Generic bifurcation of reversible vector fields on a 2-dimensional manifold

Publicacions Matem`atiques, Vol 41 (1997), 297–316. GENERIC BIFURCATION OF REVERSIBLE VECTOR FIELDS ON A 2-DIMENSIONAL MANIFOLD Marco Antonio Teixeira Abstract In this paper we deal with reversible vector fields on a 2-dimensional manifold having a codimension one submanifold as its symmetry axis. We classify generically the one parameter families of such vector fields. As a matter of fact, aspects of structural stability and codimension one bifurcation are analysed. 1. Introduction Let Mbe a C∞compact orientable two-dimensional manifold and f:M→Rbe a C∞function having 0 as regular value. Call S= {f−1(0)},M+=f−1[0,∞), M−=f−1(−∞,0]. Let ϕ:M→MbeaC∞diffeomorphism (an involution) from M onto M, such that ϕ◦ϕ=Id(ϕis an involution) and Fix ϕ=S. We say that a vector field Xon Mis ϕ-reversible (or simply reversible) if ϕ∗X=−X◦ϕ. Let Φrbe the space of the Crϕ-reversible vector fields on Mendowed with the Cr-topology (r>2). The main result of this paper says that: The set Σ0of all Crϕ-reversible vector fields in Mwhich are structurally stable is open and dense in Φr. This set is characterized in Section 3 (see Definition 3). 298 M. A. Teixeira Call Φ1=Φ r−Σ0the bifurcation set of Φr. There exists a Cr−1 immersed codimension-one submanifold Σ1of Φrsuch that: (i) Σ1is dense in Φ1(both with the relative topology); (ii) for any Xin Σ1, there exists a neighborhood B1in the intrinsic topology of Σ1such that any Yin B1is topologically equivalent to X; (iii) the part of Σ1imbedded in Φris also characterized. In [11], we have classified all the symmetric singularities of codimension 0, 1 and 2 of X∈Φr. We have presented a technique which enabled us to classify in a simple manner those singularities. In that work the treatment is local and the technique consists to make a special change of coordinates around the point and then address the analysis to the study of the contact between a general system and S. In this paper we follow those ideas and use extensively the tools of the Singularity Theory and the results contained in [1], [7], [9], [10], [11] and [12]. In our setting, the strategy is to establish a connection between a reversible system on Mand a vector defined on M+. Roughly speaking, having reduced the system to the study of vector fields defined in manifolds with boundary, the next step is to employ the results and methods contained in [10]. Denote by χrthe space of all Crvector fields on M, endowed with the Crtopology. In the class of all vector fields in M, the structural stability has been characterized by Peixoto [9]. Sotomayor [7] has characterized the structural stability of one-parameter families of vector fields in M. Devaney [3] has stated a Kupka-Smale Theorem for reversible vector fields and flows. In the class of reversible vector fields some persistent phenomena occur which cannot be destroyed by perturbations in Φrsuch as periodic orbits and saddle connections which meet the submanifold S. However, concerning non trivial recurrences no surprises arise at all. As a matter of fact, this point becomes in some sense simpler in this class. We mention for example that such reversible systems on the torus do not admit an irrational flow. We suggest that the reader see [4] for further references and connections with other problems. This paper is organized as follows. In Section 2 we give definitions and recall standard facts. Bifurcation of reversible vector fields 299 Section 3 is devoted to the study of the structural stability in Φr. Some considerations concerning structural stability in manifolds with boundary are made. In Section 4 the main result of the paper is presented. In our procedure we classify the stable one parameter families of reversible vector fields on M. Section 5 contains an appendix where the main tools required for the proof of the main theorem are recalled. 