Dynamical Systems/Ordinary Differential Equations

Dynamical Systems/Ordinary Differential Equations

Birth control in generalized Hopf bifurcations

Marc Chaperon

Institut de mathématiques de Jussieu & Université Paris 7, UFR de mathématiques, case 7012, 2, place Jussieu, 75251 Paris cedex 05, France Received 20 August 2007; accepted after revision 17 September 2007

Available online 22 October 2007 Presented by by Étienne Ghys

Abstract

We state a very general lemma ensuring, at partially elliptic rest points of families of vector fields or transformations, the birth of normally hyperbolic invariant compact manifolds. A few examples follow.To cite this article: M. Chaperon, C. R. Acad. Sci.

Paris, Ser. I 345 (2007).

©2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Contrôle des naissances dans les bifurcations de Hopf généralisées. Nous énonçons un lemme très général garantissant, aux points stationnaires partiellement elliptiques de familles de champs de vecteurs ou de transformations, la naissance de variétés compactes invariantes normalement hyperboliques. Quelques exemples suivent.Pour citer cet article : M. Chaperon, C. R. Acad.

Sci. Paris, Ser. I 345 (2007).

©2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Version française abrégée

Cette Note annonce un travail [2] fondé sur la ferme conviction que, dans le couplage d’oscillateurs, apparaissent d’autres phénomènes intéressants que des mouvements quasi-périodiques.

Hypothèses dans le cas des champs de vecteurs

Soit(u, x)→Z_u(x)∈R^mune famille assez différentiable de champs de vecteurs à paramètreu∈R^k, définie au voisinage d’un point de R^k×R^mque l’on peut supposer être 0, telle queZ₀(0)=0 et que les valeurs propres de DZ0(0)soient imaginaires pures, simples et différentes de 0, d’oùm=2n.

Les valeurs propres de partie imaginaire positive de DZ0(0) étant notées iβ1, . . . ,iβn, on suppose que, pour 1jn, l’équationβ_j =n

1(p−q)β avec(p, q)∈(Nⁿ)²et

(p+q)4 n’a que les solutions évidentes pj=qj+1 etp=qpour=j.

Un changement(u, x)→(u, gu(x)) de coordonnées locales permet de supposer queZu(0)≡0, queR^m=Cⁿ, queDZ0(0)est de la formez→(iβ1z1, . . . ,iβnzn)et queZu=Nu+Ru, oùRu s’annule à l’ordre 4 en 0 etNuest

E-mail address:chaperon@math.jussieu.fr.

1631-073X/$ – see front matter ©2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crma.2007.09.016

donnée par(1). Dans cette formule,z=(z1, . . . , zn)∈Cⁿetλj, μj, aj , bj sont des fonctions réelles différentiables telles queλ_j(0)=0 etμ_j(0)=β_j. Il en résulte que l’applicationρdeCⁿsurRⁿ₊définie parρ(z)=(|z₁|, . . . ,|z_n|) envoieN_usur

j(λ_j(u)−

a_j (u)r²)r_j∂r^∂

j.

À chaque v∈R^k on associe le champ de vecteurs X_v(x)=

jx_j(Dλ_j(0)v−

a_j (0)x²)_∂x^∂

j surRⁿ. Il est invariant sous l’action deO(1)ⁿengendrée par les symétries par rapport aux hyperplans de coordonnées. On suppose v₀∈S^k−1 tel queX_v0(x)admette une variété invariante compacte Σ_v0 normalement hyperbolique [4,6], invariante parO(1)ⁿet dont l’intersection avec l’orthant positifRⁿ₊est connexe, d’où la propriété suivante :

(P) SiR^J,J⊂ {1, . . . , n}, est le plus petit sous-espace de coordonnées contenantΣ_v0, alorsΣ_v0est transversedans R^J aux sous-espaces de coordonnées contenus dansR^J.

