In appendix C we present the geometric analysis of the resonance tongues indetail. Finally, appendix D shows that our class of networks satisfies the generic conditionguaranteeing the occurrence of generic Hopf–Neı̆mark–Sacker bifurcations.2. Complicated dynamics in feed-forward networksWe present a large class of linear feed-forward networks of the form (1) consisting of identicalcells, such that Hopf–Neı̆mark–Sacker bifurcations occur generically in the third cell. This isinline with the behaviour observed numerically in a specific example presented in [21].In our class of networks the first cell is in equilibrium and the second cell exhibits periodicbehaviour. In section 2.2 we show that the third cell then has the dynamics of a forced oscillator.In section 2.3 we continue our analysis of this class of networks by showing that, under genericconditions, the third cell undergoes generic Hopf–Neı̆mark–Sacker bifurcations. We do so bybringing the dynamics of this cell into the normal form, from which we easily derive theoccurrence of resonance tongues. The proofs and details of these results are deferred to theappendices. In section 3 we establish the feasibility of our approach by presenting a largeGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1551class of examples that satisfy the generic conditions guaranteeing the occurrence of genericHopf–Neı̆mark–Sacker bifurcations.2.1. Dynamics of the first two cellsS1-equivariance. For simplicity we assume that the function F , describing the dynamics ofeach cell as in (1), is a perturbation of an S1-equivariant function f in the sense thatF(z2, z1) = f (z2, z1) + ε P (z2, z1), (2)withf (eiθ z2, eiθ z1) = eiθf (z2, z1), (3)for all real θ . Here we identify the two-dimensional phase space of each cell with C by writingzj = xj1 + ixj2. Furthermore, ε is a small perturbation parameter. The unperturbed dynamicsof the network cells corresponds to the case ε = 0.Identity (3) is a special assumption, also used in [21] to derive the curious phenomenonof decaying amplitudes of the periodic states in the second and third cells, for a certain classof dynamic laws. However, Elmhirst and Golubitsky [19] verify that this symmetry conditionholds to third order after a change in coordinates. We have not been able to show that Hopf–Neı̆mark–Sacker bifurcations occur in more general networks, i.e. in the case the unperturbedcell dynamics is not necessarily S1-equivariant as in (3).We also assume that the dynamics of each cell depends on external parameters µ ∈ Rk , tobe specified later on. The S1-equivariance condition (3) restricts the unperturbed cell dynamicsto functions f of the formf (z2, z1, µ) = A(|z1|2, z1z2, |z2|2, µ) z1 + B(|z1|2, z1z2, |z2|2, µ) z2, (4)where A and B are complex functions depending on the parameter µ. We occasionallyexpress the dependence of f , A and B on the parameter by writing fµ, Aµ and Bµ, wherefµ(z2, z1) = f (z2, z1, µ), etc.Dynamics of the first and second cell. The S1-equivariance condition (3) implies thatfµ(0, 0) = 0. Assume that the linear part of fµ(z1, z1) at z1 = 0 has eigenvalues with anegative real part, i.e.Re (Aµ(0) + Bµ(0)) < 0, (5)for all parameter values µ in the region of our interest. Then the unperturbed dynamicsż1 = fµ(z1, z1) of the first cell has a stable equilibrium at z1 = 0.The second cell then has unperturbed dynamicsż2 = fµ(z2, 0) = B(0, 0, |z2|2, µ) z2, (6)where we use that the first cell is in its stable equilibrium. Golubitsky and Stewart [22] introducea large class of functions fµ for which the second cell undergoes a Hopf bifurcation. For thisclass of cell dynamics, and for linear feed-forward networks of increasing length, there willbe a ‘repeated Hopf’ bifurcation, reminiscent of the scenarios named after Landau–Lifschitzand Ruelle–Takens. In view of (6) the second cell undergoes a Hopf bifurcation at a parametervalue µ∗ ∈ Rk if the following additional condition holds:Re B(0, 0, 0, µ∗) = 0,DµB(0, 0, 0, µ∗) = 0.(7)In fact, (7) determines a hypersurface H∗ in parameter space Rk , such that the second cellundergoes a Hopf bifurcation at every point of H∗. In this case the periodic solutions emanating1552 H W Broer and G Vegterfrom this bifurcation are circles of the form |z2| = r(µ), where r = r(µ) is the non negativesolution of the equationRe B(0, 0, r2, µ) = 0, (8)with r(µ∗) = 0. Note that a (positive) real solution r(µ) exists (locally) only for parametervalues in one of the closed half-spaces bounded by the hypersurface H∗. Settingθ(µ) = ImB(0, 0, r(µ)2, µ),we see that the bifurcating periodic state of the second cell isz2(t) = r(µ) eiθ(µ)t ,with θ(µ∗) = ImB(0, 0, 0, µ∗). See also [21, section 5]. To guarantee that the equilibriumof the first cell and the periodic solution of the second cell persist for small values of theperturbation parameter ε we impose the conditionP(z2, 0, µ) ≡ 0, (9)where P is the perturbation term in (2). The periodic solution of the second cell will drive thedynamics of the third cell, which we consider next.2.2. Forced oscillator dynamics of the third cellThe main topic of our research is the generic dynamics of the third cell, given the simple timeevolutions of the first two cells introduced in the first part of this section. Here we like to knowwhat are the correspondences and differences with the general ODE setting. In particular, thisquestion regards the coexistence of periodic, quasi-periodic and chaotic dynamics.In co-rotating coordinates the unperturbed dynamics of the third cell, i.e. the systemż3 = fµ(z3, r(µ)eiθ(µ)t ), becomes time independent. To see this, set z3 = eiθ(µ)ty and use theS1-symmetry to deriveiθ(µ)eiθ(µ)ty + eiθ(µ)t ẏ = fµ(eiθ(µ)ty, r(µ) eiθ(µ)t )= eiθ(µ)t fµ(y, r(µ)).Therefore, the unperturbed dynamics of the third cell is given byẏ = −iθ(µ)y + fµ(y, r(µ)). (10)Equation (10) is autonomous, so the present setting might exhibit Hopf bifurcations, as notedin [17], but this is still too simple to produce resonance tongues. Indeed, all (relative) periodicmotion in (10) will lead to parallel (quasi-periodic) dynamics and the Hopf–Neı̆mark–Sackerphenomenon. Therefore, we consider the perturbed family (2). In co-rotating coordinatesy = e−iθ(µ)t z3 the perturbed dynamics of the third cell is given byẏ = −iθ(µ)y + fµ(y, r(µ)) + εe−iθ(µ)tP (y eiθ(µ)t , r(µ) eiθ(µ)t ). (11)Scaling time by puttingt = θ(µ)t (12)and denoting derivatives with respect to t by primes, we see that the third cell therefore hasforced oscillator dynamics with driving frequency 1:y ′ = g(y, µ) + ε G(y, t, µ), (13)whereg(y, µ) = −iy + f (y, r(µ), µ) (14)Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1553andG(y, t, µ) = e−itP (y eit , r(µ) eit ).In particular, G is 2π -periodic in t . For simplicity of notation we omit the bars from now on,i.e. we again denote time by t .The question about generic dynamics concerns the possible coexistence of periodic, quasi-periodic and chaotic dynamics, where the latter only is to be expected for larger values of theparameters (λ, ε), cf [14].We aim to investigate (11) for Hopf–Neı̆mark–Sacker bifurcations, which are expectedalong codimension one strata Hε in the µ-space of parameters. In section 2.3 we explain thegeneral context giving rise to the occurrence of such bifurcations. Subsequently, in section 3,we consider a specific example illustrating the procedure to verify the occurrence of suchbifurcations.2.3. Generic Hopf–Neı̆mark–Sacker bifurcations in the third cellNormal Form near the Hopf stratum. As said before, Hopf–Neı̆mark–Sacker bifurcations inperturbed forced oscillators, such as (13), may occur subordinate to a non degenerate Hopfbifurcation in the unperturbed system. So consider (y0, µ0) ∈ C×Rk such that g(y0, µ0) = 0and tr(dyg(y0, µ0)) = 0, with det(dyg(y0, µ0)) > 0. According to the implicit functiontheorem, the equation g(y, µ) = 0 has a unique solutiony = yµ, (15)for µ near µ0, coinciding with y0 for µ = µ0. Let a(µ) ± iω(µ) be the eigenvalues of thelinear part of ẏ = g(y, µ) at the equilibrium point yµ, for µ near µ0. Thena(µ) = 12 tr(dyg(yµ, µ)) and