A pseudo-normal form for planar vector fields
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Journal Name ??? , 1{32 (???) ARTICLE NO. HA-00000 A pseudo-normal form for planar vector elds Amadeu Delshams * ,Antoni Guillamon y and J. Tomas Lazaro Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal, 647, 08028 Barcelona, Spain E-mail: [email protected] c.es, [email protected] c.es, l[email protected]c.es The pseudo-normal form is presented as an alternative to the Birkho normal form for a planar vector eld with the origin as an equilibrium p oint. Its convergence is proved for non-zero critical exp onents ; , and some consequences for the center-fo cus problem are presented. Key Words : Normal Forms, integrability, Hamiltonian and reversible systems, limit cycles. 1. INTRODUCTION Given a \planar" general ordinary dierential equation (o.d.e.) of the typ e _ z = F ( z ) z 2 R 2 or C 2 _ = ! 2 T d =( R = 2 Z ) d d 0 (1) where F is analytic in z and vanishes for z =0: F (0 ) = 0 for all 2 T d , an imp ortant problem is to nd a transformation ( z ) = (( ) ), with = ( ), such that in the new variables the system takes the simpler normal form 8 < : _ = A 1 ( ) _ = A 2 ( ) _ = !: (2) * Supp orted in part by the Spanish grant BFM2000-805 and the Catalan grant 2000SGR-27. y Supp orted in part by the Spanish grant PB96-1153 and the Catalan grant 1999SGR349. 1
2 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO The existence and convergence of suchchange depends not only on the dimension d , but also on the dynamical character exhibited by the invariant torus z =0. In order to achieve an armativeanswer to this problem, we will restrict ourselves to normal forms of the type _ _ = N ( ) = A ( ) ; A ( ) _ = ! (3) that is, with A 2 = ; A 1 in (2). A system written in this wayis saidto be in Birkho normal form . It is straightforward to verify that it is Hamiltonian , with Hamilton function H ( ), where H ( u ) = R A ( u ) du .Thus, given a (Hamiltonian) system like(1), our original situation has b ecome the problem of seeking for a transformation ( ) 7! ( z ) = (( ) ) leading it into its corresponding Birkho normal form (3). The existence and convergence of the transformation to Birkho normal form has b een proved, for analytic Hamiltonian systems, in several cases: The autonomous case ( d =0) . Here, the equilibrium p oint z = 0 can be hyp erb olic or elliptic, and the rst results were due to Poincare and Birkho. The periodic case ( d =1) . In this situation, z = 0 is a p erio dic orbit = f (0 ) 2 T g . The convergence of the transformation to Birkho normal form is achieved provided is hyp erb olic, that is, with real characteristic exp onents , > 0. The dep endence of F with resp ect to the angle do es not need to be analytic, and it suces to consider F to be C 1 with resp ect to . This result was obtained by J. Moser 5] in 1956. The quasi-periodic case ( d =2) . The invariant ob ject is now a 2-dimensional torus T = f (0 ) 2 T 2 g , assumed to b e hyperb olic, that is, with 3-dimensional stable and unstable asso ciated invariant manifolds. F is assumed analytic in 2 T 2 and the frequency ! of the invariant torus is assumed to be Diophantine, that is, there exist C> 0 and 1such that j k ! j C j k j ; 8 k 2 Z 2 nf 0 g : The existence and convergence of such transformation was provided by A. Delshams et al. 2] in 1997, and can b e easily generalized for d 2. Similar results hold for analytic reversible systems. Indeed, the Birkho normal form (3) is just the form obtained when the normal form (1) is required to be invariant under the involution ( t ) ;! ( ; t ). Thus, the Birkho normal form (3) is the normal form that arises in Hamiltonian or reversible systems in a neighborhood of z =0.
