44
Catastrophe Theory in Dulac Unfoldings H.W. Broer, V. Naudot, R. Roussarie. December 8, 2004 Keywords: Hamiltonian approximation, Poincar´ e map, displacement function, bi- furcation of limit cycles and tori, scaling, Standard Catastrophe Theory, Pointed Catastrophe Theory, Dulac expansion, compensator expansion. Abstract In this paper, we study a class of families of planar vector fields. Each member of the family consists of an autonomous vector field and a non- autonomous perturbation which is periodic in time. The autonomous part is the sum of a Hamiltonian vector field which does not depend on the pa- rameters and a parameter dependent dissipative part. This setting, modulo suitable rescalings, covers for example the codimension Hopf bifurcation and several cases of (subordinate) homoclinic bifurcation. In this setting we consider the non-autonomous part as a perturbation and mainly focus on the ‘unperturbed’ autonomous family of planar vector fields. Our interest is with the geometry of the bifurcation set of limit cycles, in particular in the neighbourhood of certain homoclinic bifurcations. In our approach the cor- responding Hamiltonian vector field has a homoclinic loop to a hyperbolic saddle point. It is to be noted that the full system has the solid torus as its phase space where the ‘unperturbed’ limit cycles correspond to invari- ant 2-tori with parallel dynamics. It is to be expected that various kinds of quasi-periodic and chaotic dynamics persist for the ‘perturbed’ system. How- ever, it is known that all of this is confined to an narrow neighbourhood of the bifurcation set of limit cycles. Returning to the ‘unperturbed’ family of planar vector fields, the limit cycles are in one to one correspondence with the zeros of a displacement function. The asymptotics of this latter does not take the form of a Taylor series but is a deformation of a Dulac expansion. An exponential scaling is constructed in the parameter space. As a result we generically find an exponentially narrow fine structure in the parameter space in which the bifurcation set of limit cycles can be completely described by Standard Catastrophe Theory. Department of Mathematics and Computing Science, University of Groningen, Blauwborgje 3, 9747 AC Groningen, The Netherlands. Institut Math´ ematiques de Bourgogne, CNRS, 9, avenue Alain Savary, B.P. 47 870, 21078 Dijon cedex, France. 1

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Page 1: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

Catastrophe Theory in Dulac Unfoldings

H.W. Broer�, V. Naudot

�, R. Roussarie.

December 8, 2004

Keywords: Hamiltonian approximation, Poincare map, displacement function, bi-furcation of limit cycles and tori, scaling, Standard Catastrophe Theory, PointedCatastrophe Theory, Dulac expansion, compensator expansion.

Abstract

In this paper, we study a class of families of planar vector fields. Eachmember of the family consists of an autonomous vector field and a non-autonomous perturbation which is periodic in time. The autonomous partis the sum of a Hamiltonian vector field which does not depend on the pa-rameters and a parameter dependent dissipative part. This setting, modulosuitable rescalings, covers for example the codimension � Hopf bifurcationand several cases of (subordinate) homoclinic bifurcation. In this settingwe consider the non-autonomous part as a perturbation and mainly focus onthe ‘unperturbed’ autonomous family of planar vector fields. Our interest iswith the geometry of the bifurcation set of limit cycles, in particular in theneighbourhood of certain homoclinic bifurcations. In our approach the cor-responding Hamiltonian vector field has a homoclinic loop to a hyperbolicsaddle point. It is to be noted that the full system has the solid torus �������as its phase space where the ‘unperturbed’ limit cycles correspond to invari-ant 2-tori with parallel dynamics. It is to be expected that various kinds ofquasi-periodic and chaotic dynamics persist for the ‘perturbed’ system. How-ever, it is known that all of this is confined to an narrow neighbourhood ofthe bifurcation set of limit cycles. Returning to the ‘unperturbed’ family ofplanar vector fields, the limit cycles are in one to one correspondence withthe zeros of a displacement function. The asymptotics of this latter does nottake the form of a Taylor series but is a deformation of a Dulac expansion.An exponential scaling is constructed in the parameter space. As a result wegenerically find an exponentially narrow fine structure in the parameter spacein which the bifurcation set of limit cycles can be completely described byStandard Catastrophe Theory.

�Department of Mathematics and Computing Science, University of Groningen, Blauwborgje 3,

9747 AC Groningen, The Netherlands. Institut Mathematiques de Bourgogne, CNRS, 9, avenue Alain Savary, B.P. 47 870, 21078 Dijon

cedex, France.

1

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Contents

1 Introduction 21.1 Background and setting of the problem . . . . . . . . . . . . . . . 4

1.1.1 Classical cases of polynomial geometry . . . . . . . . . . 51.1.2 The homoclinic loop case . . . . . . . . . . . . . . . . . 61.1.3 Example: the codimension two saddle connection . . . . . 71.1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.1 The Dulac case . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 The compensator case . . . . . . . . . . . . . . . . . . . 14

1.3 Application: the codimension three saddle connection . . . . . . . 181.4 Further applications . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Proofs 222.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Proof of Theorems 3 and 4 . . . . . . . . . . . . . . . . . . . . . 37

1 Introduction

This paper contributes to the study of generic families of planar diffeomorphismsnear a degenerate fixed point [1, 17, 47, 48, 49]. Note that such diffeomorphismscan be obtained from time periodic planar vector fields, by taking a Poincare (re-turn) map. Usually, by normal form or averaging techniques [1, 12], the familybecomes autonomous to a large order. In many cases, after suitable rescaling thefamily is the sum of a Hamiltonian vector field which does not depend on the pa-rameters, a dissipative part and a non autonomous part. The latter part consists ofhigher order terms in the rescaling parameter [5, 23, 25, 26, 48]. This approach cov-ers many generic cases like the codimension � -Hopf bifurcations including degen-erate Hopf (or Chenciner) and Hopf-Takens bifurcations [17, 20, 21, 22, 47, 48, 49]but also the Bogdanov-Takens bifurcation for maps [5, 10]. Our interest is with thegeometry of the bifurcation set of limit cycles of the autonomous system whichcorrespond to invariant tori in the full system. The latter were studied system-atically by Broer, Braaksma, Huitema and Takens [9]. Limit cycles of the au-tonomous system correspond to fixed points of the Poincare map (or zeros of thedisplacement function) on one-dimensional sections. For values of the parameterthat are relatively far from the bifurcation set of limit cycles associated to the au-tonomous system, the dynamics of the non autonomous system is of Morse-Smaletype [43]. The genericity conditions on the original family often lead to certainexplicit genericity conditions on the corresponding family of scaled displacement

2

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functions [17, 20, 25, 26], which in our set-up will be translated to suitable hy-potheses. By the Division Theorem, the latter lead to structural stability of a scaleddisplacement function under contact equivalence [31, 33, 40].

Starting point of the present paper will be the family of non-autonomous sys-tems after rescaling, although our results are concerned with bifurcations of limitcycles in the autonomous truncation. In Example 1 below, we show the classi-cal way in which this format is obtained, later on more classes of examples willbe discussed. Before that, we note that for the full non-autonomous family, theautonomous bifurcation diagram provides a skeleton indicating the complexity ofthe actual dynamics [21, 51]. However, as Broer and Roussarie [17] have shown,this dynamical complexity near the bifurcation set of limit cycles, in the real ana-lytic setting, is confined to an exponentially narrow horn. For earlier results in thisdirection, see [9].

Example 1 (BOGDANOV-TAKENS BIFURCATION [5, 12, 15, 24]) Consider thefollowing unfolding of the codimension two bifurcation����� ����� � ��� ��� ����� ������� � ����� ��� � ��� � � � � !�!" � �where

�is a #%$ -periodic in the time � and represents the non autonomous part. We

assume that & ��� �'� � ���(� � � � ! & ��) � � � � � � !+*, . Under the classical rescaling [5, 15]

� �.- �0/��� ���.-21 /� � � �435-76 � �8�.- �0/� �the system above becomes

- /9�:(�<;�=�.- /� /� � -?> � /� � 3A@7! � - /� /���B- /� /� � ) � -217!DC /�FEWe see that /9 :(�G;� �H�8I � -%J :(�G;� � ) � - � ! � where

��IK� /�MLL ;N �O� /� � 3P@7!QLL ;R isHamiltonian which does not depend on the parameters

� /� � -+! and possesses a ho-moclinic loop to a hyperbolic saddle. The loop encloses a disc containing a center.Notice that the non-autonomous terms are included in the

) � - 1 !-part. See [15, 17]

for details.

This example motivates the general perturbation setting of our paper. More exam-ples will be recalled below [12, 15, 18, 25, 26] and certain new examples will betreated in detail. Our focus will be with bifurcations of limit cycles near a homo-clinic trajectory of the Hamiltonian vector field. By an explicit exponential scalingin the parameter space, an exponentially narrow horn is created, in which the bi-furcation set of limit cycles can be completely described. Indeed the structural sta-bility of a scaled displacement function under contact equivalence, leads to the fullcomplexity of Catastrophe Theory. For the original autonomous family of vectorfields, inside the horn, this yields structural stability under weak orbital equivalence[6, 27, 28], where the reparametrizations are S�T . In summary, our results reveal

3

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an exponentially narrow fine structure of the parameter space, that regards the dy-namics of the autonomous family of planar vector fields. We expect that our resultswill be of help when studying the full system on the solid torus � ������� � wherethe ‘unperturbed’ limit cycles correspond to invariant 2-tori with parallel dynam-ics. When turning to the non-autonomous ‘perturbed’ system, persistence of thesetori becomes a matter of KAM Theory, in particular of quasi-periodic bifurcationtheory [6, 11]. Indeed, the 2-tori with Diophantine frequencies form a Cantor fo-liation of hypersurfaces in the parameter space and we conjecture that most of theDiophantine tori persist, including their bifurcation pattern. The KAM persistenceresults involve a Whitney smooth reparametrization which is near the identity-mapin terms of & - &�� @ E This means that the perturbed Diophantine 2-tori correspond toa perturbed Cantor foliation in the parameter space. In the complement of this per-turbed Cantor foliation of hypersurfaces we expect all the dynamical complexityregarding Cantori, strange attractors, etc., as described in [19, 20, 21, 22, 38, 42].

