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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 113 - 118

Flows equivalencesGabriel Soler L�opez�Abstract. Given a di�erential equation on an open set O ofan n-manifold we can associate to it a pseudo-
ow, that is, a 
owwhose trajectories may not be de�ned in the entire real line. Inthis paper we prove that this pseudo-
ow is always equivalent toa 
ow with its trajectories de�ned in all R. This result extends asimilar result of Vinograd stated in the n-dimensional euclideanspace.2000 AMS Classi�cation: Primary 37E35, 37C15. Secondary 58C25.Keywords: Flow, pseudo-
ow, equivalence di�eomorphism.1. Introduction.It is well known that given a Cr-
ow (r � 1) on an n-manifold M, 	 :R � M ! M, we can associate to it a Cr�1-autonomous di�erential equationy0 = f(y). Where f maps M onto its tangent bundle, TM, in the followingway f(y) = @	dt (0;y).The converse does not work in general because the solutions of a di�erentialequation could not be de�ned in the entire real line. For example, if we takethe autonomous di�erential equation (x0;y0) = (1;1 + tan2(x)), the solutionsare de�ned for each initial condition (x0;y0) in an interval of length �. Thenwe can not associate to this autonomous di�erential equation a 
ow. However,if the manifold is compact, the converse does work [1, Theorem 4, x1.9] and [5,p.11].Let us introduce some terminology. Given two 
ows 	 and � on an n-manifold M we say that they are Cr-equivalent if there exists a Cr-di�eo-morphism h : M ! M such that h conserves the orbits of �. That is, thesubsets h(�(R;p)) and 	(R;h(p)) of M are equal for any p 2 M. Moreover theorientations of the curves 	h(p)(t) = 	(t;h(p)) and h��p(t) = h��(t;p) coincidefor any p 2 M, that is, there exists a continuous increasing map ip : R ! Rfor which h � �p(t) = 	h(p)(ip(t)). When we use the norm of a vector x 2 Rnwe are always using the norm kxk = kxk1 = maxi2f1;2;:::;ngfjx1j; jx2j; : : : ; jxnjg.�This paper has been partially supported by the D.G.I.C.Y.T. grant PB98-0374-C03-01.



114 Gabriel Soler L�opezBy R+ we denote the set of positive real numbers. As usual, given O � M,Bd(O) denote the topological boundary of the set O.From now on, when speaking of C0-di�erential equations they are supposedto be continuous and locally Lipschitz. Let r 2 N[f0g and take a Cr-di�erentialequation y0 = f(y) on an open set O � M, f : O ! TO. The classical theoryof di�erential systems assures that there exists a Cr-map  : D ! O calledpseudo-
ow, where D is an open subset of R �O and for each p 2 O the curve p(t) =  (t;p) is the solution of the equation y0 = f(y) with initial conditiony(0) = p. Analogously to the de�nition of equivalence between 
ows we say thattwo pseudo-
ows � : D � R�O ! O and  : E � R�O ! O are Cr-equivalentif there exists a Cr-di�eomorphism h : O ! O such that h conserves the orbitsof � and the orientations of the curves  h(p)(t) and h � �p(t) coincide for allp 2 O. With this terminology we will say that two autonomous di�erentialequations are Cr-equivalent if their associated pseudo-
ows are Cr-equivalent,moreover the di�eomorphism h will be called equivalence di�eomorphism.The basic question in which we are interested is to prove that for any Cr-autonomous di�erential system in an open set O, we can �nd a Cr-equivalentautonomous di�erential equation such that the associated pseudo-
ow is in facta 
ow, that is, de�ned in all R � O. This question was solved by Vinograd [4,pp. 19-21] when the phase space is Rn.Theorem 1.1 (Vinograd). Let O be an open set of Rn and let f : O ! Rnbe a Cr-map (r � 0). Then there exists a Cr-map g : O � Rn ! Rn suchthat the equations y0 = f(y) and y0 = g(y) are Cr-equivalent and the associatedpseudo-
ow to g is a 
ow. Moreover, the equivalence di�eomorphism is theidentity map. (When r = 0 we consider f and g to be locally Lipschitz)The aim of this paper is to prove the following theorem that generalizes theprevious one:Theorem 1.2 (Main Result). Let M be an n-manifold, O an open set of Mand f : O ! TO a Cr-map (r � 0) . Then there exists a Cr-map g : O ! TOsuch that the equations y0 = f(y) and y0 = g(y) are Cr-equivalent and theassociated pseudo-
ow to g is a 
ow. (When r = 0 we consider f and g to belocally Lipschitz)Section 2 is devoted to state some classical results that we need in the proofof our result. We also construct a positive C1-function that vanish only inthe boundary of O. This function will be essential in the proof of the MainTheorem in Section 3. 2. Preliminary ResultsIn the sequel we are going to use the Whitney theorem that provides a C1n-manifold M embedded in R2n+1 (see [2, x1.3]). Another Whitney theoremabout function extensions is stated and used in the proof of Lemma 2.4 toconstruct a scalar C1-function f : Rn ! R vanishing only in the boundary ofan open set O and being strictly positive in O.