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 3 In the present pap er, we will restrict ourselves to the case d = 0, that is, we will consider a general analytic two-dimensional system _ z = F ( z )where F ( z )= f ( x y ) g ( x y ) : We recall that the pro cess leading to a Birkho normal form for this system is equivalentto the existence of a transformation z =( ), close to the identityand analytic in z = ( x y ), such that the new vector eld N = F = D ] ; 1 F () is of the form given in (3). In other words, F , and N must satisfy the equation D N = F () : (4) Since the functions f and g involved in this pro cess are general (they are just assumed to be analytic in x , y ), the study of its transformation to Birkho normal form is equivalent to the problem of determining if a given system is Hamiltonian or not. Our approach, which follows an idea of D. DeLatte 1], and J. Moser, consists of lo oking for vector elds , N and B satisfying the equation D N + B = F () (5) with B ( )= b 1 ( ) b 2 ( ) : (6) Condition (5) is weaker than the one in (4) and, in particular, implies that do es not haveto be a change of variables, unless B is of the form D W . Indeed, the new system is not necessarily written in Birkho normal form. In a nave way, the remainder term B contains the obstructions of the original system to be Hamiltonian and, therefore, in the planar context, integrable. From now on, we will saythat the transformation leads system _ z = F ( z ) into its Birkho pseudo-normal form (or shorter, its pseudo-normal form) if there exists a vector eld N of the form (3) and a vector eld B of the form ( 6) such that equation (5) is satised. It is worth mentioning that, during the pro cedure of calculating the transformation and the vector elds N and B corresp onding to a given system, the connection b etween the two scalar functions b 1 and b 2 of the remainder term B is not completely determined. That is, its particular asp ect can b e established a priori with a certain degree of freedom. Thus, wemay, depending on the context, consider dierent forms for the vector
4 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO eld B like, for instance 1] B ( )= 0 b ( ) which provides an easy triangular scheme to obtain A ( )and b ( ), or B ( )= b ( ) b ( ) which will b e the one used in this pap er, since it will b e more useful to preserve the geometrical prop erties of the original system.. Wenow present our main result on the convergence of the pseudo-normal form. In its statement, as well as along the pap er, the following notation will b e used: ^ h ( x y ) will denote the terms of order equal or greater than 2 in the variables x y of a function h ( x y ) of these twovariables. Theorem 1.1 (Pseudo-Normal Form Theorem). Let us consider a general system of the form, _ z = F ( z ) z =( x y ) 2 R 2 or C 2 (7) where F ( z )= z + ^ F ( z )= 0 0 ; + ^ f ( x y ) ^ g ( x y ) is analytic in z ,with 6 =0 and such that z =0 is an equilibrium solution. Then, there exist vector elds N ( )= A ( ) ; A ( ) B ( )= b ( ) b ( ) (8) and an analytic in transformation, convergent in a neighborhoodof =0 , z =( ) =( ) leading system (7) into its pseudo-normal form , that is, satisfying D ( ) N ( )+ B ( )= F (( )) : (9) Remark 1.1. A similar result is given in 1], where the convergence of the transformation leading to the pseudo-normal form (9) is proved for the case of a non-necessarily area-preserving mapping, with real, j j > 1and B of typ e (0 b ( )). Our approach is slightly dierent since wefocus on
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 5 planar vector elds and we deal in a unied way,by means of hypothesis 6 = 0, with b oth the hyperb olic and the elliptic (linear center) equilibrium point case. There are several cases, dep ending on the vector eld F , where the remainder term B is known to b e zero, and therefore Theorem 1.1 ensures the convergence of the normal form. One of them is the case of a Hamiltonian vector eld F . Another one is the case of a reversible vector eld F . We recall that, given an involution G ,( G 2 = I d and G 6 = I d ), a system _ X = F ( X ) is called G - reversible if it is invariant under the action ( X t ) 7! ( G ( X ) ; t ). Equivalently, the transformation G conjugates F with ;F , that is G F = ;F , where G F =( D G ) ; 1 F ( G ). The involution G is called a reversing involution of the system _ X = F ( X ). In the particular case that G is linear, we will denote it by R and therefore the last equality b ecomes just R F R = ;F . For instance, R ( )=( )isareversing involution for the Birkho normal form (3). For more details about reversible systems, see 7]. Remark 1.2. The reversing involutions G of