Geometry of resonance tonguesHenk W. Broer∗ Martin Golubitsky† Gert Vegter∗April 16, 20061 IntroductionResonan e tongues arise in bifur ations of dis rete or ontinuous dynami al sys-tems undergoing bifur ations of a �xed point or an equilibrium satisfying ertainresonan e onditions. They o ur in several di�erent ontexts, depending, forexample, on whether the dynami s is dissipative, onservative, or reversible. Gen-erally, resonan e tongues are domains in parameter spa e, with periodi dynami sof a spe i�ed type (regarding period of rotation number, stability, et .). In ea h ase, the tongue boundaries are part of the bifur ation set. We mention hereseveral standard ways that resonan e tongues appear.1.1 Various contexts

Hopf bifurcation from a fixed point. Resonan e tongues an be obtained byHopf bifur ation from a �xed point of a map. This is the ontext of Se tion 2, whi his based on our paper [7℄. More pre isely, Hopf bifur ations of maps o ur wheneigenvalues of the Ja obian of the map at a �xed point ross the omplex unit ir leaway from the strong resonan e points e2πpi/q with q ≤ 4. Instead, we on entrateon the weak resonan e points orresponding to roots of unity e2πpi/q, where p andq are oprime integers with q ≥ 5 and |p| < q. Resonan e tongues themselves areregions in parameter spa e near the point of Hopf bifur ation where periodi pointsof period q exist and tongue boundaries onsist of points in parameter spa ewhere the q-periodi points disappear, typi ally in a saddle-node bifur ation. Weassume, as is usually done, that the riti al eigenvalues are simple with no othereigenvalues on the unit ir le. Moreover, usually just two parameters are varied;The e�e t of hanging these parameters is to move the eigenvalues about an openregion of the omplex plane. In the non-degenerate ase a pair of q-periodi orbitsarises or disappears as a single omplex parameter governing the system rosses

∗Institute of Mathemati s and Computing S ien e, University of Groningen. P.O. Box 800,9700 AV Groningen, The Netherlands†Department of Mathemati s, University of Houston. Houston, TX 77204-3476, USA. Thework of MG was supported in part by NSF Grant DMS-02445291

the boundary of a resonan e tongue. Outside the tongue there are no q-periodi orbits. In the degenerate ase there are two omplex parameters ontrolling theevolution of the system. Certain domains of omplex parameter spa e orrespondto the existen e of zero, two or even four q-periodi orbits.Lyapunov-S hmidt redu tion is the �rst main tool used in Se tion 2 to redu ethe study of q-periodi orbits in families of planar di�eomorphisms to the analysisof zero sets of families of Zq-equivariant fun tions on the plane. EquivariantSingularity Theory, in parti ular the theory of equivariant onta t equivalen e, isused to bring su h families into low-degree polynomial normal form, depending onone or two omplex parameters. The dis riminant set of su h polynomial families orresponds to the resonan e tongues asso iated with the existen e of q-periodi orbits in the original family of planar di�eomorphisms.Hopf bifurcation and birth of subharmonics in forced oscillators Let

dX

dt= F(X)be an autonomous system of di�erential equations with a periodi solution Y(t)having its Poin ar�e map P entered at Y(0) = Y0. For simpli ity we take Y0 = 0,so P(0) = 0. A Hopf bifur ation o urs when eigenvalues of the Ja obian matrix

(dP)0 are on the unit ir le and resonan e o urs when these eigenvalues are rootsof unity e2πpi/q. Strong resonan es o ur when q < 5. This is one of the ontextswe present in Se tion 3. Ex ept at strong resonan es, Hopf bifur ation leads tothe existen e of an invariant ir le for the Poin ar�e map and an invariant torusfor the autonomous system. This is usually alled a Naimark-Sa ker bifur ation.At weak resonan e points the ow on the torus has very thin regions in parameterspa e (between the tongue boundaries) where this ow onsists of a phase-lo kedperiodi solution that winds around the torus q times in one dire tion (the di-re tion approximated by the original periodi solution) and p times in the other.Se tion 3.1 presents a Normal Form Algorithm for ontinuous ve tor �elds, basedon the method of Lie series. This algorithm is applied in Se tion 3.2 to obtainthe results summarized in this paragraph. In parti ular, the analysis of the Hopfnormal form reveals the birth or death of an invariant ir le in a non-degenerateHopf bifur ation.Related phenomena an be observed in periodi ally for ed os illators. LetdX

dt= F(X) + G(t)be a periodi ally for ed system of di�erential equations with 2π-periodi for ing

G(t). Suppose that the autonomous system has a hyperboli equilibrium at Y0 = 0;That is, F(0) = 0. Then the for ed system has a 2π-periodi solution Y(t) withinitial ondition Y(0) = Y0 near 0. The dynami s of the for ed system near thepoint Y0 is studied using the strobos opi map P that maps the point X0 to thepoint X(2π), where X(t) is the solution to the for ed system with initial ondition2

X(0) = X0. Note that P(0) = 0 in oordinates entered at Y0. Again resonan e an o ur as a parameter is varied when the strobos opi map undergoes Hopfbifur ation with riti al eigenvalues equal to roots of unity. Resonan e tongues orrespond to regions in parameter spa e near the resonan e point where thestrobos opi map has q-periodi traje tories near 0. These q-periodi traje toriesare often alled subharmoni s of order q. Se tion 3.2 presents a Normal FormAlgorithm for su h periodi systems. The Van der Pol transformation is a tool forredu ing the analysis of subharmoni s of order q to the study of zero sets of Zq-equivariant polynomials. In this way we obtain the Zq-equivariant Takens NormalForm [37℄ of the Poin ar�e time-2π-map of the system. After this transformation,the �nal analysis of the resonan e tongues orresponding to the birth or death ofthese subharmoni s bears strong resemblan e to the approa h of Se tion 2.Coupled cell systems. Finally, in Se tion 4 we report on work in progress bypresenting a ase study, fo using on a feed-forward network of three oupled ells.Ea h ell satis�es the same dynami law, only di�erent hoi es of initial onditionsmay lead to di�erent kinds of dynami s for ea h ell. To ta kle su h systems, weshow how a ertain lass of dynami laws may give rise to time-evolutions thatare equilibria in ell 1, periodi in ells 2, and exhibting the Hopf-Ne��mark-Sa kerphenomenon in ell 3. This kind of dynami s of the third ell, whi h is revealedby applying the theory Normal Form theory presented in Se tion 3, o urs despitethe fa t that the dynami law of ea h individual ell is simple, and identi al forea h ell.1.2 Methodology: generic versus concrete systems.Analyzing bifur ations in generi families of systems requires di�erent tools thananalyzing a on rete family of systems and its bifur ations. Furthermore, if we areonly interested in restri ted aspe ts of the dynami s, like the emergen e of peri-odi orbits near �xed points of maps, simpler methods might do than in situationswhere we are looking for omplete dynami information, like normal linear behav-ior (stability), or the oexisten e of periodi , quasi-periodi and haoti dynami snear a Hopf-Ne��mark-Sa ker bifur ation. In general, the more demanding ontextrequires more powerful tools.This paper illustrates this `paradigm' in the ontext of lo al bifur ations ofve tor �elds and maps, orresponding to the o urren e of degenerate equilibriaor �xed points for ertain values of the parameter. These degenerate features areen oded by a semi-algebrai strati� ation of the spa e of jets (of some �xed order)of lo al ve tor �elds or maps. Ideally, ea h stratum is represented by a `simplemodel', or normal form, to whi h all other systems in the stratum an be redu edby a oordinate transformation (or, normal form transformation). These `simplemodels' are usually low-degree polynomial systems, equivalent to either the fullsystem or some jet of suÆ iently high order. Moreover, generi unfoldings of su hdegenerate systems also have simple polynomial normal forms. The guiding idea3