2. Preliminaries In what follows we list some known properties of a vector field X∈Φr: •The phase portrait of Xis symmetric with respect to S. So the knowledge of the phase portrait of Xin M+determines the phase portrait of Xin M. •If p∈Sand X(p)= 0 then the orbit of Xis always transverse to Sat p. •If X(p) = 0 then X(ϕ(p)) = 0. If p∈Sthen it is called a symmetric critical point of X. Otherwise it is an asymmetric critical point. Moreover ϕinterchanges the stable and unstable manifolds and a symmetric critical point cannot be an attractor or a repellor. •Any periodic orbit of Xcrossing Sis called a symmetric periodic orbit of X. Any symmetric periodic orbit is never an isolated limit cycle. If, on the other hand, a periodic orbit γof Xis away from Sit is called asymmetric and it is paired by another periodic solution given by ϕ(γ). •If Xtdenotes the flow associated to Xthen we have the following expressions: Xt◦ϕ=ϕ◦X−t,U2 t= Id and Fix{Ut}=Xt/2(Fix{ϕ}), where Ut=Xt◦ϕ. •Any orbit of Xconnecting two asymmetric hyperbolic saddles (and so meeting S, transversally) is persistent under perturbation of Xin Φr. •There is no isolated periodic orbit of Xpassing through points p∈S. •The codimension zero (generic) symmetric critical points are either a hyperbolic saddle (with real eigenvalues λand −λ)orof 300 M. A. Teixeira elliptical type (with eigenvalues ±ib). Their normal forms (see for example [11]) are: X01(x, y)=(y,x) and X02(x, y)=(−y, x). We may refer to them as generic S-singularities. •In [2] it is shown the following expressions relating the genus gof Mand the Betti number kof S: (i) k≤g+1, (ii) k=gMod(2) and (iii) g−k+1∈2N. In particular, one deduces that k= 1 or 2 provided that Mis the sphere or the torus respectively. When Mis the bitorus then k canbe1or2. We denote by T=T(S) a neighborhood of Sin M, such that ϕ(T)=T and T±=T∩M±. 3. Structural Stability 3.1. Structural stability in manifolds with boundary. Let χrbe the space of the Crvector fields on Mendowed with the Cr-topology (r>2). The results in this section will be used in the sequel. Let Z∈χrand Nbe a 2-dimensional submanifold of Mwith S=∂N. We say that p∈Sis an S-singularity of Z∈χrif either Z(p)=0or Z(p)= 0 and Zf(p)=0. It should be mention that Zf(p)=Df(p)(Z(p)) is still a real function defined on M. So we may define inductively the function Zkf(p)= Z(Zk−1f(p)) which expresses the order contact between the vector field Zand the curve Sat p. Definition 1. We say that p∈Sis a fold singularity of Zif Z(p)=0, Zf(p) = 0 and ZZf(p)= 0. In this case we say that the contact between the orbit of Zand Sat pis quadratic. A vector field Zin χris S-stable if there are neighborhoods Bof Zin χrand Vof Nin Msuch that for every Yin Bthere is an S-preserving homeomorphism h(Y):V→Vwhich is a C0equivalence between Z|V and Y|V. Such a homeomorphism will be refered to as an S-equivalence between Zand Y. The concept of S-structural stability in χris given in a natural way. Bifurcation of reversible vector fields 301 Definition 2. Let Ξ0(S) be the class of all Zin χrfor which the following conditions are satisfied: (0) Zdoes not have nontrivial recurrent trajectories; (I1) all critical points of Z|Nare hyperbolic; (I2) all periodic orbits of Z|Nare hyperbolic; (I3)Z|Ndoes not have saddle connections; (S1) all singular points of Z|Nare contained in the interior of N; (S2) all periodic orbits of Z|Nare contained in the interior of N; (S3) any tangency between a trajectory