1.1 Background and setting of the problem

The general perturbation format of the present paper is given by the family of S8Tplanar, time-periodic vector fields� � : � -�� �� ������� � : � -��%� I � -��(9� � : � (1)

where�� � :

consists of the autonomous part. The following properties hold. First�����! #"$ #%are integers. Secondly, &(')�+* is a (multi)-parameter varying

near�, while

--, �is a perturbation parameter. Thirdly, the vector field

�is #%$ -

periodic in the time � , and may depend on all the other variables and parametersas well. Finally, the vector field

� Iis Hamiltonian and does not depend on the

parameters. Notice that in Example 1, we have� � @

," � # and

% �/.. We

begin discussing the Hamiltonian function 0 . Our genericity setting implies that0 should be a stable Morse function, which leaves us with three possible cases.� I

is defined in an annulus, near a center or near a homoclinic loop, see figure 1.

Σ

Σ−

p

(a) (b) (c)

ΣΣ0

Figure 1: The Hamiltonian� I

in a annulus (a), near a center (b), near a homoclinicloop (c).

In each case, we can define a Poincare return map12 � :�354268794 � ������ 12 � : � � ! � � � - �;:��=< � : � � ! E (2)

4

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where4 6 � 4 is a transversal section and where

< � : � � ! is the scaled displace-ment function. Limit cycles are in a one to one correspondence with the zeros of< � :

. The bifurcation set � of limit cycles is defined as

� ��� � & � -%! ' � * � ���.& < � : � � ! � <�� � : � � ! � ���� �������� � ' 426�� EDenote by � :��=� ��� �7- � - 6 �

. The family9 � :

is generic in a sense to be madeprecise below and which by the Division Theorem [28, 33, 37, 40] implies S8Tstructural stability under contact equivalence of the scaled displacement function< � :

in the following sense. In the following definition the families of functionsare local, i.e., we consider germs of families.

Definition 1 (CONTACT EQUIVALENCE [6, 27, 37, 44, 50])Let � 3 � � � � ! 7 � � � � ! and � � 3 � � � � ! 7 � � � � ! be two families of functions,with & � � ' � � * � � ! E These families are � T -contact equivalent if there exist a S Tdiffeomorphism � 3 � � * � � ! 7 � � * � � ! , a S�T family of diffeomorphisms 3� � � � ! 7 � � � � ! and a S�T function ! 3 � � � � ! 7 � � � � ! with ! � � !#"� �

such that

� $ ��% ! � ! ��% !'& ��(*) �+ ��% ! EThe definition implies that sends the zero-set of � to the zero-set of � (*) �+ pre-serving multiplicity. As a consequence for small values of

-, the zero set � :

isdiffeomorphically equivalent to � 6 . Although our main interest is with the ho-moclinic loop case, we first briefly revisit the simpler cases of an annulus and acenter.

1.1.1 Classical cases of polynomial geometry

In the case of an annulus (see figure 1-(a)), by our assumption of structural stabilityunder contact equivalence and by the Division Theorem [28, 33, 37, 40],

< � :takes

the form

< � : � � ! � ! � �'� & � -%!�,.- � /0 � �'��1 � & � -%!�! E (3)

In (3), ! "� � is smooth and

, - � /0 � �'��1 � & � -%!�!F� 1 6 � & � -%! � 1 �� & � -+! � � &2&2& � 1 0 : � � & � -+! � 0 : � � � 0 � (4)

which is the Standard Catastrophe Cuspoid normal form. Here, the map

&437 � 1 6 � & � � ! � E E E ��1 0 : � � & � � !�!is a submersion. The codimension of the singularity is � 3 @

. For-

sufficientlysmall the bifurcation set � : consists of all & -values such that

,.- � /0 � �'��1 � & � -%!�!F�65 ,7-� /05 � � �'��1 � & � -+!�!Q� � �

5

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for some � near�. Since

,�- � /0 takes the form (4), we say that for sufficientlysmall

-, the geometry of the bifurcation set of limit cycles � :

is polynomial. Bystudying the zero set � 6 using Standard Catastrophe Theory [6, 30, 44, 54], werecover the generic bifurcation theory of limit cycles as it now has become standard[6, 4, 27, 30]. For instance in the case � � # , the bifurcation set is called a foldcorresponding to a saddle node of limit cycles, in the case � � .

one obtains a cuspof limit cycles.

In the case of a center (see figure 1-(b)), by similar arguments, the reduceddisplacement function takes the form

< � : ��% ! � ! ��% � & � - !�, � � /0 ��% ��� � & !�! �where

% � % � , % being the distance to the origin, with the Pointed CatastropheCuspoid normal form

, � � /0 ��% ��� !F� � 6 � & � -%! � � �� & � -%! %�� &2&2& � � 0 : �

� & � -+! % 0 : � � % 0 � %�� � E (5)

Remark that, � � /0 possesses a term of order � 3A@

while, - � /0 does not. For more

details, see [17]. Again, the map

& 37 � � 6 � & � � ! � E E E ��� 0 : � � & � � !�!is a submersion and the codimension of the singularity is � . For each

-sufficiently

small, the set � : consists of all & -values such that

, � � /0 ��% ��� � & � -%!�!F� 5 , �� /05 %

��% ��� � & � -+!�!Q� � �for some

% , �. As in the annulus case, for each sufficiently small

-, since

, � � /0takes the form (5), the geometry of the bifurcation set of limit cycles � :

is polyno-mial. � : is now described by the Pointed Catastrophe Theory. This set-up recoversall codimension � Hopf bifurcations [17, 20, 49].

1.1.2 The homoclinic loop case

Our main interest in the present paper is with the case where the Hamiltonian 0possesses a homoclinic loop � (see figure 1-(c)). Here the Poincare return map isdefined on a half section

4 6 � 4 parameterized by%�� �

. The difference betweenthe present and the above cases is that the displacement function is singular at% � �

and then can not be written as a Taylor expansion at this value. When-=� �

,the displacement function takes the form of a Dulac expansion [3, 34, 35, 45] andwhen

- "� �, the form of a compensator expansion [34, 35, 45] which deforms the

Dulac expansion. We shall be more precise in the next section.The aim of this paper is to present a Division Theorem which is adapted to

Dulac and compensator unfoldings. We shall construct a singular scaling in the pa-rameter space so that the corresponding family of displacement functions becomes

6

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structurally stable under contact equivalence, and thanks to the classical DivisionTheorem, takes the form (3). This implies that the bifurcation set � :

has polyno-mial geometry. The function ! in (3), although smooth in the interior of the horn,contains all nonregularity of the displacement function, i.e., has Dulac asymptoticsat the tip of the horn. The region in the phase space to be considered, though ar-bitrarily close to � , does not contain � . However, the region does contain all the(bifurcating) limit cycles.

1.1.3 Example: the codimension two saddle connection

To fix thoughts, before stating our results, we announce an application in the caseof a codimension 2 degenerate homoclinic orbit [17, 25, 26], as it is subordinate toa degenerate Bogdanov-Takens bifurcation.

Example 2 (DEGENERATE BOGDANOV-TAKENS BIFURCATION [17, 25, 26])Consider the following unfolding of the codimension three Bogdanov-Takens bi-furcation ��� � � � � ��� �� � � � ��� ����� � � 6 � �

�� � � � � 1 ! � E

This family is studied in [25, 26]. The bifurcation set is a topological cone withvertex at

� '�� 1 , which is transverse to the 2 sphere � � . The trace of the bifurcationset on �'� possesses, among various bifurcation points of codimension two, a pointof degenerate saddle connection. After suitable rescaling [17], the family heretakes the form��� �G;��� �G;� � � - /�� /� � - � /��� 3 @ � -�� � /� /� 6 � /� � /� /� � /� /� 1 !�!= /� EIn term of our general set-up, the family takes the form (1) with

� �4@," ���

and

0 � /��� /� ! � @# /� � 3 @

.Q/� 1 � /� EThe phase portrait of

�MIis described in figure 1-(c). The scaled displacement

function takes the form< � � : ��% ! � � 6 � � 6 � � � � -+! � 1 �

� � 6 � � � � -+! %�� : ��% ! �� %��� ��% ! � (6)

where� : ��% !

is a compensator function [12, 17, 34, 35, 36] and where� 6 ��% !��

� �� %, where

%is the distance to the saddle connection. In this example, it is generic

to assume that� "� �

and that the map� � 6 � � � ! 37 � � 6 � � 6 � � � � � ! ��1 � � � 6 � � � � � !�!is a local diffeomorphism near

�. For simplicity, we restrict to the case

- � �,

since the case- "� �

is just more complicated without being qualitatively different.Theorem 2 gives the following scaling

� 3 � 3�� � �=! � � � ��� 6 ! 7 � ��� ��� 6 ��� ! 37 � � 6 ��1 � ! �7

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where�!, �

, � 6 , � and where���� ��� 1 �� 3 � � � � �� � , �

� � !@ � � �� � �� 6 � � 6 �@ � � �� �

3 3 � � � � � � �� � � 6 � � !@ � � �� � � (7)

where,� and

� 6are S�T , see Figure 2. We claim that as an application of our

results,

<�� )�� � � � + � 6 > � � @ � � !DC � �@ � � �� �> � 6 3 � @# � ) � � � �� � !�! � � � ) � � 1 !DC �

and by the Division Theorem, we have

< � )� � � � + � 6 > � � @ � � ! C � �@ � � �� � !� ��� � 6 ��� ! > � 6 3 � �# C �

where ! is S�T and has Dulac asymptotics at � � �, see below. Therefore

< � )� � � � + � 6 > � � @ � � ! C � ! � ��� � 6 ��� ! > � 6 3 � � C � (8)

where! � �'� � 6 ��� ! � �@ � � �� � ! � ��� � 6 ��� ! �

see Definition 1.

����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0γ

τ

ψ 0B

α

0

Figure 2: Image of a rectangle� 3�� � �"! � � � ��� 6 ! under the scaling

�. On the right

hand side the corresponding region is an exponentially flat horn. The resultingbifurcation set � 6 , located in the dashed region on the right hand side, is the productof a fold bifurcation set and an interval.

The idea of the scaling in (7) is the following. Observe that the form (6) contains alogarithmic term and therefore is singular at

% � �. However, the form is no longer

singular at% � � for any � , � arbitrarily small. By putting

% � � � @ � � ! , welocalize the study of (7) near

% � � , i.e., near � � �. The map

� � )� � � � + � � ! � <�� )� � � � + � 6 > � � @ � � ! C8

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is regular and can be expanded in a Taylor series. All nonregularity of the unfoldingis now contained in the parameter dependence (7).

From (7), we have

� 6�� 3 ��� 6 3 � ! 1 � �: �� �� E

Since� 6

belongs to an interval of length # � , for fixed 1 � , we have the followingexponential estimate

� 6�� # � 1 �� � :�� � �'1 � , � E (9)

The bifurcation set of limit cycles is given by � 6 � � ��� !where

� ��� � 6 � � � �which is the product of a fold and an interval and corresponds to a saddle nodebifurcation of limit cycles.

As a consequence of the estimate (9), the image of the rescaling is an expo-nentially flat region, see Figure 2. Observe that the set of homoclinic bifurcationwhich is given by � � � � 6 � � �

is contained in the image of the rescaling intro-duced in (7). However, since the study of the displacement function is localizednear

% � � � ��� � 3 �� � � � ! , the region of the phase space under considerationis an exponentially narrow annulus, which does not contain the homoclinic orbit.Therefore, the subordinate homoclinic bifurcation cannot be studied with help ofthe form (8). Up to division by ! , the displacement function is polynomial; indeedit is the normal form of a fold [44, 54].

As mentioned before, the stability of the displacement function under contactequivalence implies weak orbital stability of the vector field unfolding with S Treparametrization.

Remark: In Example 2, we find a saddle-node of limit cycles inside the horn, cor-responding to a fold catastrophe of the scaled displacement function. We mentionthat in [10, 17] similarly stable limit cycles are discovered that correspond to scaleddisplacement functions of Morse type.

1.1.4 Outline

This paper is organized as follows. All results will be formulated in section 1.2.We first study the rescaled displacement function

< � :when

-�� �, in which case< � 6

has a Dulac expansion. In the Theorems 1 and 2 we present normal forms for< � 6 � showing that the geometry of its zero-set again is polynomial. Also we givean explicit algorithm for the normalizing rescaling. We mention that Theorem 2generalizes the example of section 1.1.3, extending it to all cuspoid bifurcations oflimit cycles of even degree. Second we treat the case

- , � � in which the scaled dis-placement function will have a so-called compensator expansion, which deforms

9

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the above Dulac case. The Theorems 3 and 4 are the analogues of Theorems 1 and2 for this compensator case.

Section 1.3 contains an example concerning a cusp bifurcation of limit cycles,as it is subordinate to a codimension 4 Bogdanov Takens bifurcation [32, 34, 35].This example is an application of Theorem 1. In section 1.4 we discuss furtherapplications of our results for the full ‘perturbed’ system, where our statements aremainly conjectural and pointing to future research. All proofs are postponed tosection 2.

1.2 Results

We briefly recall the general setting. The family of planar autonomous vector fieldto be considered has the form (1)� � : �- � � I � - � 9 � : �where

� Ipossesses a homoclinic orbit � � � 0 � � �

. We focus our studynear this homoclinic loop. For all

% % 6where

% 6is close to

�, orbits of� I

contained in ��� � � 0 � % �are supposed to be periodic, ��� tending to the

homoclinic loop � as%

tends to�

in the Hausdorff topology. The subsection4 6 �

4(see Figure 1), on which the Poincare return map is well defined, transversally

intersects each of these periodic orbits. The Poincare return map takes the form (2).The asymptotics of

< � :is given by Roussarie [45]. For convenience, we introduce

the 1-form ����� � � , which is dual to the vector field� � :

by the standard area form� on � � , i.e., ��� �� � � �.- � 5 0 � - � � � 6 � � -%! EWe have

< � 6 ��% ! � ����� � � 6 E (10)

This Abelian integral expands on a logarithmic scale, to be described below. Suchan expansion is called a Dulac expansion. In Theorem 1 and 2 a scaling in theparameter space

� 3 �0 : � � � � 7 �

0 � ��� ��� ! 37 � ��� ��� !is constructed in such a way that

< � )�� � � + � 6 � � � @ � � !�! � � � � ! ��� 6 � �� � � &2&2& ��� 0 : � � 0 : � � � ��� ��� ! � 0� ) � � 0 � � !�! � (11)

where � � � !=� ��� � �� � , in the case � � #�� 3.@and � � � !=� � �� @ � � �� � ! , in the

case � � #�� and where � � � !#"� �. The idea behind the construction of the scaling

is the following. By putting% � � � @ � � ! , the map��� � � 3 � 7 � � � 37 < � � � � 6 � � � @ � � !�!

10

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becomes S T at the origin, and we expand� � � � in a Taylor series. In the generic

case, we may identify the coefficients of the Dulac expansion with & . First, in the� � ��1 ! -space, a parametrization by � is given of the curve � defined by

� ��� � � ��1 ! '�� � & � � � � � � ! � � � � � � � � ! � &2&2&0� � ) 0 : � +� � � � � !F� � � � ) 0 +� � � � � !#"� � � EThis latter curve emanating from the origin is of ’highest degeneracy’, i.e., alongthis curve � � ) 6 � � + � � ! � � � � ! � � � � ��� ! � 0 � ) � � 0 � � !�! ESecondly, the scaling

�is defined. The image of this latter is a narrow neighbour-

hood of � and for each parameter value in this neighbourhood, the map� � )�� � � +

takes the form (11). By the Division Theorem [28, 33, 37, 40] applied to (11), weobtain � � )�� � � + � � ! � ! � �'� � ��� ! � � � ! ��� 6 � �

� � � &2&2& ��� 0 : � � 0 : � � � 0 ! �� ! � �'��� � � !�, - � /0 � ��� � ! �where ! � �'��� � � !=� � � � ! ! � ��� � ��� ! . All nonregularity of the unfolding is hiddenin the function ! and after division by this latter, the displacement function is poly-nomial. In a final step, we consider the complete compensator expansion given in[45]. In Theorem 3, respectively 4, we show that there exists an analytic deforma-tion of the scaling proposed in Theorem 1, respectively 2, such that

< � )�� � � + � : takesagain the form (11).

1.2.1 The Dulac case

We now give a precise description of the Dulac case. The logarithmic scale offunctions is given by

� � ��@ � % � �� % � % � &2&2& � %�� � %�� � � � �� % � &2&2& � EThe Abelian integral (10) expands on this scale [45, 34, 35, 36]. This means thatthere exist sequences of S T functions ��� � & ! and 1�� � & ! for integers � � �

and � @such that

< � 6 ��% ! � �

� � 6��� % � � �

��� �1�� % � � �� % ����% ! � (12)

for any � '�� EWe say that (12) is a nondegenerate Dulac unfolding of order #�� if � � "� � but

for each integer � � � � E E E � 3�@one has 1 � � �

� � ! � � � � � � � ! and if moreover, themap

&437 � � 6 � & ! ��1 � � & ! � E E E ��� � : � � & ! ��1 � � & !�!11

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is a submersion. In this case, we rewrite (12) to

< � 6 ��% ! � � : ��

� � 6� � � & ! % � � ��

� � �1�� � & ! % � � �� %��� % � ����% � ! �

where� � � � � & ! .

Similarly we say that (12) is a nondegenerate Dulac unfolding of codimension#�� 3O@if 1 � � � !#"� � but � 6 � � ! � �

for each integer � � @ � E E E � 3 @one has

1�� � � !F� � � ��� � � ! and if moreover, the map

& 37 � � 6 � & ! ��1 � � & ! � E E E ��� � : � � & ! ��1 � : � � & ! ��� � : � � & !�!is a submersion. In this case, we rewrite (12) to

< � 6 ��% ! � � : ��

� � 6��� � & ! %�� � � : ��

� � �1 � � & ! %�� � �� %��� % � � �� � � ) ��% � ! �

where� � 1 � � & ! . We introduce the following notation. Let

� 3 � � � � ! � 7 � be aDulac expansion of the form (12). We then write

� ��% ! � � %�� � �� * ��% ! � &2&2&0� � %�� � �� * ��% ! � �

) � �*+���� ) 0 � � + � * � � %

0 � �� ����% ! � (13)

where the order

on the set

� - � � ��� � ! ' � ��� � & � � � �is defined as follows�� � ! �� F� @ � � !�� � � � � F� @ ������� �� � 3 @7! �� � ! E (14)

In what follows � � denotes the set of strictly positive real numbers. We nowstate the first results.