a reversible system do not need to be linear. However,itisknown that any reversible system is conjugate, in a neighborho o d of symmetric ob ject, to a linear reversible system (Bo chner Theorem, see 4]). Moreover, having in mind that the invariance under an involution is preserved by co ordinate transformations, it is straightforward to construct families of G -reversible systems with a non-linear G . Namely, if the system _ X = F ( X ) satises R F R = ;F , where R is linear, applying a transformation of the form W = S ( X ), the new system _ W = H ( W ) is reversible with resp ect the reversing (notnecessarily linear) involution G = S R S ; 1 . Wenow summarize these results about convergence of the Birkho normal form in the following corollary of the Pseudo-Normal Form Theorem. Corollary 1.1. Given a Hamiltonian or reversible analytic system _ z = F ( z )= z + ^ F ( z ) z 2 C 2 with a diagonal matrix ( ; ) , 6 =0 , there exists an analytical change of variables z = ( ) , convergent in a neighborhood of the origin, which leads it into its Birkho Normal Form (i.e., with B 0 ). The result above simply requires an additional study of the form of the transformations of Theorem 1.1 leading to pseudo-normal form, that will be p erformed in section 5. Interestingly enough, that result for the Hamiltonian case can be derived from a Criterium of Integrability(Theorem 5.1), which establishes that the vanishing of the remainder term B in the pseudo-normal form is equivalentto the existence of an analytical
6 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO rst integral of system (7) around the origin. Therefore, for the vector eld F of system (7), its Hamiltonian character is equivalent to its reversibility and also to its integrability. For a non-integrable vector eld F ,the vector eld N in the pseudonormal form (9) plays the r^ole of its integrable part, whereas the vector eld B plays the r^ole of the obstruction to integrability. Since they have the form N ( )= A ( ) ; A ( ) B ( )= b ( ) b ( ) each one of them is determined by an analytic function A , b of the variable u = .It is worth noticing that each zero u = of the function b gives rise to a solution of system (7) given explicitly by z ( t )=( exp( tA ( u )) exp( ; tA ( u ))) : (10) If in system (7), F isarealvector eld with b oth critical exponents ; real (and non-zero: the saddle case), the transformation and the vector elds N , B can b e chosen also real, as well as the functions A , b ,whichare real analytic functions of the real variable u = . In accordance with the Criterium of Integrability, the function b ( u ) measures the non-integrability of the vector eld F . On the other hand, the non-constantcharacter of the function A ( u ) measures the aniso chronicityof the vector eld F . Indeed, given an integrable vector eld F , it will be conjugated to its normal form N , whose solutions are of the form (10), whichwillbe linear if and only if A is a constant function. On the contrary, we may sp eak ab out an (integrable) iso chronous saddle when b =0 and A is constant. Of course, this notion is simply a particular case of the one given by Christopher et al. 3], whichisalsovalid for more general critical exp onents. If the vector eld F in system (7) is the complexication of a real vector eld with b oth critical exp onents ; imaginary (and non-zero: the linear center case, with its application to the center-focus problem ), it turns out that to come backto the real variables, one has to consider , as conjugate variables (that is, ! = ), and b is a real analytic function of the variable r 2 = , whereas A is a pure imaginary function of the same variable. Alternatively,intro ducing A = ia , a is a real analytic function of the variable r 2 = , By equation (10), eachzero r 2 = of the function b gives rise to a p erio dic solution of system (7) with period T = 2 =a ( r 2 ). Unless b 0 (the integrable case, that is the origin is a center), each zero of a will b e isolated, and will give rise to a limit cycle of system (7). One has, in this way, a new to ol to lo cate limit cycles close to linear centers of analytic systems in the plane. Of course, in the case of a center ( b 0), it is clear by equation (10) the r^ole of aniso chronicityplayed by the function a .
