Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions
NNORMAL FORMS FOR MANIFOLDS OF NORMALLY HYPERBOLICSINGULARITIES AND ASYMPTOTIC PROPERTIES OF NEARBYTRANSITIONS
NATHAN DUIGNAN
Abstract.
This paper contains theory on two related topics relevant to manifolds of normallyhyperbolic singularities. First, theorems on the formal and C k normal forms for these objectsare proved. Then, the theorems are applied to give asymptotic properties of the transitionmap between sections transverse to the centre-stable and centre-unstable manifolds of somenormally hyperbolic manifolds. A method is given for explicitly computing these so calledDulac maps. The Dulac map is revealed to have similar asymptotic structures as in the caseof a saddle singularity in the plane. Introduction
Due to their persistence properties and common attributes with hyperbolic singularities, nor-mally hyperbolic manifolds have been studied and applied in great depth by many authors, seefor instance [20]. However, there appears to be little research aimed at normally hyperbolic man-ifolds consisting entirely of singular points. This is primarily a consequence of their structuralinstability under C -perturbations. Nevertheless, a general investigation of these manifolds iswarranted by recent applications in celestial mechanics [6], control theory [4], regularisation ofsingularities [7], geometric singular perturbation theory [8], and bifurcation theory [16].This work is a first venture into the properties of normally hyperbolic manifolds of singularitiesconsidered in generality. Technical results on two related topics of normal form theory areprovided. The first concerns normal form theory for these manifolds. This is studied in theformal and C k categories. The second is a study of transitions between sections transverseto the centre-stable and centre-unstable manifolds of normally hyperbolic manifolds consistingentirely of saddle singularities. We provide an extension of the work on hyperbolic saddles in R by Bonckaert and Naudot [2], and the ‘almost planar case’ of Roussarie and Rousseau [16].Moreover, the generalisation agrees with the particular application considered by Caillau et al.[4]. The Dulac maps in the general case will be shown to share many properties of the wellstudied Dulac maps in the plane.The paper begins with an investigation of normal forms in Section 2. In essence, normal formtheory aims to define the ‘simplest’ possible representation of vector field X . Two vector fieldsare said to be C k (resp. analytically, formally) conjugate if there exists a C k (resp. analytic,formal) coordinate change between them. A C k (resp. analytic, formal) normal form is a choiceof representative for each of the conjugacy classes. For this reason normal form theory plays acrucial role in understanding the local behaviour of vector fields near a hyperbolic singularity. Areasonably exhaustive account of the modern theory is given in [14].The utility of normal forms has led many authors to develop several styles of normal forms;for instance [3, 9, 1]. The most common are the semi-simple and inner-product styles. Thesemi-simple style is advantageous when the Jacobian at the singularity is semi-simple, whilst theinner-product is useful when there is some nilpotent component or when the Jacobian vanishes. a r X i v : . [ m a t h . D S ] A ug NATHAN DUIGNAN
There are no theoretical barriers to using the inner-product style, particularly the work ofStolovitch and Lombardi [12], to study normal forms for singularities in a normally hyperbolicmanifold. However, in Section 2.1, a new style of normal form will be derived which takes ad-vantage of the centre subspace. The normal form is considered through an algebraic lens, akin to[14]. The new approach provides results which are analogous to normal forms for hyperbolic sin-gularities, namely, resonance conditions which describe the irremovable monomials in Lemma 2.6,and Theorem 2.11 which categorises the formal normal form near normally hyperbolic invariantmanifolds.Normal forms are then studied in the C k category. Using a crucial theorem of Belitskii andSamavol [11], a proof is given of Corollary 2.13 on the existence of a C k transformation bringinga vector field containing a manifold of normally hyperbolic singularities into truncated normalform. In the smooth case, the result is analogous to the Sternberg-Chen Theorem for hyperbolicsingularities [18, 5]. The new style of normal form derived in Section 2.1 is crucial to the proof.With the normal form theory detailed, we then study Dulac maps near normally hyperbolicsaddles in Section 3. The investigation is motivated by the many application in [6, 16, 4].Specifically, these motivations demand asymptotic properties of the transition map betweensections transverse to the centre-stable and centre-unstable manifolds of the normally hyperbolicmanifold. All applications require only a study of the case when either the stable or unstablemanifold of each point on the normally hyperbolic manifold is of dimension 1. Thus we restrictour attention to this case.The Dulac map for families of hyperbolic saddles in the plane has been studied extensively. Foran overview see [15]. Dulac maps near a family of hyperbolic saddles in R have been treated in[2, 16]. In [4] the Dulac map near a specific manifold of normally hyperbolic saddle singularitieswas studied. The asymptotic structure of the Dulac maps in the general case is heretofore notinvestigated.In Section 3 we prove Theorem 3.8 and 3.11 on the asymptotic structure of the transitionmap. It is shown that the transition map shares properties with the familiar planar case. Inparticular, the Dulac map has a Mourtada type structure [13] and is an asymptotic series interms of the form, ω ( α, x ) = (cid:40) x − α − α , α (cid:54) = 0 − ln x α = 0 , with x some small coordinate on the section and α a parameter dependent on the eigenvalues ofthe Jacobian on the normally hyperbolic manifold.2. Normal Forms
We first give some notations. Let K be the field of real R or complex C numbers. Suppose x = ( x , . . . , x k ) ∈ K k and denote by ∂ x := ( ∂ x , . . . , ∂ x k ). Then, given a function f : K k → K k ,a vector field X on K k is defined by X = f ∂ x := f ∂ x + · · · + f k ∂ x k . Furthermore, if α = ( α , . . . , α k ) ∈ N k the multinomial notation x α will be used to represent themonomial x α . . . x α k k of degree | α | := α + · · · + α k .2.1. Formal Normal Forms
In this section the necessary theory to state and prove Theorem 2.11 on formal normal formsfor manifolds of normally hyperbolic singularities is built. Take X to be a smooth C ∞ or analytic C ω vector field on K n containing a normally hyperbolic invariant manifold N of dimension k that consists entirely of singular points. ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 3 A pre-normal form can be constructed for N from well known results in the literature. Ina neighbourhood of any point u ∈ N there exists a C ∞ transformation straightening N andaligning the stable-centre W sc ( N ) and unstable-centre W uc ( N ) manifolds with coordinate axis[20]. That is, coordinates ( x, u ) ∈ K n − k × K k local to u = 0 can be taken such that X is of theform,(2.1) X = ( A ( u ) x + f ( x, u )) ∂ x + g ( x, u ) ∂ u , f (0 , u ) = g (0 , u ) = 0 . Note that in this pre-normal form N = { x = 0 } and hence u are the centre variables. Usingthe theory in [20] further geometric properties on f, g and A can be assumed, however, for thepurposes of this paper they do not play a central role. In what follows assume that X is in thispre-normal form.In standard normal form theory one would now proceed by introducing the formal Taylorseries of X at 0 in ( x, u ) and analyse which terms can be removed by a formal, near identitycoordinate transformation ˆ φ . Much theory has been developed in this avenue. Although thesemethods can certainly be implemented here, particularly the work of [1, 12], the degeneracy ofthe flow on N enables a slight modification of the methods and leads to a normal form withmore removable terms then the standard theory.The key modification is to take a series expansion only in the normal variables x instead ofall the variables ( x, u ). This produces a series expansion about x = 0 of the form,(2.2) X ∼ X ( u ; x ) + X ( u ; x ) + . . . , X ( u ; x ) = A ( u ) x∂ x + 0 · ∂ u , where each X d ( u ; x ) is of dimension n and each component is a degree d + 1 homogeneouspolynomial in x = ( x , . . . , x n − k ) with coefficients that are functions in u . These coefficientfunctions can be considered either formal, smooth, or analytic in a neighbourhood of u = 0 if X is either formal, smooth, or analytic.With some notation identified, the algebraic structure of the series expansion (2.2) can beformulated. Definition 2.1.