Theorem 1 (DULAC CASE OF ODD CODIMENSION) Let< 3 � � � � � � � � : � 7 � ���% ��� ��1 ! 37 < � � � ��% ! � � : ��

� � 6� � % � � � : ��

� � �1 � % � � �� %����� % � � �� %�� &2&2&�� �

be a nondegenerate Dulac unfolding of codimension #�� 3 @. Then

< � )� � � + > � � @ � � ! C � � ��� ��� ! � � � �� � > , - � /� � : � � �'� � ! � ) � � � � ! C � (15)

where � ��� ��� !#"� � depends S�T on��� ��� ! , and where

� 3 � � � : � � � � 7 � � � : � � ��� ��� ! 37 > � ��� ��� ! ��1 ��� ��� ! C E12

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The map�

is S�T and of the form�� � � ��� : � � �� � � /� � ��� ��� ! � ��� � � � 3 @ �1 � � ��� : � � �� � /1 � ��� ��� ! � @ � � � � 3 @ E (16)

We further specify/1 � ��� ��� ! � 3 � � � ��� � ��� ! 3 � �� : � � /� � : � � � � � ! �/� � : � ��� ��� ! � 3 � : ��

� � ���� � : � � � � �� : � � � 0 � : � � � ! /1�� ��� ��� !� 5 � : � � �� : � � � � � : � � �� : � � 3 � ��� � : � � � � /� � : � � � !�! � �� : � � �

and for each integer � � � � E E E � � 3 # ,

/� � ��� ��� ! � 3 � : ��

� � � � ���� � � /� � ��� ��� ! 3 � : ��

��� ���� � � � � �� : � � � 0 � � � ! /1�� ��� ��� !

� 5 � � �� : � � � � � � �� : � � 3 � �� � � � � /� � � � !�! � �� : � � EFor each pair of integers

� � � ! under consideration, the coefficients� � � � , 0 � � � ,

�� � � ,� � , 5 � are real numbers. Moreover

� � is a linear map and/� � � � ! � /� � � 6 � � � �� � /� � � � � &2&2& �/� � : � � � � � ! � /� � : � � � � 6 � � � �� � /� � : � � � � � � &2&2& ETheorem 2 (DULAC CASE OF EVEN CODIMENSION) Let

< 3 � � � � � � � � 7 � ���% ��� ��1 ! 37 < � � � ��% ! � � : ��

� � 6� � % � � ��

� � �1 � % � � �� %����� % � � &2&2&�� �

be a nondegenerate Dulac unfolding of codimension #�� . Then

<�� )�� � � + > � � @ � � !DC� � ��� ��� ! ���@ � � � � �� � > ,7- � /� � � ��� � ! � ) � ��� � � � !DC � (17)

where � ��� ��� !#"� � depends S T on��� ��� ! and where

� 3 � � � : � � � � 7 � � � � ��� ��� ! 37 > � ��� ��� ! ��1 ��� ��� !DC EThe map

�is S�T and of the form�� � � � � � � : � /��� ��� ��� ! � � � � � � 3 @ �

1 � � � � : �@ � � � � �� � /1 � ��� ��� ! � @ � � � � E (18)

13

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We further specify/1 � ��� ��� ! � 3 � � � � � @ � � � � � � �� � ! : ��� � ��� !3 � � �� � , � � : � � � ! �for each integer � � @ � E E E � � 3 @

,/1 � ��� ��� ! � 3 � � � �� � � @ � � � � � � �� � ! : � � � ��� ! � � � ��� ! 3 � � �3 � � �� � � � � , � � � : � � � ! � � � � � �� � � , � � : � � � ! 3 � � �� � , � � � : � � � ! �/� � : � ��� ��� ! � 3 � �� �@ � � � � �� � ��

� � ���� � : � � � � �� : � � � 0 � : � � � ! /1�� ��� ��� !3 ��� � � 6 � � � : � � @ � � � � �� � ! : � � � � �� � � � : � � � ! �

and for each integer � � � � E E E � � 3 # �/� � ��� ��� ! � 3 � : ��

� � � � � �� � � /� � ��� ��� !

3 � �� �@ � � � � �� � ����� �

��� � � � � �� : � � � 0 � � � ! /1�� ��� ��� !3 ��� � � 6 � � � � @ � � � � �� � ! : � 3 � � �� � � � � � ! E

For each pair of integers� � � ! under consideration,

� � � � , 0 � � � ,�� � � , � � , are real

numbers. Moreover, � � is a linear map and

, � � � ! � , � � 6 � � � �� � , � � � � &2&2& �� � � � ! � � � � 6 � � � �� � � � � � � &2&2& E1.2.2 The compensator case

In the case- "� �

an asymptotic expansion for< � :

in [45] was found to be< � : ��% ! � � 6 � 1 �

� % � � -�����% � % � ! � � � �

� %�� - ����% � % � ! � � &2&2&

(19)� � � � % � � - � � ��% � % � ! � � 1 � � � � % � � � � � -�� � � � ��% � % � ! �� � � ��% � & � -%! �where ��� and 1�� � � for

��� �� � , depend S�T on� & � -+! . Either

� � � � � � !F� 1 � � �� � � � ! � � � � � � � � 3 @ � ��� � � � � � � � !#"� � �

or

� � � � � � ! � 1 � � �� � � � ! � � � � � � � !Q� � � �� � � � 3 # �������.1 � � � � � !#"� � E

14

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Furthermore,� � � ��% !

is a compensator function of the form

� ��% ! � % :��� 3 @- 1 � � - 1 � � �� ��% ! � � �� %� - 1 � � �

where- 1 �K@"3 ��� � !

and where��� � !

is the absolute value of the ratio of thenegative eigenvalue of the linear part of

� : � at the saddle point by the positive one.

Moreover, the functions� � � � � are polynomial in

%and

% � ��% !, taking the form

� � ��% � %�� ��% !�! � � � � 6 % � � &2&2& � � � ��% � %�� ��% !�!F� � � � 6 % � � � � � � � ��% ! � &2&2& �where

� � � 6 and� � � 6 are real numbers. Note that the compensator unfolding deforms

the dulac unfolding and is obtained from the latter by replacing� �� ��% !

by� ��% !

.The remainder

� � is of class S � and flat of order � at% � � E An expansion of the

form (19) is said to be a nondegenerate compensator unfolding of codimension �if for

-�� �, it is a nondegenerate Dulac unfolding of codimension � . Theorem 1

and 2 can be extended for compensator unfoldings as follows.

Theorem 3 (COMPENSATOR CASE OF ODD CODIMENSION) Let

< 3 � � � � � � � � : � 7 � ���% ��� ��1 ! 37 < � � � ��% ! � � : ��

� � 6� � � % � � &2&2&�� � � : ��

� � �1 � � %�� � ��% ! � &2&2& �

� ��� % � � ��% ! � &2&2&��

be a nondegenerate compensator unfolding of codimension #�� 3 @. Then

< � )�� � � + > � � @ � � !DC� � ��� ��� � -+! � � � � � ! > ,7- � /� � : � � ��� � ! � ) � � � � !DC �where � ��� ��� � -%!#"� � depends S T on

��� ��� � -%! and where

� 3 � � � : � � � � 7 � � � : � � ��� ��� ! 37 > � ��� ��� ! ��1 ��� ��� ! C EThe map

�is S�T and of the form

��� � � � : � � � � � ! /� � � � � � -+! � � � � � � 3 @ �1�� � � � : � � � � ! /1 � � � � � � -%! � @ � � � � 3A@ E

We further specify

/1 � � � � � � -%! � 3 ���� � � � � -+! ��� � : � � � � � � � � -+! 3 � � � � ! : � /� � : � � � � � � -+! �15

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where for each integer � � � � E E E � #�� 3 # ,�� � is linear in

�with coefficients de-

pending analytically on-

and � � � � � ! . Furthermore

/� � : � � � � � � -+! � 3 � : ��

� � �� �� � : � � � � � � -+! � : � � � ! � �0 � : � � � � � � -+! /1 � � � � � � -+!� �5 � : � � � � -+! � : � � � ! � � � : � � : � � � !3 � � �� � : � � � � � � -%! � /� � : � � � � -%!�! � : � � � ! �/��� � � � � � -+! � 3 � : ��

� � � � ��

�� � � � � � -+! /� � � � � � � -+!

3 � : ��

� � �� �� � � � � � � -+! � : � � � ! � �0 � � � � � � -+!�! /1�� � � � � -+!

� �5 � � � � -+! � : � � � ! � � � � : � � � ! 3 � � �� � � � � � � -+! � /� � � � !�! � : � � � ! �� � � � E E E � � 3 # E

For each pair of integers� � � ! under consideration,

�� � � � , �0 � � � , �

�� � � , �� � , �5 � , /� � and/� � , are analytic functions in

-and in � � � � � ! .

Theorem 4 (COMPENSATOR CASE OF EVEN CODIMENSION) Let< 3 � � � � � � � � 7 � ���% ��� ��1 ! 37 < � � � ��% ! � � : ��

� � 6��� � % � � &2&2&�� � ��

� � �1 � � % � � ��% ! � &2&2& �

� � � % � � &2&2& �be a nondegenerate compensator unfolding of codimension #�� . Then

< � )�� � � + > � � @ � � !DC.� � ��� ��� � -+! ���@ � � � � � � -+! � � � ! > ,7- � /� � � ��� � ! � ) � � � � � � !DC �where � ��� ��� � -%!#"� � depends S T on

��� ��� � -%! , and where

� 3 � � � : � � � � � ��� ��� ! 37 > � ��� ��� ! ��1 ��� ��� ! C EThe map

�is S�T and of the form

��� � � � : � /��� � � � � � -+! � ��� � � � 3 @ �1 � � ��� : �@ � � � � � � ! /1 � � � � � � -+! � @ � � � � E

We further specify/1 � � � � � � -+! � 3 � � � � � � -+! �.� @ � � � � � � -%! � � � !�! : � �� � � � � � � -+!3 � � � � !�, � � : � � � � -+! �16

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for each integer � � @ � E E E � � 3 @/1 � � � � � � -+! � 3 � � � � � -%! � � � ! � @ � � � � � � -+! � � � !�! : � �� � � � � � � -%!� �

� � � � � � � -%! 3 � � � � � � -%!3 � � � � � ! � � , � � � : � � � � -+! � � � � � � -+! � � � � � !�, � � : � � � � -%!3 � � � � !�, � � � : � � � � -+! �where for each integer � �4@ � E E E � � 3 @

, � � is linear in�

with coefficients depend-ing analytically on

-and � � � � � ! . Furthermore

/� � : � � � � � � -+! � 3 � � � !@ � � � � � � -%! � � � ! ��� � �

�� � : � � � � � � -+! � : � � � ! /1�� � � � � � -%!3 � � � !@ � � � � � � -%! � � � ! ��

� � �

�0 � : � � � � � � -%! /1�� � � � � � -+!3 � �� � � 6 � � � -+! � � � : � � @ � � � � � � -+! � � � !�! : �� � � � � ! � � : � � � � -%! �and for each � �4@ � E E E � � 3 # ,

/��� � � � � � -+! � 3 � : ��

� � � � ��

�� � � � � � -+! /� � � � � � � -+!