is that the interesting features of the system are mu h more easily extra ted fromthe normal form than from the original system.Singularity Theory provides us with algebrai algorithms that ompute su hsimple polynomial models for generi (families of) fun tions. In Se tion 2 weapply Equivariant Singularity Theory in this way to determine resonan e tongues orresponding to bifur ations of periodi orbits from �xed points of maps. How-ever, before Singularity Theory an be applied we have to use the Lyapunov-S hmidt method to redu e the study of bifur ating periodi orbits to the analysisof zero sets of equivariant families of fun tions on the plane. In this redu tion weloose all other information on the dynami s of the system.To over ome this restri tion, we apply Normal Form algorithms in the ontextof ows [4℄, [36℄ yielding simple models of generi families of ve tor �elds (possiblyup to terms of high order), without �rst redu ing the system a ording to theLyapunov-S hmidt approa h. Therefore, all dynami information is present inthe normal form. We follow this approa h to study the Hopf-Ne��mark-Sa kerphenomenon in on rete systems, like the feedforward network of oupled ells.1.3 Related workThe geometri omplexity of resonan e domains has been the subje t of manystudies of various s opes. Some of these, like the present paper, deal with quiteuniversal problems while others restri t to interesting examples. As opposed tothis paper, often normal form theory is used to obtain information about thenonlinear dynami s. In the present ontext the normal forms automati ally areZq-equivariant.Chenciner’s degenerate Hopf bifurcation. In [17, 18, 19℄ Chen iner onsid-ers a 2-parameter unfolding of a degenerate Hopf bifur ation. Strong resonan esto some �nite order are ex luded in the `rotation number' ω0 at the entral �xedpoint. In [19℄ for sequen es of `good' rationals pn/qn tending to ω0, orrespondingperiodi points are studied with the help of Zqn -equivariant normal form theory.For a further dis ussion of the odimension k Hopf bifur ation ompare Broer andRoussarie [12℄.The geometric program of Peckam et al. The resear h program re e ted in[30, 31, 33, 35℄ views resonan e `tongues' as proje tions on a `traditional' parameterplane of (saddle-node) bifur ation sets in the produ t of parameter and phasespa e. This approa h has the same spirit as ours and many interesting geometri properties of `resonan e tongues' are dis overed and explained in this way. Wenote that the earlier result [34℄ on higher order degenera ies in a period-doublinguses Z2 equivariant singularity theory.Parti ularly we like to mention the results of [35℄ on erning a lass of os- illators with doubly periodi for ing. It turns out that these systems an have oexisten e of periodi attra tors (of the same period), giving rise to `se ondary'4

saddle-node lines, sometimes en losing a ame-like shape. In the present, moreuniversal, approa h we �nd similar ompli ations of traditional resonan e tongues, ompare Figure 1 and its explanation in Se tion 2.4.Related work by Broer et al. In [15℄ an even smaller universe of annulus mapsis onsidered, with Arnold's family of ir le maps as a limit. Here `se ondary'phenomena are found that are similar to the ones dis ussed presently. Indeed,apart from extra saddle-node urves inside tongues also many other bifur ation urves are dete ted.We like to mention related results in the reversible and symple ti settingsregarding parametri resonan e with periodi and quasi-periodi for ing terms byAfsharnejad [1℄ and Broer et al. [5, 6, 8, 9, 10, 11, 13, 14, 16℄. Here the methodsuse Floquet theory, obtained by averaging, as a fun tion of the parameters.Singularity theory (with left-right equivalen es) is used in various ways. Firstof all it helps to understand the omplexity of resonan e tongues in the stabilitydiagram. It turns out that rossing tongue boundaries, whi h may give rise toinstability po kets, are related to Whitney folds as these o ur in 2D maps. Theseproblems already o ur in the linearized ase of Hill's equation. A question iswhether these phenomena an be re overed by methods as developed in the presentpaper. Finally, in the nonlinear ases, appli ation of Z2- and D2-equivariant sin-gularity theory helps to get dynami al information on normal forms.2 Bifurcation of periodic points of planar diffeomor-

phisms

2.1 Background and sketch of resultsThe types of resonan es mentioned here have been mu h studied; we refer to Tak-ens [37℄, Newhouse, Palis, and Takens [32℄, Arnold [2℄ and referen es therein. Formore re ent work on strong resonan e, see Krauskopf [29℄. In general, these worksstudy the omplete dynami s near resonan e, not just the shape of resonan etongues and their boundaries. Similar remarks an be made on studies in Hamil-tonian or reversible ontexts, su h as Broer and Vegter [16℄ or Vanderbauwhede[39℄. Like in our paper, in many of these referen es some form of singularity theoryis used as a tool.The problem we address is how to �nd resonan e tongues in the general setting,without being on erned by stability, further bifur ation and similar dynami alissues. It turns out that onta t equivalen e in the presen e of Zq symmetry is anappropriate tool for this, when �rst a Liapunov-S hmidt redu tion is utilized, seeGolubitsky, S hae�er, and Stewart [23, 25℄. The main question asks for the num-ber of q-periodi solutions as a fun tion of parameters, and ea h tongue boundarymarks a hange in this number. In the next subse tion we brie y des ribe howthis redu tion pro ess works. Using equivariant singularity theory we arrive atequivariant normal forms for the redu ed system in Se tion 2.3. It turns out that5