Define the following algebraic objects:i. ˆ C ∞ ( u ) , C ∞ ( u ) , C ω ( u ) the local rings of formal power series, smooth functions, and analyticfunctions in u ∈ K k defined in a neighbourhood of 0. Denote all three by C .ii. C P d the free C -module generated by the set of degree d + 1 monomials in x .iii. C H d the free C -module given by n copies of C P d . Consider each element of C H d as an n -dimensional vector space with components homogeneous polynomials of degree d + 1 in x and whose coefficients are C functions in u .iv. C H the Lie algebra of n dimensional formal vector fields in x with coefficients in C . Wetake the usual Lie bracket [ · , · ] for vector fields.v. C F the associated Lie group of C H .With these definitions, (2.2) can now be seen as identifying X with a formal vector fieldˆ X ∈ C H and decomposing ˆ X into X d ( u ; x ) ∈ C H d . In what follows, vector fields ˆ X ∈ C H areconsidered in order to produce a result on formal normal forms. This provides a succinct Liealgebraic approach to the theory. In Section 2.2, properties about the actual vector field X arerecovered.As detailed in [14], formal, near identity transformations ˆ φ ∈ C F can be constructed via agenerating vector field U ∈ C H by taking ˆ φ the time 1 flow of U . Moreover, one can pull backˆ X ∈ C H to produce the transformed vector field ˜ X through the relation,(2.3) ˜ X = exp( L U ) ˆ X, L U := [ U, · ] . Note that ˆ φ is in general a divergent series in x and thus only a formal transformation.However, one can write the expansion so that the coefficients of the x terms are functions in NATHAN DUIGNAN C ( u ). Using exp( L U ) is particularly useful to preserve a Hamiltonian structure, see for instance[17], but it is being used here in the general sense.In line with the usual normal form theory, a cohomological equation on each C H d will nowbe constructed from (2.3). A consequent examination of the cohomological equations will revealwhich monomial vector terms in ˆ X can be removed by a formal transformation ˆ φ .Let U d ∈ C H d and transform ˆ X by the generated transformation ˆ φ d to obtain,˜ X = exp( L U d ) ˆ X = ( Id + L U d + . . . )( X + X + · · · + X d + . . . )= ( X + X + · · · + X d + [ U d , X ] + . . . ) . The first terms influenced by the transformation ˆ φ d is at order d and produces the equation(2.4) [ X , U d ] = X d − ˜ X d . However, if U d ∈ C H d it is not necessarily true that so too is [ X , U d ]. To see this, let a vectorfield X act on a vector field U by treating X as a derivation on each coordinate function and let U = U x ∂ x + U u ∂ u . Then,[ X , U d ] = X ( U d ) − U d ( X )= ( A ( u ) x∂ x ) ( U xd ∂ x + U ud ∂ u ) − ( U xd ∂ x + U ud ∂ u ) ( A ( u ) x∂ x )= ( A ( u ) x∂ x ( U xd ) − U xd ∂ x ( A ( u ) x )) ∂ x + ( A ( u ) x∂ x U ud ) ∂ u − U ud ∂ u ( A ( u )) x ) ∂ x . The terms ˜ L d ( U xd ∂ x ) := [ X , U xd ∂ x ] = ( A ( u ) x∂ x ( U xd ) − U xd ∂ x ( A ( u ) x )) ∂ x and X ( U ud ∂ u ) = ( A ( u ) x∂ x U ud ) ∂ u are both in C H d . The final term U ud ∂ u ( A ( u ) x ) ∂ x is in C H d +1 . If this final term is pushed into the higher order terms of the expansion, then theeffect of U d on ˆ X has first influence at degree d and is quantified by the modified cohomologicalequation (2.5) ˆ L d ( U d ) = X d − ˜ X d , with ˆ L d := ˜ L d ⊕ X , ˜ L d ∈ End( C H xd ) , X ∈ End( C H ud )and C H xd , C H ud are the submodules with vanishing u and x components respectively. Remark 2.2.
It is worth pointing out the difference between the modified cohomological equa-tion and the usual homological equation in the normal form theory using the semi-simple orinner-product styles. The usual cohomological equation is of the form, L d ( U d ) = X d − ˜ X d , with L d := [ X , · ]. In the usual styles one has each X d ∈ H d , the vector space of degree d + 1 homogeneous vector fields. With this grading L d : H d → H d . The fact that L d is anendomorphism on H d is crucial to constructing an iterative scheme on the degree d , which in turnconstruct the normal form. However, in the new approach of this paper, we have decomposed thevector field X through the grading X d ∈ C H d , the C -module of vector fields homogeneous in x only. In the above calculation, it is shown that L d ( U d ) produces a term U ud ∂ u ( A ( u ) x ) ∂ x ∈ C H d +1 .Thus, L d acting on C H d is not an endomorphism. Ignoring the higher order term U ud ∂ u ( A ( u ) x ) ∂ x produces the endomorphism ˆ L d as desired. ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 5
Remark 2.3.
A choice of ordering of the degree d +1 monomials vectors x α := x α . . . x α n − k , | α | := α + · · · + α n − k = d +1 creates a basis for C P d . Then, by ordering each vector component ∂ x i , ∂ u i together with the ordering of C P d , a basis for C H d can be obtained. Let the dimension of C H d be D ( d ). As C H d is a free module over C , we have C H d ∼ = ( C ) D ( d ) . Thus, with a choice of basis,one can consider ˆ L d as a D ( d ) square matrix with entries in C , that is, End( C H d ) ∼ = M D ( d ) ( C ).With the modified cohomological equation derived, terms in X d removable by some formaltransformation ˆ φ ∈ C F can now be determined. In fact, it should be evident that all terms of X d that are in Im( ˆ L d ) can be removed by a choice of U d , and conversely, any component of X d in C H d \ Im( ˆ L d ) are irremovable. By taking ˜ X d equal to the sum of these irremovable terms, itcan be assured that X d − ˜ X d ∈ Im( ˆ L d ) and the modified cohomological equation at order d canbe solved. Formally, one takes the quotient modulecoker( ˆ L d ) := C H d (cid:46) Im( ˆ L d )and a choice of representatives ˜ X d of elements [ ˜ X d ] ∈ coker( ˆ L d ). In the terminology introducedby Murdock [14], this choice of representative is considered a normal form style .In summary, it has been shown that a formal normal form for ˆ X can be constructed throughan iterative procedure. Assuming ˆ X has been normalized to order d −
1, generate a formal, nearidentity transformation φ d from a vector field U d ∈ C H d . The pull-back of ˆ X by φ d leaves termsof order d − d the modified cohomological equation. Then,one removes all terms from X d that are contained in Im( ˆ L d ) and the normalized terms becomea choice of representative from coker( ˆ L d ). The procedure is repeated for d + 1. The followingcentral theorem has thus been proved. Theorem 2.4.
Let X be a C ∞ vector field containing a manifold of normally hyperbolic singu-larities N and let ˆ X be the corresponding formal series of X at 0. Then there exists a sequence oftransformations φ d generated by homogeneous vector fields U d ∈ C H d which formally conjugates ˆ X to the normal form, (2.6) ˜ X = X + (cid:88) d ≥ ˜ X d , with ˜ X d a representative of [ ˜ X d ] ∈ coker( ˆ L d ) . Whilst Theorem 2.4 gives the algebraic structure of the normal form for a vector field X ,it does little to give a more concrete explanation of what terms ˜ X d look like or how to findand choose the precise representative. Crucially, we want to know in what situations it can beassumed that ˜ X d = 0, that is, we want to know a simple way of determining when X d ∈ Im( ˆ L d ).Answers are provided in the case A ( u ) is diagonalisable. In this case it may assumed that A ( u ) = diag ( λ ( u ) , . . . , λ n − k ( u )) and by hyperbolicity each Re λ i (0) (cid:54) = 0. Lemma 2.5 follows. Lemma 2.5.
Suppose X = A ( u ) ∂ x and A ( u ) = diag( λ ( u ) , . . . , λ n − k ( u )) . Then each modifiedhomological operator ˆ L d ∈ End( C H d ) is diagonal. More precisely, if α ∈ N n − k , | α | = d + 1 , λ ( u ) := ( λ ( u ) , . . . , λ n − k ( u )) , and (cid:104)· , ·(cid:105) is the usual dot product on K n − k , then (2.7) ˆ L d ( x α ∂ x i ) = ( (cid:104) λ ( u ) , α (cid:105) − λ i ( u )) x α ∂ x i , ˆ L d ( x α ∂ u i ) = (cid:104) λ ( u ) , α (cid:105) x α ∂ u i . Proof.