3 � � � !@ � � � � � � -%! � � � ! ��� � �

� �� � � � � � � -+! � : � � � ! � �0 � � � � � � -+!�! /1�� � � � � � -+!3 � �� � � 6 � � � -+! � � � � @ � � � � � � -%! � � � !�! : � 3 � � � � ! � � � � � -+! E

For each pair of integers� � � ! under consideration,

�� � � � , �0 � � � , �

�� � � , � � , , � and

� �are analytic functions in

-and � � � � � ! .

Remarks:

- Concerning the topology of the bifurcation set of limit cycles, Mard ��sic

[34, 35] shows that the following properties hold. In the Dulac expan-sion case (

- � �), the bifurcation diagram is homeomorphic to the bi-

furcation diagram of the model, � � /0 ��% ��� ! in (5) and in the compensator

case (-�, �

), the bifurcation diagram is homeomorphic to the product ofthe interval

� � � - 6 ! and the bifurcation diagram of the polynomial model,, � � /0 ��% ��1 ! . If we restrict the parameter space to the image of the map�

given in Theorem 1-4, these latter properties also directly follow from ourapproach. Mard �

�sic’s approach is rather topological and his results are not

suitable for more analytical studies. We emphasize here the explicit char-acter of our results that are preparations for further applications. Indeed in[16], we apply these results to study quasi periodic bifurcation of invarianttori for the full system

� � : � - � �. For more detailed see sub-section 1.4.

17

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- An interesting observation concerns the image of the scaling and its order ofcontact with the hyperplane

� � 6 � � �of homoclinic bifurcation. In the case

of odd codimension, if a point� � ��1 ! belongs to the boundary of the image

of the scaling. From (16) it follows that for all% , �

there exists a constant� ,�@

such that for each integer � �4@ � E E E � � 3A@ �� & 1 � : � & � : � � � � � � & � � & � & 1 � & � � � & 1 � : � & � : � : � �

and� & 1 � : � & � � & � 6 & � � & 1 � : � & � : � E

This implies that the contact is � -flat, compare with Example 3. However,in the case of even codimension, from (18) it follows that for all

% , �there

exists a constant S ,�@such that

& 1 � & � S ��� � 3 � 3 �& 1 � : � & ! � � �4@ � E E E � � 3 # �& � � & � S ��� � 3 �& 1 � : � & ! � � � � � E E E � � 3 @ E

This implies that the contact is infinitely flat and that the bifurcation set islocated in an exponentially narrow horn, compare with Example 2. Further-more, the region in the phase space under consideration is exponentially thinannulus. This observation reveals a significant difference between the oddand the even codimension case.

- Concerning the geometry of the bifurcation set of limit cycles, from formulae(15) and (17) we clearly obtain the hierachy of the Standard CatastropheTheory in terms of the parameter

�and this hierachy is independant of � .

In this sense we can say that along the curve � , the singularity has beenremoved. From this, using formulas (16) and (18), we can deduce a hierachyfor the bifurcation sets in the parameters

� � ��1 ! which now depends on � ina non regular manner when � approaches

�. It would be of interest, at least

in the odd codimension case, to further study the geometry of the bifurcationset of limit cycles, for instance the order of contact between the differentsubordinate 5 3 fold bifurcations in a family of cuspoids of degree � , 5 .However, when working in the range of the scaling, it is appropriate to workin terms of the parameter

�and the variable � .

1.3 Application: the codimension three saddle connection

Consider the following unfolding of the codimension four Bogdanov-Takens bifur-cation ���� � � ��� � � � ����� � � � 6 � �

� � � � � � � � � 6 ! E (20)

18

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This family was partially studied in [25, 32], for more background see [4, 24, 26,30, 51]. After the rescaling� �435- 6 � � 6 �.-�� & 6 � �

��.-�� & � � � � �.- 6 & � � � � - � /�'� ���.- 1 /� �

and then omitting all bars, we end up with the following 3-parameter family ofvector fields�� � �� � - � �

�� � - > � � � 3 @7! � - � � � & 6 � & � � � & � � 1 � � 6 ! � ) � - �7! C E (21)

In terms of our general system (1) we have� �4@

and" ���

,% ,��

, i.e.,

� � : �.-7� I � -�� 9 � : �where

0 � /��� /� ! � @# /� � 3 @

. /� 1 � /� EThe scaled displacement function takes the form

< � : ��% ! � � 6 � 1 �� % � ��% ! � &2&2&�� � � �

� %�� &2&2&�� ���� % � � ��% ! � &2&2&�� � ) ��% 1 ! EFor simplicity, we again restrict to the case

-"� �. The scaling

� ��� ��� ! �K> � ��� ��� ! ��1 ��� ��� !DCof Theorem 1 takes the explicit form

1 �� 3 . � � 3 # � � � �� � � � � /� � � � ! �

� 6 � � � � � �� � � � 6 � � � �� � � �� � � � �� � � � � /� 6 � � ! � (22)

� �� . � � � �� � � # � � � �� � � � �

� �� �� � � � /� � � � ! E

Therefore

<�� )�� � � + � 6 > � � @ � � ! C � � � � �� � ! � ��� � ��� ! ��� 6 � �� � � � 1 !� ! � �'� � ��� !�,7- � /1 � �'� � ! �

where ! is given by the Division Theorem. The latter function, although smoothif � , �

, has Dulac asymptotics at � � �. Up to a division by ! , the scaled

displacement function is the normal form of a cusp. The bifurcation set of limitcycles � is given by � � � ��� !

where

� ��� � � � 3�. ��� � # � 1 ! � � , � �?� '�� � �19

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which is the product of a cusp and an interval, see Figure 4. From (22) we observethat if

� � 6 ��1 � ��� � ! belongs to the bifurcation set of limit cycles, the followingestimates hold

& � � & � & 1 � & � & � � & � : � � & � � & � � & � 6 & � @� & � � & � : � �

for any% , �

. This implies that contrary to the even codimension case, see Ex-ample 2, the order of contact between the bifurcation set of limit cycles and thehyperplane

� � 6 � � �of homoclinic bifurcation is finite.

00.002

0.0040.006

0.0080.01

0

0.05

0.1–0.1

–0.05

0

0.05

0.1

0.01

0.02

0.03

0.04

0.00050.001

0.00150.002

0.00250.003

0.0035

–1e–05–5e–06

1e–051.5e–05

Figure 3: Image of the cube� � � � E � @ � � � 3 � E @ � � E @ � � � 3 � E @ � � E @ � under the scaling

(22). On the right hand side, the resulting set is bounded by surfaces which meetat the origin with finite order.

±2

±1

0

1

2

±3±2.5

±2±1.5

±1±0.5

0

±1

0

1

PSfrag replacements�

���

���

-4

-2

0

2

-2

-1

0

1

-0.5

0

0.5

1

1.5

-4

-2

0

2

-2

-1

0

1

PSfrag replacements

� �

���

��

Figure 4: The bifurcation set of limit cycles before the scaling (�

on the left handside) and after the scaling ( � on the right hand side). This latter is the topologicalproduct of a cusp and an interval squeezed in a narrow horn.

1.4 Further applications

We summarize as follows. Given a family of vector fields of the form (1)� � : � �'� � ! � - � ��� ��� � ��� � & � -%! �with �'� � ��� '�� � &�'�� * and where

- � �is a real perturbation parameter. The

autonomous part� � : � - � �8I � - � 9� � :

is such that��I

is Hamiltonian with

20

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Hamilton function 0 , which is of Morse type and9 � :

a dissipative family. More-over,

� Ipossesses a homoclinic loop � . The dissipative part is generic in an

appropriate sense. In this setting we focus our study near the homoclinic loop �and our concern is with the autonommous family

�� � :. By an explicit scaling, a

narrow fine structure for the bifurcation set of limit cycles is found in the parameterspace � * ��� & � E We show that the corresponding geometry is polynomial where ascaled displacement function can display the full complexity of the cuspoid family

,7- � /0 � ��� � ! � � 6 � �� � � &2&2& � � 0 : � � 0 : � � � 0 � (23)

where� � � � & ! is appropriate, compare with (4).

We now discuss certain consequences the above results have for the full non-autonomous family

� � : � - � �, see (1), where

�is assumed time-periodic of

period #%$ and therefore has the solid torus � ��� ��� as its phase space. Here thelimit cycles of the ‘unperturbed’ part

� � :correspond to invariant 2-tori with par-

allel dynamics [9, 11]. By normal hyperbolicity these tori persist as long as themulti-parameter & remains outside an exponentially narrow neighbourhood of thebifurcation set of limit cycles found before, compare with [17] for details.

