Singular splitting of separatrices for the perturbed McMillan map
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SINGULAR SPLITTING OF SEPARATRICES FOR THE PERTURBED MCMILLAN MAP AMADEU DELSHAMS y RAFAEL RAM IREZ-ROS DepartamentdeMatematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona (Espa ~na). The problem In this lecture, we consider the following family of planar standard-likemaps F ( x y )= y ; x + 2 0 y 1+ y 2 + "V 0 ( y ) ! where V ( y )= P n 1 V n y 2 n is an even entire function. Provided that 0 + V 1 " > 1, the origin O = (0 0) is a hyp erb olic xed point with Sp ec d F ( O )] = n e h o , and its characteristic exponent h> 0isgiven bycosh h = 0 + V 1 " . For " =0, F is an integrable map (called McMil lan map ), whose stable and unstable invariant curves to the origin coincide, giving rise to a separatrix .Thus, the map F can be considered as a p erturbation of the McMillan map, " b eing the perturbation strength . These two parameters, h> 0and " , will b e considered the intrinsic parameters of the map F under study. Our goal is to show that for " 6 = 0 and for a general p erturbation, the stable and unstable invariant curves of the p erturb ed map intersect transversally along exactly two primary homoclinic orbits in the rst quadrant in particular, the unp erturb ed separatrix splits. The term primary means that the homo clinic orbits p ersist for all " small enough. The pieces of the p erturb ed invariantcurves b etween two consecutive homo clinic p oints enclose a region called lobe .Our measure of the splitting size will be the area of this lob e. This lobe area is a homo clinic symplectic invariant, that is, it do es not dep end on the symplectic co ordinates used, and all the lob es have the same area. Lob e areas also measure the ux along the homo clinic tangle, which is related to the study of transp ort. Both parameters, h > 0and " , will be small \enough", but the exact interpretation of this sentence is crucial for understanding the dierent kinds of results to b e presented. Sp ecically,we are going to deal with the following situations: 1. The regular case : xed h> 0, and " ! 0. 2. The singular case : h ! 0 + . In its turn this case sub divides in two sub-cases: (a) The non-perturbative case: " xed and h ! 0 + . (b) The perturbative case: " = O ( h p )and h ! 0 + , for some p> 0. The analytical results here presented are expressed in terms of the Melnikov potential of the problem, which gives explicit formulae for our map. This is the reason for our choice of the p erturb ed McMillan map as a mo del, instead of more celebrated maps like the Henon map or the Taylor-Chirikov map. (See 6] for results concerning those maps.) The name \singular" for the case h ! 0 + ,is due to the fact that the lob e areas are exp onentially small in h . The measure of suchsmall quantities requires a very careful treatment, b oth from a numerical and an analytical p ointofview.
The mo del The family of standard-like maps under study is given by F ( x y )=( y ; x + U 0 ( y )) U ( y )= 0 log(1 + y 2 )+ "V ( y ) (1) where V ( y )= P n 1 V n y 2 n is an even entire function. For := 0 + "V 1 > 1 the origin O = (0 0) is a hyp erb olic xed point with Sp ec d F ( O )] = n e h o ,where the characteristic exponent h> 0is determined by cosh h = . We will consider the characteristic exp onent h and the perturbation strength " as the intrinsic parameters of our mo del. Accordingly, for every h > 0and every real " , we rewrite the map (1)intheform F ( x y )= ( y ; x + U 0 ( y )) U ( y )= U 0 ( y )+ "U 1 ( y ) U 0 ( y )= log(1 + y 2 ) U 1 ( y )= V ( y ) ; V 1 log(1 + y 2 ) : (2) From now on, the subscript \0" will denote an unp erturb ed quantity, that is, " =0, and the following notations will b e used without further comment: = cosh h =sinh h: Setting " = 0 in (2), we obtain the so-called McMil lan map F 0 ( x y )=( y ; x + U 0 0 ( y )) = y ; x + 2 y 1+ y 2 ! which is an integrable map, with a p olynomial rst integral given by I 0 ( x y )= x 2 ; 2 xy + y 2 + x 2 y 2 : The phase space asso ciated to F 0 is rather simple, since it is foliated by the level curves of the rst integral I 0 , which are symmetric with resp ect to the origin. As > 1, the zero level of I 0 is a lemniscate, whose lo ops are separatrices to the origin. From nowon,we will concentrate on the separatrix in the quadrant f x y > 0 g ,whichcan be parameterized by z 0 ( t )=( x 0 ( t ) y 0 ( t )) = ( 0 ( t ; h= 2) 0 ( t + h= 2)) 0 ( t )= sech t: (3) This parameterization is called natural since F 0 ( z 0 ( t )) = z 0 ( t + h ), a fact that can be checked simply by noting that 0 ( t ) is a homo clinic solution of the dierence equation 0 ( t + h )+ 0 ( t ; h )= U 0 0 ( 0 ( t )) : A natural parameterization is unique except for a translation in the indep endentvariable. To determine it,itisworth lo oking at the reversors of the map. Indeed, the involution R + 0 ( x y ):=( y x )isa reversor of the McMillan map F 0 , that is, F ; 1 0 = R + 0 F 0 R + 0 .The separatrix is R + 0 -symmetric, i.e., R + 0 =, and intersects transversely the xed set C + 0 := f z : R + 0 z = z g of R + 0 in one p oint z + 0 . The parameterization (3) ofhas been chosen to satisfy z 0 (0) = z + 0 .