This is a simple calculation using the definition of ˆ L d . (cid:3) Let v denote x i or u i . Then C H d admits submodules C H α,v , each defined as the free moduleover x α ∂ v and all of which are isomorphic to C . Hence, Lemma 2.5 reduces the problem of NATHAN DUIGNAN describing Im( ˆ L d ) into a study of the endomorphisms L α,v ∈ End( C H α,v ) ∼ = End( C ) and theirimages. These endomorphisms act by mere multiplication of f α,v ( u ) on C , where f α,v ( u ) is givenby the coefficient of x α ∂ v in (2.7). Finding a representative of coker( ˆ L d ) is reduced to findingrepresentatives of coker( L α,v ) = C H α,v (cid:46) Im( L α,v ) . The image Im( L α,v ) is equivalent to the ideal generated by f α,v , namely (cid:104) f α,v (cid:105) . It follows, if f α,v has a multiplicative inverse, that is, f α,v is a unit, then Im( L α,v ) = C H α,v . Consequently,coker( L α,v ) = 0 and the unique representative 0 can be chosen. The following lemma is analogousto the usual resonance conditions for normal forms of hyperbolic singular points. Lemma 2.6.
Suppose A ( u ) = diag( λ ( u ) , . . . , λ n − k ( u )) . Then all terms of the form, (2.8) f ( u ) x α ∂ x i , (cid:104) α, λ (0) (cid:105) − λ i (0) (cid:54) = 0 f ( u ) x α ∂ x i , (cid:104) α, λ (0) (cid:105) (cid:54) = 0 do not appear in the normal form ˜ X .Proof. From Theorem 2.4 a normal form transformation can be found which brings the coefficientof x α ∂ v to a representative of [ f ( u )] ∈ coker( L α,v ). If it can be shown that f α,v is a unit then theremarks of the proceeding exposition show this representative can be taken as 0. The units of C are easily described as the functions g ( u ) such that g (0) (cid:54) = 0. Now, f α,v ( u ) = (cid:104) α, λ ( u ) (cid:105) − λ i ( u )when v = x i and (cid:104) α, λ ( u ) (cid:105) when v = z i , thus the lemma can be concluded. (cid:3) Definition 2.7.
The vector monomials in the union of the sets,(2.9) Res x := { x α ∂ x i | (cid:104) α, λ (0) (cid:105) − λ i (0) = 0 } , Res u := { x α ∂ u i | (cid:104) α, λ (0) (cid:105) = 0 } , Res d := { x α ∂ v ∈ Res x ∪ Res u | | α | = d + 1 } are called resonant . Moreover, the free C -submodule over the set Res d is denoted by C Res d andcalled the resonant submodule of order d .The final problem to be resolved concerns these resonant terms. They can not a priori beremoved and a choice of representative must be made. A concrete explanation of the problem ofchoosing a representative is, given a function F ( u ) ∈ C , finding q ( u ) , r ( u ) ∈ C such that F ( u ) = r ( u ) + q ( u ) f α,v ( u ) . In the normal form procedure, F ( u ) is the coefficient of x α ∂ v in X d and choosing an r ( u ) amountsto choosing a representative of [ F ( u )] ∈ coker( L α,v ). The question is now, is it possible to dothis quotient? Of course, one can always take r ( u ) = F ( u ) and q ( u ) = 0, but this may not be the‘simplest’ form of r ( u ). For instance, if F ( u ) = f ( u ), clearly a better choice is r ( u ) = 0 , q ( u ) = 1.The following divisibility theorem provides what may be called the simplest form of r ( u ). Theorem 2.8 (Weierstrass/Mather Division Theorem [10]) . Let f be a smooth (resp. analyticor formal) K -valued function defined on a neighbourhood of in K × K k − such that f ( u ,
0) = u m g ( u ) where g (0) (cid:54) = 0 and g is smooth (resp. analytic or formal) on some neighbourhood of in K . Then given any smooth (resp. analytic or formal) real-valued function F defined on aneighbourhood of in K × K k − , there exist smooth (resp. analytic or formal) functions q and r such that(i) F = r + qf on a neighbourhood of in R × R k − , and(ii) r ( u ) = (cid:80) m − i =0 r i ( u , . . . , u k ) u i . ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 7
Remark 2.9.
When f (cid:54) = 0 is a formal or analytic function on R k then, possibly after a linearchange of u , there is always an m and a u i such that f ( u i ,
0) = u mi g ( u i ). The value of m is givenby the first non-zero m -jet of f . Moreover, it is shown in [10] that q, r are unique. Algebraically,this means a unique representative of each element in coker( ˆ L d ) can be taken for C = ˆ C ∞ or C ω . Remark 2.10.
Uniqueness of the functions r, q fails when f is C ∞ . The issue is the existenceof f (cid:54) = 0 such that the ∞ -jet is 0, so called flat functions . A counterexample is given in [10].Take f polynomial, F = 0, and G flat. Then both r = 0 = q and r = G, q = − G/f satisfy F = r + qf and are smooth. Algebraically, this means a unique representative of each elementin coker( ˆ L d ) when ˆ L ∈ End( C ∞ H d ) can not be be given by Theorem 2.8. However, a choice ofrepresentative can be made by decomposing F = ˆ F + ¯ F , f = ˆ f + ¯ f where ˆ · , ¯ · represent the formaland flat part respectively. r can be chosen as the unique formal function given by Theorem 2.8and satisfying ˆ F = ˆ r + q ˆ f . The flat terms can then be added to get an r = ˆ r + ¯ r, ¯ r = ¯ F − q ¯ f .For the counterexample, this forces the choice of r = q = 0.The main theorem for diagonalisable A ( u ) has thus been proved. Theorem 2.11.
Let X be a vector field of class C = ˆ C ∞ , C ∞ , or C ω containing a manifoldof normally hyperbolic singularities N and let ˆ X ∈ C H be the corresponding formal series of X . Then there exists a sequence of transformations φ d generated by homogeneous vector fields U d ∈ C H d which formally conjugates ˆ X to the normal form, (2.10) ˜ X = X + (cid:88) d ≥ ˜ X d , with ˜ X d ∈ C Res d whose coefficients are of the form r ( u ) given in Theorem . In particular, if X is analytic or formal then r ( u ) is polynomial in at least one of the u i . C k -Normal Forms Theorem 2.11 provides a formal normal form ˜ X for a given vector field X near a point u ofa normally hyperbolic manifold of singularities N . The theorem states the existence of a formaltransformation ˆ φ bringing ˆ X into its normal form ˜ X . However, the statement is only formal,meaning that ˜ X ∼ ˆ φ ∗ X where ∼ is equivalence of the series expansion at 0 in one of the forms(2.2). There are three questions worth addressing:(1) Can ˆ φ be taken smooth or analytic?(2) Can the formal conjugacy be replaced with smooth or even analytic conjugacy?(3) If ˜ X K := X + (cid:80) d ≤ K ˜ X d is the normal form of X truncated at degree K , does thereexist an integer k and φ ∈ C k which conjugates X to ˜ X K ?Due to a lemma of Borel [10, pg. 98, Lemma 2.5], the first question is partially answered.The lemma guarantees, for any formal series ˆ φ , the existence of a smooth function φ ∼ ˆ φ . As aconsequence, there is a smooth transformation φ such that ˜ X ∼ φ ∗ X .If φ can be taken analytic then both proposed questions are answered. A substantial amountof work in the literature has already addressed the potential analyticity of φ for a hyperbolicsingularity, for an overview see [19]. In this context, provided the eigenvalues of the Jacobian atthe singularity satisfy the Bruno conditions, analyticity is guaranteed. The condition also holdsfor families of vector fields. Analyticity is not of concern in this paper, but due to the similarityin the resonance conditions between normal forms for hyperbolic singularities and normal formsfor normally hyperbolic sets of singularities, we conjecture an analogous condition holds. NATHAN DUIGNAN
The question remains, if φ can only be assumed smooth in general, whether formal conjugacycan be replaced by smooth conjugacy. In the case of a purely hyperbolic singularity, the questionis answered positively by the Sternberg-Chen Theorem [18, 5].So far it has been shown, for a point u ∈ N , that ˜ X = φ ∗ X + τ ( u ; x ) where τ ( u ; x ) is flat in x . If the flat term τ can be removed, then smooth conjugacy follows. A more general problemis, given two vector fields X, ˜ X with identical K ( k )-jet at 0, when can it be guaranteed X, ˜ X are C k conjugate for some function K ( k ). The most general theorem in this direction has beenproved for maps by Samovol and for vector fields by Belitskii. Theorem 2.12 (Belitskii-Samovol [11]) . For any k ∈ N and any tuple λ ∈ C n there existsan integer K = K ( k, λ ) such that the following holds. Suppose two germs of vector fields at asingularity with the spectrum of linearization equal to λ have a common centre manifold, and theirjets of order K coincide at all the points of this manifold. Then these germs are C k equivalent. If X contains an N and is in pre-normal form, then Theorem 2.12 can be applied providedthe K ( k )-jets of X and ˜ X agree along x = 0 in a neighbourhood of ( x, u ) = 0. But indeed thisis true for φ ∗ X + τ ( u ; x ) and ˜ X as τ is flat only in x . Hence, the following key corollary on the C k -normal form theorem near points in N has been shown. Corollary 2.13.