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 7 Indeed, from the series of a : a ( r 2 )= a 0 + a 1 r 2 + a 2 r 4 + a 3 r 6 + convergent for, say, r 2 <u 0 and where = ia 0 , one obtains straightforwardly the so-called periodconstants T m of the series expansion T =2 =a ( r 2 )=1+ T 1 r 2 + T 2 r 4 + T 3 r 6 + We summarize these results in the following corollary. Corollary 1.2. For any planar vector eld (7) under the assumptions of Theorem 1.1, with a function b in (8) not identical ly zero, we have ( i ) For each zero u = of b , there exists a solution of the form (10) of system (7). ( ii ) In the linear center case ( = ia 0 , a 0 > 0 ), for each zero r 2 of the analytic function b ,there exists a limit cycle of period T =2 =a ( r 2 ) , with A = ia in (8). Coming back to the non-center case, the series expansion of b b ( r 2 )= b 1 r 2 + b 2 r 4 + b 3 r 6 + gives rise to a kind of focal values .In contrast with the center case, where the constants a m are uniquely determined, in the non-center case the constants b m (and therefore a m ) are not uniquely determined, but it turns out that each of them is uniquely determined modulo the ideal generated by the previous ones, as the so-called Lyapunov constants do. This common prop erty gives strong evidences that the constants b m are, in fact, the Lyapunov constants mo dulo some constant term. The rest of the pap er is organized as follows. In the next section, the pro of of the Pseudo-Normal Form Theorem is b egun, at least at a formal level. Later on, in section 3 we detail the inductive pro cess, and nally,in section 4, the pro of of Theorem 1.1 is nished. Section 5 deals with the Criterium of Integrability and the recursive computation of the constants b m . As a corollary, the pro of of the Corollary 1.1 is readily p erformed. 2. PSEUDO-NORMAL FORM THEOREM: FORMAL SOLUTION In order to solve, formally, the so-called homological equation D N + B = F () (11)
8 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO we assume that the remainder function B begins with terms of order at least 2 in , .Intro ducing N = I d + ^ N , = I d + ^ , B = ^ B and F = I d + ^ F (see the notation intro duced b efore Theorem 1.1), we obtain D ( I d + ^ ) ( I d + ^ N )+ ^ B =+ ^ F () which is equivalentto D ^ N ; ^ = ^ F () ; ^ N ; ^ B: Thus, by dening the functional operator, L N ":= D " N ; " equation (11) is equivalentto L N ^ = ^ F () ; ^ N ; ^ B: (12) 2.1. The homological equation Before dealing with the resolution of (12), we study rst the formal solution of the linear equation L N ^ = H (13) that is, the formal invertibility of the functional op erator L N . First of all, notice that L N " is of order equal or greater than 2in =( ) if " is. This means that we can consider H = ^ H =( ^ h 1 ( ) ^ h 2 ( )). Intro ducing ^ =( ^ ( ) ^ ( )), we write the series expansions for the comp onents of ^ and ^ H , ^ ( )= X j + k 2 jk j k ^ ( )= X j + k 2 jk j k and ^ h m ( )= X j + k 2 h ( m ) jk j k , m =1 2. Hence, L N ^ = D ^ N ; ^ = ^ ^ ^ ^ A ( ) ; A ( ) ; 0 0 ; ^ ^ = ( ^ ; ^ ) A ( ) ; ^ ( ^ ; ^ ) A ( )+ ^ = ^ h 1 ( ) ^ h 2 ( ) :
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 9 Equating the rst comp onents of the last expression weget ^ h 1 ( )= X j + k 2 h (1) jk j k = X j + k 2 f ( j ; k ) A ( ) ; g jk j k which, using that A ( )= + ^ A ( )= + X m 1 m ( ) m is equal to X j + k 2 ( j ; k ; 1) jk j k + X m 1 X j + k 2 m ( j ; k ) jk j + m k + m = X j + k 2 ( j ; k ; 1) jk j k + X m 1 X j + k 2( m +1) m ( j ; k ) j ; mk ; m j k : We see that the corresp onding co ecient jk of a given order in , is computed, iteratively, as a function of the co ecient h (1) jk of the same order and co ecients j k of alower order, (which are known from previous steps of the pro cess) provided that j 6 = k +1. The terms of the expansion of ^ that we are not able to determine from our system are of the typ e X k 1 k +1 k ( ) k and are known as resonant terms . So, at the end, following this iterative scheme, we are able to determine the co ecients jk , of the function ( ) if j 6 = k + 1, with arbitrary values for k +1 k , k 0. Analogously, for the second comp onent, such co ecients jk can be obtained, provided that k 6 = j + 1 and for any xed value of jj +1 , j 0. It is worth stressing that the only condition we need to carry out this (formal) pro cedure is that do es not vanish. 2.2. Denition of the pro jections In solving the linear equation (13), wehave seen that it is only p ossible to compute the co ecients of the series expansion of = ( ) corresp onding to non-resonant terms. This fact leads to the following Definition 2.1. Given a function f of the form f ( x y )= X j + k 1 f jk x j y k