of Z|Nand Sis quadratic; (S4)Z|Ndoes not have tangency connections; (S5)Z|Ndoes not admit a connection between a saddle critical point and a tangency point. Theorem 3.1 (Andronov-Pontryaguin and Peixoto). (a) Ξ0(S) is open and dense in χr; (b) Zis S-structurally stable in χrif only if it belongs to Ξ0(S). Assume that Z∈Ξ0(S). A separatrix of Zis an orbit which connects either two saddle critical points or two tangency points between the vector field and Sor a tangency point and a saddle critical point. Any equivalence between two vector fields in χrmust preserve such objects. Consider Z∈χr, and p∈N. The positive (resp. negative) limit set of an orbit γ(p)ofZ|Nis the set L+(p) (resp. L−(p)) of points q∈N which are limit points of sequences of the form φX(p, tn) with tntending to ω(p) (resp. α(p)). 3.2. Structural stability of reversible vector fields. Let X∈φr. The coming construction will be useful in the sequel. 3.2.1. A construction. On a small tubular neighborhood Tiof each connected component Si of Sin M, consider the following coordinates (θi,ρ i) with Si={ρi=0} and 0 ≤θi≤1 and |ρi|<ε. Let ϕibe the restriction of ϕon Ti. In the above coordinates, ϕi(θi,ρ i)=(αi(θi,ρ i),β i(θi,ρ i)) satisfies: ρiβi(θi,ρ i)<0 and ϕi(θi,0) = (θi,0). 302 M. A. Teixeira Define now the germ of a C∞mapping at Si, Fi:Ti,S i→ 2 by Fi(θi,ρ i)=(θi+αi(θi,ρ i),θ iαi(θi,ρ i)+ρiβi(θi,ρ i)). Proposition 3.2. Each Fiis a fold mapping at (θi,0) and Fi◦ϕi=Fi. Proof: Neglecting the subscripts we have: Dϕ(θ, 0) = 1b 0−1 where b=αρ(θ,0). By a straightforward calculation we get DF(θ,0) = 2b 2θbθ . The function (θ, ρ) = Det(DF(θ, ρ)) satisfies: (θ,0) = 0 and θ(θ,0) = −4−b2<0. So the curve K= Kern(DF(θ, 0)) = {(u, v); 2u+bv =0}is transverse to the curve Γ(F)={(u, v); (u, v)=0}. It is immediatee to get F◦ϕ=F. This finishes the proof. On each Ticonsider the coordinates (θ, ρ) and ϕ=(α, β) given above. On the half “plane ” ρ>0, let u=θ+αand v=θα +ρβ. In the new coordinates, Sis expressed by v=u 22, and Xis transformed in X∗(u, v) in such a way that X∗u, u 22= 0. We finally define the vector field H=H(X) on the region F(Ti)by H(u, v)= X∗ v−u 22. Bifurcation of reversible vector fields 303 3.2.2. Local settings. i) Let p∈Sand X∈Φr. It is well known (Montgomery-Bochner Theorem in [6]) that the involution ϕat p,isC∞conjugated to ϕ(x, y)=(x, −y)atp0= 0. In this section we carry out the analysis on 2,0 and fix f(x, y)=y. Observe that in local coordinates around a point p∈Swe have the following expressions: S={(x, y); y=0}and ϕ(x, y)=(x, −y). We may choose F(x, y)=(x, y2) and from the reversibility properties of the vector field, it takes the form X(x, y)=ya(x, y2),b(x, y2) 2. Moreover by u=xand v=y2we derive that: H(u, v)=(a(u, v),b(u, v)) for v>0. ii) In these coordinates the trajectory of Xpassing through a regular point is always “orthogonal” to S. At a tangency point (resp. critical point), the contact between the orbit through p(resp. an invariant manifold) and Sdecays by a factor of 1 2in comparison with the orbit or invariant manifold of H(X) passing through the same point. Following these considerations, we denote by H(s) a trajectory of H(X) corresponding to a trajectory sof X. Remark 3.3. Observe that Hcan be C∞extended to a full neighborhood of p. Moreover, due to the symmetry properties of X, we deduce that the behavior of H(X) near Sdetermines completely the behavior of Xin a small neighborhood Tof S. It follows that in ρ>0, Xis topologically equivalent to H(X). This leads us to analyse the S-stability of Hon the region H(T) with boundary S. Moreover, outside a small neighborhood Vof Sin T+,Xand H(X) are Crconjugated. Remark 3.4. It should be mentioned that if p∈Sis a fold point of H(X) then it is a codimension 0 (generic) critical point of X.Itis of saddle type (resp. elliptical type) provided that it is a internal (resp. external) tangency. 