0.20.10

-0.4

-0.3

-0.2

-0.1

00.20.10

0.20.10

-0.3

-0.2

-0.1

0

-0.1

0.20.10

-0.3

-0.2

-0.1

0

-0.1

0 0.1 0.2 0.3

-0.6

-0.4

-0.2

0

0.2

0 0.1 0.2 0.3

Figure 1: Resonan e tongues with po ket- or ame-like phenonmena near a de-generate Hopf bifur ation through e2πip/q in a family depending on two omplexparameters. Fixing one of these parameters at various (three) values yields a fam-ily depending on one omplex parameter, with resonan e tongues ontained inthe plane of this se ond parameter. As the �rst parameter hanges, these tongueboundaries exhibit usps (middle pi ture), and even be ome dis onne ted (right-most pi ture). The small triangle in the rightmost pi ture en loses the region ofparameter values for whi h the system has four q-periodi orbits.the standard, nondegenerate ases of Hopf bifur ation [2, 37℄ an be easily re ov-ered by this method. When q ≥ 7 we are able to treat a degenerate ase, wherethe third order terms in the redu ed equations, the `Hopf oeÆ ients', vanish. We�nd po ket- or ame-like regions of four q-periodi orbits in addition to the re-gions with only zero or two, ompare Figure 1. In addition, the tongue boundaries ontain new usp points and in ertain ases the tongue region is blunter than inthe nondegenerate ase. These results are des ribed in detail in Se tion 2.4.2.2 Reduction to an equivariant bifurcation problemOur method for �nding resonan e tongues | and tongue boundaries | pro eedsas follows. Find the region in parameter spa e orresponding to points wherethe map P has a q-periodi orbit; that is, solve the equation Pq(x) = x. Usinga method due to Vanderbauwhede (see [39, 40℄), we an solve for su h orbitsby Liapunov-S hmidt redu tion. More pre isely, a q-periodi orbit onsists of qpoints x1, . . . , xq where

P(x1) = x2, . . . , P(xq−1) = xq, P(xq) = x1.Su h periodi traje tories are just zeroes of the mapP(x1, . . . , xq) = (P(x1) − x2, . . . , P(xq) − x1).6

Note that P(0) = 0, and that we an �nd all zeroes of P near the resonan e pointby solving the equation P(x) = 0 by Liapunov-S hmidt redu tion. Note also thatthe map P has Zq symmetry. More pre isely, de�neσ(x1, . . . , xq) = (x2, . . . , xq, x1).Then observe that

Pσ = σP.At 0, the Ja obian matrix of P has the blo k formJ =

A −I 0 0 · · · 0 0

0 A −I 0 · · · 0 0...0 0 0 0 · · · A −I

−I 0 0 0 · · · 0 A

where A = (dP)0. The matrix J automati ally ommutes with the symmetry σand hen e J an be blo k diagonalized using the isotypi omponents of irredu iblerepresentations of Zq. (An isotypi omponent is the sum of the Zq isomorphi representations. See [25℄ for details. In this instan e all al ulations an be doneexpli itly and in a straightforward manner.) Over the omplex numbers it ispossible to write these irredu ible representations expli itly. Let ω be a qth rootof unity. De�ne Vω to be the subspa e onsisting of ve tors[x]ω =

x

ωx...ωq−1x

.A short al ulation shows thatJ[x]ω = [(A − ωI)x]ω.Thus J has zero eigenvalues pre isely when A has qth roots of unity as eigenvalues.By assumption, A has just one su h pair of omplex onjugate qth roots of unityas eigenvalues.Sin e the kernel of J is two-dimensional | by the simple eigenvalue assumptionin the Hopf bifur ation | it follows using Liapunov-S hmidt redu tion that solvingthe equation P(x) = 0 near a resonan e point is equivalent to �nding the zerosof a redu ed map from R

2 → R2. We an, however, naturally identify R

2 withC, whi h we do. Thus we need to �nd the zeros of a smooth impli itly de�nedfun tion

g : C → C,where g(0) = 0 and (dg)0 = 0. Moreover, assuming that the Liapunov-S hmidtredu tion is done to respe t symmetry, the redu ed map g ommutes with the7

a tion of σ on the riti al eigenspa e. More pre isely, let ω be the riti al resonanteigenvalue of (dP)0; theng(ωz) = ωg(z). (1)Sin e p and q are oprime, ω generates the group Zq onsisting of all qth rootsof unity. So g is Zq-equivariant.We propose to use Zq-equivariant singularity theory to lassify resonan e tonguesand tongue boundaries.

2.3 Zq singularity theoryIn this se tion we develop normal forms for the simplest singularities of Zq-equivariant maps g of the form (1). To do this, we need to des ribe the formof Zq-equivariant maps, onta t equivalen e, and �nally the normal forms.The structure of Zq-equivariant maps. We begin by determining a uniqueform for the general Zq-equivariant polynomial mapping. By S hwarz's theorem[25℄ this representation is also valid for C∞ germs.Lemma 1 Every Zq-equivariant polynomial map g : C → C has the form

g(z) = K(u, v)z + L(u, v)zq−1,where u = zz, v = zq + zq, and K, L are uniquely de�ned omplex-valuedfun tion germs.Zq contact equivalences. Singularity theory approa hes the study of zeros of amapping near a singularity by implementing oordinate hanges that transform themapping to a `simple' normal form and then solving the normal form equation. Thekinds of transformations that preserve the zeros of a mapping are alled onta tequivalen es. More pre isely, two Zq-equivariant germs g and h are Zq- onta tequivalent if

h(z) = S(z)g(Z(z)),where Z(z) is a Zq-equivariant hange of oordinates and S(z) : C → C is a reallinear map for ea h z that satis�esS(γz)γ = γS(z)for all γ ∈ Zq.

Normal form theorems. In this se tion we onsider two lasses of normalforms | the odimension two standard for resonant Hopf bifur ation and one moredegenerate singularity that has a degenera y at ubi order. These singularitiesall satisfy the nondegenera y ondition L(0, 0) 6= 0; we explore this ase �rst.8

Theorem 2 Suppose thath(z) = K(u, v)z + L(u, v)zq−1where K(0, 0) = 0.1. Let q ≥ 5. If KuL(0, 0) 6= 0, then h is Zq onta t equivalent to

g(z) = |z|2z + zq−1with universal unfoldingG(z, σ) = (σ + |z|2)z + zq−1. (2)2. Let q ≥ 7. If Ku(0, 0) = 0 and Kuu(0, 0)L(0, 0) 6= 0, then h is Zq onta tequivalent to

g(z) = |z|4z + zq−1with universal unfoldingG(z, σ, τ) = (σ + τ|z|2 + |z|4)z + zq−1, (3)where σ, τ ∈ C.