ω(µ) =√det(dyg(yµ, µ)) − a(µ)2. (16)The system y ′ = g(y, µ), corresponding to the unperturbed dynamics of the third cell,undergoes a non degenerate Hopf bifurcation near µ0, if the following (versality) conditionholds:Versal unfolding of linear part condition:The map Rk → R2 : µ → (a(µ), ω(µ)) has rank two at µ0. (17)If this condition holds, the non degenerate Hopf bifurcation occurs when the parameterµ passes a codimension one hypersurface H0 consisting of parameters µ at which a(µ) = 0.Note that the zero set of a is a regular hypersurface in parameter space near µ0. When µcrosses H0 a periodic orbit of period ω(µ) bifurcates (i.e. is born or dies) from the equilibriumpoint yµ.Subharmonics may emerge or disappear if the parameters are close to a resonance point, apoint on the Hopf stratum at which the normal frequency is rational. Since the versal unfoldingof linear part condition guarantees that the normal frequency ω(µ) is non-constant as µ rangesover the Hopf stratum H0, there is an abundance of such points on H0. Select µ0 ∈ H0 suchthat ω(µ0) = pq , where p and q are relatively prime. In other words, the system exhibits ap : q resonance for µ = µ0. We only consider weak resonances, i.e. we assume that q 5.See [1] and [33] for a discussion of strong and weak resonances.We are now ready to state one of our main results on the normal form for the forcedoscillator dynamics of the third cell. This normal form will reveal the generic Hopf–Neı̆mark–Sacker bifurcations in the dynamics of this cell.1554 H W Broer and G VegterFigure 3. Resonance zones for forced oscillator families: the Hopf–Neı̆mark–Sacker phenomenon(sliced and presented from two different viewpoints for enhanced visualization).Theorem 1 (normal Form to order q). Suppose the forced oscillator (13) satisfies the versalunfolding of linear part condition, for µ near µ0, and the normal frequency at µ0 is equal toω0 = pq . Then the system can be brought into the normal formZ′ = iω0Z + (α + iδ) Z + Z F(|Z|2, ν, ε) + γ (ν, ε) ε Zq−1 eipt + O(|Z|q), (18)locally uniformly in (ν, ε). Here F(|Z|2, ν, ε) is a complex polynomial of degree at most q −1with F(0, ν, ε) ≡ 0. The O(|Z|q) terms are 2π -periodic in t . Furthermore, Z ∈ C, andν ∈ Rk is a new parameter obtained by a reparametrization Rk → Rk , with ν1 = α andν2 = δ. Let ν0 ∈ Rk correspond to the parameter value µ0 under this reparametrization.Generically, γ (ν0, 0) = 0, and in this case a further reparametrization brings the system intothe formZ′ = iω0Z + (α + iδ) Z + Z F(|Z|2, ν, ε) + ε Zq−1 eipt + O(|Z|q). (19)The normal form (19) is Zq-symmetric and even S1-symmetric in the unperturbed casecorresponding to ε = 0. Note that the linear part is a versal unfolding of the linear partZ′ = iω0Z of the central system corresponding to (ν, ε) = (ν0, 0).The proof of theorem 1 is deferred to appendix B. Theorem 3 gives a more preciseexpression for the constant γ (ν0, 0), showing that, generically, this constant is indeed nonzero. Here we note that the normal form transformation also involves a change in parameters.The hyperplane ε = 0 in parameter space is invariant under this change of parameters, so ε isagain a small perturbation parameter for the normalized system.Resonance tongues. The geometry of the resonance tongues can be deduced from the normalform (19) of the forced oscillator dynamics of the third cell. The following theorem impliesthat, under generic additional conditions, the order of tangency at the tongue tips is generic.Here the Hopf coefficient Fu(0, 0, 0) is the partial derivative of F(u, µ, ε) with respect to u. Itis the coefficient of z2z in (18) for (ν, ε) = (ν0, 0). Note that, for ε = 0, the system undergoesa generic Hopf bifurcation if the real part of this coefficient is non zero, cf [24, chapter 3.4].Theorem 2 (geometry of the resonance tongues). Consider system (19). If Fu(0, 0, 0) = 0and γ (ν0, 0) = 0 in (18), then the p : q resonance tongues of the forced oscillator (19) aresubsets of parameter space bounded by hypersurfaces of the formδ = ±ε (−α)(q−2)/2 + O(ε2). (20)The boundary of the resonance tongues is shown in figure 3, for a three-parameter family,in the case q = 5. In fact, the essential parameters of the forced oscillator (18) are α, δGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1555Figure 4. Takens normal form vector field of the Poincaré map of the periodic system ż =iω0z + (α + iδ) z + z |z|2 + ε z4 eit , corresponding to system (19) in the weakly resonant casep = 1 and q = 7, and fixed ε = 0. The Poincaré map itself is obtained by composing thetime-1-flow of this vector field by a rotation over 2π/7. See appendix C and [33]. The normallyhyperbolic invariant circle of the Poincaré map corresponds to an invariant torus in the 3D phasespace C × R/(2πZ). The dynamics on the invariant circle undergoes a saddle-node bifurcationwhen the parameters pass the boundary of the resonance tongue, corresponding to the birth (ordeath) of a pair of periodic orbits of period q.and ε. Together, these tongue boundaries form a cuspidal surface. At this surface the Hopf–Neı̆mark–Sacker bifurcation should occur, since here the Floquet exponents of the linear partof the forced oscillator cross the complex unit circle. This bifurcation gives rise to an invariant2-torus in the 3D phase space C × R/(2πZ); see figure 4. For a more detailed bifurcationanalysis of generic and mildly degenerate Hopf–Neı̆mark–Sacker families we refer to ourrecent work [8]. More specifically, resonances occur when the eigenvalues of the Poincarétime-2π -map cross the unit circle at roots of unity e2π ip/q . The normal form of the Poincarémap and, hence, the bifurcation diagram shown in figure 4 are similar to those obtained byGambaudo in [20, equation (3.1) and figure 2]. Also, the ‘resonance horn’ of [20, figure 1] issimilar to the resonance set shown in figure 3. ‘Inside’ the tongue the 2-torus is phase-lockedto subharmonic periodic solutions of order q [14, 16].Remark. In a feed-forward network with more cells, in the higher order cells we expectthe generic and standard Hopf–Landau–Lifschitz–Ruelle–Takens scenarios with respect to thecoexistence of periodicity, quasi-periodicity and chaos [12, 16, 30, 31].1556 H W Broer and G VegterGenericity of the normal form. The generic conditions guaranteeing that the constant γ (ν0, 0)in (18) be non zero, and hence that the normal form of the dynamics of the third cell is ofthe form (19), can be made explicit in specific examples of cell dynamics. These conditionsare expressed in terms of the coefficients of a low-degree jet of f and the coefficients ofcertain resonances in the perturbation term P of the forced oscillator dynamics (2) of the thirdcell. In section 3 we present a class of examples of cell dynamics of the form (13), wherethe unperturbed system is a low-degree polynomial system, and show what these genericconditions reduce to for this class of systems.To derive the conditions guaranteeing γ (ν0, 0) = 0, cf theorem 1, we first introduce somenotation. The Jordan normal form of the linear part of ẏ = g(y, µ0) at y = y0 is of the formż = (a(µ) + iω(µ))z, cf (16). This linear part, denoted by Lµ, is of the formLµ(w) = α(µ) w + β(µ) w.Note thata(µ) = Re α(µ).Writing α(µ) = a(µ) + ib(µ) and β(µ) = c(µ) + id(µ), we observe thatω(µ) =√det(Lµ) − a(µ)2 =√b(µ)2 − c(µ)2 − d(µ)2.A straightforward computation shows that the transformation sµ, bringing Lµ into the Jordannormal form, i.e. such thats−1µ ◦ Lµ ◦ sµ(Z) = (iω(µ) + a(µ))Z,is given bysµ(Z) = σ(µ)Z + τ(µ)Z, (21)whereσ(µ) = 12 (b(µ) − d(µ) + ω(µ)) + 12 ic,τ (µ) = 12 (b(µ) − d(µ) − ω(µ)) + 12 ic.