Par stabilité de l’hyperbolicité normale, il existe un voisinage ouvertV_v0 dev₀dans S^k−1tel que chaqueX_v avec v∈V_v0 ait une variété invariante normalement hyperboliqueΣ_vdifféomorphe à Σ_v0 et proche de celle-ci, unique, doncO(1)ⁿ-invariante et possédant la propriété(P )avec le mêmeJ. Il en résulte queS0,v:=ρ⁻¹(Σv)⊂C^J est une sous-variété compacteU(1)ⁿ-invariante, de mêmes différentiabilité et codimension queΣv.

Théorème 1(lemme de naissance). Sous ces hypothèses, il existe un ouvertU_v0 de R^k adhérent à 0dont le cône tangent¹ en0est un cône ouvert de sommet0contenantR^∗₊V_v0, tel que chacun des Z_u avecu∈U_v0 admette une variété invariante compacte normalement hyperboliqueS_udifféomorphe àS_0,v0, de mêmes indice et co-indice que la variété invarianteΣ_v0 deX_v0, dépendant de manière au moinsC^1+α deuet tendant vers{0}quandu→0. Plus précisément, pour tout chemin lisseγ:(R₊,0)→(R^k,0)vérifiantγ (0)˙ ∈V_v0, on aγ (ε)∈U_v0pourε >0assez petit etlimε→0ε^−1/2S_{γ (ε)}=S_{0,γ (0)˙} au sens au moinsC¹.

Exemple 0.SiJ= ∅et doncΣ_v0= {0}, l’hypothèse signifie que lesDλ_j(0)v0sont tous non nuls et lesS_u= {0}sont des zéros hyperboliques.

Exemple 1 (bifurcations de Hopf). Pour J = {}, l’hypothèse signifie que l’on a Dλ_j(0)v0=0 pour j =, a(0)Dλ(0)v0>0, Σ_v0 = {x∈R^J: |x| =

Dλ(0)v0

a(0) }etS_0,v0 = {z∈C^J: |z| =

Dλ(0)v0

a(0) }. Les S_u sont donc des orbites périodiques et l’on obtient une version un peu faible de la bifurcation de Hopf.

Exemple 2 (naissance de tores invariants de dimensiond n).Pour #J =d >1, l’hypothèse est vérifiée avec dimΣv₀ =0 si et seulement si l’on a les trois conditions suivantes : la matriceA=(aj )j,∈J est inversible, le pointy_v0 :=A⁻¹(Dλ_j(0)v0)_j∈J ∈R^J est à coordonnées strictement positives et le zérox_v0 ∈R^J deX_v0 défini par x_v0j= √y_v0j,j∈J, est hyperbolique. Dans ce cas,Σ_v0 =O(1)ⁿx_v0 etS_0,v0=ρ⁻¹(x_v0)est un toreT^d ainsi (donc) que lesS_u.

Exemple 3(naissance de sphères invariantes de dimension 2d−1, 2dn).Pour #J =d >1, il peut arriver [7,8,2] queΣ_v0 soit une sphèreS^d−1entourant l’origine dansR^J, cede manière structurellement stable pourkn (et mêmekd) [2]. LesSu sont alors des sphèresS^2d−1. Sid=n, elles entourent l’origine dansCⁿet le résultat ressemble plus à la bifurcation de Hopf que la naissance de tores invariants.

Exemple 4(naissance d’autres variétés de López de Medrano–Meersseman [9,1] invariantes).Le lemme précédent permet (entre autres) de traiter facilement les cas où Mathilde Kammerer-Colin de Verdière avait mis en évidence la naissance de produits de sphères [7] et ceux où nous avions mis en évidence avec elle la naissance de sommes connexes de tels produits [3].

Remarque.Ces exemples ne s’excluent pas mutuellement. Ils seront développés dans [2], qui contiendra aussi la preuve du lemme de naissance. Celui-ci vaut pour les applications : voir ci-après.

1 Ensemble des vitessesγ (0)˙ de chemins dérivablesγ:(R,0)→(R^k,0)vérifiantγ (ε)∈Uv₀pourε >0.

This is an account of work [2] based on the firm belief that, when coupling oscillators, interesting phenomena occur beyond quasiperiodic motions.

1. The birth lemma for vector fields

Hypotheses.Let(u, x)→Z_u(x)∈R^mbe a smooth enough family of vector fields with parameteru∈R^k, defined in the neighbourhood of a point ofR^k×R^mwhich we may assume to be 0, such thatZ₀(0)=0 and that the eigenvalues ofDZ₀(0)are pure imaginary, simple and different from 0, hencem=2n.