Remark. Normal forms for the ases q = 3 and q = 4 are slightly di�erent.See [7℄ for details.2.4 Resonance domainsWe now ompute boundaries of resonan e domains orresponding to universalunfoldings of the form

G(z) = b(u)z + zq−1. (4)By de�nition, the tongue boundary is the set of parameter values where lo albifur ations in the number of period q points take pla e; and, typi ally, su hbifur ations will be saddle-node bifur ations. For universal unfoldings of the sim-plest singularities the boundaries of these parameter domains have been alledtongues, sin e the domains have the shape of a tongue, with its tip at the reso-nan e point. Below we show that our method easily re overs resonan e tongues inthe standard least degenerate ases. Then, we study a more degenerate singularityand show that the usual des ription of tongues needs to be broadened.Tongue boundaries of a p : q resonan e are determined by the following systemzG = 0det(dG) = 0.

(5)This follows from the fa t that lo al bifur ations of the period q orbits o ur atparameter values where the system G = 0 has a singularity, that is, where therank of dG is less than two. Re alling thatu = zz v = zq + zq w = i(zq − zq),we prove the following theorem, whi h is independent of the form of b(u).9

Theorem 3 For universal unfoldings (4), equations (5) have the form|b|2 = uq−2

bb′+ bb ′ = (q − 2)uq−3To begin, we dis uss weak resonan es q ≥ 5 in the nondegenerate ase orrespond-ing to the situation of Theorem 2, part 1. where a q

2−1 usp forms the tongue-tipand where the on ept of resonan e tongue remains un hallenged. Using Theo-rem 3, we re over several lassi al results on the geometry of resonan e tonguesin the present ontext of Hopf bifur ation. Note that similar tongues are found inthe Arnold family of ir le maps [2℄, also ompare Broer, Sim�o and Tatjer [15℄.We �nd some new phenomena in the ase of weak resonan es q ≥ 7 in themildly degenerate ase orresponding to the situation of Theorem 2, part 2. Herewe �nd `po kets' in parameter spa e orresponding to the o urren e of fourperiod-q orbits.

The nondegenerate singularity when q ≥ 5. We �rst investigate the non-degenerate ase q ≥ 5 given in 2. Hereb(u) = σ + uwhere σ = µ + iν. We shall ompute the tongue boundaries in the (µ, ν)-plane inthe parametri form µ = µ(u), ν = ν(u), where u ≥ 0 is a lo al real parameter.Short omputations show that

|b|2 = (µ + u)2 + ν2

bb′+ bb ′ = 2(µ + u).Then Theorem 3 gives us the following parametri representation of the tongueboundaries:

µ = −u +q − 2

2uq−3

ν2 = uq−2 −(q − 2)2

4u2(q−3).In this ase the tongue boundaries at (µ, ν) = (0, 0) meet in the familiar q−2

2 usp

ν2 ≈ (−µ)q−2.See also Figure 2. It is to this and similar situations that the usual notion ofresonan e tongue applies: inside the sharp tongue a pair of period q orbits existsand these orbits disappear in a saddle-node bifur ation at the boundary.10

PSfrag repla ements 2 0µ

ν

Figure 2: Resonan e tongue in the parameter plane. A pair of q-periodi orbitso urs for parameter values inside the tongue.Tongue boundaries in the degenerate case. The next step is to analyze amore degenerate ase, namely, the singularity

g(z) = u2z + zq−1,w hen q ≥ 7. We re all from 3 that a universal unfolding of g is given by G(z) =

b(u)z + zq−1, whereb(u) = σ + τu + u2.Here σ and τ are omplex parameters, whi h leads to a real 4-dimensional param-eter spa e. As before, we set σ = µ + iν and onsider how the tongue boundariesin the (µ, ν)-plane depend on the omplex parameter τ. In [7℄ we �nd an expli itparametri representation of the tongue boundaries in (σ, τ)-spa e for q = 7.Cross-se tions of these resonan e tongues of the form τ = τ0 are depi ted in Fig-ure 1 for several onstant values of τ0. A new ompli ation o urs in the tongueboundaries for ertain τ, namely, usp bifur ations o ur at isolated points of thefold (saddle-node) lines. The interplay of these usps is quite interesting and hal-lenges some of the traditional des riptions of resonan e tongues when q = 7 andpresumably for q ≥ 7.

3 Subharmonics in forced oscillatorsAs indi ated in the introdu tion, subharmoni s of order q (2qπ-periodi orbits) orrespond to q-periodi orbits of the Poin ar�e time-2π-map. However, sin e thePoin ar�e map is not known expli itly, applying the method of Se tion 2 dire tly isat best rather umbersome, if not ompletely infeasible in most ases, espe iallysin e we are after a method for omputing resonan e tongues in on rete systems.Therefore we follow an other, more dire t approa h by introdu ing a Normal FormAlgorithm for time dependent periodi ve tor �elds. This method is expli it, andin prin iple omputes a Normal Form up to any order.11

3.1 A Normal Form AlgorithmFirst we present the Normal Form Algorithm in the ontext of autonomous ve -tor �elds. Our approa h is an extension of the well-known methods introdu edin [36℄, and aimed at the derivation of an implementable algorithm. This pro e-dure transforms the terms of the ve tor �eld in `as simple a form as possible', upto a user-de�ned order. It does so via iteration with respe t to the total degreeof these terms. In on rete systems we determine this normal form exa tly, i.e.,making the dependen e on the oeÆ ients of the input system expli it. To thisend we have to ompute the transformed system expli itly up to the desired order.The method of Lie series turns out to be a powerful tool in this ontext. We �rstpresent the key property of the Lie series approa h, that allows us to omputatethe transformed system in a rather straightforward way, up to any desired order.Lie series expansion. For nonnegative integers m we denote the spa e of ve tor�elds of total degree m by Hm, and the spa e of ve tor �elds with vanishingderivatives up to and in luding order m at 0 ∈ C by Fm. Note that Fm =∏

k≥mHk.Proposition 4 Let X and Y be ve tor �elds on C, where X is of the form

X = X(1) + X(2) + · · · + X(N) modFN+1, (6)with X(n) ∈ Hn, and Y ∈ Hm, with m ≥ 2. Let Yt, t ∈ R, be the one-parametergroup generated by Y, and let Xt = (Yt)∗(X). ThenXt = X +

⌊ N−1m−1

⌋∑

k=1

(−1)k

k!tk ad(Y)k(X) modFN+1 (7)

= X +

N∑

n=1

⌊N−nm−1

⌋∑

k=1

(−1)k

k!tk ad(Y)k(X(n)) modFN+1. (8)Proof. We follow the approa h of Takens [36℄ and Broer et.al. [3, 4℄ to obtainthe Taylor series of Xt with respe t to t in t = 0 using the basi identity

∂

∂tXt = [Xt, Y] = − ad(Y)(Xt).Using this relation, we indu tively prove that:∂k

∂tkXt = (−1)k ad(Y)k(Xt).Using the latter identity for t = 0, we obtain the formal Taylor series