Let X contain a manifold of normally hyperbolic singularities N and assumeit in pre-normal form. Then there exists a function K ( k ) : N → N such that K ( k ) → ∞ as k → ∞ , and such that X is C k -conjugate to the normal form X K ( k ) in a neighbourhood of anypoint p ∈ N . Asymptotic Properties of the Transition Map Near Some NormallyHyperbolic Saddles
In this section we derive the asymptotic properties of transitions near a manifold N of nor-mally hyperbolic singularities and provide a method to compute them. We assume that at eachpoint u ∈ N the eigenvalues are real and there is at least one pair of eigenvalues of oppositesign, that is, N contains normally hyperbolic saddles. Ideally asymptotic properties would beshown for arbitrary dimensions of the centre-stable W sc ( N ) and centre-unstable W uc ( N ) man-ifolds. However, a derivation is given only when the unstable manifold at each point u ∈ N isone dimensional. Moreover, for clarity, focus is given only on manifolds N of co-dimension 3.All methods introduced naturally extend to the higher co-dimension cases. Remarks are giventhroughout for the case N is co-dimension 2.Let X be a smooth vector field defined in a neighbourhood of a co-dimension 3 manifold N of normally hyperbolic saddle singularities. Let the dimension of N be k . Without loss ofgenerality assume that X is in the pre-normal form (2.1) with ( x, y, z ) ∈ R so that N is givenby ( x, y, z ) = 0 and the centre variables are given by u ∈ R k . By a time rescaling it can beassumed that for all u ∈ N the eigenvalues of DX u restricted to the normal space of N are givenby (1 , − α ( u ) , − β ( u )) and satisfy, − α (0) ≤ − β (0) < . Choose coordinates x, y, z so that the linearisation of the normal space is given by x∂ x − α ( u ) ∂ y − β ( u ) z∂ z . Note that if − α ( u ) = − β ( u ) then DX λ (0) may have some nilpotent component pre-venting this diagonalisation. This case is dealt with in the proceeding theory simply by treatingthe additional z∂ y term as a higher order term.Before discussing the transitions of interest in this paper, it is useful to first classify the form ofvector fields X in a neighborhood of N . This was accomplished in the previous section throughnormal form theory. The following proposition is an application of this work. ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 9
Proposition 3.1. i) Suppose that, α (0) = p q ∈ Q , β (0) = p q ∈ Q , α (0) β (0) / ∈ N with both p , q and p , q co-prime. Let U y = x p q y, U z = x p q z. Under these resonance conditions the normal form of X is given by, (3.1) ˙ x = x ˙ y = − α ( u ) y + y (cid:88) n + n ≥ α n ,n ( u ) U q n y U q n z ˙ z = − β ( u ) z + z (cid:88) n + n ≥ β n ,n ( u ) U q n y U q n z , ˙ u i = (cid:88) n + n ≥ δ in ,n ( u ) U q n y U q n z , i = 1 , . . . , k, with n , n ∈ N . If α (0) / ∈ Q (resp. β (0) / ∈ Q ) then there is no U y (resp. U z ) dependency.ii) If additionally α (0) β (0) ∈ N then there exists m, p, q ∈ N with p, q co-prime such that α (0) = m pq , β (0) = pq . Let U y = x mpq y, U z = z pq y. Under these resonance conditions the normal form is given by, (3.2) ˙ x = x ˙ y = − α ( u ) y + y (cid:88) n ≥− qn − mn ≥ α n ,n U n y U qn − mn z ˙ z = − β ( u ) z + z (cid:88) n ≥ qn − mn ≥− β n ,n U n y U qn − mn z ˙ u i = (cid:88) n ≥ qn − mn ≥ δ in ,n U n y U qn − mn z , i = 1 , . . . , k, with n , n ∈ N . If α (0) , β (0) / ∈ Q then there is no U y , U z dependency.Proof. As stated, the proposition is a direct consequence of Theorem 2.11 on the normal form neara point in N . It has been assumed that A ( z ) is diagonalised so that A ( u ) = diag(1 , − α ( u ) , − β ( u )).Then by Theorem 2.11 and Corollary 2.13 we are guaranteed, in a neighbourhood of ( x, y, z, u ) =0, a smooth transformation φ conjugating X to a vector field˜ X = X + (cid:88) d ≥ ˜ X d , with ˜ X d ∈ C ∞ Res d . From Lemma 2.6 each vector field in Res d consists of linear combinationsof resonant monomial vector fields, x n y n z n ∂ x such that n − α (0) n − β (0) n − ,x n y n z n ∂ y such that n − α (0) n − β (0) n + α (0) = 0 ,x n y n z n ∂ z such that n − α (0) n − β (0) n + β (0) = 0 ,x n y n z n ∂ u i such that n − α (0) n − β (0) n = 0 , for n , n , n ∈ N and n + n + n ≥
2. Having a complete description of these resonantmonomials will give the normal form. We derive the resonant monomials only for the y componentas the other components follow almost identically. If α (0) = p q ∈ Q , β (0) = p q ∈ Q , α (0) β (0) / ∈ N with both p , q and p , q co-prime, then asolution to n − α (0) n − β (0) n + α (0) = 0 is given by n = k p + k p , n = 1 + k q , n = k q , for k , k ∈ N with k + k ≥
1. This produces the monomial of the form y ( x p y q ) k ( x p z q ) k ∂ y = yU q k y U q k z ∂ y as desired. If α (0) / ∈ Q then we must have k = 0, hence, the resonant monomialhas no U y dependence. Similarly if β (0) / ∈ Q , then k = 0 and there is no U z dependence. Theseresults conclude case 1 of the proposition.Alternatively, if α (0) β (0) ∈ N , then there exists m, p, q ∈ N with p, q co-prime such that α (0) = m pq , β (0) = pq . In such a case, a solution to n − α (0) n − β (0) n + α (0) = 0 is given by n = pk , n = 1 + k , n = qk − mk , for k , k ∈ Z such that k ≥ − , ≤ qk − mk . This produces the monomial of the form y ( x p z q ) k ( yz − m ) k ∂ y = U qk − mk z U k y ∂ y as desired. If α (0) / ∈ Q then it must be that β (0) / ∈ Q .In this instance, k = k = − x components of the vector field. Througha time rescaling, all these can be moved from the x component to the other components. (cid:3) Remark 3.2.
The difference between the normal forms (3.1) and (3.2) comes from the additionalresonance α (0) /β (0) ∈ N . Geometrically, this is represented by the fact that y = 0 , z = 0 areinvariant in (3.1) whilst the resonant terms with coefficients α − ,n , β n , − in (3.2) prevent onefrom performing a smooth transformation to have the axis invariant. Remark 3.3.
The case when N is co-dimension 2 is significantly simpler. The normal form isgiven by restricting to z = 0 in system (3.1). A qualitative depiction of the co-dimension 2 caseis given in Figure 3.1. Σ y N W sc ( N ) W uc ( N ) Σ x D Figure 3.1.
Diagram of the case N is co-dimension 2 in R The normal form in Proposition 3.1 gives a classification of vector fields X near a manifoldof normally hyperbolic saddle singularities N . Hence, by studying the flow of (3.1) and (3.2) ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 11 we are able to ascertain properties of all flows near these objects. In particular, we seek anunderstanding of hyperbolic transitions near N .Consider the section Σ = [0 , × [ − , × R k defined in the normal form coordinates of (3.1)or (3.2). A representation of Σ in relation to N is given in Figure 3.2 for the case N is dimension0 inside R and in Figure 3.1 for the case N is co-dimension 2.Σ + z Σ x D N Figure 3.2.