16 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO and x a value for P such that P # (2) =0. Then, h R # (1) I d i = F 2 h R # (2) I d i = R n ^ F ( (1) ) o 3] + R ( DF 2 # (1) ) # N (1) +# B (1) = P F 3 + P ( DF 2 # (1) ) and, for K 3, h R # ( K ) I d i = R n ^ F ( ( K ; 1) ) o K +1] + R ( DF 2 # ( K ; 1) ) ; K D ( R # (1) ) # N ( K ; 2) where K =0if K is even and K = 1 for K odd. Moreover, ab out N and B ,ifwe write K =2 M ; 1 with M> 1, wehave # N (2 M ; 1) +# B (2 M ; 1) = P DF 2 # (2 M ; 1) + P n ^ F ( (2 M ; 1) ) o 2 M +1] : Wewant to stress the fact that, b esides the simplicity of the scheme ab ove, its solution at any step is given by a linear equation. Concretely,wemust solve h R # ( K ) I d i = R # H ( K ) (22) where the term on the right-hand side is known from the previous steps of the pro cess. Thus, if we write # ( K ) = ( ) ( ) # H ( K ) = h 1 ( ) h 2 ( ) where ( )= X j + k = K +1 j 6 = k +1 jk j k ( )= X j + k = K +1 k 6 = j +1 jk j k and h ` ( )= X j + k = K +1 j 6 = k +1 h ( ` ) jk j k ` =1 2 the explicit solution for (22) is given by jk = h (1) jk ( j ; k ; 1) where j + k = K + 1 and j 6 = k +1
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 17 jk = h (2) jk ( j ; k +1) where j + k = K + 1 and k 6 = j +1 : 4. PROOF OF THE CONVERGENCE 4.1. Denition of the norms Let us consider the following domain around the origin D r = z =( z 1 z 2 :::z n ) 2 C n : k z k 1 = max j =1 :::n j z j j r and let f ( z ) be an analytic function f ( z )= X 2 N n f z : Writing z = j z j e i' we can also express it in multi-index notation as f ( z )= f ( j z j ' ( z )) = X 2 N n f j z j e i ' (23) b eing ' = 1 ' 1 + 2 ' 2 + + n ' n and ' = ' ( z )=arg z . At rst, for such kind of functions we consider the supremum norm k f k 1 r =sup z 2D r j f ( z ) j : However, this is not the norm we are going to deal with. This new norm, closely related to the ` 2 -norm, will b e dened thanks to the following result. Lemma 4.4. Given a positive real number r ,there exits a unique r 0 such that r = r e r . Hence, we dene k f k r = k f ( j z j ' ( z )) k r =sup j z j j j r 1 (2 ) n Z T n j f ( j z j ' + i ) j 2 d' 1 = 2 with ' = ' ( z ) = arg z and where r satises r = r e r . Notice that it is well-dened, since j z e i ( ' + i ) j j z j e j j r e r = r . Moreover, we can express the Fourier co ecients of f ( j z j ' + i ) in terms of the corresp onding ones of f ( j z j ' ). Namely, applying a shift ' 7;! ' + i
18 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO with j j r , it follows that f j z j = 1 (2 ) n Z T n f ( j z j ' ) e ; i ' d' = 1 (2 ) n Z T n f ( j z j ' + i ) e ; i ( ' + i ) d' = e 1 (2 ) n Z T n f ( j z j ' + i ) e ; i ' d' and then 1 (2 ) n Z T n f ( j z j ' + i ) e ; i ' d' = f j z j e ; : Using the isometry b etween L 2 and ` 2 norms, wehave Z T n j f ( j z j ' + i ) j 2 d' = X 2 N n j f j 2 j z j 2 e ; 2 so sup j z j j j r 1 (2 ) n Z T n j f ( j z j ' + i ) j 2 d' 1 = 2 = 1 (2 ) n X 2 N n j f j 2 r 2 e 2 j j r ! 1 = 2 where j j = j 1 j + j 2 j + + j n j . Indeed we can write the norm dened ab ove, in a equivalentway, as an slightly weighted ` 2 -norm, k f k r = k f ( j z j ' ( z )) k r = 1 (2 ) n X 2 N n j f j 2 r 2 e 2 j j r ! 1 = 2 with r = r e r and ' ( z )=arg z . Such norm satises some useful prop erties collected in the following lemma, whose pro of is standard. Lemma 4.5. ( i ) For any positive number r we have k f k r k f k 1 r : ( ii ) Given a positive number R , for any , r satisfying 0 <<r R , the fol lowing estimate k f k 1 c 0 ( r ; ) n= 2 k f k r
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 19 holds, where c 0 = c 0 ( n ) is a constant which depends on n and R . ( iii ) Let us consider an analytic function ": D 7!D s , satisfying that k " k s . Then, if s<r we have k f " k c 0 ( r ; s ) n= 2 k f k r : ( iv ) Let f M ( z ) be an homogeneous polynomial of order M , f M ( z )= X j j = M f z : Then, if 0 r R , we have the fol lowing bound k f M k r r R M k f M k R where s means the unique positive real number satisfying s = s e s . This norm can b e easily extended to a norm for vector elds. More precisely,wehave Definition 4.1. Let us consider a vector eld F ( z ) = ( f ( z ) g ( z )), analytic in D r . Then, we dene k F k r = ; k f k 2 r + k g k 2 r 1 = 2 : It is straightforward to verify that this norm satises analogous results to the ones achieved in the previous lemma. 4.2. Convergence of the iterativescheme The scheme we will followto provethe convergence of , N and B is supp orted on simple ideas. Namely,since = I d + P K 1 # ( K ) , N = I d + P K 2 # N (2 K ; 1) and ^ B = P K 2 # B (2 K ; 1) ,we will get estimates for k # ( K ) k , k # N (2 K ; 1) k and k # B (2 K ; 1) k in a suitable domain D abit smaller than the original one, D r 0 . More precisely, we will obtain b ounds of typ e a 2 M ; 1 k 1 " 2 M + k 2 "a 2 M ; 3 a 2 M ; 2 k 3 " 2 M ; 1 + k 4 "a 2 M ; 3 where a K := k # ( k ) k and 0 <"< 1, k 1 :::k 4 are suitable constants. It will b e derived from these expressions the convergence of k k . The corresp onding estimates for N and B come from the fact that k k ma jorates them.