304 M. A. Teixeira 3.2.3. The manifolf Σ0. Denote by X+the restriction of Xto Cl{M+}. Definition 3. We say that X∈Φris simple if the following conditions are satisfied: (0) Xdoes not have nontrivial recurrent trajectories; (i) all asymmetric critical points of Xare hyperbolic; (ii) all asymmetric periodic orbits of Xare hyperbolic; (iii) X+does not have saddle connections. (iv) all symmetric singularities of Xare of codimension 0. Remark 3.5. As pointed out, if pis a saddle critical point of Xwe have to distinguish in our analysis the cases where pis in Sor not. In the first case, it corresponds to a generic (interior) contact between H(X) and S, and the eigenvalues associated to DX(p) are ±λ. Remark 3.6. If Xis simple we may find, from [10], a tubular neighborhood T(S)ofSin Msuch that X∗=X|M+−T(S)satisfies the following conditions: (0) X∗does not have nontrivial recurrent trajectories; (i) all critical points and periodic orbit of X∗are hyperbolic and contained in the interior of M+−T(S); (ii) X∗does not have saddle connections; (iii) if X∗(p) is tangent to ∂T(S) then this contact is quadratic. In other words, we mean that X∗satisfies the conditions of structural stability of vector fields defined on manifolds with boundary given in [8]. This auxiliar vector field is very useful in the proof of the main results of this paper. The following result is an immediate consequence of Theorem 3.1 and Proposition 3.2. Theorem 3.7. X∈Σ0if and only if Xis simple. Moreover, Σ0is open and dense in Φr. Bifurcation of reversible vector fields 305 Proof: First of all, we observe that the above conditions (0), (i), (ii), (iii) coincide exactly with the characterization of the Morse-Smale Vector Fields on M. It should be mentioned that those saddle separatrices, asymmetric orbits and asymmetric critical points appear in pairs (each one in a different connected component of M−S). The symmetric singularities of Xare studied by the auxiliar vector field H(X) (see Proposition 3.2). As a matter of fact they correspond to quadratic tangencies between H(X) and S. Due to the symmetry properties of X, it is clear that all the analysis can be performed via X+. The results and techniques, contained in [8] and [10], involving the generic contact between H(X) and Smust be used here. We observe that the question involving recurrences is answered exactly in the same way as in the usual theory (see [5]). So it is straightforward, from Theorem 3.1 and Proposition 3.2 to get that Xis structurally stable provided that it is a simple vector field. So the genericity of Σ0in Φrbecomes evident. Assume for instance that Xviolates some condition given in Definition 3. Once again we appeal to Theorem 3.1 and to Proposition 3.2 and conclude that Xcannot be structurally stable. In fact: a) If Xhas non-hyperbolic critical points or periodic orbits outside Sthen we use standart techniques to approximate it by Yin Φr having just hyperbolic critical points or periodic orbits outside S. We proceed similarly when we have a saddle connection off S. b) Assume now that Xhas a non codimension zero critical point p∈S. This implies that H(X) and Shave a degenerated contact at p. Again, we perturb Xby getting Y, such that H(Y) has in a neighborhood of pin M+just generic contact and/or hyperbolic critical points outside S(see [10] and [11]). This finishes the proof. 4. Bifurcation set 4.1. Generic bifurcation in manifolds with boundary. Let χ1be the complement of Ξ0(s)inχr. Assume that Z∈χ1. Definition 4. We say that p∈Sis a cusp singularity of Zif Z(p)=0, Zf(p)=0. 