Diagram for when N is co-dimension 3 in R .The interior of Σ is an isolating neighbourhood of N in the region x ≥ x = 0 and y = z = 0 respectively. Now,decompose Σ into its various faces,Σ x := Σ ∩ { x = 1 } , Σ ± y := Σ ∩ { y = ± } , Σ ± z := Σ ∩ { z = ± } and note that, due to the fact that x = 0 is the centre-stable manifold, points p ∈ Σ ± y ∪ Σ ± z must flow into the interior of Σ. Provided that p / ∈ { x = 0 } , that is p is not in the centre-stablemanifold of N , we are guaranteed that p is eventually flowed out of the interior of Σ. For p takensufficiently close to W sc ( N ), the flow of p will intersect Σ x . It follows that there is a naturaldiffeomorphism, ˜ D : Σ ± y ∪ Σ ± z \ W sc ( N ) → Σ x . Moreover, ˜ D admits a continuous extension tothe map D : Σ ± y ∪ Σ ± z → Σ ± x . The primary achievement of this section is to obtain an explicit asymptotic series of D near x = 0.Note that the choice of section Σ is arbitrary. However, the transition for any other choice ofsection, provided it is transverse to both the stable and unstable manifolds of N , can be obtainedby simply flowing points on Σ to the new section. This transition is smooth, and thus, does notinfluence the asymptotic structure of D . The particular choice of Σ made in this paper has historical precedent. Due to its relevanceto Hilbert’s 16 th problem, the case when ˙ u = 0 and N is co-dimension 2 has been well studied; areview is given in [15]. As ˙ u = 0, this case can be considered as a family of hyperbolic singularitiesin the plane. In this context D is referred to as the Dulac map . Before proceeding to the generalcase, it is worth mentioning some properties of the Dulac map in the planar case.As per remark 3.3, the normal form for the planar case can be deduced from Proposition 3.1by considering u a parameter and restricting to z = 0 in case i). Explicitly, the normal form is˙ x = x ˙ y = − α ( u ) y + y (cid:88) n ≥ α n ( x p y q ) n , with α (0) = p/q ∈ Q . The Dulac map is the transition D : Σ + y = { y = 1 } → Σ x = { x = 1 } .There are two key results known for Dulac maps in the planar case. First, if x ∈ Σ + y then theDulac map near u = 0 is asymptotic to the series, D ( x ) ∼ x α ( u )0 (cid:88) i ≥ g i ( u, x ) x ip , where g i ( x ) is polynomial in the function, ω ( α , x ) = (cid:40) x − α − α , α (cid:54) = 0 − ln x α = 0 , and α ( u ) := α ( u ) − α (0). See [15, sec. 5.1] for details.The other key result in the due to Mourtada [13]. Setting g ( u, x ) = (cid:80) g i ( u, x ) x ip is hasbeen shown that, lim x → + x n d n gdx n = 0 , for all n ∈ N and uniformly in u . Functions that exhibit this behaviour are known as Mourtadatype functions .Outside of the planar case little is known. Roussarie and Rousseau [16] investigated the socalled ‘almost planar case’. They treat a family of hyperbolic saddles in R with the specificeigenvalue β (0) = 1 and with α (0) / ∈ Q to avoid resonance conditions of Proposition 3.1. Inthe framework of this paper this case corresponds to an N of co-dimension 3 and with u aparameter, that is, ˙ u = 0. They explicitly computed the asymptotic structure of the Dulac mapand showed it shares properties with the planar case, namely, its components are Mourtada typefunctions, and the asymptotic series again contains these ω functions. However, by assumingthe non-resonance conditions, in particular the case α (0) /β (0) ∈ N , they did not investigate acrucial difference between the planar case and the co-dimension 3 case.To see this, take α (0) , β (0) / ∈ Q . From Proposition 3.1 the normal form is simply,(3.3) ˙ x = x ˙ y = − α ( u ) y + α − , ( u ) z m ˙ z = − β ( u ) z ˙ u = 0 ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 13 with α − , ( u ) = 0 if α (0) /β (0) / ∈ N . Let ( x , y , z , u ) ∈ Σ ± y ∪ Σ ± z with y = ± , z = ± ± y ∪ Σ ± z respectively and take ( y , z , u ) ∈ Σ x . Then system (3.3) can be integrated to yield,(3.4) t = − ln x y = x α ( u )0 ( y + α − , ( u ) z m ω ( γ ( u ) , x )) z = x β ( u )0 z u = u with γ ( u ) = α ( u ) − mβ ( u ).The introduction of the term ω ( γ , x ) due to the resonance α (0) /β (0) prevents the Dulac mapfrom having the same properties as in the planar case. However, for the case ˙ u = 0, Bonackertand Naudot [2] were able to show, even in the resonant case, that the Dulac map will alwayshave the form (3.4) to leading order. Specifically they showed, for D : Σ + z → Σ x ,(3.5) y = x α ( u )0 ( y + α − , ( u ) ω ( γ ( u ) , x ) + f ( x , y )) z = x β ( u )0 (1 + g ( x , y )) , with f, g functions of Mourtada type. No investigation was made to show the asymptotic struc-tures of f, g or the case when ˙ u (cid:54) = 0.In the remainder of the section we treat each of case i) and ii) from Proposition 3.1 in thegeneral case with ˙ u (cid:54) = 0. The structure of f, g will be given in Theorem 3.8 and Theorem 3.11.The approach taken in the proof of each theorem depends on whether the normal form (3.1) or(3.2) is considered. The two approaches are similar in concept, but differ in some details.3.1. Case 1: α (0) /β (0) / ∈ N We proceed by first considering the case α (0) /β (0) / ∈ N but α (0) = p q and β (0) = p q with p , q and p , q pairs of co-prime positive integers. The normal form is given by (3.1).Introduce as coordinates U y = x p /q y, U z = x p /q z and let α ( u ) = p q + α ( u ) , β ( u ) = p q + β ( u ) , where α , β are O ( u ). Under this coordinate transform the normal form (3.1) is brought intothe vector field,(3.6) ˙ x = x ˙ U y = − α ( u ) U y + U y (cid:88) α n ,n ( u ) U q n y U q n z ˙ U z = − β ( u ) U z + U z (cid:88) β n ,n ( u ) U q n y U q n z ˙ u = (cid:88) δ n ,n ( u ) U q n y U q n z The introduction of these coordinates brings the centre-stable manifold x = 0 to the invariantmanifold U y = U z = 0.We follow [15] by considering variations of the solutions on U y = U z = 0 , u = u . Moreexplicitly, we consider a variation of each orbit ( U y , U z , u ) = (0 , , u ) by a small displacement in U y , U z denoted by U y , U z respectively. This variation can be written as a power series of the form,(3.7) U y ( U y , U z , u ; t ) = U (1) y ( u , t ) U y + U y (cid:88) U ( n ,n ) y ( u , t ) U q n y U q n z U z ( U y , U z , u ; t ) = U (1) z ( u , t ) U z + U z (cid:88) U ( n ,n ) z ( u , t ) U q n y U q n z ,u ( U y , U z , u ; t ) = u + (cid:88) u ( n ,n ) ( u , t ) U q n y U q n z with, U (1) y (0) = U (1) z (0) = 1 , U ( n ,n ) y ( u ,
0) = U ( n ,n ) z ( u ,
0) = u ( n ,n ) ( u ,