(22)We require that the linear transformation sµ is invertible, i.e. (µ) := Det sµ = |σ(µ)|2 −|τ(µ)|2 = 0. Its inverse is then given bys−1µ (Y ) = (µ)−1(σ (µ)Y − τ(µ)Y ).To simplify notation we put r0 = r(µ0), with r given by (8), y0 = y(µ0), σ0 = σ(µ0),τ0 = τ(µ0) and 0 = (µ0).Theorem 3 (coefficient of rotationally non-symmetric normal form term). Suppose theconditions of theorem 1 hold. Then the coefficient γ (ν0, 0) of the non-symmetric term ε zq−1 eiptin the normal form (18) is non zero for generic f and P , cf (2). Furthermore, this coefficientis of the formγ (ν0, 0) = c0(f ) + c1(f ) C1(P ) + c2(f ) C2(P ) + C(P ), (23)where• c0(f ) depends on the (q − 1)-jet of y → f (y, r0, µ0) at y = y0;• c1(f ) and c2(f ) depend linearly on f , and c1(f ) = c2(f ) = 0 if f is a polynomial ofdegree less than q in z, z;Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1557• C1(P ) and C2(P ) are linear in P ;• C(P ) is given byC(P ) = 12π (q − 1)!∫ 2π0∂q−1∂zq−1∣∣∣∣z=0e−ipt(σ0 P (z, t) − τ0 P (z, t))dt, (24)whereP (z, t) = −10 e−it P ((y0 + σ0 z + τ0 z) eit , r0 eit ). (25)In other words, the contribution C(P ) of the perturbation term P is the coefficient of the termzq−1 eipt in the Taylor–Fourier expansion of the function (z, t) → σ0 P (z, t) − τ0 P (z, t).In general, it is hard to determine the constant c0(f ) explicitly, although in principle thenormal form procedure described in appendix B.1 gives an explicit method for doing so. Onthe other hand, the constant C(P ) can often be computed more easily, as we will show in thecase study in section 3. Furthermore, for generic perturbation terms P we have C(P ) = 0.If f is a polynomial of degree less than q in z, z, we have γ (ν0, 0) = c0(f ) + C(P ) = 0 forgeneric P . We will analyse this situation in more detail in our case study.A proof of theorem 3 is given in appendix D.3. Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward networks: a case studyWe present a class of explicit examples that satisfy the generic conditions guaranteeing theoccurrence of generic Hopf–Neı̆mark–Sacker bifurcations.Cell dynamics in the unperturbed setting. Condition (3) of S1-symmetry, i.e.f (eiθ z2, eiθ z1) = eiθf (z2, z1),restricts the unperturbed cell dynamics to functions f of the form (4), i.e.f (z2, z1, µ) = A(|z1|2, z1z2, |z2|2, µ) z1 + B(|z1|2, z1z2, |z2|2, µ) z2.In our case study we simply takeA = −1 and B = λ + i + (1 + κ i)|z1|2 − |z2|2.Here κ is a second parameter, so from now on µ = (λ, κ) ∈ R2. With these choices we havefµ(z2, z1) = −z1 + (λ + i + (1 + κi)|z1|2 − |z2|2) z2. (26)With this choice of f condition (5) is satisfied, so the first cell has a stable equilibrium atz1 = 0. The dynamics of the second cell isż2 = fµ(z2, 0) = (λ + i − |z2|2) z2,so the second cell undergoes a supercritical Hopf bifurcation at λ = 0, cf (8), with frequencyθ(µ) ≡ 1. In other words, in this case study we do not need to rescale time as in (12). Thehypersurface H∗, defined by (7), is of the form H∗ = {(0, κ) | κ ∈ R}. The stable periodicsolution, occurring for λ > 0, has the formz2(t) =√λeit ,in other words, r(µ) = √λ, in terms of (8). In particular, we restrict the parameters to the halfspace λ > 0 from now on. In view of (13) the dynamics of the third cell is of the formẏ = g(y, µ) + ε e−itP (y eit ,√λ eit ), (27)withg(y, µ) = −iy + fµ(y,√λ) = −√λ + (2 + iκ)λy − |y|2y. (28)1558 H W Broer and G VegterWe first determine the curve in the (λ, κ)-plane corresponding to the occurrence of Hopfbifurcations of the unperturbed dynamics of the third cell (i.e. with ε = 0). Subsequently, wepresent several examples of perturbed dynamics of this cell for which generic Hopf–Neı̆mark–Sacker bifurcations occur.Generic Hopf bifurcation of the unperturbed system. Before specifying the perturbation termin (11), we determine the Hopf stratum for the unperturbed dynamics.Lemma 4. The Hopf stratum of the unperturbed dynamics of the third cell, corresponding tothe family (27) with ε = 0, is the curve H0 in the (λ, κ) plane given byλ = 1√κ2 + 1, with |κ| > 1.If µ = (λ, κ) ∈ H0, the Hopf bifurcation takes place at the pointy(µ) = 1 − κ i(1 + κ2)34,and the normal frequency of the Hopf bifurcation along this Hopf curve is given byω(κ) =√κ2 − 1κ2 + 1.Proof. The Hopf stratum is the curve given by the equationsg(y, µ) = 0 and tr(dyg(y, µ)) = 0,with det(dyg(y, µ)) > 0. Since dyg(y, µ)w = α w + β w, withα = (2 + iκ)λ − 2|y|2 and β = −y2, (29)and tr(dyg(y, µ)) = 2Re α, these equations reduce to{−√λ + (2 + iκ)λy − |y|2y = 0,λ − |y|2 = 0.This set of equations gives the expressions for λ and y in terms of κ . We also have to guaranteethat det(dyg(y, µ)) > 0 at a point (y, µ) ∈ H0. A short calculation shows thatdet(dgµ) = |2(λ − |y|2)2 + iκλ|2 − |y|4 = λκ2 λ2 − |y|4 = κ2 − 1κ2 + 1,so the frequency along the Hopf curve H0 is given by the expression for ω(κ) stated in thelemma, subject to the condition |κ| > 1. This result shows that the normal frequency varies monotonically from 0 to 1 along theHopf curve H0. We study Hopf–Neı̆mark–Sacker bifurcations near a Hopf point µ0 at whichω(µ0) = pq , where p and q are relatively prime and 0 < p < q. According to lemma 4, thishappens at the points µ0 = (λ0, κ0), withκ0 = ±√1 − ω(µ0)21 + ω(µ0)2= ±√q2 − p2q2 + p2, λ0 =√12(1 − p2q2).If the system satisfies the versal unfolding of linear part condition, theorem 1 yields the normalform (18).Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1559Lemma 5. The system y ′ = g(y, µ), with g given by (28), satisfies the versal unfolding oflinear part condition, i.e. the map(λ, κ) → (a(λ, κ), ω(λ, κ))with a and ω given by (16) is invertible, locally near any point of the Hopf stratum H0.Proof. Let (λ0, κ0) be a point on the Hopf stratum H0. We have to prove that∂(a, δ)∂(λ, κ)∣∣∣∣(λ0,κ0)= 0.Putting Y = |yλ,κ |2 we get a = 2(λ − Y ) and δ = λ2κ2 − Y 2 − ω2N . Therefore∂(a, δ)∂(λ, κ)=∣∣∣∣∣∣∣∣2(1 − ∂Y∂λ)−2∂Y∂κ2(λκ2 − Y ∂Y∂λ)2(λ2κ − Y ∂Y∂κ)∣∣∣∣∣∣∣∣= 4(λ2κ − λ2κ ∂Y∂λ+ λκ2∂Y∂κ− Y ∂Y∂κ).So it remains to compute the values of ∂Y∂λand ∂Y∂κat (λ0, κ0). Now we pass to real coordinatesby putting y = u + iv, with u, v ∈ R. Denote the real and imaginary parts of g(u + iv) bygr(u, v) and gi(u, v), respectively. Since g(yλ,κ) = 0, we have Y = u2 + v2, where u, v is thesolution of the system of equationsgr(u, v) = gi(u, v) = 0. (30)Note that u and v depend on λ and κ and that∂Y∂λ= 2(u∂u∂λ+ v∂v∂λ)and∂Y∂κ= 2(u∂u∂κ+ v∂v∂κ). (31)Differentiating each of the equations in (30) with respect to λ and κ we obtain∂gr∂u∂gr∂v∂gi∂v∂gi∂v ·∂u∂λ∂u∂κ∂v∂λ∂v∂κ +∂gr∂λ∂gr∂κ∂gi∂λ∂gi∂κ =(0 00 0).From this identity we compute the values of ∂u∂λ, ∂u∂κ, ∂v∂λand ∂v∂κat (λ0, κ0), using Mathematica.Plugging these values into (31), and usingλ0 = (1 + κ20 )−12 , u0 = (1 + κ20 )−34 and v0 = −κ0 (1 + κ20 )−34 ,we obtain∂(a, δ)∂(λ, κ)∣∣∣∣(λ0,κ0)= 16κ0κ40 − 1> 0.From this inequality we conclude that the reparametrization is locally invertible at points(λ0, κ0) of the Hopf curve. The latter lemma allows us to conclude that the dynamics of the third cell can be brought intothe normal form (18). Moreover, it turns out that we can explicitly compute the coefficientγ (ν0, 0) in this normal form. Hence, theorem 1 guarantees that the normal form can be reducedeven further to the simpler form (19).1560 H W Broer and G VegterThe Hopf coefficient. In our case study, the second condition for the occurrence of genericHopf–Neı̆mark–Sacker bifurcations also holds, i.e. the real part of the Hopf coefficient is nonzero.Lemma 6. In the case study (27), the real part of the Hopf coefficient Fu(0, 0, 0), or,equivalently, the real part of the coefficient of z2z in the normal form (18) for (µ, ε) = (µ0, 0),is non zero.Proof. We first bring the linear part of the system ẏ = g(y, µ0) into the Jordan normal form,with g given by (28) with λ = (κ2 + 1)−1/2 and |κ| > 1, cf lemma 4. To this end, first bringthe Hopf singular point to the origin by the translation Y = y − y0, where y0 = y(µ0). Thistranslation transforms the system ẏ = g(y, µ0) into the formẎ = B10Y + B01Y + B20Y 2 + B11YY + B02Y 2 − Y 2Y , (32)withB10 = λ0(2 + iκ0) − 2|y0|2, B01 = −y20 ,B20 = −y0, B11 = −2y0, B02 = 0.Subsequently, we bring the linear part of (32) into the Jordan normal form Ż = iω0Z via (21),i.e. we putY = σ0Z + τ0Z,where σ0 and τ0 are given by (22) for µ = µ0. In view of (29) we havea0 = Re α(µ0) = 0, b0 = Im α(µ0) = κ0λ0,c0 = Re β(µ0) = −Re y20 , d0 = Im β(µ0) = −Im y20 , (33)where y0 = (1 − κ0 i)(κ20 + 1)−3/4 and λ0 = (1 + κ20 )−1/2, cf lemma 4. A straightforwardcomputer-assisted calculation shows that the transformed system is of the formŻ = iω0Z + A20Z2 + A11 Z Z + A02Z2 + A30Z3 + A21Z2 Z + A12Z Z2 + A03 Z3, (34)whereA20 = κ02(1 + κ20 )5/4(1 − 2i√κ20 − 1),A11 = κ0(1 + κ20 )5/4(−1 + 2i√κ20 − 1),A02 = κ02(1 + κ20 )5/4(2 − κ20 − 2i√κ20 − 1),A21 = κ20 − κ40(1 + κ20 )2.