Near the bifurcation set of limit cycles we can apply quasi-periodic bifurcationtheory as developed by [2, 7, 8, 9, 11, 13, 20, 39, 52, 53]. To this end we consider‘unperturbed’ tori which are Diophantine. This means that the two frequencies� @ � � � & !�! are such that

����� � & ! 3 � " ���� ��� " : � (24)

for certain� , �

and� , # E For fixed

�and

�formula (24) inside � * � � & � de-

fines a Cantor foliation of hypersurfaces. At the intersection of this foliation withthe bifurcation set, the ‘unperturbed’ family

�� � :displays a cuspoid bifurcation

pattern of 2-tori as can be directly inferred from the above. In the ‘perturbed’ case� � : � - � �, we conjecture persistence of many of these Diophantine tori, includ-

ing the bifurcation pattern, provided that & - & � @ E Here� � �'� -%! � � -%!

hasto be chosen appropriately. For instance whenever the Cantor foliation intersectsa fold-hyperplane transversally, we expect a quasi-periodic saddle-node bifurca-tion of 2-tori to occur [9, 11]. These KAM persistence results involve a Whitneysmooth reparametrization which is near the identity-map in terms of & - & � @ E Thismeans that the perturbed Diophantine 2-tori correspond to a perturbed Cantor foli-ation in the parameter space � * ��� & � E In the complement of this perturbed Cantorfoliation of hypersurfaces we expect all the dynamical complexity regarding Can-tori, strange attractors, etc., as described in [19, 20, 21, 22, 38, 42]. This programwill be the subject of [16] where we aim to apply [9, 11, 52, 53] to establish theoccurrence of quasi-periodic cuspoid bifurcations in the three cases (a), (b) and (c)of Figure 1.

A second topic is the generalization of the approach of the present paper tocases where the Hamiltonian 0 in (1) no longer is a (stable) Morse function, butundergoes simple bifurcations.

21

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Acknowledgements: We thank Pavao Mard ��sic, Carles Simo, Floris Takens, Re-

nato Vitolo and Florian Wagener for their help and their stimulating discussion.

2 Proofs

We first give a proof of Theorem 1 and 2. Theorem 3 and 4 generalize Theorem1 and 2 to compensator unfoldings and are obtained by a perturbation method.Before we distinguish between the odd and even codimension case we first describethe main idea. Consider a nondegenerate Dulac unfolding of the form

< � � � ��% ! � ����� 6

� � %��Q� ����� �

1�� %�� � �� %�� 0 ��% ! � (25)

where 0 ��% ! � � % � � � � �� % � &2&2&. Put

% � � � @ � � ! ��� � < � � � > � � @ � � !DC� � � � � � � ! EUsing the Taylor formula we get

� � � � � � ! � � ��� � 6

� � � � � ) � � � � � � ! E (26)

Since the Dulac unfolding is nondegenerate we can see the coefficients of the un-folding as independent. The goal of this section is to show that independent co-efficients of the Dulac unfolding yield independent coefficients for the monomials� � in (26). It will also be proven that the non zero leading term in (25) ( � � foreven and 1 � for odd codimension) leads to a non zero leading term in (26). Toillustrate this, we consider the case of codimension � � #�� 3 @

, i.e., where 1 � "� � EThis case is covered by Theorem 1 and in (11) we have � � � ! � ��� � �� � . It followsthat indeed the leading term in (25) is of degree #�� 3 @

; compare with Example3 for a special case. In the case of codimension � � #�� covered by Theorem 2,we similarly have � � � ! � � ��� � @ � � �� � ! in (11); compare with Example 2 for aspecial case.

To be precise, for each integer � � � � E E E � #�� , in (25) we have

� � � � ) � + � � !���

� � ����< ) � +� � � � � ! (27)

and together with (25) we get

� � � � ����

����� 6

� � ��% � ! ) � +� � � � � ����

��� � �

1�� ��% � � �� % ! ) � +� � � � � ���� 0 ) � + � � ! E

22

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Observe that the following equalities hold

� ���� � �

��% � ! ) � +� � � � � � � � � ���� � 3 � ! � ! � � � � �

� ���� � �

��% � ! ) � +� � � � � � � � �� ���� 1 �

��%�� � �� % ! ) � +� � � � � 3�@7! � : � : � � � � 3 3 @7! ���� 1 � � � � � �

and

� �� � 0 ) � + � � ! � � � � � � �� � � � � � ! �

where � � � � ! � � � � 6 �B� � � � � � �� � � &2&2& EWe use the Leibniz rule to compute

��% � � �� % ! ) � +� � � , when � � and we get

� ���� 1 �

��%�� � �� % ! ) � +� � � � 1 � ���

��- � 6

> ��

C ��% � ! ) � : - +� � � � � �� % ! ) - +� � � �� 1�� �

�� �� �� �

�� 3 � ! �� 1�� ��� �

��- � �

> ��

C ��% � ! ) � : - +� � � � � �� % ! ) - +� � � �� 1�� �

�� �� �� �

�� 3 � ! �� 1�� ��� �

��- � �

> ��

C � 3�@7! - : � � � 3A@7! � �� 3 � � � ! � EObserve that

� ����

��% � � �� % ! ) � +� � � � � ����� �� �

�� 3 � ! � � � ����

��% � � �� % ! ) � +� � � �where @

�����%�� � �� % ! ) � +� � � � ��

- � �> �

C � 3�@7! - : � � � 3 @7! � ���� � 3 � � � ! � �

� ��- � �

� 3�@7! - : � �� � � 3 � ! � � � 3 � ! � � E23

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We then introduce the following notation. Let

� � ��� � � ! � � 6 ������� � � � ��� 6 ������� � � ��� � ��� � � � ! � � 6 ������� � � � ��� � ������� � ��K� � 0 � � � ! � � 6 ������� � � � ��� � ������� � ��� ��� � �� � � � ! � � � � � ������� � � � � � � � ������� � � �

be respectively the � � @ � � � @, � � @ � � , � � @ � � and � � � matrices defined

as follows

�� � � � 0 � � � �K>

�C � � � �

�� � � � 0 � � � � � � � � �

� � � � � � 3�@7! � : � : �� �

� � � 3 3 @7! � � � � � �� � � � � @

���> % � � �� % C ) � +� � � �

��- � �

� 3�@7! - : � �� � � 3 � ! � � � 3 � ! � � � � � � �� � � � � � 3�@7! � : � � �

� � � � � 3 3 @7! � E

Finally H� � � � � � ! � � � ������� � � � : � � � � � ������� � � is defined as follows

� � � � �4@ � � � � � � � � � � � � !#"� � � � � ! EWe also introduce the following matrices

� : � ��:� � � ! � � 6 ������� � � : � � � � 6 ������� � � : � � � �2 � �

:� � � �

�� � �%�

� : � � 0 :� � � ! � � 6 ������� � � : � � � � � ������� � � � �� �2 � 0 :� � � � 0 � � �and

� : � ��� :� � � ! � � 6 ������� � � : � � � � � ������� � � � �� �2 � � :� � � � � � � �%�With the above notations we get

� � � ��� � 6 �

� � � � � � �F� ��� � �

> 0 � � � � �� � � � � � � C 1 � � � (28)

� � � � � � �� � � � � � ! �if � � � and

� � � ����� �

1���� �� 3�@7! � : � : �

��� � � � 3 3 @7! � � � � � � � �� � � � � � ! � (29)

� ����� �

� � � �21�� � � � � � � � � �� � � � � � ! �if � , � . We state the following lemma.

24

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Lemma 1 Let

� � � � � � � ! � � � ������� � � � : � � ��� � ������� � � � � � � � � � � � � � � , � � � � � � � � � � � �� � � � � � � ! � � � ������� � � � : � � ��� � ������� � � : � � � � � � � � � � � �

�O� � < � � � ! � � � � � ������� � � � : � � � � � ������� � � : � � < � � � ��� � � � EThen

�,

�, �

and � are invertible.

PROOF: We show that��

is invertible. The non degeneracy of the three othermatrices follows exactly from the same argument. First of all, we claim that foreach integer

@ � � � �� � ,� � � � , � . To show this, recall that

� � � � � @���

> % � � �� % C ) � +� � � EIt then follows that

� � � � � � � � � @� � � @7! � > � � @7! % � � �� %�� % � C ) � +� � � �� � @� � @ � � � � � �� � � @7! � � 3 � ! � E (30)

In particular, for each integer@ � � � � ,

� � � � � ��0 � �

@� E

Since for each integer@ � � � �

��� - �K> % - � �� % C �� � � � @ �

it turns out that for each integer � � @ � E E E � � ,��� - is always positive. With (30) it

then follows that for each integer@ � � � �� � ,

� � � � ’s is always positive.By definition

� � � � � ! � � � ������� � � � : � � ��� � ������� � � � @���

> ��% � � �� % ! ) � +� � � C � � � ������� � � � : � � � � � ������� � � EThus

� � �7� � � � � ! � @� � � : �� � � ��� �

� � > ��%�� � �� % ! ) � +� � � C EFor each integer

� @ � E E E � � , we put � � ��% ! � ��% � � �� % ! ) � + . Then

� � �7� � � � � ! � @� � � : �� � � ���

� � � � � E E E � � � ! � @7! �25

Page 26: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

where

� � � � � E E E � � � ! ��% ! ����������

� ���% ! E E E � � ��% !� ����% ! E E E � �� ��% !...

......

� � : ����% ! E E E � � : �� : � ��% !

���������

ELet � 3 � 7 � � % 37 � ��% ! be a S T function. We have that

� � �*� � � E E E � �*� � ! ��% ! � � � � � � � � E E E � � � ! ��% ! EThis latter property can be shown by induction on � using the fact that the deter-minant is multilinear. Take

� ��% !F� � 3�@ � ! % � : � � � � 3 @7! � EWe then get

� � �*� � � E E E � �*� � ! ��% ! �����������

@ � �% � � % � &2&2& � � : � % � : � � � 3�@7! � � � �� %� � � # � � % &2&2& � � 3 @7! � � : � % � : �� � # � � &2&2& � � 3 @7! � � 3 # ! � � : � % � : 1E E E E E E E E E E E E E E E� � � E E E � � : � � � 3 @7! �

����������

�where for each integer

�4@ � E E E � � 3 @,

� � � � 3�@7! �� � 3 @7! � ��%�� � �� % ! �� � � � � 3�@7! � �� � 3 @7! � � � � � "� � EThis implies that

� � �*� � � E E E � �*� � ! � @7! � � : ��

� � �

� � � "� � �this ends the proof of Lemma 1. �

From now on we need to distinguish between the odd-codimension case (de-veloped in the next sub-section) and the even-codimension case (developed in thelast sub-section).