312 M. A. Teixeira (b) Assume that p1∈Sand p2∈ Sin such a way that Wu(p−1) = Ws(p2)=ν+(X). Choose neighborhoods B1,U1and U2of X in Φr,p1in M,{p1,p 2}in Mrespectively, such that Yhas a unique saddle point in U1∩S, unique saddle point in U2∩M+. Furthermore Wu(p−1) (resp. Ws(p−2)) is transverse to ∂U1 at m1∈Int(M+) (resp. ∂U1at n1∈Int(M+)) and transverse to ∂U2at m2∈Int(M+) (resp. ∂U2at n2∈Int(M+)). Hence, Y∈Σ1(3) if and only if m1=n1. Now standart techniques (see in [10]) allows us to finish the proof of the lemma in the present case. (c) if both points p1and p2are in S, the proof is very similar to the above case. Remark 4.9. As an illustration assume for instance that X∈Σ1(3)− Σ0 1(3) in such a way that there is a homoclinic orbit ν(X)atpXand a saddle separatrix s converging to this orbit. Then we may find a sequence Xn∈Σ1(3) converging to Xin Φ1(see Figure 6). Observe that such pX has to be necessarily in Int(M+). XnX0 ss Figure 6. Asymmetric semi-stable periodic orbit. Bifurcation of reversible vector fields 313 The following result is obtained from [11] by using the same techniques of the proof of Lemma 4.8 which have been developed in [10]. Lemma 4.10. Σ1(4)is a Cr−1imbedded codimension one submanifold of Φr. Moreover, every X∈Σ1(4)has a neighborhood Bin Φ1such every Y∈Bis C0equivalent to X. Theorem 4.11. i) Σ1is dense in Φ1(both with the relative topology); ii) for any Xin Σ1, there exists a neighborhood B1in the intrinsic topology of Σ1, such that any Yin B1is topologically equivalent to X; iii) Σ0 1is the part of Σ1imbedded in Φr; iv) In the space of one parameter families of vector fields in Φr, let Θbe the collection of elements ξ(λ)(with |λ|<ε), such that: a) ξ(λ)⊂Σ0∪Σ0 1;b)ξis transversal to Σ0 1. Then any family ξ is structurally stable if and only if ξ∈Θ. Proof: This proof follows from the definitions of Σ1and Σ0 1and from the Lemmas 4.4, 4.6, 4.8 and 4.10. Corollary 4.12. In the space of one parameter families of vector fields in Φr, let Θbe the collection of elements ξ(λ)(with |λ|<ε), such that: a) ξ(λ)⊂Σ0∪Σ0 1;b)ξis transversal to Σ0 1. Then any family ξis structurally stable if and only if ξ∈Θ. Appendix In this section we recall some aspects of the main result in [10] concerning the structural stability of one parameter families of vector fields defined in manifolds with boundary. The bifurcation set χ1in χris the union of the following sets of vector fields χ1(I1), χ1(I2), χ1(I3), χ1(S1), χ1(S2), χ1(S3), χ1(S4) and χ1(S5) where the conditions I1), I2), I3), S1), S2), S3), S4) and S5) given in the definition of the set Ξ0,are violated respectively. Call Ξ1(I1) the set of vector fields Xin χ1(I1) such that Z|Nhas a unique non-hyperbolic critical point outside S. Moreover it is a codimension one critical point of Z(i.e. a saddle-node or a generic Hopf singularity) and all the other conditions I2), I3), S1), S2), S3), S4) and S5) in Definition 1 are satisfied. 314 M. A. Teixeira Call Ξ1(I2) the set of vector fields Zin χ1(I2) such that Z|Nhas a unique non-hyperbolic periodic orbit γ(Z). Moreover it is a codimension one periodic orbit of Z(i.e. a semi stable periodic orbit) and all the other conditions I1), I3), S1), S2), S3), S4) and S5) in Definition 1 are satisfied. We denote by Ξ0 1(I2) the subset of Ξ1(I2) constituted by the elements Zwhich satisfy: i) there exists no q∈N−γ(Z), such that L+(q)= L−(q)=γ(Z); b) there exists no saddle points siin N,i=1,2 such that L+(Wu(s1)) = L−(Ws(s2))=γ(Z); c) associated to Zthere exists no (s, q)∈N×N, where sis a saddle point of Z,q∈S, and Z(q)is tangent to Sat this point, with L+(q)=L−(q)=γ(Z); there exist no pi∈Swith