20 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO Let us b e more precise. First, assume that the following quantities are nite on D r 0 L 0 := k ^ F k r 0 L 1 := k DF 2 k r 0 Moreover, consider r j = r j ; 1 ; , j =1 2 3, intermediate radii and dene 0 < := r 3 r 0 < 1 with r 3 and r 0 the unique p ositivenumbers satisfying r 3 = r 3 e r 3 and r 0 = r 0 e r 0 , respectively.Since the nal domain of convergence will b e D r 3 ,we dene a K := k # ( K ) k r 3 : Since # (2 K ; 1) = O 2 K ] ,wehavethat P # (2 K ; 1) = 0 and, consequently, R # (2 K ; 1) =# (2 K ; 1) . Furthermore, remind that during the iterative scheme the value of the pro jection P ^ can b e chosen arbitrary,sowewill need to imp ose some conditions on this term. Concretely,wewantittobe absolutely convergentin such norm and b ounded by the norm of the vector eld ^ F , more precisely, X K 1 kP # ( K ) k r 0 c 3 k ^ F k r 0 (24) where c 3 is a p ositive constant (that could b e large). Let us fo cus our attention on the former estimates. From equation R # (1) I d = F 2 it turns out that j j a 1 L 0 . From h R # (2) I d i = R F 3 + R DF 2 # (1) and hyp othesis (24), one gets j j a 2 ( c 3 j j +1) L 0 + L 1 a 1 . Finally, k # N (1) +# B (1) k r 3 L 0 + L 1 a 1 .It is now imp ortant to remark the concrete shap e of the vector elds N and B . Since N ( )=( A ( ) ; A ( )) and B ( )=( b ( ) b ( )), it is not dicult to check that k # N (2 K ; 1) k r 3 k # B (2 K ; 1) k r 3 c 2 k # N (2 K ; 1) +# B (2 K ; 1) k r 3 where c 2 is a constant dep ending only on r 0 . Therefore, the previous b ounds b ecome k # N (1) k r 3 L 0 + L 1 a 1 and k # B (1) k r 3 L 0 + L 1 a 1 . Our aim is to get recurrent estimates on a 2 K ; 1 . To this purp ose let us consider the equation h R # (2 M ; 1) I d i = R n ^ F ( (2 M ; 2) ) o 2 M ] + R ( DF 2 # (2 M ; 2) ) ; D ( R # (1) ) # N (2 M ; 3) :
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 21 A rst nave estimate reads j jkR # (2 M ; 1) k r 3 k n ^ F ( (2 M ; 2) ) o 2 M ] k r 3 + k DF 2 k r 3 k # (2 M ; 2) k r 3 + k D ( R # (1) ) k r 3 k # N (2 M ; 3) k r 3 (25) but can b e rened by dealing with its terms separately. Namely, ( i ) Since D ( R # (1) )= O 1] , applying the last lemma, it follows that k D ( R # (1) ) k r 3 k D ( R # (1) ) k r 0 2 L 0 j j : ( ii ) Wehave k F (2 M ; 2) k r 3 c 1 k F k r 1 provided k (2 M ; 2) k r 3 <r 2 . So, using again the same lemma, it turns out k n ^ F ( (2 M ; 2) ) o 2 M ] k r 3 2 M k n ^ F ( (2 M ; 2) ) o 2 M ] k r 2 c 1 L 0 2 M : ( iii ) Having in mind that DF 2 = O 1] , it follows that k DF 2 k r 3 k DF 2 k r 0 L 1 : ( iv ) Since k # N (2 J ; 1) +# B (2 J ; 1) k r 3 k DF 2 k r 3 k # (2 J ; 1) k r 3 + kP n ^ F ( (2 J ; 1) ) o 2 J ; 1] k r 3 wehave k # N (2 M ; 3) k r 3 c 1 c 2 L 0 2 M ; 1 + c 2 L 1 a 2 M ; 3 : ( v ) From h R # (2 M ; 2) I d i = R n ^ F ( (2 M ; 3) ) o 2 M ; 1] + R DF 2 # (2 M ; 3) and assumption (24), it is deduced that j j a 2 M ; 2 c 1 + c 3 j j L 0 2 M ; 1 + L 1 a 2 M ; 3 :