2.1 Proof of Theorem 1

In the odd case � � #�� 3 @, the parameters are � 6 � E E E ��� � : � ��1 � � E E E ��1 � : � . More-

over we have 1 � � � "� �is a constant. This implies that

< � � � ��% ! � � : ��

� � 6� � % ��� � : ��

� � �1 � %�� � �� % �� % � � �� %�� % � � ��% ! �

where � ��% ! � � 6 � ��% � �� % � &2&2& E

26

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In this situation, the term � � % � in (25) is included in the term% � � ��% !

. By putting� � � � @ � � ! and< � � � > � � @ � � ! C � � � � � � � ! , we get

� � � � � � ! � � 6 � �� � � E E E � � � � : � � � � : � � � � � : � � � � : � � ) � � � � ! E (31)

With (27), (28) and (29) the following equalities hold

���

� 6...� � : �

���� � � :

�����

� 6� � �...

� � : � � � : ������� � � �� � � :

�����

1 � �...1 � : � ��� : �� � �

������ (32)

� � :�����

1 � �...1 � : � � � : �� ���

������ �

�����

� 6 � � !...� � : � � � !� � : � � � !

������ �

�����

� �...� � � : �� � � : �

������ � > � � � �� � C

�����

1 � �...1 � : � ��� : �� � �

������ �

�����

� � � � !...� � � : � � � !� � � : � � � !������ � (33)

where for each integer � ' � � � � � @ � E E E � #�� 3 @ �,� � � � ! takes the form� � � � ! � � � /� � � � ! � � � � /� � � 6 � � � �� � /� � � 6 � &2&2&�! E

We have the following strategy. We aim to find a scaled reparametrization of theform

� ��� ��� ! � � � ��1 !linking

� � ��� 6 � E E E � � � � : 1 ! and � with� � ��1 ! . For each integer � �4@ � E E E � � 3A@ �

we require�

to have the components

��� � � � : � � �� � � /��� ��� ��� ! ��1 � � � � : � � �� � /1 � ��� ��� ! � (34)

leading to� � )�� � � + � � ! � � � � ! ��� 6 � �� � � &2&2& � � � � : 1 ��� � : 1 � � ��� ��� ! � � � : � � ) � ��� � !�! E

We first determine � � � ! and show that the leading term � ��� ��� ! is such that � � � � � !#"� � .Note that the set

� � � ��� ! parametrizes the curve � of ’highest degeneracy’ of� � )�� � � + . We then obtain the desired expression for� � )� � � + by solving

� � � � � � ! � � � � � @ � E E E � #�� 3 # E27

Page 28: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

To be more precise, we start solving the equation

� � � � � � � � � E E E � #�� 3 # EWith (33) and the definition of the matrix

�, we get

� � ���

� �...� � � : ����� � �

��� 1 � �...1 � : � ��� : �

���� ��� � � � �� �

�����

@�...�

������ (35)

� � � ���� � � � �...� � � : � � �

���� � ��

�� � � � !...� � � : � � � !

���� E

By Lemma 1, the matrix�

is invertible. Let

� : � � ��� � � � ! � � � ������� � � : � � � � � ������� � � : �be its inverse. After multiplication by

� : � on the left hand side and the right handside of (35) we get��

� 1 � �...1 � : � ���

���� � 3 � � � � �� �

���

����...� � : � � �

���� � � �

��� /� � � � !.../� � � : � � � !

���� � (36)

where��� /� � � � !.../� � � : � � � !

���� �43 � � : �

��� � � � �...� � � : � � �

���� 3 � : �

��� /� � � � !.../� � � : � � � !

���� E (37)

Observe that for each integer � �4@ � E E E � � 3 # ,

/� � � � ! � /� � � 6 � � � �� � /� � � 6 � &2&2& EWe now set

� � �in (34). With (35) and the definition of the matrix

�we have��

�� � � � � � ��� !...� � � : � � � ��� !

���� � �

��� 1 � �...1 � : � ��� : �

���� (38)

� � � ���� � � � � � �...� � � : � � �

���� � ��

�� � � � � � !...� � � : � � � !

���� �

28

Page 29: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

� � � � �� � �

��� /1 � � � ��� !.../1 � : � � � ��� !

����

� � � ���� � � � � � � � /� � � � � � !...� � � : � � � � /� � : � � � !

���� E

Since� : � is invertible we have

������ � E E E � � � : � � � !#"� � . Therefore after division by

��� � �� � of both sides of (36), we get� /1 � � � ��� ! � E E E � /1 � : � � � ��� !�!#"� � � � E E E � � ! EMoreover, since

�is non degenerate, with (38) we must have

��� � � � � � � � � : � & � � � � ��� ! & � 3 � � & � � � ! & � �� � �where � � � !#"� �

. But� � � � � � � ��� ! � E E E � � � � : � � � ��� !�! � � � � E E E � � ! �which implies that � � � : � � � ��� !F� � � � ! � � � �� � �with

� � � ! � � : ��

��� �

< � : � � � /1�� � � ��� ! ��� � �� : � � � � � � : � � � � � � : � � � !�! EAs we announced before, we set � � � !�� ��� � �� � E We now develop the scaling.For each integer � � � � E E E � #�� 3 # � we put

� � ��� ��� ! � � � � � � �� � � �� � � #�� 3 . �� � � : � ��� ��� !�� � EFrom (35) and (37) we then get

��� 1 � �...1 � : � ��� : �

���� � 3 � � � � �� �

�����

@�...�

������ � � � �

��� /� � � � !.../� � � : � � � !

���� (39)

� � � � �� ������

� �...� � � : 1������� E

29

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After multiplication of both sides of (39) by��: � we get��

� 1 � �...1 � : � � � : �

���� � 3 � � � � �� �

���

����...� � � �

���� � � �

��� /� � � � !.../� � � : � � � !

���� (40)

� � � � �� � � : �

�����

� �...� � � : 1������� E

Next, with (32) one gets

� � : � � � : � � 3 � : ��

� � ���� � : � � � � � �� � 0 � : � � � ! 1�� � � 3 � � � � �� � 0 � : � � �� � � : � � � � �� � 3 � � � � � : � � � 3 � � /� � : � � � ! �

and for each � �4@ � E E E � � 3 # ���� � � � 3 � : ��

� � � � ���� � � � ��� � 3 � : ��

� � ���� � � � � � �� � 0 � � � ! 1���� �3 � � � � �� � 0 � � � � � � � � � �� � 3 � � � � � � � 3 � � /� � � � ! E

Recall that ���� � � � : � � �� � � /� ��� ��� ! � �� � � � 3A@ �1 � � ��� : � � �� � /1 � ��� ��� ! � @ � � � � 3 @ E (41)

Putting �����

����� !...� � : � ��� !� � : � ��� !

������ � � : �

�����

� �...� � � : 1������� � � � � � � � � � 3 � 0 � � � � 5 � �

by (40), this implies that/1 � ��� ��� ! � 3 � � � 3 � �� : � � /� � : � � � � � ! ��� � ��� ! � @ � � � � 3 @ � (42)

and

/� � : � ��� ��� ! � 3 � : ��

��� ���� � : � � � � �� : � � � 0 � : � � � ! /1�� ��� ��� !3 � ��� � : � � � � /� � : � � � !�! � �� : � � � 5 � : � � �� : � � � � � : � � �� : � � �/� � ��� ��� ! � 3 � : ��

��� � � � �� � � /� � ��� ��� ! 3 � : ��

��� ���� � � � � �� : � � � 0 � � � ! /1 � ��� ��� ! (43)

3 � ��� � � � � /� � � � !�! � �� : � � � 5 � � �� : � � � � � � �� : � � � �� � � � 3 # E30

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Observe that (42), (43) together with (41) define a scaling in the parameter space.In the new parameters

��� ��� ! , the displacement function takes the form

� � )�� � � + � � ! � � � � �� � > � 6 � �� � � &2&2& � � � � : 1 ��� � : 1 � � ��� ��� ! ��� � : � � ) � ��� � !DC �

where � � � � � ! � � � � !#"� �. By a rescaling of the form � � � 3 � � ��& � ��� ��� ! & and after

removing the tildes, we get

� � )�� � � + � � ! �43 & � ��� ��� ! & � � � �� � > ,.- � /� � : � � � ! � ) � ��� � !DC EThis ends the proof of Theorem 1. �

In the last subsection we prove Theorem 2. Though the idea is similar, thereare differences in the computations that deserve to be mentioned.

2.2 Proof of Theorem 2

We now turn to the even case � � #�� . The strategy is the same as in the previoussection. For � � #�� , the expression in (25) holds with � � � � "� �

is a constant andwe have

� � � � � � ! � � 6 � �� � � &2&2& � � � � : � � � � : � � � � � � � � � ) � � � � � � ! � (44)

where���

� 6...� � : �

���� � � :

�����

� 6� � �...

� � : � ��� : ������� � � � : � � �� � � : !

��� 1 � �...1 � ���

����

� � � � � � �� ������

� 6 � � !��� � !...� � : � � � !

������ � (45)

�����

� �...� � � : �� � � : �

������ � > � � � �� � C

�����

1 � �...1 � : � � � : �1 � ���

������ (46)

� � � ������

@�...�

������ � � � � � � �� �

�����

� � � � !...� � � : � � � !� � � : � � � !������ �

31

Page 32: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

and �����

� � � �...� � � : �� � �

������ � �

�����

1 � �...1 � : � � � : �1 � ���

������ � � � � � � �� �

�����

� � � � � � !...� � � : � � � !� � � � � !������ E (47)

From (46), we get

� � � � � � � � �� � 1 � � � � ����� �

� � � � 1�� � � � � � � � � �� � � � � � ! �and for each integer � � � � @ � E E E � #�� 3 @

,

� � � ��� � �

� � � � 1���� �F� � � � � � �� � � � � � ! EBy Lemma 1, the matrix �

is invertible. Putting

� : � � � � � � � ! � � � ������� � � � � � � ������� � � �and after multiplication by � : � of both sides of (46), we get

� : �

�����

� �...� � � : �� � � : �

������ � >���� � � �� � � : � C

�����

1 � �...1 � : � � � : �1 � � �

������ (48)

� � � ������

� � � �...� � : � � �� � � �

������ � � � � � � �� � �� : �

�����

� � � � !...� � � : � � � !� � � : � � � !������ E

Observe that

� : � ������� E E E � � � � �� E E E � � � � �...