0) = 0 , so that at t = 0, ( U y , U z , u ) = ( U y , U z , u ).Each of the coefficient functions U ( n ,n ) y , U ( n ,n ) z , u ( n ,n ) , referred to as the variation co-efficients , can be computed through the variational equations. These equations are derived bysubstituting (3.7) into system (3.6) and equating coefficients of U n y U n z . The first order equationsare given by, ddt U (1) y = − α ( u ) U (1) y , U (1) y (0) = 1 ,ddt U (1) z = − β ( u ) U (1) z U (1) z (0) = 1 , Both equations are linear and hence admit explicit solutions,(3.8) U (1) y = e − α ( u ) t , U (1) z = e − β ( u ) t . The higher order variational equations are given for each ( n , n ) ∈ N by,(3.9) ddt U ( n ,n ) y = − α ( u ) U ( n ,n ) y + R ( n ,n ) y , U ( n ,n ) y (0) = 0 ,ddt U ( n ,n ) z = − β ( u ) U ( n ,n ) z + R ( n ,n ) z , U ( n ,n ) z (0) = 0 ,ddt u ( n ,n ) = R ( n ,n ) u , u ( n ,n ) (0) = 0 , with R ( n ,n ) y , R ( n ,n ) z , R ( n ,n ) u polynomial in U (˜ n , ˜ n ) z , U (˜ n , ˜ n ) y , u (˜ n , ˜ n ) for ˜ n + ˜ n < n + n .The equations are linear, thus admit solutions,(3.10) U ( n ,n ) y = e − α ( u ) t (cid:90) t e α ( u ) τ R ( n ,n ) y ( τ ) dτU ( n ,n ) z = e − β ( u ) t (cid:90) t e β ( u ) τ R ( n ,n ) z ( τ ) dτu ( n ,n ) = (cid:90) t R ( n ,n ) u ( τ ) dτ. A more precise form of the variation coefficients can be given. Take β ∈ R and similar to theworks on bifurcation theory, for instance [15], introduce the function(3.11) Ω( β, t ) := (cid:90) t e βτ dτ = (cid:40) e βt − β , β (cid:54) = 0 ,t β = 0 . Note that lim β → Ω( β, t ) = Ω(0 , t ) so that Ω( β, t ) can be considered as a family of smoothfunctions continuous in β . Definition 3.4. (1) Denote by O the ring of functions smooth in u in a neighbourhood of 0 and rational in α, β ∈ R . ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 15 (2) Denote by R α,β the polynomial ring over O with indeterminates Ω( ± α, t ) , Ω( ± β, t ) , t .That is, R α,β := O [Ω( ± α, t ) , Ω( ± β, t ) , t ] . (3) Define the subring ¯ R α,β of elements P ( α, β ; t ) ∈ R α,β such thatlim α,β → P ( α, β ; t ) =: P (0 , t ) exists,For example, α − Ω( α, t ) is in R α,β but not in ¯ R α,β , whilst α − (Ω( α, t ) − t ) is in both.The following lemmas give essential properties of R α,β . Lemma 3.5. R α,β , ¯ R α,β are closed under the operators, I t ( P ) := (cid:90) t P ( α, β ; τ ) dτ, D t ( P ) := ddt P ( α, β ; τ ) . Moreover I t : ¯ R dα,β → ¯ R d +1 α,β and D t : ¯ R d +1 α,β → ¯ R dα,β .Proof. If the result can be shown for R then by the dominated convergence theorem it is auto-matically guaranteed for ¯ R .From the definition of Ω in (3.11) one easily computes any function P ∈ R α,β can be writtenas a linear combination of functions of the form t j e ( n α + n β ) t , for some j, n , n ∈ Z . Through the linearity of the integral, it follows I t ( P ) will be a linearcombination of integrals K j := (cid:90) t τ j e ( n α + n β ) τ dτ. Each of these integrals has the recurrence formula K j = 1 n α + n β t j e ( n α + n β ) t − jn α + n β K j − . The recurrence formula, together with the fact that e n αt = (1 + α Ω( α, t )) n , gives closure of R α,β under integration.Similarly, D t ( t j e ( n α + n β ) t ) = ( jt j − + ( n α + n β ) t j )e ( n α + n β ) t . Hence, the closure under D t is guaranteed. (cid:3) Lemma 3.6.
Let P ( α, β ; t ) ∈ ¯ R α,β . Then P (0 , t ) is polynomial in t .Proof. P ( α, β ; t ) can be written as a linear combination of functions of the form, f ( α, β ) t j e ( n α + n β ) t where f is a rational function. As f is rational then by definition there exists p, q polynomial in α, β with f ( α, β ) = p ( α, β ) /q ( α, β ). Let d p , d q be the degree of p, q respectively. If d p − d q > α,β → f ( α, β ) = 0.Now, if P ∈ ¯ R α,β we must have lim α,β → d k dt k P ( α, β ; t ) = d k dt k P (0 , t ). The derivative d/dtf ( α, β ) t j gives the function ( jt j − +( n α + n β ) t j )e ( n α + n β ) t which is the sum of a functionof one degree less in t and a function with coefficient ( n α + n β ) p ( α, β ) /q ( α, β ). The coeffi-cient is again rational with sum of degrees d p − d q + 1. Hence, there exists k < ∞ such that,for all ˜ k > k , d ˜ k dt ˜ k P ( α, β ; t ) contains only terms with coefficients f = p/q with sum of degrees d p − d q >
0. Taking the limit α, β → d ˜ k dt ˜ k P (0 , t ) = 0 for all ˜ k > k . It follows that P (0 , t ) is polynomial in t . (cid:3) With the definition of R α,β given and the preceding lemmas, we have the following propositionon the form of the variation coefficients. Proposition 3.7.
For all ( n , n ) ∈ N there exists functions ˜ U y ( n ,n ) ( u , t ) , ˜ U z ( n ,n ) ( u , t ) , ˜ u ( n ,n ) i ( u , t ) ∈ ¯ R α ,β , i = 1 , . . . , k, such that, U ( n ,n ) y ( u ; t ) = e − α ( u ) t ˜ U y ( n ,n ) ( u , t ) U ( n ,n ) z ( u ; t ) = e − β ( u ) t ˜ U z ( n ,n ) ( u , t ) u ( n ,n ) ( u ; t ) = ˜ u ( n ,n ) ( t ) with ˜ u ( n ,n ) := (cid:16) u ( n ,n )1 , . . . , ˜ u ( n ,n ) k (cid:17) . Moreover:i) Each ˜ U y ( n ,n ) , ˜ U z ( n ,n ) , ˜ u ( n ,n ) is polynomial in α ˜ n , ˜ n , β ˜ n , ˜ n , δ ˜ n , ˜ n for ˜ n + ˜ n ≤ n + n .ii) If α n ,n (resp. β n ,n , δ in ,n ) vanish for n + n ≤ n ∈ N then U ( n ,n ) y ( t ) (resp. U ( n ,n ) z , U ( n ,n ) u i )vanish for n + n ≤ n. Proof.
The proposition will be proved by induction on k = n + n . From (3.8) it is known that U (0 , y = U (1) y = e − α ( u ) t · , U (0 , z = U (1) z = e − β ( u ) t · , u (0 , = u . As 1 and each component of u are elements of ¯ R α ,β the result is true for k = 0.Now assume true for all n , n ∈ N such that n + n < k . Take any n , n ∈ N with n + n = k and let K represent each of U y , U z , u . It was shown that each K ( n ,n ) are given by the solutionsto the variational equations computed in (3.10). As remarked before (3.10), each R ( n ,n ) K is apolynomial in K (˜ n , ˜ n ) for ˜ n + ˜ n ≤ n + n = k , and as such, if each K (˜ n , ˜ n ) ∈ ¯ R α ,β byassumption, then R ( n ,n ) K ∈ ¯ R α ,β . Furthermore, e κt = (1 + κ Ω( κ, t )) for κ = α , β , − α , − β .Hence, e α ( u ) t R ( n ,n ) y , e β ( u ) t R ( n ,n ) z , R ( n ,n ) u are all elements of ¯ R α ,β .By Lemma 3.5 ¯ R α ,β is closed under integration. Thus we can set˜ U y ( n ,n ) := (cid:90) t (1 + α Ω( α , τ )) R ( n ,n ) y dτ, ˜ U z ( n ,n ) := (cid:90) t (1 + β Ω( β , τ )) R ( n ,n ) z dτ, ˜ u ( n ,n ) := (cid:90) t R ( n ,n ) u dτ, to conclude the proposition.The fact that ˜ U y ( n ,n ) , ˜ U z ( n ,n ) ( t ) , ˜ u ( n ,n ) i ( t ) are polynomial in α n ,n , β n ,n , δ n ,n is aconsequence of the polynomial nature of R y , R z , R u . Property ii) follows from the fact that theremainder terms vanish if there are no lower order non-linear terms in (3.6). (cid:3) At last we return to the Dulac map D . The time to go from Σ ± y ∪ Σ ± z to Σ x can be computedfrom ˙ x = x as simply t = − ln x . The transition maps can be derived from the solution tothe variational equations using at t = 0, ( U y , U z ) = ( x p /q y , x p /q z ) and at t = − ln x , ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 17 ( U y , U z , u ) = ( y , z , u ). That is,(3.12) y = U y ( x p /q y , x p /q z , u , − ln x ) ,z = U z ( x p /q y , x p /q z , u , − ln x ) ,u = u ( x p /q y , x p /q z , u , − ln x ) , with y = ± z = ± ± y , Σ ± z respectively.Define(3.13) ω ( α, x ) = x − α − α α (cid:54) = 0 ω (0 , x ) = − ln x . The function ω is related to Ω by ω ( α, x ) = Ω( α, − ln t ) . By taking t = − ln x in the definition of R α,β , ¯ R α,β there are induced rings R ωα,β , ¯ R ωα,β .At last, we have the following theorem on the asymptotic structure of the Dulac map. Theorem 3.8.