Moreover, the involution R ; 0 := F 0 R + is another reversor of F 0 .The separatrix is also R ; 0 -symmetric and intersects transversely the xed set C ; 0 of R ; 0 in one p oint z ; 0 , and it turns out that z 0 ( h= 2) = z ; 0 .The asso ciated orbits O + 0 := f z 0 ( nh ) : n 2 Z g , O ; 0 := f z 0 ( h= 2+ nh ): n 2 Z g , are called symmetric homo clinic orbits, since R 0 O 0 = O 0 . For " 6 = 0, the phase p ortrait of the exact map (2) lo oks more intricate. The origin is a hyp erb olic xed p oint with the same characteristic exp onent h , since the p erturbation "U 0 1 ( y ) = O ( y 3 ) do es not not contain linear terms at the origin. We denote by W u s its unstable and stable invariant curves with resp ect to F . Since the map (2) is odd, the invariantcurves are symmetric with resp ect to the origin, so that we concentrate only on the p ositive quadrant f x y > 0 g . By the form of the p erturbation, R + := R + 0 is also areversor of F , as well as the involution R ; := FR + , which is given by R ; ( x y ) = ( x ; y + U 0 ( x )). Their xed sets C = f z : R z = z g are imp ortant b ecause R W u = W s . Consequently, any point in the intersection C \W u is a homo clinic p oint, and gives rise to a symmetric homo clinic orbit. Since the separatrix intersects transversely the unp erturb ed curve C 0 at the p oint z 0 , there exists a point z = z 0 + O ( " ) 2 C \W u and, therefore, there exist at least two symmetric homo clinic orbits on the quadrant f x y > 0 g ,for j " j small enough. They are called primary since they exist for arbitrary small j " j . The Melnikov theory Wenow recall some p erturbative results 1, 2] to detect the existence of transverse primary homo clinic orbits for exact maps. For simplicity, we will assume that all the ob jects are smo oth and we shall restrict the discussion to maps on the plane with the usual symplectic structure: the area. Given the symplectic form ! = d x ^ d y on the plane R 2 , a map F : R 2 ! R 2 is called exact if there exists some function S : R 2 ! R suchthat F ( y d x ) ; y d x = d S . The function S is called the generating function of F and, except for an additive constant, it is uniquely determined. Let F 0 : R 2 ! R 2 be an integrable exact dieomorphism with a separatrix to a hyp erb olic xed p oint z 1 . Next, consider a family of exact dieomorphisms F " = F 0 + "F 1 + O ( " 2 ), as a general p erturbation of the situation ab ove, and let S " = S 0 + "S 1 + O ( " 2 ) be the generating function of F " . Weintro duce the Melnikov potential of the problem as the smo oth real-valued function L : ! R L ( z )= X n 2 Z b S 1 ( z n ) z n = F 0 n ( z ) z 2 (4) where b S 1 : R 2 ! R is dened by b S 1 = S 1 ; y d x ( F 0 ) F 1 ]. (In comp onents, writing F 0 =( X 0 Y 0 ), F 1 =( X 1 Y 1 ), b S 1 is simply given by b S 1 = S 1 ; Y 0 X 1 .) In order to get an absolutely convergent series (4), b S 1 is determined by imp osing b S 1 ( z 1 )=0. The dierential of L is a geometrical ob ject whichgives the O ( " )-distance b etween the p erturb ed invariant curves W u s . More precisely, let ( t e )be some cotangent coordinates adapted to |that is, in these co ordinates the separatrix is given lo cally by f e = 0 g and the symplectic form ! reads as d t ^ d e |and let f e = E u s ( t ) g be apart of W u s . (Let us recall that cotangent co ordinates can be dened in neighb orho o ds of Lagrangian sub-manifolds.) Then, in 2]itisshown that E u ( t ) ; E s ( t )= "L 0 ( t )+ O ( " 2 )