Denoting by iβ1, . . . ,iβnthe eigenvalues with positive imaginary part, we assume that, for 1jn, the equation β_j=_n

1(p−q)βwith(p, q)∈(Nⁿ)² and

(p+q)4 admits only the obvious solutionsp_j=q_j+1 and p=qfor=j.

Via a local change of coordinates(u, x)→(u, g_u(x)), we can suppose thatR^m=Cⁿ,Z_u(0)≡0, thatL:=DZ₀(0) is of the formLz=(iβ1z₁, . . . ,iβnz_n)and thatZ_u=N_u+R_u, whereR_u vanishes to order 4 at the origin andN_uis a (real) polynomial vector field of degree 3 onCⁿwhich commutes withLand therefore (by the above hypothesis) is invariant under the natural action ofU(1)ⁿ. In other words,

N_u(z)=

z_j

λ_j(u)+iμj(u)− n

=1

a_j (u)+ibj (u)

|z|²

1jn

, (1)

wherez=(z1, . . . , zn)∈Cⁿ andλj, μj, aj , bj are differentiable real functions withλj(0)=0 andμj(0)=βj. In particular, the mappingρ:Cⁿ→Rⁿ₊ defined byρ(z)=(|z1|, . . . ,|zn|)sendsNu onto the vector field

j(λj(u)−

a_j (u)r²)r_j∂r^∂

j.

To eachv∈R^k we associate the vector fieldXv(x)=

jxj(Dλj(0)v−

aj (0)x²)_∂x^∂

j onRⁿ. It is invariant under the action ofO(1)ⁿgenerated by the symmetries with respect to coordinate hyperplanes. We assumev₀∈S^k−1 such thatX_v0(x)admits a compact normally hyperbolic [4,6] invariant manifoldΣ_v0, invariant byO(1)ⁿand whose intersection with the non-negative orthantRⁿ₊is connected, hence the following property:

(P) IfR^J,J⊂ {1, . . . , n}, is the smallest coordinate subspace which containsΣ_v0, thenΣ_v0 is transversalinR^J to the coordinate subspaces lying inR^J.

As normal hyperbolicity is open, there exists an open neighbourhood V_v0 of v₀ inS^k−1 such that every X_v with v∈V_v0 has a normally hyperbolic invariant manifoldΣ_vdiffeomorphic toΣ_v0 and close to it, unique and therefore O(1)ⁿ-invariant, such thatΣ_v satisfies (P ) with the sameJ. It follows that S_0,v :=ρ⁻¹(Σ_v)⊂C^J is a compact U(1)ⁿ-invariant submanifold with the same smoothness and codimension asΣ_v.

Remarks.For positiveε, the submanifoldε^1/2Σ_v0 is a normally hyperbolic invariant manifold ofX_εv0.Thus, if the mappingu→(λ₁(u), . . . , λ_n(u))is a submersion at 0 (which is generically the case ifk=n), the existence ofΣ_v0 is equivalent to that ofν∈Rⁿsuch that the vector field

jx_j(ν_j−

a_j (0)x²)_∂x^∂

j has a compact normally hyperbolic invariant manifold as before.

Using diagonal changes of coordinates inCⁿ, one can see that, for each, the functionsaj +ibj are defined up to multiplication by the same positive function ofu.

Theorem 1(birth lemma). Under those hypotheses, there exists an open subsetU_v0 ofR^k, whose closure contains 0 and whose tangent cone²at0is an open cone with vertex0containingR^∗₊V_v0, such that every Z_u withu∈U_v0 has a compact normally hyperbolic invariant manifoldS_udiffeomorphic toS_0,v0, with the same index and co-index as the invariant manifoldΣ_v0 ofX_v0, depending at leastC^1+α onuand tending to{0}whenu→0. More precisely, for each smooth path γ:(R₊,0)→(R^k,0)withγ (0)˙ ∈V_v0, one hasγ (ε)∈U_v0 for small enough positiveεand limε→0ε^−1/2S_{γ (ε)}=S_{0,γ (0)˙} in the at leastC¹sense.