Xt =∑

k≥0

1

k!tk ∂k

∂tk

∣∣∣∣t=0

Xt

=∑

k≥0

(−1)k

k!tk ad(Y)k(X).12

Sin e Y ∈ Hm, the operator ad(Y)k in reases the degree of ea h term in its ar-gument by k(m − 1). Sin e the terms of lowest order in X are linear, we seethat ad(Y)k(X) = 0 modFN+1,if 1 + k(m − 1) > N. Therefore,Xt =

⌊ N−1m−1

⌋∑

k=0

(−1)k

k!tk ad(Y)k(X) modFN+1

= X +

⌊ N−1m−1

⌋∑

k=1

(−1)k

k!tk ad(Y)k(X) modFN+1,whi h proves (7). In view of (6) the latter identity expands to

Xt =

N∑

n=1

⌊ N−1m−1

⌋∑

k=0

(−1)k

k!tk ad(Y)k(X(n)) modFN+1. (9)Sin e ad(Y)k(X(n)) ∈ Hn+k(m−1), we see thatad(Y)k(X(n)) = 0 modFN+1,for k > N−n

m−1. Therefore, for �xed index n, the inner sum in (9) an be trun atedat k = ⌊N−n

m−1⌋, whi h on ludes the proof of (8). �

The Normal Form Algorithm. Consider a ve tor �eld X having a singularpoint with semisimple linear part S. Our goal is to design an iterative algorithmbringing X into normal form, to some pres ribed order N.Lemma 5 (Normal Form Lemma [36])The ve tor �eld X an be brought into the normal form

X = S + G(2) + · · · + G(m) modFm+1,for any m ≥ 2, where G(i) ∈ Hi belongs to Ker ad(S).Proof. Assume that X is of the formX = S + G(2) + · · · + G(m−1) + X(m) modFm+1, (10)where X(m) ∈ Hm, and G(i) ∈ Hi belongs to Ker ad(S). If Y ∈ Hm and Xt =

(Yt)∗(X), thenXt = X − t ad(Y)(X(1)) modFm+1. (11)This is a dire t onsequen e of Proposition 4. See also Takens [36℄ and Broeret.al. [3, 4℄. Sin e S is semisimple, we know that

Hm = Ker ad(S) + Imad(S),13

so we write X(m) = G(m) + B(m), where G(m) ∈ Ker ad(S) and B(m) ∈ Imad(S). Ifthe ve tor �eld Y satis�es the homologi al equationad(S)(Y) = −B(m), (12)it follows from (10) and (11) that X1 is in normal form to order m, sin eX1 = S + G(2) + · · · + G(m−1) + G(m) modFm+1.

�Our �nal goal, namely bringing X into normal form to order N, is a hieved byrepeating this algorithmi step with X repla ed by the transformed ve tor �eldX1, bringing the latter ve tor �eld into normal form to order m + 1. Sin e thehomologi al equation involves the homogeneous terms of X1 of order m+1, we useidentity (8) to ompute these terms. Furthermore, we enfor e uniqueness of thesolution Y of (12) by imposing the ondition Y ∈ Im ad(S). However, omputingjust the homogeneous terms of X1 of order m + 1 is not suÆ ient if m + 1 < N,sin e subsequent steps of the algorithm a ess the terms of even higher order inthe transformed ve tor �eld. Therefore, we use (8) to ompute these higher orderterms.These steps are then repeated until the �nal transformed ve tor �eld is innormal form to order N + 1. This pro edure is expressed more pre isely in thenormal form algorithm in Figure 3.Algorithm (Normal Form Algorithm)

Input: N, S, X[2..N], satisfying1. S is a semisimple linear ve tor �eld2. X = S + X[2] + · · · + X[N] modFN+1,with X[n] ∈ Hn

(∗ X is in normal form to order 1 ∗)

for m = 2 to N do

(∗ bring X into normal form to order m ∗)determine G ∈ Ker ad(S) ∩Hm and B ∈ Imad(S) ∩Hm su h thatX[m] = G + Bdetermine Y, with Y ∈ Im ad(S) ∩Hm, su h thatad(S)(Y) = −B

(∗ ompute terms of order m + 1, . . . ,N of transformed ve tor �eld ∗)for n = 1 to N do

for k = 1 to ⌊N−nm−1

⌋ do

X[n + k(m − 1)] := X[n + k(m − 1)] +(−1)k

k!ad(Y)k(X[n])Figure 3: The Normal Form Algorithm.14

3.2 Applications of the Normal Form AlgorithmThe Hopf bifur ation o urs in one-parameter families of planar ve tor �elds hav-ing a nonhyperboli equilibrium with a pair of pure imaginary eigenvalues withnonzero imaginary part. In this bifur ation a limit y le emerges from the equilib-rium as the parameters of the system push the eigenvalues o� the imaginary axis.See also Figure 4.In this ontext it is easier to express the system in oordinates z, z on the omplex plane. The linear part of the ve tor �eld at the point of bifur ation isthen _z = iωz. To apply the Normal Form Algorithm, we �rst derive an expressionfor the Lie bra kets of real ve tor �elds with in these oordinates.The Lie-subalgebra of real vector fields. We identify R

2 with C, by asso i-ating the point (x1, x2) in R2 with x1 + ix2 in C. The real ve tor �eld X, de�nedon R

2 byX = Y1

∂

∂x1

+ Y2∂

∂x2

, orresponds to the ve tor �eldX = Y

∂

∂z+ Y

∂

∂z, (13)on C, where Y = Y1 + iY2.

Example. Taking Y(z, z) = czk+1zk, with c a omplex onstant, the ve tor �eldX given by (13) is SO(2)-equivariant. Writing c = a + ib, with a, b ∈ R, andz = x1 + ix2, we obtain its real form via a straightforward omputation:

X = (x21 + x2

2)k(a(x1

∂

∂x1

+ x2

∂

∂x2

) + b(−x2

∂

∂x1

+ x1

∂

∂x2

)).In parti ular, the real ve tor �eld ωN(−x2

∂

∂x1

+x1

∂

∂x2

) orresponds to the omplexve tor �eld S = iωN(z∂

∂z− z

∂

∂z).We denote the ∂

∂z- omponent of a real ve tor �eld X by XR, so:

X = XR

∂

∂z+ XR

∂

∂z.The real ve tor �elds form a Lie-subalgebra of the algebra of all ve tor �elds on

C. The following result justi�es this laim.Lemma 6 Let X and Y be real ve tor �elds on C, and let f : C → C be asmooth fun tion. Then