Flows equivalences 115Theorem 2.1 (Whitney). Let M be an n-manifold of class Cr, r � 1. Thenthere exists a Cr-embedding f : Mn ! R2n+1 such that f(M) is a closedC1-submanifold of R2n+1.We will use another less known Whitney's Theorem. Its proof can be foundcombining [7, p. 177,Th. 4] and [8]. We introduce some necessary terminologyfor its statement: if � 2 (f0g [ N)n, y 2 Rn and f is a map de�ned on anopen subset of Rn, we denote �! = �1!�2! : : :�n!, j�j = �1 + �2 + � � � + �n, y� =y�11 y�22 : : :y�nn and D�f(y) = @�1+�2+���+�n@x�11 @x�22 :::@x�nn f(y). As usual we mean D0f = f.Theorem 2.2 (Whitney). Let C � Rn be a closed set (as a subset of Rn).Then the following statements hold.(1) Let f0 : C ! Rm be a bounded Lipschitz map. Then there is a boundedLipschitz map f : Rn ! Rm such that f(x) = f0(x) for any x 2 C.(2) Let 1 � k � 1 and let f� : C ! Rm be arbitrary maps for any� 2 (f0g [ N)n with 0 � j�j � k. Let F
;r : C � C ! Rm be de�ned byF
;r(x;y) = f
(y) � P0�j�j�r f
+�(x)(y�x)��!kx � ykrif x 6= y and F
;r(x;x) = 0otherwise, for any 
 2 (f0g [ N)n and 0 � r < 1 with j
j + r � k.Suppose that all maps F
;r are continuous. Then there is a Ck mapf : Rn ! Rm such that D�f = f� for any � 2 (f0g [ N)n, 0 � j�j � k.The following result is an easy consequence of the previous Theorem:Corollary 2.3. Let C � Rn be a closed set decomposed into disjoint sets Aand B, C = A [ B. Given two real numbers a and b de�ne f� : C ! Rm asfollows: f�(x) = a for any x 2 A and f�(x) = b for any x 2 B. Then there isa C1-map f : Rn ! R such that f(x) = f�(x) for any x 2 C and any partialderivate of f is equal to 0 in C.Proof. Take for each � 2 (f0g[N)n, f� : C ! R with f� � 0 for any 0 < j�j <1 and f0 = f�. It is clear that the functions f� satisfy the conditions of part2 of Theorem 2.2. Then there exists a C1-function f : Rn ! R that extendsf0 and whose derivates are 0 in C. �We also need some previous lemmas:Lemma 2.4. Let O � Rn be a nonclosed set. There exists a C1-map f : Rn ![0;1[ such that f(x) = 0 for any x 2 Bd(O) and f(x) 2]0;1[ for any x 2 O.Proof. We are going to construct the C1-map as the sum of a function series.Thus we are going to construct C1-functions fi : Rn ! [0;1[ for every i 2 N.De�ne Cj = ? for j 2 ZnN, C1 = fx 2 O : 1 < d(x;Bd(O)g and for j 2 Nnf1gconsider Cj = fx 2 O : 1j < d(x;Bd(O)) < 1j�1g (eventually Cj = ? for j



116 Gabriel Soler L�opezsmall). Let A = Ci and B = [j>i+1Cj S[j<i�1Cj SBd(O). It is clear thatA \ B = ? and A and B are closed sets. We de�negi�(x) = (1 if x 2 A0 if x 2 B:Using that R and ]�2;2[ are C1-di�eomorphic (being 0 a �xed point of thedi�eomorphism) and the precedent Corollary, we extend gi� to Rn obtainingthe C1-function gi : Rn !]�2;2[ that vanishes in Bd(O) and veri�es gi(x) = 1if x 2 Ci. De�ne fi = g2i4 and obtain the C1-function fi : Rn ! [0;1[ withfi(x) = 0 for every x 2 Bd(O).Let us de�ne f(x) = P1i=0 1i2 fi(x) which is clearly uniformly convergentbecause it is bounded by the series of real number P1i=0 1i2 . Moreover, thefunction f is C1 in each point x 2 O because, in a neighborhood of x thefunction f is at most the sum of three C1-functions fj di�erent of 0. Then fis C1. Obviously the function f can not vanish in the set O. That concludesthe proof. �Finally we state a Theorem about maximal solutions of di�erential equations.Find the proof, e.g., in [3, p.195, Th. 25.9].Theorem 2.5. Let f : Rn ! Rn be a continuous bounded map. Then everymaximal solution of y0 = f(y) is de�ned in the entire real line.3. Proof of The Main ResultTheorem (Main Result). Let M be an n-manifold, O an open set of M andf : O ! TO a Cr-map (r � 0) . Then there exists a Cr-map g : O ! TO suchthat the equations y0 = f(y) and y0 = g(y) are Cr-equivalent and the associatedpseudo-
ow to g is a 
ow. (When r = 0 we consider f and g to be locallyLipschitz)Proof. As M is embedded in R2n+1 thanks to Whitney theorem, i : M !R2n+1, we can see the manifold M and the vector �eld f in R2n+1. Thus wehave f : O � M ! R2n+1. Denote by fj the j-th component of f, fj : M ! R.Let � : R2n+1 ! R be a C1-function equal to 0 in Bd(O) and strictlypositive outside (see precedent section, Lemma 2.3). De�ne 
 : O ! R as
(x) = 1exp(P2n+1j=0 fj(x)2) and G : M ! R2n+1 asG(x) = (
(x)�(x)f(x) si x 2 O,0 si x 62 MnO:G is bounded by 1p2ek
(x)�(x)f(x)k1 � k
(x)f(x)k1� supa2R � aea2 �� 1p2e