22 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO Applying together b ounds ( i ) ; ( v )onto inequality (25), we arriveat the following estimate j j a 2 M ; 1 L 0 c 1 j j +2 c 1 c 2 L 0 + c 1 L 1 + c 3 j j L 1 j j 2 M + L 1 j j ( L 1 +2 L 0 ) 2 a 2 M ; 3 whichinvolves only o dd terms of the sequence f a K g K . Rening the constants we reach the nal expression a 2 M ; 1 K 1 j j 2 2 M + K 2 j j 2 2 a 2 M ; 3 where K 1 = j j ( c 1 + c 3 L 1 )+2 c 1 c 2 L 0 + c 1 L 1 K 2 = L 1 ( L 1 +2 L 0 ) dep end on k F k r 0 , k DF 2 k r 0 and j j . In the same way,from( v ), a 2 M ; 2 K 3 j j 2 M ; 1 + K 4 j j a 2 M ; 3 with K 3 = c 1 + c 3 j j and K 4 = L 1 also dep end only on k F k r 0 , k DF 2 k r 0 and j j . Cho osing suchthat j j 2 L 1 ( L 1 +2 L 0 ) the inequality satised by a 2 M ; 1 ,for M 2, becomes a 2 M ; 1 K 1 j j 2 2 M + a 2 M ; 3 : (26) The convergence of the series P M 1 a 2 M ; 1 (and, therefore, of P M 1 a 2 M ), comes directly from the application of the following result. Lemma 4.6. Let us consider a sequence f a k g k ,with a k 0 8 k , such that the fol lowing recurrent inequalities are satised, for m 2 and 0 < "< 1 , a 2 m ; 1 k" 2 m + "a 2 m ; 3 a 2 m ; 2 k 1 " 2 m ; 1 + k 2 "a 2 m ; 3 : (27)
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 23 Then, X m 1 a m is convergent. Proof. We start bychecking that P m 1 a 2 m ; 1 is convergent. Tothis end, we apply recursively the rst equation in (27), getting a 2 m ; 1 k ; " 2 m + " 2 m ; 1 + + " m +1 + " m ; 1 a 1 km" m +1 + " m ; 1 a 1 : So, X m 1 a 2 m ; 1 a 1 + k X m 2 ( m ; 1) " m +1 + a 1 X m 2 " m ; 1 which is not dicult to see that is equal to a 1 1 ; " + k " 3 +2 " (1 ; " ) (1 ; " ) 2 : Finally, the convergence of the even part comes from X m 2 a 2 m ; 2 k 1 X m 2 " 2 m ; 1 + k 2 " X m 2 a 2 m ; 3 : Notice that this lemma gives the convergence of k k r 3 ,provided wetake " = , with satisfying condition (26). Concretely, k k r 3 kI d k r 3 + X m 1 k # ( m ) k r 3 = r 3 p 2 + X m 1 a m : Moreover, b ecause of the restriction imp osed by ( ii )on k (2 M ; 2) k r 3 we cho ose in suchawaythat P a m is less than r 2 (= r 0 ; 2 ). The convergence, in k k r 3 -norm, of N and B is easily derived from the estimates, k # N (2 M ; 1) k r 3 k # B (2 M ; 1) k r 3 c 1 L 0 2 M +1 + L 1 a 2 M ; 1 : In this waywe get analiticity of the transformation and the vector elds N and B in a domain D r 3 , where r 3 = r 3 e r 3 and r 3 = r 0 .This concludes the pro of of the Pseudo-Normal Form Theorem. 5. INTEGRABILITY AND PSEUDO-NORMAL FORMS Integrability is closely related to an sp ecial pseudo-normal form of the system. Precisely, the existence of a rst integral will depend on the fact that b ( )vanishes.