X(f) = X(f), (14)and[X, Y] = 〈X, Y〉 ∂

∂z+ 〈X, Y〉 ∂

∂z,where the bilinear antisymmetri form 〈·, ·〉 is de�ned by

〈X, Y〉 = X(YR) − Y(XR).15

Derivation of the Hopf Normal Form. To derive the Hopf Normal Form, weapply the Normal Form Algorithm to a ve tor �eld with linear partS = iωN(z

∂

∂z− z

∂

∂z).The adjoint a tion of S on the Lie-subalgebra of real ve tor �elds is given by:ad(S)(X) = 〈S,X〉 ∂

∂z+ 〈S,X〉 ∂

∂z,with

〈S,X〉 = iωN(z∂XR

∂z− z

∂XR

∂z− XR).In parti ular, if Y = YR

∂

∂z+ YR

∂

∂zwith YR = zkzl, then

〈S, Y〉 = iωN (k − l − 1) zkzl.Therefore, the adjoint operator ad(S) : Hm → Hm has non-trivial kernel for modd. If m = 2k + 1, this kernel onsists of the monomial ve tor �eld Y withYR = z|z|2k. These observations lead to the following Normal Form.Corollary 7 If a ve tor �eld on C has linear part S = iωN(z

∂

∂z− z

∂

∂z), thenthe Normal Form Algorithm brings this ve tor �eld into the form_z = iωz +

m∑

k=1

ckz|z|2k + O(|z|2m+3). (15)The nondegenerate Hopf bifurcation. An other appli ation of the NormalForm Algorithm is the omputation of the �rst Hopf oeÆ ient c1 in (15). Let Xbe given by_z = iωz + a0z

2 + a1zz + a2z2 + b0z

3 + b1z2z + b2zz

2 + b3z3 + O(|z|4).The Normal Form Algorithm omputes the following normal form for this system:_z = iωz +

(b1 −

i

3ω(3a0a1 − 3|a1|

2 − 2|a2|2)

)z2z + O(|z|4).This result an also be obtained by a tedious al ulation. See, for example, [26,page 155℄1To analyze the emergen e of limit y les we rewrite the Hopf Normal Form_w = iωNw + wb(|w|2, µ) + O(n + 1)in polar oordinates as:_r = r Reb(r2, µ) + O(n + 1)_ϕ = ωN + Imb(r2, µ) + O(n + 1)1The term |hww |2 in identity (3.4.26) of [26, page 155℄ should be repla ed by |hww |2.16

Limit y les are obtained by solving r = r(µ) from the equation Reb(r2, µ) = 0.The frequen y of the limit y le is then of the form ω(µ) = ωN + Imb(r(µ)2, µ).A non-degenerate Hopf bifur ation o urs if the �rst Hopf oeÆ ient c1 in (15)is nonzero. Consider, e.g., the simple ase b(u, µ) = µ + u. Putting µ = a + iδ wesee that the limit y le orresponds to the traje tory wwa,δ(t) =

√−aei(ωN+δ)t (a ≤ 0)This limit y le exists for a < 0, and is repulsive in this ase.

Figure 4: Birth or death of a limit y le via a Hopf bifur ation.Hopf-Neımark-Sacker bifurcations in forced oscillators. We now studythe birth or death of subharmoni s in for ed os illators depending on parameters.In parti ular, we onsider 2π-periodi systems on C of the form_z = F(z, z, µ) + εG(z, z, t, µ), (16)obtained from an autonomous system by a small 2π-periodi perturbation. Hereε is a real perturbation parameter, and µ ∈ R

k is an additional k-dimensionalparameter. Subharmoni s of order q may appear or disappear upon variation ofthe parameters if the linear part of F at z = 0 satis�es a p : q-resonan e onditionwhi h is appropriately detuned upon variation of the parameter µ.The Normal Form Algorithm of Se tion 3.1 an be adapted to the derivation ofthe Hopf-Ne��mark-Sa ker Normal form of periodi systems. Consider a 2π-periodi for ed os illator on C of the form_z = XR(z, z, t, µ),whereXR(z, z, t, µ) = iωNz + (α + iδ)z + zP(z, z, µ) + εQ(z, z, t, µ). (17)Here µ ∈ R

k, and ε is a small real parameter. Furthermore we assume that Pand Q ontain no terms that are independent of z and z (i.e., P(0, 0, µ) = 0 andQ(0, 0, t, µ) = 0), and that Q does not even ontain terms that are linear in zand z. Any system of the form (16) with linear part _z = iωNz an be broughtinto this form after a straightforward initial transformation. See [3℄ for details.Subharmoni s of order q are to be expe ted if the linear part satis�es a p : q-resonan e ondition, in other words, if the normal frequen y ωN is equal to p

q(with p and q relatively prime). 17

Theorem 8 (Normal Form to order q)The system (17) has normal form_z = iωNz + (α + iδ)z + zF(|z|2, µ) + d ε zq−1 eipt + O(q + 1), (18)where F(|z|2, µ) is a omplex polynomial of degree q − 1 with F(0, µ) = 0, andd is a omplex onstant.Proof. To derive a normal form for the system (17) we onsider 2π-periodi ve tor �elds on C × R/(2πZ) of the form

X = XR

∂

∂z+ XR

∂

∂z+

∂

∂t,with `linear' part

S = iωN(z∂

∂z− z

∂

∂z) +

∂

∂t.For nonnegative integers m we denote the spa e of 2π-periodi ve tor �elds oftotal degree m in (z, z, µ) with vanishing ∂

∂t- omponent by Hm. As before Fm =

∏k≥mHk.The adjoint a tion of S on the Lie-subalgebra of real 2π-periodi ve tor �eldswith zero ∂

∂t- omponent is given by:ad(S)(X) = 〈S,X〉R

∂

∂z+ 〈S,X〉R

∂

∂z,with

〈S,X〉R = iωN(z∂XR

∂z− z

∂XR

∂z− XR) +

∂XR

∂t.If XR = µσzkzleimt, with |σ| + k + l = n, then

〈S,X〉R = (iωN(k − l − 1) + im)XR.Therefore, the normal form ontains time-independent rotationally symmetri terms orresponding to k = l + 1 and m = 0. Sin e for ε = 0 the system isin Hopf Normal Form, all non-rotationally symmetri terms ontain a fa tor ε, so|σ| > 0 for these terms. It is not hard to see that for n ≤ q and |σ| > 0 the onlynon-rotationally symmetri term orresponds to k = 0, l = q − 1, m = p, and|σ| = 1. Therefore, this non-symmetri term is of the form

d ε zq−1 eipt,for some omplex onstant d. �

18

3.3 Via covering spaces to the Takens Normal Form

Existence of 2πq-periodic orbits. The Van der Pol transformation. Sub-harmoni s of order q of the 2π-periodi for ed os illator (17) orrespond to q-periodi orbits of the Poin ar�e time 2π-map P : C → C. These periodi orbitsof the Poin ar�e map are brought into one-one orresponden e with the zeros ofa ve tor �eld on a q-sheeted over of the phase spa e C × R/(2πZ) via the Vander Pol transformation, f [16℄. This transformation orresponds to a q-sheeted overingΠ : C × R/(2πqZ) → C × R/(2πZ),

(z, t) 7→ (zeitp/q, t (mod 2πZ)) (19)with y li De k group of order q generated by(z, t) 7→ (ze2πip/q, t − 2π).The Van der Pol transformation ζ = ze−iωNt lifts the for ed os illator (16) to thesystem_ζ = (α + iδ)ζ + ζP(ζeiωNt, ζe−iωNt, µ) + εQ(ζeiωNt, ζe−iωNt, t, µ) (20)on the overing spa e C × R/(2πqZ). The latter system is Zq-equivariant. Astraightforward appli ation of (20) to the normal form (18) yields the followingnormal form for the lifted for ed os illator.