(35)The values of A30, A12 and A03 are irrelevant, in view of the following result: the system (34)is brought into the normal form (18), which for (κ, λ, ε) = (κ0, (κ20 + 1)−1/2, 0) is given byż = iω0z + c1z2z + O(|z|4),where the Hopf coefficient c1 is given byc1 = i2ω0(2A20A11 − 2|A11|2 − 43 |A02|2) + A21. (36)For a derivation of this result, see [27, lemma 3.6], especially (3.18) in the proof of thislemma, or [24, appendix to section 3.4], especially (3.4.11) and (3.4.26). In fact, the normalGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1561form procedure presented in appendix B.1 yields the same expression for the Hopf coefficient.Substituting the values of Aij given in (35) into (36) yieldsc1 = − κ201 + κ20+κ20 − 6κ403(κ20 + 1)2√κ20 − 1i,and hence Re c1 = 0. The perturbation term. We continue our case study with a closer analysis of the perturbationterm in (27). We show that, for a general class of perturbation terms P in (2), the constantC(P ), introduced in theorem 3, can be chosen arbitrarily. Since the coefficient γ (ν0, 0) of therotationally non-symmetric term in the normal form (18) is equal to γ (ν0, 0) = Cf + C(P ),this implies that a suitable choice of the perturbation term P yields γ (ν0, 0) = 0. In view oftheorem 2, this implies the occurrence of generic Hopf–Neı̆mark–Sacker bifurcations in thedynamics of the third cell.Consider a perturbation term of the monomial formP(z2, z1) = P0 zk1 z1l zm2 z2q−1−m,where P0 is a non zero complex constant. Note that this term satisfies condition (9),guaranteeing the persistence of the equilibrium of the first cell and the periodic solution of thesecond cell.Then, in the notation of theorem 3,P (z, t) = P0 λ12 (k+l)0 ei(k−l+2m−q)t (y0 + σ0 z + τ0 z)m (y0 + σ0 z + τ0 z)q−1−m= P0 λ12 (k+l)0 ei(k−l+2m−q)t (τm0 σ0q−1−m zq−1 + σm0 τ0q−1−m zq−1)+ z z Rq−3(z, t) + Rq−2(z, t),where Rk(z, t) is a polynomial of degree k in z and z, with coefficients that are 2π -periodicin t . Since the term z z Rq−3(z, t) + Rq−2(z, t) does not contain a term of the form zq−1 eipt , itdoes not contribute to the coefficient C(P ), cf theorem 3. More precisely,C(P ) =P0 λ12 (k+l)0 σ0q−m τm0 , if k − l + 2m − q = p,−P0 λ12 (k+l)0 σ0m τq−m0 , if k − l + 2m − q = −p,0, if k − l + 2m − q = ±p.Given the value of Cf , we take k − l + 2m − q = ±p and choose P0 such that γ (ν0, 0) =Cf + C(P ) = 0. This choice of P0 is possible since λ0, σ0 and τ0 are non zero. Indeed,λ0 = (κ20 + 1)−1/2 = 0 and Im σ0 = Im τ0 = c0 = 0, cf (22) and (33).Remark. Observe that C(P ) = 0 if the degree of P in z2, z2 is less than q − 1. Therefore,our example above is the ‘simplest’ yielding non zero C(P ).Consequently, for this choice of P the dynamics exhibits Hopf–Neı̆mark–Sackerbifurcations corresponding to the birth and death of subharmonics of order q, for parametervalues (λ, κ, ε) near the point (λ0, κ0, 0) on the Hopf line in the plane ε = 0 in parameterspace. This phenomenon even occurs for generic perturbation terms P .Lemma 7. For generic choices of the perturbation term P in (27) the coefficient γ (ν0, 0) ofthe rotationally non-symmetric term in the normal form (18) for (ν, ε) = (ν0, 0) is non zero.1562 H W Broer and G VegterProof. Since C(P ) is linear in P , we have C(P ) = 0 for generic P . To see this, decomposeP as P = P ′ + P ′′ + P ′′′, with P ′ of degree less than q − 1, P ′′ homogeneous of degree q − 1and P ′′′ of degree greater than q − 1 in z2, z2. Then theorem 3 implies C(P ′) = 0. Computec0(f )+C(P ′′′) and take the coefficient(s) of P ′′ in such a way that c0(f )+C(P ′′)+C(P ′′) = 0.Given arbitrary, but fixed P ′′′, the latter inequality holds for generic P ′′. This proves thatc0(f ) + C(P ) = 0 for generic P . Lemmas 4–7 show that in our case study the conditions of theorem 1 and 2 are met forgeneric perturbation terms. Note that generic means open and dense with respect to theZariski topology. More precisely, the preceding discussion is summarized in the followingresult.Theorem 8. Consider a three-cell feed-forward network with cell dynamics given by the three-parameter familyFλ,κ,ε(z1, z2) = fλ,κ(z2, z1) + ε P (z2, z1),where fλ,κ is given by (26). For q > p > 0, with p and q relatively prime, the dynamics ofthe third cell has a p : q resonance point at(λ, κ, ε) =√12(1 − p2q2), ±√q2 − p2q2 + p2, 0 .For generic perturbation terms P the dynamics of the third cell undergoes a generic Hopf–Neı̆mark–Sacker bifurcation at this point. The resonance tongues in (λ, κ, ε)-space emanatefrom a cuspidal edge in the (λ, κ) plane given byλ = 1√κ2 + 1, ε = 0.The tongue boundaries exhibit the standard generic geometry, given by the local model (20).See also figure 3.4. Conclusion and future workWe have explored a framework for detecting the occurrence of generic Hopf–Neı̆mark–Sackerbifurcations of families of perturbed forced oscillators. Applying this framework, we havebeen able to show that such bifurcations occur in a large class of linear feed-forward networksof coupled cells, each with the same dynamics. We have shown in detail how to apply ourframework to concrete examples of such cell dynamics.Our approach is based on fine-tuning existing normal form techniques, which, applied toconcrete systems, provide the generic conditions for the occurrence of Hopf–Neı̆mark–Sackerbifurcations. Furthermore, this approach allowed us to show that these bifurcations give riseto resonance tongues with ‘standard’ geometry.In [5] we have developed a method, based on Liapunov–Schmidt reduction and Zqequivariant singularity theory, to find period q resonance tongues in a Hopf bifurcation fordissipative maps. We recovered the standard non degenerate results, but also illustrated themethod on the degenerate case (q 7). We are currently analysing the resonance tonguesof such mildly degenerate systems, which are (singular) hypersurfaces in four-dimensionalparameter space.We also plan to investigate the geometry of resonance tongues in families of cell dynamicsexhibiting mildly degenerate Hopf–Neı̆mark–Sacker bifurcations. This phenomenon onlyoccurs in systems depending on more than three parameters, so the analysis is typically moreGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1563complicated than in this paper, especially since the resonance tongues are (singular) surfacesin parameter space of dimension at least four.AcknowledgmentsThis work is a continuation of Broer, Golubitsky and Vegter [5] and [6]. The authors thankMarty Golubitsky for his support and helpful comments, Sijbo Holtman for preparing figure 4and the referees for suggesting some improvements and a further reference.Appendix A. Versal unfolding of the linear part: Jordan normal formIn this appendix we show how to bring the linear part of the forced oscillator (13) into theJordan normal form at the equilibrium y = yµ. By applying a sequence of transformations,as already described in section 2.2, we arrange that for all parameter values in the region ofinterest the dynamics of the third cell has an equilibrium z = 0 and that the linear part of thisequilibrium is in the Jordan normal form. The following lemma describes this sequence oftransformations.Lemma 9. Suppose the forced oscillatorẏ = g(y, µ) + ε G(y, t, µ) (37)satisfies the generic Hopf bifurcation condition (17), for µ near µ0, and the normal frequencyat µ0 is equal to ω0 = pq . Then there is a transformation of the formz = Z + εϕ(t, ε, µ), (38)which is 2π -periodic in t , such that successive application of the transformations Y = y −yµ,cf (15), Z = s−1µ (Y ), cf (21) and (38), brings the perturbed system (37) into the formż = (iω(µ) + a(µ))z + h(z, µ) + εH(z, t, ε, µ), (39)where1. z → h(z, µ) has zero 1-jet at z = 0, i.e. h(0, µ) = 0, dzh(0, µ) = 0 and h(z, µ) dependslinearly on g;2. H(z, t, ε, µ) is 2π -periodic in t , H(0, t, ε, µ) = 0, and H(z, t, ε, µ) depends linearly onG (and