24 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO Theorem 5.1 (Criterium of integrability). Let us consider a system _ z = F ( z )= z + ^ F ( z ) (28) with = diag ( ; ) and 6 =0 , verifying that thereexistvector elds N , B and a transformation z =( ) , N ( )= A ( ) ; A ( ) B ( )= ^ B ( )= b ( ) b ( ) leading it to its pseudo-normal form, i.e. satisfying D N + B = F () . Then, _ z = F ( z ) has a rst integral h ( z ) if and only if B 0 .Moreover, if h ( z ) = h ( x y ) is a rst integral of this system, then it has the form h = ~ h ; 1 ,where ~ h ( )= ~ h ( ) , depending only on the product . Remark 5.1. Since B ( )=( b ( ) b ( )), this result also holds for the scalar function b ( ). Proof. It is clear that if B 0, this is, our system can b e written in Birkho normal form, any function of the form ~ h ( ) is a rst integral. Moreover, this is the unique kind of rst integrals it has. Then, it is straightforward to obtain one for the initial system. To prove the theorem in the other sense, we apply some ideas given by C.L. Siegel and J.K. Moser (see 8, x 30]). Indeed, assuming that system (28) has a rst integral h and that B 6 =0we arrive at a contradiction. Performing the transformation z =( ), system (28) b ecomes _ = N ( )+ D ( )] ; 1 B ( ) : (29) It is easy to verify that h ( z )is arstintegral of _ z = F ( z )ifandonly if ( h )( )isarstintegral of (29). This rst integral ~ h = h can be written as ~ h ( )= ~ h M ( )+ ~ h M +1 ( )+ with ~ h M ( ) 6 = 0, M 1, ~ h J ( )b eing homogeneous p olynomials of order J in , . Then, since ~ h is a rst integral, wehave that the equation D ~ h N + D ] ; 1 B =0 (30) holds for any order in the variables , .On the other hand, we know that b egins with the identity and that B is of order greater or equal than 3 in , . Therefore, the homogeneous p olynomial of minimal order we get from the left-hand side of (30) comes from D ~ h N @ @ ~ h M ( ) @ @ ~ h M ( ) + ; ; (31)
A PSEUDO-NORMAL FORM FOR PLANAR VECTOR FIELDS 25 where ( ; ) is the linear part of N ( )=( A ( ) ; A ( )). If we write ~ h M ( )= X j + k = M h ( M ) jk j k equations (30) and (31) lead to X j + k = M ( j ; k ) h ( M ) jk j k =0 so, if j 6 = k , we have that h ( M ) jk =0. In other words, if ~ h ( ) is a rst integral of system (29), then it starts with a term h m ( ) m ,where m = M= 2 and h m 6 =0. Once we know how the rst integral ~ h begins, we seek for the term of typ e ( ) s on the left-hand side of equation (30), having minimal order in , . First, notice that D ~ h ( ) N ( )doesnotcontribute to this kind of terms, b ecause if ~ h ( )= + c ` ( ) ` + + d jk j k with j 6 = k , it follows that D ~ h ( ) N ( ) = ; + `c ` ( ) ` ; 1 + + jd jk j ; 1 k + + `c ` ( ) ` ; 1 + + kd jk j k ; 1 + A ( ) ; A ( ) = + c ` ( ` ; ` )( ) ` + + d jk ( j ; k ) j k + = d jk ( j ; k ) j k + Concerning the second part of (30), since we are assuming B 6 =0, there must exist a non-zero constant ` ,in such a waythat b ( )= ` ( ) ` + h.o.t. Moreover, since D ] ; 1 B = B + , the ( ) s -term of minimal order provided by D ~ h D ] ; 1 B comes from ; mh m ( ) m ; 1 + mh m ( ) m ; 1 + ` ( ) ` + ` ( ) ` + =2 mh m ` ( ) m + ` + where + means terms of higher order. From equation (30), it follows that mh m ` = 0, which is a contradiction since m , h m and ` do not vanish.
32 A. DELSHAMS, A. GUILLAMON AND J.T. L AZARO 7. Sevryuk, M.B. Reversible systems . Volume 1211 of Lecture Notes in Mathematics , Springer, Berlin, 1986. 8. Siegel, C.L. and Moser, J.K. Lectures on Celestial Mechanics . Springer, Berlin, Heidelb erg, New York, 1971.