Theorem 9 (Equivariant Normal Form of order q)On the overing spa e, the lifted for ed os illator has the Zq-equivariant nor-mal form: _ζ = (α + iδ)ζ + ζF(|ζ|2, µ) + d ε ζq−1

+ O(q + 1), (21)where the O(q + 1) terms are 2πq-periodi .Resonance tongues for families of forced oscillators. Bifur ations of q-periodi orbits of the Poin ar�e map P on the base spa e orrespond to bifur ationsof �xed points of the Poin ar�e map ~P on the q-sheeted overing spa e introdu edin onne tion with the Van der Pol transformation (19). Denoting the normalform system (18) on the base spa e by N , and the normal form system (21) of thelifted for ed os illator by ~N , we see that

Π∗~N = N .The Poin ar�e mapping ~P of the normal form on the overing spa e now is the

2πq-period mapping ~P = ~N 2πq + O(q + 1),where ~N 2πq denotes the 2πq-map of the (planar) ve tor �eld ~N . Following theCorollary to the Normal Form Theorem of [16, page 12℄, we on lude for theoriginal Poin ar�e map P of the ve tor �eld X on the base spa e thatP = R2πωN

◦ ~N 2π + O(q + 1),19

where R2πωNis the rotation over 2πωN = 2πp/q, whi h pre isely is the TakensNormal Form of P at (z, µ) = (0, 0), see [37℄.Our interest is with the q-periodi points of Pµ, whi h orrespond to the �xedpoints of ~Pµ. This �xed point set and the boundary thereof in the parameter spa e

R3 = {a, δ, ε} is approximately des ribed by the dis riminant set of

(a + iδ)ζ + ζ~F(|ζ|2, µ) + εdζq−1

,whi h is the trun ated right hand side of (21). This gives rise to the bifur ationequation that determine the boundaries of the resonan e tongues. The followingtheorem implies that, under the onditions that d 6= 0 6= Fu(0, 0), the order oftangen y at the tongue tips is generi . Here Fu(0, 0) is the partial derivative ofF(u, µ) with respe t to u.Theorem 10 (Bifurcation equations modulo contact equivalence)Assume that d 6= 0 and Fu(0, 0) 6= 0. Then the polynomial (21) is Zq-equivariantly onta t equivalent with the polynomial

G(ζ, µ) = (a + iδ + |ζ|2)ζ + εζq−1

. (22)The dis riminant set of the polynomial G(ζ, µ) is of the formδ = ±ε(−a)(q−2)/2 + O(ε2). (23)Proof. The polynomial (22) is a universal unfolding of the germ |ζ|2ζ + εζ

q−1under Zq onta t equivalen e. See [7℄ for a detailed omputation. The tongueboundaries of a p : q resonan e are given by the bifur ation equationsG(ζ, µ) = 0,det(dG)(ζ, µ) = 0.As in [7, 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. A ording to (the proof of) [7, Theorem 3.1℄, the system ofbifur ation equations is equivalent to

|b|2 = ε2uq−2,

bb′+ bb ′ = (q − 2)ε2uq−3,where b ′ =

∂b

∂u(u, µ). A short omputation redu es the latter system to theequivalent

(a + u)2 + δ2 = ε2uq−2,

a + u =1

2(q − 2)ε2uq−3.Eliminating u from this system of equations yields expression (23) for the tongueboundaries. �20

PSfrag repla ementsa

εε

δ PSfrag repla ements aa

εε

δ

Figure 5: Resonan e zones for for ed os illator families: the Hopf-Ne��mark-Sa kerphenomenon.The dis riminant set of the equivariant polynomial (22) forms the boundary ofthe resonan e tongues. See Figure 5. At this surfa e we expe t the Hopf-Ne��mark-Sa ker bifur ation to o ur; here the Floquet exponents of the linear part of thefor ed os illator ross the omplex unit ir le. This bifur ation gives rise to aninvariant 2-torus in the 3D phase spa e C×R/(2πZ). Resonan es o ur when theeigenvalues ross the unit ir le at roots of unity e2πip/q. `Inside' the tongue the2-torus is phase-lo ked to subharmoni periodi solutions of order q.4 Generic Hopf-Neımark-Sacker bifurcations in feed

forward systems?

Coupled Cell Systems. A oupled ell system is a network of dynami al sys-tems, or ells, oupled together. This network is a �nite dire ted graph withnodes representing ells and edges representing ouplings between these ells. See,e.g., [22℄.We onsider the three- ell feed-forward network in Figure 6, where the �rst ell is oupled externally to itself. The network has the form of a oupled ellPSfrag repla ements1 2 3Figure 6: Three- ell linear feed-forward networksystem _x1 = f(x1, x1)_x2 = f(x2, x1)_x3 = f(x3, x2)with xj ∈ R

2.Under ertain onditions these networks have time-evolutions that are equilib-ria in ell 1 and periodi in ells 2 and 3. In [20℄ Elmhirst and Golubitsky des ribe21

a urious phenomenon: the amplitude growth of the periodi signal in ell 3 is tothe power 16rather than to the power 1

2with respe t to the bifur ation parameterin the Hopf bifur ation. See also Se tion 3.2.For te hni al reasons we assume that the fun tion f, des ribing the dynami sof ea h ell, is S

1-symmetri in the sense thatf(eiθz2, e

iθz1) = eiθf(z2, z1), (24)for all real θ. Here we identify the two-dimensional phase spa e of ea h ell withC by writing zj = xj1+ ixj2. Identity (24) is a spe ial assumption, that we will tryto relax in future resear h. However, in [20℄ Elmhirst and Golubitsky verify thatthis symmetry ondition holds to third order after a hange of oordinates. Wealso assume that the dynami s of ea h ell depends on external parameters λ, µ,to be spe i�ed later on.Dynamics of the first and second cell. The S