also on g, though not necessarily linearly).Remark. The proof of this lemma is a straightforward calculation. In fact, we will findan explicit expression for the transformation (38), which, in turn, will allow us to determineexplicit expressions for the terms h(z, µ) and H(z, t, ε, µ) in (39). From these expressionswe will then be able to conclude that if g is a polynomial of degree m in z, z, then1. h(z, µ) is a polynomial of degree m in z, z;2. H(z, t, 0, µ) = K(z, t, µ) − K(0, t, µ) + Pm−1(z, t, µ),where K is given byK(Z, t, µ) = 1(σ G( yµ + σ Z + τ Z, t, µ ) − τ G( yµ + σ Z + τ Z, t, µ )), (40)where Pm−1(z, t, µ) is a polynomial of degree at most m − 1 in z, z and where σ and τ aregiven by (22) and = |σ |2 −|τ |2. In particular, K(Z, t, µ) is 2π -periodic in t . These explicitexpressions will be needed in the proof of theorem 3 in appendix D.1564 H W Broer and G VegterProof. The strategy of the proof is as follows. We first show that successive application of thetransformations Y = y −yµ and Z = s−1µ (Y ) brings the linear part of the unperturbed system,i.e. (37) with ε = 0, into the Jordan normal form. Then we apply these two transformationsto the perturbed system (37), which reintroduces a time dependent constant term. We showthat a third transformation of the form (38) eliminates this constant, in other words, brings thesystem into the 2π -periodic form (39), which has a 2π -periodic orbit z = 0.Recall that we first arrange that the equilibrium of the unperturbed system is at the originby introducing a localized variable Y , defined near 0 ∈ C byy = yµ + Y,where yµ is the equilibrium given by (15). In this way the unperturbed dynamics ẏ = g(y, µ)is transformed into the formẎ = Lµ(Y ) + g1(Y, µ), (41)where Lµ is the linear part of Y → g(yµ + Y, µ) at Y = 0. In particular, g1(Y, µ) has noconstant and linear terms in Y and Y , i.e. g1(0, µ) = 0 and dY g1(0, µ) = 0.The second transformation is the inverse of sµ given by sµ(Z) = σ Z +τ Z, cf (21), whereσ and τ are given by (22). PuttingZ = s−1µ (Y ) = −1(σ Y − τ Y ),with = |σ |2 − |τ |2, brings the unperturbed system (41) into the Jordan normal form, i.e.Ż = (iω(µ) + a(µ))Z + h(Z, µ),whereh(Z, µ) = −1 ( σ g1( σ Z + τ Z, µ ) − τ g1( σ Z + τ Z, µ ) ). (42)In particular, the two-jet of Z → h(Z, µ) vanishes at Z = 0 for µ near µ0.We now apply the transformations Y = y − yµ and Z = s−1µ (Y ) to the perturbed system(37). In other words, we puty = yµ + σ Z + τ Z,which brings (37) into the Jordan normal formŻ = (i ω + a) Z + h(Z, µ) + ε K(Z, t, µ),where h(Z, µ) is given by (42) and K(Z, t, µ) by (40), i.e.K(Z, t, µ) = −1(σ G(yµ + σ Z + τ Z, t, µ) − τ G(yµ + σ Z + τ Z, t, µ)).Denoting the time derivative of ϕ by ϕ̇, and observing that the inverse of the thirdtransformation (38) is given by Z = z − εϕ(t, ε, µ), we see that the transformed systemis of the formż = Ż + εϕ̇ = (z, ϕ, t, ε, µ) + εϕ̇, (43)where(z, ϕ, t, ε, µ) = (iω + a)(z − εϕ) + h(z − εϕ, µ) + ε K(z − εϕ, t, µ).Here we suppress the dependence of a and ω on µ and the dependence of ϕ and its derivativeϕ̇ on t .Our goal is to find a function ϕ = ϕ(t, ε, µ) such that the terms on the right-hand sideindependent of z and z vanish, i.e. to reduce (43) to a system of the form ż = O(|z|), uniformlyGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1565in (t, ε, µ). Since (43) is of the formż = (0, ϕ, t, ε, µ) + εϕ̇ + O(|z|),we have to solve the differential equation(0, ϕ, t, ε, µ) + εϕ̇ = 0 (44)for ϕ = ϕ(t, ε, µ). PuttingK0(t, µ) = K(0, t, µ) and K1(ϕ, t, ε, µ) = K(−εϕ, t, µ) − K(0, t, µ)yields(0, ϕ, t, ε, µ) = −ε ( (iω + a) ϕ + K0(t, µ) ) + h(−εϕ, µ) + ε K1(ϕ, t, ε, µ).Since the two-jet of z → h(z, µ) vanishes at z = 0, we have h(−εϕ, µ) = O(ε2). Similarly,K1(ϕ, t, ε, µ) = O(ε). Therefore, the division lemma guarantees the existence of a smoothfunction K2 : C × C × R × Rk → R such thath(−εϕ, µ) + ε K1(ϕ, t, ε, µ) = ε2 K2(ϕ, t, ε, µ).Note that also K2(ϕ, t, ε, µ) is 2π -periodic in t . Therefore, (43) is of the formż = ε (ϕ̇ − (iω + a)ϕ + K0(t, µ) + ε K2(ϕ, t, ε, µ)) + O(|z|).Observe that e2π(iω+a) ∈ Z for µ near µ0, since a(µ0) = 0 and e2π iω(µ0) ∈ Z. Therefore, thereis a unique function ϕ : R × R × Rk → R, such that ϕ(t, ε, µ) is 2π -periodic in t , andϕ̇ − (iω + a)ϕ + K0(t, µ) + ε K2(ϕ, t, ε, µ) = 0,and such that ϕ0(t, µ) := ϕ(t, 0, µ) is of the formϕ0(t, µ) = e(iω(µ)+a(µ))t(c0 −∫ t0e−(iω(µ)+a(µ))s K0(s, µ) ds), (45)withc0 = (e2π(iω(µ)+a(µ)) − 1)−1∫ 2π0e−(iω(µ)+a(µ))s K0(s, µ) ds.With this choice of ϕ, transformation (38) reduces the system to one of the forms ż = O(|z|).To prove that the reduced system is of the form (39) it remains to analyse the right-hand sideof this system in more detail. To this end, (43) and (44) imply that the transformed system isof the formż = (z, ϕ, t, ε, µ) − (0, ϕ, t, ε, µ)= (iω + a) z + h(z − εϕ, µ) − h(−εϕ, µ) + ε (K(z − εϕ, t, µ) − K(−εϕ, t, µ)).The Taylor expansion with respect to ε shows thath(z − εϕ, µ) − h(−εϕ, µ) = h(z, µ) + ε h1(z, t, ε, µ), (46)where h1 : C × R × R × Rk → R is a smooth function of the formh1(z, t, ε, µ) = −ϕ0(t, µ) hz(z, µ) − ϕ0(t, µ) hz(z, µ) + O(ε),uniformly in (z, t, µ), which is 2π -periodic in t . Taking z = 0 in (46) shows thath1(0, t, ε, µ) ≡ 0. (47)PuttingH(z, t, ε, µ) = h1(z, t, ε, µ) + K(z − εϕ, t, ε, µ) − K(−εϕ, t, ε, µ)), (48)1566 H W Broer and G Vegterwe see that the transformed system is of the form (39). The function H(z, t, ε, µ) is2π -periodic in t , and, in view of (47), it satisfies H(0, t, ε, µ) = 0. Furthermore, (48)and (40) show that H(z, t, ε, µ) depends on g and linearly G. This completes the proof of thelemma. We have given the expressions for H in detail, since these will be used in the proof oftheorem 3 to be presented in appendix D. In fact, we will need the following result.Lemma 10. For ε = 0, the function H(z, t, ε, µ), given by (48), satisfiesH(z, t, 0, µ0) = −ϕ0(t, µ0) hz(z, µ0) − ϕ0(t, µ0) hz(z, µ0) + K(z, t, µ0) − K(0, t, µ0),(49)withK(z, t, µ0) = σ0 P (z, t) − τ0 P (z, t), (50)h(z, µ0) = σ0 λf (z) − τ0 λf (z) + aff(z, z), (51)where aff(z, z) is an affine function of z, z, where P (z, t) is defined by (25) and whereλf (z) = −10 f (y0 + σ0 z + τ0 z). (52)Proof. Expressions (49) and (50) are straightforward consequences of (48) and (46). In viewof the defining relations (42) and (14) of h in terms of g and of g in terms of f , respectively,identity (51) follows from the observation thatg1(σ0 z + τ0 z, µ0) = g(y0 + σ0 z + τ0 z, µ0) − Lµ0(σ0 z + τ0 z)= f (y0 + σ0 z + τ0 z, µ0) + aff(z, z). Appendix B. Derivation of normal formsMethods for bringing the continuous dynamical system into the polynomial normal formare well known from the literature; See, e.g. [3, 27, 32]. In this appendix we show how tofine-tune this standard normal form procedure in such a way that the normal form providesadditional information revealing the occurrence of generic Hopf–Neı̆mark–Sacker bifurcations.In section B.1 we show that the terms of order i in the normal form depend linearly on the termsof order i in the original system and polynomially on the terms of lower order. Section B.2presents the normal form procedure for periodic systems, and in section B.3 we use these normalform results to prove theorem 1. The results presented in this appendix are of independentinterest, since their scope is much wider than the class of forced oscillators appearing in thedynamics of the third cell.Appendix B.1. A normal form procedureIn this section we derive a normal form for the larger class of 2π -periodic systems of the form(39) where the system is in p : q resonance, i.e. ω(µ0) = pq for some Hopf parameter µ0, withp and q relatively prime. The normal form corresponds to the rotationally symmetric Hopfnormal form for ε = 0. It turns out that the full normal form is not rotationally symmetric, butexhibits a Zq-symmetry.Our approach to the computation of normal forms is a slight extension of the well-knownmethods introduced in [32]. This procedure transforms the terms of the vector field in ‘assimple a form as possible’, up to a user-defined order. It does so via iteration with respect toGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1567the total degree of these terms. Before focusing on periodic systems in the plane we reviewthe method for computing normal forms in a general Lie algebra setting and isolate some of itsfeatures that are crucial for our subsequent study of subharmonics in the forced oscillator (39).Lie series expansion. For non negative integers m we denote the space of vector fields (onsome Euclidean space of arbitrary dimension) of total degree m by Hm and the space of vectorfields with vanishing derivatives up to and including order m at 0 ∈ C by Fm. Note thatFm =∏km Hk .Lemma 11. Let X and Y be vector fields on C, where X is of the formX = X(1) + X(2) + · · · + X(N) mod FN+1, (53)with X(n) ∈ Hn and Y ∈ Hm, with m 2. Let Yt , t ∈ R, be the one-parameter groupgenerated by Y and let Xt = (Yt )∗(X). ThenXt = X + N−1m−1 ∑k=1(−1)kk!tkad(Y )k(X) mod FN+1 (54)= X +N∑n=1 N−nm−1 ∑k=1(−1)kk!tkad(Y )k(X(n)) mod FN+1. (55)Proof. We follow the approach of Takens [32] and Broer et al [3,4] to obtain the Taylor seriesof Xt with respect to t in t = 0 using the basic identity∂∂tXt = [Xt, Y ] = −ad(Y )(Xt).Using this relation, we inductively prove that∂k∂tkXt = (−1)kad(Y )k(Xt).Using the latter identity for t = 0, we obtain the formal Taylor seriesXt =∑k01k!tk∂k∂tk∣∣∣∣t=0Xt=∑k0(−1)kk!tk ad(Y )k(X).Since Y ∈ Hm, the operator ad(Y )k increases the degree of each term in its argument byk(m − 1). Since the terms of lowest order in X are linear, we see thatad(Y )k(X) = 0 mod FN+1,if 1 + k(m − 1) > N . Therefore,Xt = N−1m−1 ∑k=0(−1)kk!tkad(Y )k(X) mod FN+1= X + N−1m−1 ∑k=1(−1)kk!tkad(Y )k(X) mod FN+1,1568 H W Broer and G Vegterwhich proves (54). In view of (53) the latter identity expands toXt =N∑n=1 N−1m−1 ∑k=0(−1)kk!tkad(Y )k(X(n)) mod FN+1. (56)Since ad(Y )k(X(n)) ∈ Hn+k(m−1), we see thatad(Y )k(X(n)) = 0 mod FN+1,for k > N−nm−1 . Therefore, for fixed index n, the inner sum in (56) can be truncated at k = N−nm−1 ,which concludes the proof of (55). Computing normal forms. Consider a vector field X having a singular point with semisimplelinear part S. Our goal is to bring X into the normal form to some prescribed order N . SinceS is semisimple we know thatHm = Ker ad(S) + Im ad(S).Here ad(S) : Hm → Hm is the restriction to Hm of the adjoint action of the linear part S of X.The linear operators Gm : Hm → Hm and Bm : Hm → Hm are the corresponding projectionsonto Ker ad(S) and Im ad(S), respectively. So, X(m) = Gm(X(m))+Bm(X(m)), for X(m) ∈ Hm.Lemma 12 (normal form lemma [32]). Consider a vector field X = S + X(2) + · · · +X(m) mod Fm+1, with X(i) ∈ Hi . There is a transformation bringing the vector field Xinto the normal form∗(X) = S + G(2) + · · · + G(m) mod Fm+1,for any m 2, where G(i) ∈ Hi belongs to Ker ad(S). Furthermore, G(i) is of the formG(i) = Gi(X(i)) + i(S, X(2), . . . , X(i−1)),where i(S, X(2), . . . , X(i−1)) is a polynomial in the coefficients of S, X(2), . . . , X(i−1).The first part of the lemma is well known, cf [32]. The second part follows from a carefulanalysis of the normal form transformation, which we give now.Proof. The normal form is determined in an iterative process, which successively brings theterms of order 2, . . . , N into the normal form. So assume that X has been brought into theformX̃ = S + G(2) + · · · + G(m−1) + X̃(m) + · · · + X̃(N) mod FN+1, (57)where the homogeneous parts of X̃ satisfy the following conditions.P1(m): G(i) ∈ Hi belongs to Ker ad(S), for 2 i < m, and is of the formG(i) = Gi(X(i)) + i , where i depends polynomially on the coefficients ofS, X(2), . . . , X(i−1).P2(m): X̃(i) ∈ Hi is of the form X̃(i) = X(i) + i , for m i N , where i dependspolynomially on the coefficients of S, X(2), . . . , X(i−1);Note that (57) is trivially true for m = 1, since the linear part is already in the normal form.We determine a normal form transformation bringing the vector field into the form (57) withm replaced by m + 1, such that P1(m + 1) and P2(m + 1) hold for this incremented value of m.By induction, this will prove the lemma.The normal form transformation will be the time-1-map of a vector field Y ∈ Hm, to bedetermined in such a way that the term of order m of the transformed vector field is in theGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1569normal form. Let Yt , t ∈ R, be the one-parameter group generated by Y and let Xt = (Yt )∗(X̃).To determine the homogeneous components of X1, we apply lemma 11, replacing X with X̃and taking into account that X̃(1) = S and X̃(i) = G(i), for 2 i < m. Splitting off the termfor n = 1 in (55) we see thatX1 = X̃ − ad(Y )(S) + N−1m−1 ∑k=2(−1)kk!ad(Y )k(S)+N∑n=2 N−nm−1 ∑k=1(−1)kk!ad(Y )k(X̃(n)) mod FN+1. (58)Recall that X̃(m) = G(m) +B(m), with G(m) = Gm(X̃(m)) ∈ Ker ad(S) and B(m) = Bm(X(m)) ∈Im ad(S). Now there is a unique Y ∈ Im ad(S) satisfying the homological equationad(S)(Y ) = −B(m). (59)Since X̃ − ad(Y )(S) = X̃ + ad(S)(Y ) = X̃ − B(m) = S + G(2) + · · · + G(m) mod Fm+1,identities (58) and (59) imply that X1 is in the normal form to order m. To see this, observethat ad(Y )k(X̃(n)) ∈ Hn+k(m−1). Therefore, both sums on the right-hand side of (58) consist ofterms in Fm+1. In other words,X1 = S + G(2) + · · · + G(m−1) + G(m) mod Fm+1.Observe that G(m) = Gm(X̃(m)) + m, where m = Gm(m). Since m is a polynomial inX(2), . . . , X(m), the same thing holds for m. This proves P1(m + 1).To prove P2(m + 1), observe that the solution Y of the homological equation (59)depends linearly on the coefficients of X̃(m) and, hence, in view of P2(m), polynomiallyon the coefficients of S, X(2), . . . , X(m). Therefore, each of the terms ad(Y )k(X̃(n)) in (58)depends polynomially on these coefficients. In other words, the homogeneous term oforder n, m < n N , of the transformed vector field X1 is obtained by adding a finitenumber of polynomials in the coefficients of S, X(2), . . . , X(m) to X̃(n). Together with P2(m)this observation implies P2(m + 1) and, hence, completes the proof of the normal formlemma. Appendix B.2. Normal form of periodic systemsThe Lie algebra of planar periodic systems. When dealing with real 2π -periodic vector fieldson R2 we identify R2 with C, by associating the point (x1, x2) in R2 with x1 + ix2 in C. Thereal 2π -periodic vector fields correspond to systems on C × R/(2πZ) of the formX = XR ∂∂z+ XR∂∂z̄+ XS∂∂t, (60)where XR : C × R/(2πZ) → C and XS : C × R/(2πZ) → C are 2π -periodic functions, i.e.XR(z, t + 2π) = XR(z, t) and XS(z, t + 2π) = XS(z, t). Recalling that∂∂z= 12(∂∂x1− i ∂∂x2)and∂∂z̄= 12(∂∂x1+ i∂∂x2),we see that the real vector field X, defined on R2 × (R/2πZ) byX = Y1 ∂∂x1+ Y2∂∂x2+ Y3∂∂t,1570 H W Broer and G Vegtercorresponds to the vector fieldX = Y ∂∂z+ Y∂∂z̄+ Y3∂∂t,on C × R/(2πZ), where Y = Y1 + iY2. These vector fields form a Lie subalgebra of thealgebra of vector fields on C × R/(2πZ). This is justified by the fact that the Lie bracket oftwo real 2π -periodic vector fields X and Y on C × R/(2πZ) is given by[X, Y ] = 〈X, Y 〉R∂∂z+ 〈X, Y 〉R∂∂z̄+ 〈X, Y 〉S∂∂t, (61)where the R-bilinear antisymmetric forms 〈·, ·〉R and 〈·, ·〉S are defined by〈X, Y 〉R = X(YR) − Y (XR) and 〈X, Y 〉S = X(YS) − Y (XS). (62)The proof consists of a straightforward computation. Therefore, we omit the details.The adjoint action in a periodic context. Let Hm, m 1, be the space of real vector fieldson C × R/(2πZ) with vanishing ∂∂tcomponent, cf (60), whose real components XR(z, t) arehomogeneous polynomials of degree m in (z, z) with 2π -periodic coefficients. In other words,XR(z, t) =m∑k=0fk(t) zk zm−k, (63)where fk is 2π -periodic. The Fourier coefficients of a 2π -periodic function f are denoted bycn(f ), i.e. for n ∈ Z:cn(f ) = 12π∫ 2π0e−ins f (s) ds.To extend the context of Hopf bifurcations we consider 2π -periodic systems with the linearpart of the formS = iω(z∂∂z− z ∂∂z̄)+∂∂t, (64)where ω is a real number. The following result describes the action of ad(S) on the space of2π -periodic vector fields.Lemma 13 (adjoint action ad(S)). The operator ad(S), associated with (64), leaves thesubspace of vector fields with zero ∂∂t-component invariant. The restriction of this operator toHm is semisimple, i.e. for m 2 there is a decompositionHm = Ker ad(S) + Im ad(S). (65)If Gm : Hm → Hm is the corresponding projection operator onto Ker ad(S), then, for a vectorfield X with XR of the form (63), the image Gm(X) has a∂∂z-component of the formGm(X)R =∑kc(m+1−2k) ω(fk) ei(2k−m−1)ωt zk zm−k, (66)where k ranges over the index set {k | 0 k m and (2k − m − 1) ω ∈ Z}.Proof. If XS = 0, then (62) implies 〈S, X〉S = 0, so ad(S) leaves the subspace of vector fieldswith a zero ∂∂t-component invariant. Furthermore, a straightforward computation showsad(S)XR =m∑k=0(iω(2k − m − 1) fk(t) + f ′k(t))zk zm−k.Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1571Table 1. The basis for the kernel of ad(S) : Hm → Hm, for 0 m q − 1, in the case ω = pq ,and the projection Gm(X) ∈ Ker ad(S), where XR is of the form (63).Basis of Ker ad(S) Gm(X)R0 m < q − 1, m even ∅ 00 m < q − 1, m = 2n + 1 {Y 0}, with Y 0R= z |z|2n c0(fn+1) Y 0Rm = q − 1, m even {Y 1}, with Y 1R= zq−1 eipt c−p(fq−1) Y 1Rm = q − 1 = 2n + 1 {Y 0, Y 1}, with Y 0R= z |z|2n c0(fn+1) Y 0Rand Y 1R= zq−1 eipt + c−p(fq−1) Y 1RTherefore, ad(S)X = 0 iff iω(2k − m − 1) fk(t) + f ′k(t) = 0, for 0 k m. Ifω(2k − m − 1) ∈ Z, the only 2π -periodic solution is fk = 0. If ω(2k − m − 1) ∈ Z,then any 2π -periodic solution is of the form fk(t) = fk(0) eiω(m+1−2k)t . Therefore, the kernelof ad(S) consists of all vector fields X with the real part XR belonging to the spaceSpan {eiω(m+1−2k)t zk zm−k | 0 k m and (2k − m − 1) ω ∈ Z}.Let Gm : Hm → Hm be the projection of X onto Ker ad(S), defined by (66). A straightforwardcomputation shows that X − Gm(X) ∈ Ker ad(S). Therefore, Gm gives rise to the splitting(65), so ad(S) is semisimple. Remark. In our study of Hopf–Neı̆mark–Sacker bifurcations, we will focus on periodic vectorfields with ω = pq. Using (66) we see that in this case the term without rotational symmetryof lowest order is of degree q − 1 in z, z. More precisely, Table 1 presents the kernel of ad(S)and the corresponding projection (66) for m < q. This result will be used in the next sectionto derive the normal form to order q. Note that in [15] we obtained normal forms for periodicsystems by averaging over time. The two approaches are closely related, as explained in [3].Appendix B.3. Normal form to order q: proof of theorem 1The normal form for parametrized planar periodic systems of the form (39) can be determinedusing the methods of section B.1. More precisely, we consider a parametrized 2π -periodicsystem X with XR of the formXR(z, t, µ) = (iω(µ) + a(µ))z + h(z, µ) + εH(z, t, µ), (67)where µ ∈ Rk and ε is a small real parameter, such that, for µ0 ∈ Rk ,1. a(µ) ∈ R and ω(µ) ∈ R satisfy a(µ0) = 0 and ωN := ω(µ0) = pq , with p and qrelatively prime;2. h and H contain no terms independent of z and z (i.e. h(0, µ) = 0 and H(0, t, µ) = 0);3. h does not contain terms that are linear in z and z.Lemma 14 (normal form to order q). The parametrized system (67), satisfying properties 1,2 and 3, has the normal formż = iωNz +(α(µ, ε) + iδ(µ, ε))z + zF (|z|2, µ, ε) + ε γ (µ, ε) zq−1 eipt + O(|z|q), (68)uniformly in (ε, µ), where the O(|z|q) terms are 2π -periodic in t . Here F(|z|2, µ, ε) is acomplex polynomial of degree at most q − 2 with F(0, µ, ε) = 0. Furthermore,• α and δ are real-valued functions such that α(µ, 0) = a(µ) and δ(µ, 0) = ω(µ)−ω(µ0).In particular, α and δ vanish at (µ, ε) = (µ0, 0).1572 H W Broer and G Vegter• γ (µ, ε) ∈ C, andγ (µ0, 0) = d + h,where d is the coefficient of the term ε zq−1 eipt in the Taylor–Fourier expansion ofH(z, t, µ0) at z = 0 and h is a polynomial in the coefficients of the (q − 1)st Taylorpolynomial of h(·, µ0) in z, z at z = 0.Proof. To separate the contribution of h and H to the normal form and to the normal formtransformation, we regard the normal form procedure as consisting of two successive steps.Step 1. Determine the normal form to order q of the unperturbed system, i.e. of system (67) withε = 0, and apply the corresponding normal form transformation to the complete (perturbed)system (67). Since the unperturbed system is time independent, it follows from lemma 13 thatthis first step yields an intermediate vector field of the formż = (iω(µ) + a(µ)) z + zh̃(|z|2, µ) + O(|z|q), (69)uniformly in µ, where h̃(|z|2, µ) is a time-independent polynomial of degree less than q in zand z. Indeed, the normal form terms of degree m belong to the kernel of ad(S) : Hm → Hm,cf lemma 12. According to lemma 13, the time independent terms in this kernel are 0, if m isodd, and multiples of the vector field Y with YR = z |z|2k , if m = 2k + 1. This proves (69).Step 2. Applying the normal form procedure of step 1 to the full system (67) yields a systemof the formż = (iω(µ) + a(µ)) z + zh̃(|z|2, µ) + ε H̃ (z, t, µ) + O(|z|q), (70)uniformly in (t, µ, ε). Since ω(µ0) = pq , we apply the normal form procedure presented insection B.1, where the linear part is given byS = iωN(z∂∂z− z ∂∂z̄)+∂∂t,with ωN = ω(µ0) = pq . For 2 m q − 1, the kernel of ad(S) : Hm → Hm is given intable 1. Therefore, the normal form of X is of the form (68). Since this normal form coincideswith (69) for ε = 0, it follows that δ(µ, 0) = ω(µ) − ωN and α(µ, 0) = a(µ).To prove that γ (µ0, 0) = d + h, we consider the system, given by (70) for µ = µ0, andderive its normal form. Then γ (µ0, 0) is the coefficient of the term ε zq−1 eipt in this normalform.For µ = µ0 system (69) has the rotationally symmetric formX0 = S +q−1∑n=2G(n)0 + O(|z|q),whereG(n)0 ={0, if n is even,akzk+1zk, if n = 2k + 1. (71)Here the complex number ak is a constant. Let X be system (70) for µ = µ0, then X is of theformX = X0 + εq−1∑n=1X(n) + O(|z|q + ε2),uniformly in t . Here X(n) is homogeneous of degree n in z, z and 2π -periodic in t , butindependent of the parameters µ and ε. Since the normal form transformation of step 1 is aGeneric Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1573near identity transformation of the form z → z+O(|z|2), where the higher order terms dependpolynomially on the coefficients of the (q − 1)-jet of h at z = 0, it follows that the coefficientof ε zq−1 eipt in X is of the form d + h, with d and h as in the statement of the lemma. Weshall prove that further normal form transformations leave this coefficient invariant.Again we derive the normal form of X inductively with respect to the degree of the termsin z, z. To this end we introduce the following predicate, for 0 m q:P(m): there is a transformation m on C × R/(2πZ) bringing X into the form(m)∗(X) = X0 + εm−1∑n=1G(n)1 + εq−1∑n=mX̃(n) + O(|z|q + ε2), (72)where G(n)1 belongs to the kernel of ad(S) : Hn → Hn. Furthermore, the coefficients ofε zq−1 eipt in X and in (m)∗(X) are equal.To see that P(q) implies the statement of the lemma, recall that the kernel of ad(S) isgiven by table 1. Therefore, the ∂∂z-component of (q)∗(X) is of the formż = iωNz +∑k1,2k+2q(ak + εbk) z |z|2k + ε (d + h) zq−1 eipt + O(|z|q + ε2),which implies the statement of the lemma.We now prove inductively that P(m) holds. Taking 0 equal to the identity map we seethat P(0) holds. So assume P(m) holds, for 0 m < q. Denote (m)∗(X) by X̃. As in theproof of lemma 12, we find the transformation bringing X̃ into the form (72), with m replacedwith m + 1, by solving the homological equationad(S)(Y ) = −ε B(m), (73)where the right-hand side is the component of εX̃(m) in the image of ad(S), up to sign, i.e.X̃(m) = G(m)1 + B(m). Then the degree m term of (Y1)∗(X) is equal to G(m)0 + ε G(m)1 , which isin the normal form.The terms in the transformed vector field (Y1)∗(X) of order greater than m are determinedusing the Lie series expansion (55). Since (73) implies Y = O(ε), it follows thatad(Y )k(X̃(n)) = O(εk). Similarly, ad(Y )(ε G(n)1 ) = O(ε2), for 1 n < m, andad(Y )(ε X̃(n)) = O(ε2), for m n < q. Putting Y = ε Ỹ , we see that the transformedvector field is of the form(Y1)∗(X̃) = X̃ − ad(Y )(S) + εq−1∑n=2ad(Ỹ )(G(n)0 )= X0 + εm−1∑n=1G(n)1 + ε (X̃(m) + ad(Ỹ (S))+ ε( q−1∑n=m+1X̃(n) +q−1∑n=2ad(Ỹ )(G(n)0 ))+ O(|z|q + ε2)= X0 + εm∑n=1G(n)1 + εq−1∑n=m+1˜̃X(n)+ O(|z|q + ε2),where˜̃X(n)= X̃(n) + ad(Ỹ )(G(n−m+1)0 ). (74)1574 H W Broer and G VegterTaking m+1 = Y1 ◦m we see that the first part of P(m) holds. To prove that the coefficientsof ε zq−1 eipt in (Y1)∗(X̃) and in X are equal, we consider two cases.Case 1. m < q − 1. Identity (74) implies that ˜̃X(q−1)= X̃(q−1) + ad(Ỹ )(G(q−m)0 ). The normalform term is zero if q − m is even, otherwise it contains a factor z2, since q − m 3, cf (71).In the latter case ad(Ỹ )(G(q−m)0 ) contains a factor z, so the coefficients of zq−1 eipt in ˜̃X(q−1)and X̃(q−1) are equal. Therefore, the transformation Y1 leaves the coefficient of ε zq−1 eiptinvariant.Case 2. m = q − 1. Now the coefficient of ε zq−1 eipt in (Y1)∗(X) is equal to the coefficientof zq−1 eipt in G(q−1). Now G(q−1) = Gq−1(X̃(q−1)), so in view of table 1 the coefficient ofzq−1 eipt in G(q−1) is equal to the coefficient of zq−1 eipt in G(q−1) in X̃(q−1). According to theinduction hypothesis P(q − 1), the latter coefficient is equal to the coefficient of ε zq−1 eipt inX. This concludes the proof of lemma 14. Remarks. 