L+(p1)=L−(p2)=γ(Z). Denote by Ξ1(I3) the set of vector fields Zin χ1(I3) such that X|Nhas a unique saddle connection ν(Z). Moreover all the other conditions I1), I2), S1), S2), S3), S4) and S5) in Definition 1 are satisfied. We denote by Ξ0 1(I3) the subset of Ξ1(I3) constituted by the elements Xsuch that the saddle connection is a homoclinic orbit. Moreover no trajectory of Z|Nwhich is either tangent to Sor a saddle separatrix tends to ν(Z). Denote by Ξ1(S1) the set of vector fields Zin χ1(S1) such that Z|N has a unique critical point pZin Sand: a) pZis S-hyperbolic and of nodal type, b) there is no separatrix which is the strong manifold of pZ and c) all the other conditions I1), I2), I3), S2), S3), S4) and S5)in Definition 1 are satisfied. Denote by Ξ1(S2) the set of vector fields Zin χ1(S2) such that Z|N has a unique periodic orbit γ(Z) tangent to S, while all the other conditions I1), I2), I3), S1), S3), S4) and S5) in Definition 1 are satisfied. We denote by Ξ0 1(S2) the subset of Ξ1(S2) constituted by the elements Zsuch that γ(Z) is neither the αnor the 8limit, of either the saddle separatrices or the trajectories tangent to S(with respect to Z|N). Denote by Ξ1(S3) the set of vector fields Zin χ1(S3) such that Z|N has a unique orbit tangent to Sand this tangency point is at a cusp point. Moreover all the other conditions I1), I2), I3), S1)S2), S4) and S5) in Definition 1 are satisfied. Denote by Ξ1(S4) the set of vector fields Zin χ1(S4) such that Z|N has a unique orbit tangent to Sin more than one point. Moreover this orbit contains exactly 2 tangency points and all the other conditions I1), I2), I3), S1), S2), S3) and S5) in Definition 1 are satisfied. Bifurcation of reversible vector fields 315 Denote by Ξ1(S5) the set of vector fields Zin χ1(S5) such that Z|N has a unique saddle separatrix tangent to S. Moreover all the other conditions I1), I2), I3), S1), S2), S3) and S4) in Definition 1 are satisfied. We define Ξ1as the union of the sets Ξ1(Ij) and Ξ1(Sl) for j=1,2,3 and k=1,2,3,4,5. The part of Ξ1imbedded in χris: Ξ0 1=Ξ 1(I1)∪Ξ0 1(I2)∪Ξ0 1(I3)∪Ξ1(S1)∪Ξ0 1(S2)∪Ξ1(S3)∪Ξ1(S4)∪Ξ1(S5). References 1. A. Andronov, E. Leontovich, I. Gordon and A. Maier, Theory of bifurcations of dynamical systems on a plane, Israel Program for Sc. Translations, Jerusalem (1971). 2. R. Cruz, Reflections on closed manifolds, Preprint IMECC/ UNICAMP (1996). 3. R. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc. 218 (1976), 89–113. 4. J. S. W. Lamb, Reversing Symmetries in Dynamical Systems, Thesis, University of Amsterdam (1994). 5. W. de Melo and J. Palis,“Geometric Theory of Dynamical Systems, an introduction,” Springer Verlag, NY, 1982. 6. D. Montgomery and L. Zippin,“Topological Transformations Groups,” Interscience, NY, 1955. 7. J. Sotomayor, Generic one parameter families of vector fields on 2-dimensional manifolds, Publ. IHES 43 (1974), 5–46. 8. M. C. Peixoto and M. M. Peixoto, Structural stability in the plane with enlarged conditions, An. Acad. Brasil. Ciˆenc. 31 (1959), 136–160. 9. M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1(1962), 101–120. 10. M. A. Teixeira, Generic bifurcation in manifolds with boundary, J. Differential Equations 25(1) (1977), 65–89. 11. M. A. Teixeira, Singularities of reversible vector fields, Phys. D 100 (1997), 101–118. 12. S. M. Vishik, Vector fields near the boundary of a manifold, Vestnik 316 M. A. Teixeira Moskov. Univ. Ser. Mat. Mekh. 27(1) (1972), 21–28. 1991 Mathematics subject classifications: 58F14, 34C23 IMECC-UNICAMP Caixa Postal 6065 13081-970 Campinas BRASIL e-mail: [email protected] Primera versi´o rebuda el 30 de Novembre de 1996, darrera versi´o rebuda el 21 d’Abril de 1997