and that the construction ab ove do es not dep end on the cotangent co ordinates used. The following theorem is astraightforward corollary of this geometric construction. Theorem 1 Under the above notations and hypotheses, the non-degenerate critical points of L are associated to perturbed transverse homoclinic orbits. Moreover, when al l the critical points of L are non-degenerate, al l the primary homoclinic orbits arising from are found in this way. Final ly, if z and z 0 are consecutive (in the internal order of the separatrix) non-degenerate critical points of L , their associated perturbed homoclinic orbits determine a lobe with area A = " L ( z ) ; L ( z 0 )] + O ( " 2 ) : The regular case We are now ready to apply the theory ab ove to our mo del. Along this section, the characteristic exp onent h> 0willbe considered xed ,andthen " ! 0. It is worth noting that the knowledge of the natural parameterization (3) of the unp erturb ed separatrix will be the crucial point to compute explicitly the Melnikov potential (4). The map F = F 0 + "F 1 + O ( " 2 )given in (2) is exact with generating function S ( x y )= ; xy + U 0 ( y )+ "U 1 ( y ). Writing its expression in comp onents F 0 =( X 0 Y 0 ), F 1 =( X 1 Y 1 ), it turns out that X 1 =0, and consequently b S 1 ( x y )= S 1 ( x y )= U 1 ( y ). The parameterization (3) allows us to write the Melnikov p otential (4) of our problem as L ( t ):= L ( z 0 ( t )) = X n 2 Z U 1 ( y 0 ( t + hn )) = X n 2 Z f ( t + hn ) ; g ( t + hn )] where f ( t ):= V ( 0 ( t + h= 2)) and g ( t ):= V 1 log (1 + 0 ( t + h= 2) 2 ). We are now confronted to the computation of L ( t ). Let P n 2 Z v n ( h ) 2 n b e the Laurent expansion around =0 of the function 7! f ( ; h= 2+ i = 2 ; i h ), and 0 ( h ):=8 X n 1 (2 ) 2 n ; 1 (2 n ; 1)! v ; n ( h )=8 b V (2 )+ O ( h 2 ) (5) where b V ( ):= P n 1 V n 2 n ; 1 = (2 n ; 1)! is the so-called Borel transform of V ( y ). If V ( y )is a p olynomial, 0 ( h ) can b e explicitly computed in a nite numb er of steps. For instance, 0 ( h )= ( 8 2 2 h ; 2 for V 0 ( y )= y 8 3 2 4 h ; 2 1 + 2 h ; 2 ]for V 0 ( y )= y 3 : It turns out that 0 ( h ) is an even entire function such that L ( t )=constant+ e ; 2 =h cos(2 t=h ) h ; 0 ( h ) = 2+ O ( e ; 2 2 =h ) i : (6) We refer to 3] for the details. From the formula (5), it is clear that if b V (2 ) 6 = 0 and h is small enough, the set of critical p oints of the Melnikov p otential (6) is h Z = 2. All of them are non-degenerate, and parameterize the two unp erturb ed, symmetric, primary homo clinic orbits O 0 .Now, the following result is a corollary of theorem 1.
Theorem 2 Assume that b V (2 ) 6 =0 . Then, for any smal l enough (but xed) characteristic exponent h > 0 , there exists a positive constant " = " ( h ) such that the map (2) has exactly two transverse, symmetric, primary homoclinic orbits O in the quadrant f x y > 0 g ,for 0 < j " j <" . These orbits determine a lobe with area A = "A Mel + O ( " 2 ) , where the rst order in " approximation A Mel is given by A Mel = L ( h= 2) ; L (0) = e ; 2 =h h 0 ( h )+ O ( e ; 2 2 =h ) i : We note that "A Mel is the dominant term for the Melnikov formula of the lob e area A only if j " j < " ( h ) = o (exp( ; 2 =h )). Otherwise, in the case " = O ( h p ), the Melnikov theory as describ ed is not useful, since it only gives the very coarse estimate A = O ( h 2 p ), and not the desired exp onentially small asymptotic b ehavior. The singular non-p erturbative case The limit h ! 0 + in (2) is highly singular, since all the interesting dynamics is contained in a O ( h ) neighborhood of the origin, which b ecomes a parab olic point of the map for h =0. To see clearly this behavior, we p erform the following linear change of variables: z = Cw C = h ; 1 = 2 1 = 2 1 = 2 ; 1 = 2 ! z =( x y ) w =( u v ) that is, we diagonalize the linear part of (2) at the origin and we scale by a factor h . Then, C ; 1 ( F ( Cw )) = w + hX 0 ( w )+ O ( h 2 ), where X 0 ( u v )= u ; ( u + v ) 3 ; v + ( u + v ) 3 =1 ; ( V 1 +2 V 2 ) " (7) is a Hamiltonian vector eld, with asso ciated Hamiltonian H 0 ( u v )= uv ; ( u + v ) 4 = 4 : (8) This shows clearly that C ; 1 FC is O ( h )-close to the identity, and that, after the change of variables z = Cw ,the map (2) asymptotes to the Hamiltonian ow asso ciated to the vector eld (7) when h ! 