Flows equivalences 117and it is locally lipstchitz in M: it is clear that G is locally Lipschitz insideand outside of O, let us now show that G is locally Lipschitz in Bd(O).Take y 2 Bd(O) and x in a neighborhood Uy of y. ThenkG(x) � G(y)k = k
(x)�(x)f(x)k= k
(x)�(x)f(x)k� k
(x)f(x)kk�(x)k� 1p2ek�(x)k:Since � is C1 it will be locally Lipschitz and taking Ux small enough we willhave kG(x) � G(y)k � 1p2ek�(x)k � 1p2eMkx � yk:Therefore G is locally Lipschitz in M.By Theorem 2.1 M is closed in R2n+1. Now we can use Theorem 2.2 forextending G to a locally Lipschitz function G1 : R2n+1 ! R2n+1 with G1jM =G. Hence the autonomous di�erential equation y0 = G1(y) has uniqueness ofsolutions and using Theorem 2.5, the solutions are de�ned in the entire realline.Denote by  the pseudo-
ow associated to f and by � the restriction of the
ow associated to G1 to R � O. Notice that � is the pseudo-
ow associated toy0 = g(y) where g : O ! TO is de�ned by g(y) = G1(y). We must see that �and  are Cr-equivalent with equivalence di�eomorphism Id : O ! O, that is,we must prove that the orbits of  are orbits of � with the same orientation.Let y : I = (a;b) ! O be an orbit of  , that is y0(t) = f(y(t)). Consider thereal function s : I = (a;b) ! R de�ned by s(t) = c + R tc 1
(y(u))�(y(u)) du withc 2 I. As s0(t) = 1
(y(t))�(y(t)) > 0, s is strictly increasing and there exists itsinverse t : s(I) ! (a;b). De�ne z : s(I) ! O as z(s) = y(t(s)) and notice thatz0(s) = y0(t(s)) 1s0(t(s)) = f(y(t(s)))
(y(t(s)))�(y(t(s))) = G(z(s)) = g(z(s))and z(c) = y(t(c)) = y(c). Thus the orbits of  and � coincide and also theirorientations because s is strictly increasing. �References[1] A. Andronov and E. Leontovich, Qualitative theory of second-order dynamic systems,(John Wiley And Sons, New York, 1973).[2] M.W. Hirsch, Di�erential topology. (Springer Verlag, New York, 1976).[3] V. Jim�enez L�opez, Ecuaciones diferenciales, (Universidad de Murcia, Murcia, 2000).[4] V. Nemytskii and V. Stepanov, Qualitative theory of di�erential equations. (PrincetonUniversity Press, Princeton, 1960).[5] Jacob Palis and Wellington De Melo, Geometric theory of dynamical systems. (Springer-Verlag, New-York, 1982).[6] Jorge Sotomayor, Li�c~oes de equa�c~oes diferenciais ordin�arias. (Instituto De Matem�aticaPura e Aplicada Rio De Janeiro, Rio de Janeiro, 1979).[7] E. M. Stein, Singular integrals and di�erentiability properties of functions. (PrincetonUniv. Press, Princeton, 1970).[8] H. Whitney, Analytic extensions of di�erentiable functions de�ned in closed sets, Trans.Amer. Math. Soc. 36 (1934), 63-89.



118 Gabriel Soler L�opez Received February 2001Revised version April 2001
Gabriel Soler L�opezDepartamento de Matem�atica Aplicada y Estad��sticaUniversidad Polit�ecnica de CartagenaPaseo Alfonso XIII 5230203-CartagenaSpainE-mail address: gabriel.soler@upct.es