2 Set of all velocitiesγ (0)˙ of differentiable pathsγ:(R,0)→(R^k,0)satisfyingγ (ε)∈Uv₀for all positiveε.

Example 1.0.IfJ= ∅and thereforeΣv₀= {0}, the hypothesis means that everyDλj(0)v0is non-zero, and the lemma does select values ofufor whichS_u= {0}is a hyperbolic rest point ofZ_u.

Example 1.1(Hopf bifurcations).IfJ = {}, the hypothesis means that one hasa(0)Dλ(0)v0>0,Dλ_j(0)v0=0 forj=,Σv₀= {x∈R^J:|x| =

Dλ(0)v₀

a(0) }andS0,v₀= {z∈C^J:|z| =

Dλ(0)v₀

a(0) }. Therefore, theSu’s are periodic orbits and we get a somewhat weak version of the Hopf bifurcation.

Example 1.2 (birth of invariant tori of dimension d n). For #J =d > 1, the hypothesis is satisfied with dimΣ_v0 =0 if and only if the following three conditions hold: the matrix A=(a_j )_j,∈J is invertible, the point y_v0:=A⁻¹(Dλ_j(0)v0)_j∈J ∈R^Jhas all its coordinates positive and the zerox_v0∈R^JofX_v0 defined byx_v0j= √y_v0j, j∈J, is hyperbolic. Then,Σ_v0=O(1)ⁿx_v0 andS_0,v0=ρ⁻¹(x_v0)is ad-dimensional torus, hence so are theS_u’s.

Example 1.3(birth of invariant spheres of dimension 2d−1,2d n).For #J =d >1, the submanifold Σ_v0 can be [7,8,2] a sphereS^d−1around the origin inR^J, a situationwhich can be structurally stable forkn(in fact, kd) [2]. Then, theS_u’s are spheresS^2d−1. Ifd=n, they lie around the origin inCⁿand the result is much more reminiscent of the Hopf bifurcation than the birth of invariant tori.

In contrast with the latter case, the existence of such a sphere Σ_v0 is non-trivial. To introduce more familiar objects, one can remark that the change of variablesy=π(x):=(x₁², . . . , x_n²)sendsX_v0 onto the Lotka–Volterra fieldY_v0(y)=2

jy_j(Dλ_j(0)v0−

a_j (u₀)y)_∂y^∂

j onRⁿ₊. The sphereΣ_v0 is simply the inverse image byπ of the carrying simplexσv₀ ofYv₀, whose existence can follow from Hirsch’s theorem [5]. For example, ifσv₀ is the standard simplex

yj=1 ofRⁿ₊, thenΣv₀=π⁻¹(σv₀)is the unit sphere

x_j²=1 ofRⁿandS0,v₀ is the unit sphere |z_j|²=1 ofCⁿ. Details will be given in [2].

Example 1.4(birth of other invariant López de Medrano–Meersseman manifolds [9,1]).The birth lemma enables one (among other things) to treat easily the cases where Mathilde Kammerer-Colin de Verdière had established the birth of products of spheres [7] and those where we had established together the birth of connected sums of such products [3].

Note that those examples do not exclude each other.

2. The birth lemma for maps

Hypotheses.LetR^k×R^m(u, x)→h_u(x)∈R^mbe a smooth enough family of maps, defined in the neighbourhood of a point ofR^k×R^mwhich we may assume to be 0, such thath0(0)=0 and that the eigenvalues ofDh0(0)are of modulus 1, simple and different from 1, hence eitherm=2nwith no real eigenvalue, orm=2n−1 with the sole real eigenvalue−1.

Denoting the eigenvalues by e^±iβ1, . . . ,e^±iβn, 0< β1<· · ·< βnπ, we assume that, for 1jn, the equation e^iβj = ⁿ1e^i(p−q)β with(p, q)∈(Nⁿ)² and

(p+q)4 admits only the obvious solutionsp_j =q_j+1 and p=qfor=j.