Suppose that α (0) /β (0) / ∈ N . Then the Dulac map D is asymptotic to the series (3.14) y ∼ x α ( u )0 y (cid:88) n + n ≥ ¯ U y ( n ,n ) ( u ; x )( x p y q ) n ( x p z q ) n z ∼ x β ( u )0 z (cid:88) n + n ≥ ¯ U z ( n ,n ) ( u ; x )( x p y q ) n ( x p z q ) n u ∼ u + (cid:88) n + n ≥ ¯ u ( n ,n ) ( u ; x )( x p y q ) n ( x p z q ) n with y = ± , z = ± when mapping from Σ ± y , Σ ± z respectively. Each coefficient K ( n ,n ) =¯ U y ( n ,n ) , ¯ U z ( n ,n ) or ¯ u ( n ,n ) i , i = 1 , . . . , k, has the properties:i) K ( n ,n ) ∈ ¯ R ωα ,β .ii) If α ( u ) , β ( u ) are constant then K ( n ,n ) is polynomial in ln x .iii) K ( n ,n ) is polynomial in α ˜ n , ˜ n , β ˜ n , ˜ n , δ ˜ n , ˜ n for ˜ n + ˜ n ≤ n + n with vanishing constantterm.iv) If α n ,n (resp. β n ,n , δ in ,n ) vanish for n + n ≤ n ∈ N then ¯ U ( n ,n ) y ( t ) (resp. ¯ U ( n ,n ) z , ¯ U ( n ,n ) u i )vanish for n + n ≤ n. Proof.
The proof is primarily a consequence of Proposition 3.7 and the form of D given in (3.12).The explicit computation is given for y with the z , u following analogously. It is given that, y = U y ( x p /q y , x p /q z , u , − ln x ) . An asymptotic expansion for U y is given by the variation of U y in (3.7), that is, U y ( U y , U z , u ; t ) ∼ U (1) y ( u , t ) U y + U y (cid:88) U ( n ,n ) y ( u , t ) U q n y U q n z . Then, from Proposition 3.7 each of the variational coefficients U ( n ,n ) y ( u , t ) has the structure, U ( n ,n ) y ( u , t ) = e − α ( u ) t ˜ U y ( n ,n ) ( t ) with ˜ U y ( n ,n ) ( t ) ∈ R α ,β . By substituting t = − ln x , it follows, U ( n ,n ) y ( u , − ln x ) = x α ( u )0 ˆ U y ( n ,n ) ( x ) , for some ˆ U y ( n ,n ) ( x ) ∈ R ωα ,β . Hence, y ∼ x α ( u )0 U y + U y (cid:88) x α ( u )0 ˆ U y ( n ,n ) ( u ; x ) U q n y U q n z = x α ( u )0 x p /q y + x p /q y (cid:88) x α ( u )0 ˆ U y ( n ,n ) ( u ; x )( x p /q y ) q n ( x p /q z ) n = x p /q + α ( u )0 y (cid:16) (cid:88) ˆ U y ( n ,n ) ( u ; x ) y n q z n q x n p + n p (cid:17) . The desired asymptotic form of the y component of D follows.Properties i), iii) and iv) follow immediately from Proposition 3.7. If α ( u ) , β ( u ) are constantthen α ( u ) = β ( u ) = 0. The form can be computed by taking lim α ,β → ˆ U y ( n ,n ) ( u ; x ).As ˆ U y ( n ,n ) ∈ ¯ R α ,β then Lemma 3.6 gives property ii). (cid:3) Remark 3.9.
Setting z = 0 , y = 1 gives the Dulac map of a co-dimension 2 manifold ofnormally hyperbolic saddle singularities. If it is further assumed that u is merely a parameter,that is ˙ u = 0, then Theorem 3.8 gives the asymptotic structure of the transition near a family ofplanar hyperbolic saddles. This result agrees with [15].3.2. Case 2: α (0) /β (0) ∈ N In this section we treat the case α (0) /β (0) ∈ N . The general approach is the same as in theprevious section, however some minor care needs to be taken when dealing with the coefficients α − ,n , β n , − in the normal form (3.2).To make summation symbols less cumbersome, define the following subsets of N ,(3.15) N := (cid:8) ( n , n ) ∈ N (cid:12)(cid:12) n ≥ − , qn − mn ≥ , ( n , n ) (cid:54) = 0 (cid:9) N := (cid:8) ( n , n ) ∈ N (cid:12)(cid:12) n ≥ , qn − mn ≥ − , ( n , n ) (cid:54) = 0 (cid:9) N := (cid:8) ( n , n ) ∈ N (cid:12)(cid:12) n ≥ , qn − mn ≥ , ( n , n ) (cid:54) = 0 (cid:9) . Then, introduce as coordinates U y = x mp/q y, U z = x p/q z, and define α , β through, α ( u ) = m pq + α ( u ) , β ( u ) = pq + β ( u ) . In these new coordinates the normal form (3.2) is transformed to the vector field,(3.16) ˙ x = x ˙ U y = − α ( u ) U y + U y (cid:88) ( n ,n ) ∈ N α n ,n ( u ) U n y U qn − mn z ˙ U z = − β ( u ) U z + U z (cid:88) ( n ,n ) ∈ N β n ,n ( u ) U n y U qn − mn z ˙ u = (cid:88) ( n ,n ) ∈ N δ n ,n ( u ) U n y U n z The crucial achievement of the coordinate transform is to decouple U y , U z , u from x . ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 19
The centre-stable manifold x = 0 has been brought to U z = U y = 0. Similar to Section 3.1,we consider variations of the solutions U y = U z = 0. More explicitly, we consider a variation ofthe form,(3.17) U y ( U y , U z , u ; t ) = U (1) y ( u , t ) U y + U y (cid:88) ( n ,n ) ∈ N U ( n ,n ) y ( u , t ) U n y U qn − mn z U z ( U y , U z , u ; t ) = U (1) z ( u , t ) U z + U z (cid:88) ( n ,n ) ∈ N U ( n ,n ) z ( u , t ) U n y U qn − mn z ,u ( U y , U z , u ; t ) = u + (cid:88) ( n ,n ) ∈ N u ( n ,n ) ( u , t ) U n y U qn − mn z with, U (1) y (0) = U (1) z (0) = 1 , U ( n ,n ) y ( u ,
0) = U ( n ,n ) z ( u ,
0) = u ( n ,n ) ( u ,
0) = 0 , so that at t = 0, ( U y , U z , u ) = ( U y , U z , u ).The following proposition gives the structure of the variation coefficients. Proposition 3.10.
There exists functions ˜ U ( n ,n ) y , ˜ U ( n ,n ) z , ˜ u ( n ,n ) i ∈ ¯ R α ,β such that, U ( n ,n ) y ( u ; t ) = e − α ( u ) t ˜ U y ( n ,n ) ( t ) U ( n ,n ) z ( u ; t ) = e − β ( u ) t ˜ U z ( n ,n ) ( t ) u ( n ,n ) ( u ; t ) = ˜ u ( n ,n ) ( t ) with ˜ u ( n ,n ) := (cid:16) u ( n ,n )1 , . . . , ˜ u ( n ,n ) k (cid:17) . Moreover:i) Each ˜ U y ( n ,n ) , ˜ U z ( n ,n ) , ˜ u ( n ,n ) is polynomial in α ˜ n , ˜ n , β ˜ n , ˜ n , δ ˜ n , ˜ n for ˜ n + q ˜ n − m ˜ n ≤ n + qn − mn with zero constant term. .ii) If α n ,n (resp. β n ,n , δ in ,n ) vanish for n + qn − mn ≤ n ∈ N then U ( n ,n ) y ( t ) (resp. U ( n ,n ) z , U ( n ,n ) u i ) vanish for n + qn − mn ≤ n. The proof is omitted as it is almost identical to Proposition 3.7, namely, using induction on n , n to show that the integral solution to the variational equations gives the desired functions˜ U y ( n ,n ) ( t ) , ˜ U z ( n ,n ) ( t ) , ˜ u ( n ,n ) ( t ).Returning to the Dulac map, one again computes the time to go from Σ ± y ∪ Σ ± z to Σ x assimply t = − ln x . We have the relation,(3.18) y = U y ( x mp/q y , x p/q z , u , − ln x ) ,z = U z ( x mp/q y , x p/q z , u , − ln x ) ,u = u ( x mp/q y , x p/q z , u , − ln x ) . The theorem on the asymptotic structure of the Dulac map follows.