0 + . When such situation takes place, it is known that the map has homo clinic p oints to its (weakly) hyp erb olic xed p ointfor h ! 0 + , if and only if the limit Hamiltonian ow has a homo clinic orbit to its hyp erb olic equilibrium p oint. From the expression (8), we see that the zero level f H 0 ( u v )= 0 g contains homo clinic connections to the origin if and only if > 0, i.e., if and only if ( V 1 +2 V 2 ) "< 1 : (9) Assuming > 0, the homo clinic orbit of the Hamiltonian (8) is given by w 0 ( t )= ; 1 = 2 cosh t ; sinh t 2 cosh 2 t cosh t + sinh t 2 cosh 2 t ! which is analytic on the strip f t 2 C : j= t j <d := = 2 g .In this situation, it is also wellknown 5] that the splitting size is O (exp( ; =h )), for all < 2 d = 2 . We summarize this result in the following theorem. Theorem 3 For any real " verifying (9), and any 2 (0 2 ) ,thereexists a constant N = N ( " ) 0 such that the area of the lobe between the invariant curves of the map (2) satises: j A j N e ; =h ( " xed h ! 0 + ) :
The singular p erturbative case The previous theorem gives only an upp er b ound for the lob e area and not an asymptotic one (the constant N ( " ) can blow up when ! 2 ). On the other hand, it do es not exclude the case A = 0, that is, it cannot detect eective splitting of separatrices. In the p erturbativecase " = O ( h p ), for p> 6, the following result gives an asymptotic expression for the lob e area in terms of the Melnikov p otential, and establishes transversal splitting of separatrices. Theorem 4 Assume that " = O ( h p ) , p > 6 . Then, if b V (2 ) 6 = 0 ,there exists h > 0 such that the map (2) has exactly two transverse, symmetric, primary homoclinic orbits in the rst quadrant, for al l 0 <h<h . Moreover, they enclose a lobe with area A = " e ; 2 =h h 8 b V (2 )+ O ( h 2 ) i ( h ! 0 + ) : If b V (2 )=0 , there may exist more primary homoclinic orbits, but the areaofanylobe is O ( "h 2 e ; 2 =h ) . The pro of of this theorem is contained in 3, 4]. It is based on the study of the p erturb ed invariant curves of the maps (1) for complex values of the discret time t , as close as p ossible to the singularities of the unp erturb ed natural parameterization z 0 ( t ) given in (3). This approachwas suggested by V.I. Lazutkin several years ago, for the case of the Taylor-Chirikov map. To the b est of our knowledge, this theorem is the rst analytical result ab out asymptotics for singular separatrix splitting for a map with a complete and rigorous pro of. Anumerical study for singular cases can b e found in 4]. The numerical results suggest that the lob e area A is given by A = " e ; 2 =h h " ( h )+ O ( e ; 2 2 =h ) i ( " xed h ! 0 + ) where " ( h )isaneven Gevrey-1 function such that the radius of convergence of its Borel transform is 2 2 , and " ( h )= 0 ( h )+ O ( " ), uniformly in h 2 0 1]. We nish this lecture by remarking that the numerical computation of the lob e areas for singular cases requires the use of an exp ensive multiple-precision arithmetic and to expand the invariant curves W u s up to an optimal order, whichis very large, see 4]. References 1] A. Delshams and R. Ram"rez-Ros, \Poincare-Melnikov-Arnold metho d for analytic planar maps", Nonlinearity , 9 , (1996), 1{26. 2] A. Delshams and R. Ram"rez-Ros, \Melnikov p otential for exact symplectic maps", to app ear in Comm. Math. Phys. , (1997). 3] A. Delshams and R. Ram"rez-Ros, \Exp onentially small splitting of separatrices for p erturb ed integrable standard-like maps", to app ear in J. Nonli. Sci. , (1997). 4] A. Delshams and R. Ram"rez-Ros, \Singular separatrix splitting and Melnikov metho d: An exp erimental study", submitted to Experiment. Math. , (1997). 5] E. Fontich and C. Simo, \The splitting of separatrices for analytic dieomorphisms", Ergod. Th. & Dynam. Sys. , 10 ,(1990), 295{318. 6] V.G. Gefreich, V.F. Lazutkin and M.B. Tabanov, \Exp onentially small splitting in Hamiltonian systems", Chaos , 1 , (1991), 137{142.