Via a local change of coordinates(u, x)→(u, g_u(x)), we can suppose thatR^m=CⁿorCⁿ⁻¹×R,h_u(0)≡0, that Dh₀(0)is the mapz→(e^iβ1z₁, . . . ,e^iβnz_n)and thath_u=σ◦g_N¹

u+R_u, whereR_uvanishes to order 4 at the origin, σ (z)=

z ifm=2n, (z₁, . . . , z_n−1,−z_n) ifm=2n−1, andg_N^t

uis the flow of a vector fieldN_uof the form(1)withλ_j(0)=0 and μ_j(0)=β_j ifm=2n,

μ_j(0)=β_j forj < nand (of course)μ_n(u)=b_n(u)=0 ifm=2n−1.

As in Section 1, we associate toN_u the vector fieldsX_v onRⁿ, and we assume that X_v0 has a compact normally hyperbolic invariant manifoldΣ_v0, invariant byO(1)ⁿ and such thatΣ_v0∩Rⁿ₊is connected. With the notation of Section 1, we have the same result and similar examples:

Theorem 2(birth lemma). Under those hypotheses, there exists an open subsetUv₀ ofR^k, whose closure contains0 and whose tangent cone at0is an open cone with vertex0containingR^∗₊Vv₀, such that everyhu withu∈Uv₀ has a compact normally hyperbolic invariant manifoldS_u diffeomorphic to S_0,v0, with the same index and co-index as the invariant manifoldΣ_v0 ofX_v0, depending at least C^1+α onuand tending to {0}whenu→0. More precisely, for each smooth path γ:(R₊,0)→(R^k,0)withγ (0)˙ ∈V_v0, one hasγ (ε)∈U_v0 for small enough positiveεand limε→0ε^−1/2S_{γ (ε)}=S_{0,γ (0)˙} in the at leastC¹sense.

Example 2.0.IfJ= ∅, the lemma selects values ofufor whichSu= {0}is a hyperbolic rest point ofhu.

Example 2.1 (Poincaré–Andronov and “Hopf ” bifurcations). Of course, if #J =1, the hypothesis has the same meaning forX_v0 as in example 1.1.

IfJ= {n},m=2n−1, thenS_0,v0 =Σ_v0 = {z∈Cⁿ⁻¹×R: z₁= · · · =z_n−1=0,|z_n| =

Dλn(0)v0

ann(0) }. Therefore, theSu’s are 2-periodic orbits and we get a (bad) version of the period doubling bifurcation.

IfJ = {}(with < nifm=2n−1), the hypothesis implies thatS0,v₀= {z∈C^J: |z| =

Dλ(0)v₀

a(0) }. Therefore, theS_u’s are invariant closed curves and we get the “Hopf” bifurcation for maps.

Example 2.2(birth of invariant tori of dimensiondn).For #J=d >1 and dimΣv₀=0, as in Example 1.2, the hypothesis implies thatΣv₀ =O(1)ⁿxv₀. It follows that theSu’s ared-dimensional tori except if m=2n−1 and n∈J, in which case they consist of a pair of(d−1)-tori exchanged byh_u.

Example 2.3(birth of invariant spheres).For #J =d >1, ifΣ_v0 is a sphereS^d−1around the origin inR^J, then the S_u’s are(2d−1)-spheres except form=2n−1 andn∈J, in which case they are(2d−2)-spheres. Ifd=n, we get a very natural generalization of the Hopf bifurcation for maps.

Example 2.4 (birth of other invariant López de Medrano–Meersseman manifolds).As for vector fields, the birth lemma enables one (among other things) to establish quite easily the birth of products of spheres [7,8] and that of connected sums of such products [3,8].

Again, those examples do not exclude each other. The advantage of maps is that form=3 one can draw pictures.

The content of the present Note will be developed in the forthcoming paper [2].

Acknowledgements

I thank Jean Diebolt for asking me whether the invariant circles of the Hopf bifurcation can generalize as invariant spheres of codimension 1. This work is part of a research project with Santiago López de Medrano. It owes much to conversations with Mathilde Kammerer-Colin de Verdière and Alain Chenciner. I am grateful to François Laudenbach, Robert Moussu and Eduard Zehnder for their encouragement.

References

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