1. It is not hard to prove that h is a non-constant polynomial, which is identically0 if h is rotationally symmetric.2. Note that the hyperplane ε = 0 is mapped onto the hyperplane ε̃ = 0 in (µ̃, ε̃)-space.Furthermore, if µ0 = (µ01, . . . , µ0k), then the bifurcation point (µ, ε) = (µ0, 0) correspondsto (µ̃, ε̃) = (0, 0, µ03, . . . , µ0k, 0).3. We continue our study of the birth or death of subharmonics of order q by analysing thenormal form (18) further. In the following we will drop the tildes from our notation.Appendix C. Geometry of resonance tongues: proof of theorem 2Existence of 2πq-periodic orbits: the Van der Pol transformation. Subharmonics of orderq of the 2π -periodic forced oscillator (19) correspond to q-periodic orbits of the Poincarétime 2π -map P : C → C. These periodic orbits of the Poincaré map are brought into one–one correspondence with the zeros of a vector field on a q-sheeted cover of the phase spaceC × R/(2πZ) via the Van der Pol transformation, cf [15,24]. This transformation correspondsto a q-sheeted covering : C × R/(2πqZ) → C × R/(2πZ),(z, t) → (zeitp/q, t (mod 2πZ)) (75)with cyclic Deck group of order q generated by(z, t) → (ze2π ip/q, t − 2π).The Van der Pol transformation ζ = ze−iωN t lifts the forced oscillator (67) to the systemζ̇ = (α + iδ) ζ + ζ h(ζeiωN t , µ) + ε H(ζeiωN t , t, µ) (76)on the covering space C × R/(2πqZ). The latter system is Zq-equivariant. A straightforwardapplication of (76) to the normal form (18) yields the following normal form for the liftedforced oscillator.Lemma 15 (equivariant normal form of order q). On the covering space, the lifted forcedoscillator has the Zq-equivariant normal form:ζ̇ = (α + iδ) ζ + ζ F (|ζ |2, µ, ε) + ε ζ q−1 + O(|z|q), (77)where the O(|z|q) terms are 2πq-periodic.Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1575Resonance tongues for families of forced oscillators. Bifurcations of q-periodic orbits ofthe Poincaré map P on the base space correspond to bifurcations of fixed points of thePoincaré map P̃ on the q-sheeted covering space introduced in connection with the Van derPol transformation (75). Denoting the normal form system (68) on the base space by N andthe normal form system (77) of the lifted forced oscillator by Ñ , we see that∗Ñ = N .The Poincaré mapping P̃ of the normal form on the covering space now is the 2πq-periodmappingP̃ = Ñ 2πq + O(|z|q),where Ñ 2πq denotes the 2πq-map of the (planar) vector field Ñ . Following the corollary tothe normal form theorem of [15, p 12], we conclude for the original Poincaré map P of thevector field X on the base space thatP = R2πωN ◦ Ñ 2π + O(|z|q),where R2πωN is the rotation over 2πωN = 2πp/q, which precisely is the Takens normalform [33] of P at (z, µ) = (0, 0).Our interest is in the q-periodic points of Pµ, which correspond to the fixed points of P̃µ.From now on we assume that our parameter space is three-dimensional and that µ = (α, δ, ε).Lemma 15 implies that the fixed point set of P̃µ and the boundary thereof in the parameterspace is approximately described by the discriminant set of(a + iδ)ζ + ζ F (|ζ |2, µ) + ε ζ q−1, (78)which is the truncated right-hand side of (77). This gives rise to the bifurcation equationthat determines the boundaries of the resonance tongues. The following theorem implies that,under the conditions that Fu(0, 0) = 0, the order of tangency at the tongue tips is generic.Here Fu(0, 0) is the partial derivative of F(u, µ) with respect to u.Lemma 16 (bifurcation equations modulo contact equivalence). Assume thatFu(0, 0) = 0.Then the polynomial (78) is Zq-equivariantly contact equivalent with the polynomialG(ζ, µ) = (a + iδ + |ζ |2)ζ + εζ q−1. (79)The discriminant set of the polynomial G(ζ, µ) is of the formδ = ±ε(−a)(q−2)/2 + O(ε2). (80)For a definition and similar use of contact equivalence we refer to [5]. Here it is sufficient tokeep in mind that contact equivalent families have diffeomorphic (parametrized) zero sets anddiffeomorphic bifurcation sets. Therefore, theorem 2 follows from lemma 16.Proof. Polynomial (79) is a universal unfolding of the germ |ζ |2ζ + εζ q−1 under Zq contactequivalence. See [5] for a detailed computation. The tongue boundaries of a p : q resonanceare given by the bifurcation equationsG(ζ, µ) = 0,det(dG)(ζ, µ) = 0.As in [5, theorem 3.1] we put u = |z|2 and b(u, µ) = a + iδ + u. Then G(ζ, µ) =b(u, µ)ζ + εζq−1. According to (the proof of) [5, theorem 3.1], the system of bifurcation1576 H W Broer and G Vegterequations is equivalent to|b|2 = ε2uq−2,bb′+ bb′ = (q − 2)ε2uq−3,where b′ = ∂b∂u(u, µ). A short computation reduces the latter system to the equivalent(a + u)2 + δ2 = ε2uq−2,a + u = 12 (q − 2)ε2uq−3.Eliminating u from this system of equations yields expression (80) for the tongueboundaries. The discriminant set of the equivariant polynomial (79) forms the boundary of the resonancetongues. Also see figure 3. This concludes the proof of theorem 2.Appendix D. The non-symmetric term of order q: proof of theorem 3We now prove theorem 3 by deriving a precise expression for the coefficient γ (ν0, 0) ofε zq−1 eipt in the normal form (18) for ε = 0. According to lemma 14, this coefficient is of theformγ (µ0, 0) = d + h,where d is the coefficient of the term zq−1 eipt in the Taylor–Fourier expansion of H(z, t, 0, µ0)at z = 0 and h is a polynomial in the coefficients of the (q −1)-jet of h(·, µ0) in z, z at z = 0.Claim 1. d is equal to the right-hand side of (23).To prove this claim, we apply lemma 10 to conclude that the coefficient of d of zq−1 eiptin the Taylor–Fourier expansion of H(z, t, 0, µ0) is of the formd = c1(f ) C1(P ) + c2(f ) C2(P ) + C(P ),with• c1(f ) is the coefficient of zq−1 in the Taylor expansion of hz(z, µ0) and c2(f ) is thecoefficient of zq−1 in the Taylor expansion of hz(z, µ0). In view of (51) and (52) thesecoefficients depend linearly on f and can be computed explicitly. Moreover, if f is apolynomial of degree less than q, we have c1(f ) = c2(f ) = 0.• C1(P ) is the coefficient of eipt in the Fourier expansion of −ϕ0(t, µ0) and C2(P ) is thecoefficient of eipt in the Fourier expansion of −ϕ0(t, µ0). In view of (45) and (50), thesecoefficients depend linearly on P , and can be computed explicitly.• C(P ) is the coefficient of zq−1 eipt in the Taylor–Fourier expansion of K(z, t, µ0).According to (9), this coefficient is given by (24) and depends linearly on P .Moreover, the functional C is non zero and independent of C1 and C2. Therefore, for fixed fthe coefficient d is non zero for generic P .Generic Hopf–Neı̆mark–Sacker bifurcations in feed-forward systems 1577Claim 2. h only depends on the (q − 1)-jet of y → f (y, µ0) at y = y0.To prove this claim, recall that, according to lemma 9, h depends linearly on g. Therefore,the (q − 1)-jet of h(·, µ0) in z, z at z = 0 depends only on the (q − 1)-jet of g at z = 0. Sinceg depends linearly on f , cf (14) and (41), it follows that the claim holds.Taking c0(f ) = h concludes the proof of theorem 3. 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