1-symmetry (24) implies thatfλ,µ(0, 0) = 0. Note that from now on we make the dependen e of f on theparameters expli it in our notation. Assume that the linear part of fλ,µ(z1, z1) atz1 = 0 has eigenvalues with negative real part. Then the �rst ell has a stableequilibrium at z1 = 0.The se ond ell has dynami s_z2 = fλ,µ(z2, z1) = fλ,µ(z2, 0),where we use that the �rst ell is in its stable equilibrium. The paper [24℄ intro-du es a large lass of fun tions fλ,µ for whi h the se ond ell undergoes a Hopfbifur ation. For this lass of ell dynami s, and for linear feed-forward networksof in reasing length, there will be `repeated Hopf' bifur ation, reminis ent of thes enarios named after Landau-Lifs hitz and Ruelle-Takens.To obtain more pre ise information on the Hopf bifur ation in the dynami sof the se ond ell we onsider a spe ial lass of fun tions fλ,µ satisfying (24). Inparti ular, we require that

fλ,µ(z2, 0) = (λ + i − |z2|2) z2, (25)giving a super riti al Hopf bifur ation at λ = 0. The stable periodi solution,o urring for λ > 0, has the form

z2(t) =√

λeit.

Dynamics of the third cell. The main topi of our resear h is the generi dynami s of the third ell, given simple time-evolutions of the �rst two ells. Herewe like to know what are the orresponden es and di�eren es with the generalODE setting. In parti ular this question regards the oexisten e of periodi , quasi-periodi and haoti dynami s. 22

In o-rotating oordinates the dynami s of the third ell be omes time-independent.To see this, set z3 = eity, and use the S1-symmetry to derive

ieity + eit _y = = fλ,µ(eity,√

λ eit)

= eit fλ,µ(y,√

λ).Therefore, the dynami s of the third ell is given by_y = −iy + fλ,µ(y,√

λ). (26)Equation (26) is autonomous, so the present setting might exhibit Hopf bifur a-tions, but it is still too simple to produ e resonan e tongues. Indeed, all (relative)periodi motion in (26) will lead to parallel (quasi-periodi ) dynami s and theHopf-Ne��mark-Sa ker phenomenon. Therefore, we now perturb the basi fun tionf = fλ,µ(z2, z1), to

Fλ,µ,ε(z2, z1) := fλ,µ(z2, z1) + εP(z2, z1).In ells 1 and 2 any hoi e of the perturbation term P(z2, z1) gives the dynami s_z1 = Fλ,µ,ε(z1, z1) = fλ,µ(z1, z1) + εP(z1, z1),_z2 = Fλ,µ,ε(z2, 0),with the same on lusions as before, namely a steady state z1 = 0 in ell 1 and aperiodi state z2 =√

λeit in ell 2 (when λ > 0). For these two on lusions it issuÆ ient thatP(z2, 0) ≡ 0.Turning to the third ell we again put y = e−itz3, and so get a perturbed redu edequation _y = −iy + fλ,µ(y,√

λ) + εe−itP(y eit,√

λ eit). (27)The third ell therefore has for ed os illator dynami s with driving frequen y 1.The question about generi dynami s regards the possible oexisten e of periodi and quasi-periodi dynami s. We aim to investigate (27) for Hopf-Ne��mark-Sa kerbifur ations, whi h are expe ted along urves Hε in the (λ, µ)-plane of parameters.We expe t to �nd periodi tongues (See also Figure 5) and quasiperiodi hairs,like in [15℄. This is the subje t of ongoing resear h. The ma hinery of Se tion 3.2should provide us with suÆ iently powerful tools to investigate this phenomenonfor a large lass of oupled ell systems.5 Conclusion and future workWe have presented several ontexts in whi h bifur ations from �xed points of mapsor equilibria of ve tor �elds lead to the emergen e of periodi orbits. For ea h ontext we present appropriate normal form te hniques, illustrating the generalparadigm of `simplifying the system before analyzing it'. In the ontext of generi 23

families we apply generi te hniques, based on Lyapunov-S hmidt redu tion andZq-equivariant onta t equivalen e. In this way we re over standard results onresonan e tongues for nondegenerate maps, but also dis over new phenomena inunfoldings of mildly degenerate systems. Furthermore, we present an algorithm forbringing on rete families of dynami al systems into normal form, without losinginformation in a preliminary redu tion step, like the Lyapunov-S hmidt method.An example of su h a on rete system is a lass of feedforward networks of oupled ell systems, in whi h we expe t the Hopf-Ne��mark-Sa ker-phenomenon to o ur.With regard to further resear h, our methods an be extended to other on-texts, in parti ular, to ases where extra symmetries, in luding time reversibility,are present. This holds both for Lyapunov-S hmidt redu tion and Zq equivariantsingularity theory. In this respe t Golubitsky, Marsden, Stewart, and Dellnitz [21℄,Knoblo h and Vanderbauwhede [27, 28℄, and Vanderbauwhede [38℄ are helpful.Furthermore, there is the issue of how to apply our results to a on rete familyof dynami al systems. Golubitsky and S hae�er [23℄ des ribe methods for obtain-ing the Taylor expansion of the redu ed fun tion g(z) in terms of the Poin ar�emap P and its derivatives. These methods may be easier to apply if the systemis a periodi ally for ed se ond order di�erential equation, in whi h ase the om-putations again may utilize parameter dependent Floquet theory. We also plan toturn the Singularity Theory methods of Se tion 2 into e�e tive algorithms, alongthe lines of our earlier work [9℄.Finally, in this paper we have studied only degenera ies in tongue boundaries.It would also be interesting to study low odimension degenera ies in the dynami sasso iated to the resonan e tongues. Su h a study will require tools that are moresophisti ated than the singularity theory ones that we have onsidered here.References[1℄ Z. Afsharnejad. Bifur ation geometry of mathieu's equation. Indian J. PureAppl. Math., 17:1284{1308, 1986.[2℄ V.I. Arnold. Geometri al Methods in the Theory of Ordinary Di�erentialEquations. Springer-Verlag, 1982.[3℄ B.L.J. Braaksma, H.W. Broer, and G.B. Huitema. Toward a quasi-periodi bifur ation theory. In Mem. AMS, volume 83, pages 83{175. 1990.[4℄ H.W. Broer. Formal normal form theorems for ve tor �elds and some onse-quen es for bifur ations in the volume preserving ase. In Dynami al Sys-tems and Turbulen e, volume 898 of LNM, pages 54{74. Springer-Verlag,1980.[5℄ H.W. Broer, S.-N. Chow, Y. Kim, and G. Vegter. normally ellipti hamilto-nian bifur ation. ZAMP, 44:389{432, 1993.24

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