Theorem 3.11.
Suppose that α (0) /β (0) ∈ N and set γ = α − mβ . Then the Dulac map isasymptotic to the series, (3.19) y ∼ x β ( u )0 y + α − , ( u ) z m ω ( γ , x ) + y (cid:88) ( n ,n ) ∈ N ¯ U ( n ,n ) y ( u ; x )( x mp y q ) q n ( x p z q ) n − mq n z ∼ x α ( u )0 z + z (cid:88) ( n ,n ) ∈ N ¯ U ( n ,n ) z ( u ; x )( x mp y q ) q n ( x p z q ) n − mq n u ∼ u + (cid:88) ( n ,n ) ∈ N ¯ u ( n ,n ) ( u ; x )( x mp y q ) q n ( x p z q ) n − mq n with y = ± , z = ± when mapping from Σ ± y , Σ ± z respectively. Each coefficient K ( n ,n ) =¯ U y ( n ,n ) , ¯ U z ( n ,n ) or ¯ u ( n ,n ) i , i = 1 , . . . , k, has the properties:i) K ( n ,n ) ∈ ¯ R ωα ,β .ii) If α ( u ) , β ( u ) are constant then K ( n ,n ) is polynomial in ln x .iii) K ( n ,n ) is polynomial in α ˜ n , ˜ n , β ˜ n , ˜ n , δ ˜ n , ˜ n for ˜ n + q ˜ n − m ˜ n ≤ n + qn − mn withzero constant term.iv) If α n ,n (resp. β n ,n , δ in ,n ) vanish for n + qn − mn ≤ n ∈ N then ¯ U ( n ,n ) y ( t ) (resp. ¯ U ( n ,n ) z , ¯ U ( n ,n ) u i ) vanish for n + qn − mn ≤ n. Proof.
The proof is almost identical to the proof of Theorem 3.8, namely, using equation (3.18),Proposition 3.10 and substituting t = − ln x into the solution to the variational equations to getthe asymptotic structure. The only difference is showing the additional α − , z m ω ( γ , x ) term inthe y component of the Dulac map D . This comes from the variational coefficient U ( − , y ( u , t ).The coefficient must solve the variational equation ddt U ( − , y ( u , t ) = − α ( u ) U ( − , y ( u , t ) + α − , ( u ) U (1) z ( u , t ) . By Proposition 3.10 it is known that U (1) z ( u , t ) = e − β ( u ) t . It follows that, U ( − , y ( u , t ) = α − , Ω( α − mβ , t ) = Ω( α − mβ, t ) . Finally, U ( − , y is the coefficient of U mz in the U y variation. Substituting U z = x p/q z as perequation 3.18 yields the desired term in the asymptotic expansion of y . (cid:3) Remark 3.12.
Due its applicability to problems in celestial mechanics, especially [6], it is worthisolating the case when α, β take constant values on N . In the co-dimension 2 case, one obtainsthe asymptotic series by setting z = 0 , y = 1 in Theorem 3.8 and invoking property ii) to get,(3.20) y ∼ x α (cid:88) n ≥ ˆ U ( n ) y ( u ; ln x ) x np u ∼ u + (cid:88) n ≥ ˆ u ( n ) ( u ; ln x ) x np , for functions ˆ U ( n ) y , ˆ u ( n ) polynomial in ln x and smooth in u .It is now evident that the asymptotic structure of the higher dimensional Dulac maps D share similar properties to the well known planar case. In the planar case the coefficients ANIFOLDS OF NORMALLY HYPERBOLIC SINGULARITIES 21 functions g i ( u, x ) are known to be polynomial in the functions ω ( α , x ). This is mirroredin the present case with each of the coefficients K ( n ,n ) ∈ R ωα ,β , the ring of polynomialsin ω ( ± α , x ) , ω ( ± β , x ). The Mourtada property of the higher order asymptotic terms, firstshown in the case ˙ u = 0 in [2], should also be evident. Acknowledgment
The author would like to thank Holger Dullin for all the discussions and constructive criticismsgiven throughout the writing of this paper.
References [1] Genrikh R. Belitskii. C ∞ -normal forms of local vector fields. Acta Applicandae Mathematicae , 70(1):23–41,2002.[2] Patrick Bonckaert and Vincent Naudot. Asymptotic properties of the Dulac map near a hyperbolic saddlein dimension three.
Annales de la Facult´e des Sciences de Toulouse. S´erie VI. Math´ematiques , 10, January2001.[3] Alexander D. Bruno.
Local Methods in Nonlinear Differential Equations . Springer Series in Soviet Mathe-matics. Springer-Verlag, Berlin, 1989.[4] Jean-Baptiste Caillau, Jacques Fejoz, Micha¨el Orieux, and Robert Roussarie. Singularities of min time affinecontrol systems. preprint, February 2018.[5] Kuo-Tsai Chen. Equivalence and Decomposition of Vector Fields About an Elementary Critical Point.
Amer-ican Journal of Mathematics , 85(4):693–722, 1963.[6] Nathan Duignan and Holger R. Dullin. On the c / -regularisation of simultaneous binary collisions in thecollinear 4-body problem. to appear.[7] Nathan Duignan and Holger R. Dullin. Regularisation for planar vector fields. Nonlinearity , 2019,arXiv:1901.08701. in press.[8] Freddy Dumortier and Robert Roussarie. Smooth Normal Linearization of Vector Fields Near Lines of Sin-gularities.
Qualitative Theory of Dynamical Systems , 9(1):39–87, November 2010.[9] C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet, and G. Iooss. A simple global characterization for normalforms of singular vector fields.
Physica D: Nonlinear Phenomena , 29(1):95–127, November 1987.[10] Marty Golubitsky and Victor Guillemin.
Stable Mappings and Their Singularities . Graduate Texts in Math-ematics. Springer-Verlag, New York, 1973.[11] Yulij Ilyashenko and Weigu Li.
Nonlocal Bifurcations , volume 66 of
Mathematical Surveys and Monographs .American Mathematical Society, Providence, Rhode Island, October 1998.[12] Eric Lombardi and Laurent Stolovitch. Normal forms of analytic perturbations of quasihomogeneous vectorfields: Rigidity, invariant analytic sets and exponentially small approximation.
Annales Scientifiques del’ ´Ecole Normale Sup´erieure , pages 659–718, 2010.[13] A Mourtada. Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan mise sous formenormale.
Bifurcations of Planar Vector Fields , pages 272–314, 1990.[14] James Murdock.
Normal Forms and Unfoldings for Local Dynamical Systems . Springer, 2006.[15] Robert Roussarie.
Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem , volume 164 of
Progress in Mathematics . Birkh¨auser Verlag, Basel, 1998.[16] Robert Roussarie and Christiane Rousseau. Almost Planar Homoclinic Loops in R3.
Journal of DifferentialEquations , 126(1):1–47, March 1996.[17] Carl L Siegel and J¨urgen K Moser.
Lectures on Celestial Mechanics: Reprint of the 1971 Edition . Springer,2012.[18] Shlomo Sternberg. On the Structure of Local Homeomorphisms of Euclidean n-Space, II.
American Journalof Mathematics , 80(3):623–631, 1958.[19] Sebastian Walcher. Symmetries and convergence of normal form transformations.
Monograf´ıas de la RealAcademia de Ciencias Exactas, F´ısicas, Qu´ımicas y Naturales de Zaragoza , 25, January 2004.[20] Stephen Wiggins.
Normally Hyperbolic Invariant Manifolds in Dynamical Systems . Applied MathematicalSciences. Springer-Verlag, New York, 1994.(Nathan Duignan). Applied MathematicalSciences. Springer-Verlag, New York, 1994.(Nathan Duignan)