......

...� E E E � � � � �

������ E

Now, we aim to find a scaled reparametrization of the form� ��� ��� ! � � � ��1 !

linking� � ��� 6 � E E E � � � � : � ! and � with

� � ��1 ! . We require�

to have the compo-nents

� � � � � : � /� � ��� ��� ! � �� � � � 3A@ �1 � � � � : �@ � � � � �� � /1�� ��� ��� ! � @ � � � � �

32

Page 33: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

where � � is a constant, leading to� � )�� � � + � � ! � � � � ! ��� 6 � �� � � &2&2& � � � � : � ��� � : � � � ��� ��� ! � � � � ) � ��� � � � !�! E

We first determine � � � ! and show that the leading term � ��� ��� ! is such that � � � � � !#"� � .We then obtain the desired expression for

� � )� � � + by solving

� � � � � � ! � �D� � � @ � E E E � #�� 3 @ ETo be more precise, we start solving the equation

� � � � � � � � � � E E E � #�� 3 @ � ��� � � � � "� � EWith (48) and for each integer � �4@ � E E E � � , we get

1�� � � � � �� � � � � � 1 � � � �� � � � � � � � � � � � � �� � , � � � : � � � ! � � � (49)

where ���

, � � � !..., � � : � � � !���� � � : �

���

� � � � !...� � � : � � � !���� E

Again for each integer � � � � E E E � #�� 3 @,

, � � � ! � , � � 6 � , � � � � � �� � � &2&2& EBy putting � � � in (49), we get

1 � �43 � � � � � � � � �� � , � � : � � � !@ � � � � � � �� �and for each � �4@ � E E E � � 3 @

,

1 � � � 3 � � � � � � � � � � � � �� � � � � � � � � � � � �� � � , � � : � � � !@ � � � � � � �� �� 3 � � � � � � � 3 � � � � � �� � , � � � : � � � ! �which implies that

1�� � � � � � � � � � � � � � � � �� � � � � � � � � � � � �� � � , � � : � � � !@ � � � � � � �� �� 3 � � � � � ��� 3 � � � � � � � � � ��� � �� �@ � � � � � � �� � 3 � � � � � �� � , � � � : � � � ! �1 � � � � 3 � � � � � ���@ � � � � � � �� � � � � � � ��� � � � �� � � , � � : � � � !@ � � � � � � �� �3 � � � � � �� � , � � � : � � � !� 3 � � � � � ���@ � � � � � � �� � � � � � � � �� � � /, � � � !@ � � � � � � �� � �

33

Page 34: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

where /, � is of the form/, � � � ! � /, � � : � � �� : � � � /, � � 6 � /, � � � � � �� � � &2&2& EThis also implies that��

� 1 � �...1 � ���

���� � 3 � � �@ � � � � � � �� �

��� � � � �...� � � �

���� � � � � � � �� � �@ � � � � � � �� �

��� /, � � � !.../, � � � !

���� E

With (47) we get���

� � � � � � ��� !...� � � � � ��� !���� � 3 � � �@ � � � � � � �� � �

��� � � � �...� � � �

����

� ��� � � � �� � �@ � � � � � � �� � � E��� /, � � � !.../, � � � !

���� E

By Lemma 1, both � : � and � are invertible. It follows that

� : �

�����

@�...�

������ � ��

�����...� � � �

���� "� � � ��� � �

��� � � � �...� � � �

���� "� � �

thus� � � � � � � ��� ! � E E E � � � � � � ��� ! ! does not vanish. Since for each � � � � E E E � #�� 3 @

,� � � � , this implies that

� � � � � ��� ! � /� � � ! � �@ � � � � � � �� � � ������ /� � � !#"� � EWe now develop the scaling. Putting

� � ��� ��� ! � � � � �@ � � � � � � �� � � � � � � #�� 3 # � � � � : � ��� ��� ! � � �with (48) it follows that for each integer � �4@ � E E E � � ,

1�� � � � � �� � � � � � 1 � � � (50)� � � � � � � � � � � � � � �� � , � � � : � � � ! � � � : ��

��� � � � � � � � ���@ � � � � � � �� � EIn particular we have

1 � � � � � �� � � � � � 1 � � � � 3 � � � � � � � � � � : ��

��� � � � � � � � � �@ � � � � � � �� �3 � � � � � �� � , � � : � � � ! E34

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Thus

1 � � 3 � � � � �� @ � � � � � � �� � ! � @� @ � � � � � � �� � ! � � : ��

� � �� � � � � � � �@ � � � � � � �� � (51)

3 � � �� � , � � : � � � ! @@ � � � � � � �� � EWe now put

��� � � � : � /��� ��� ��� ! �'1�� � ��� : �@ � � � � �� � /1�� ��� ��� ! E (52)

Write �����

� ���� !

...� � : � ��� !� � ��� !

������ � �� : �

�����

� �...� � � : �������� E

With (50), (51) and (52), it follows that/1 � ��� ��� ! � � � � � /1 � ��� ��� ! � �� � �� � � � � � @ � � � � � � �� � ! (53)� � � @ � � � � � � �� � ! � �� � , � � � : � � � ! � � � : ��

� � � � � � � � �%�and we get/1 � ��� ��� ! � @ � � � � � � �� � ! � 3 � � � � � � @ � � � � � � �� � !� � � : ��

� � � � � � � � � 3 � � @ � � � � � � �� � ! � �� � , � � : � � � ! �/1 � ��� ��� ! � 3 � � � � � @ � � � � � � �� � ! : � � � : ��

��� �� � � � � �3 � � �� � , � � : � � � ! �� 3 � � � � � � � @ � � � � � � �� � ! : ��� � ��� !3 � � �� � , � � : � � � ! E

Moreover, with (53) and (54), for each � �4@ � E E E � � 3 @ � we get

/1 � ��� ��� ! � � � � � � �� � > 3 � � � � � ��� @ � � � � � � �� � ! : ��� � ��� ! 3 � � �� � , � � : � � � !DC� 3 � � � � � � � � : ��

��� � � � � � � � 3 � � �� � , � � � : � � � !3 � � � � � � � � � � �� � 3 � � �� � � � � � � , � � � : � � � ! �35

Page 36: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

/1 � ��� ��� ! � 3 � � � 3 � � � �� � � @ � � � � �� � ! : � � � ��� ! � � � ��� !3 � � �� � , � � � : � � � ! (54)3 � � �� � � � � , � � � : � � � ! � � � � � �� � � , � � : � � � ! �where � � � � � � � . Now with (45), we get

� � : � � � : � � � � : � ���@ � � � � � � �� � 3 �� � : � � 6 � �3 ����� �

��� � : � � � � � �� � 0 � : � � � ! 1���� � 3 � � � � � �� � � � : � � � ! �thus

/� � : � ��� ��� ! � 3 � � � � � : � � @ � � � � �� � ! : � (55)3 � � �� �@ � � � � �� � ! ��� � �

��� � : � � � � �� : � � � 0 � : � � � ! /1�� ��� ��� !3 � � �� � � � : � � � ! �where

� � � � � 6 and for each � �4@ � E E E � � 3 # ,

� � � � � 3 � � � � 3 � : ��

��� � � � �� � � � � � � ��� � � � � @ � � � � � � �� � ! : �

3 ����� �

��� � � � � � �� � 0 � � � ! 1���� � 3 � � � � � �� � � � � � ! �thus

/��� ��� ��� ! � 3 � � 3 � : ��

� � � � � �� � � /� � ��� ��� ! � � � � @ � � � � � � �� � ! : � (56)

3 � � �� �@ � � � � �� � ! ����� �

��� � � � � �� : � � � 0 � � � ! /1�� ��� ��� ! 3 � � �� � � � � � ! EObserve that the expressions given in (54), (55) and (56) together with (52) definea scaling. In the new parameters

��� ��� ! , the displacement function takes the form

� � )�� � � + � � ! � ���@ � � � � �� � > � 6 � �� � � &2&2& � � � � : � � � � : � � � ��� ��� ! � � � � ) � � � � � C �

where � � � � � !F� /� � � !#"� �. By a rescaling of the form � � � 3 � � ��& � ��� ��� ! & and after

removing the tildes, we get

� � )�� � � + � � ! � 3 & � ��� ��� ! & � �@ � � � � �� � > ,.- � /� � � � ! � ) � ��� � � � !DC EThis ends the proof of Theorem 2. �

36

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2.3 Proof of Theorems 3 and 4

To construct the scaling, we first recall that

< � : ��% ! � � 6 � 1 �� % � � -��

���% � % � ! � � � �

� %�� - ����% � % � ! � � &2&2&� � � : � � % � : � � - � � : � ��% � %�� ! � � 1 � � % � � � -�� � ��% � % � ! �� � � ��% � & � -%! E

For simplicity we assume that, 1 � � � "� �and we are treating an unfolding of odd

codimension. In the even codimension case, the proof is completely similar.Let � , � , and put

% � � � @ � � ! where � close to�. Recall that

� ��% ! � % : �� 3 @- 1 EIt follows that

� � � � @ � � ! � � @- 1 > � : �� � @ � � ! :��� 3 � @ � � ! : �� � � @ � � ! : �� 3 @ C �� � � � ! � @ � � ! : �� ��� � @ � � ! � (57)� � � � ! � @ � -���� ��� -+!�! � � �� � @ � � ! � -�� � � ��� -+! �

where both�� and

� � are analytic functions. Consider the maps

% 37�� � ��% ! � 1�� ��% � � ��% ! � -�� � ��% � %�� ��% !�!�! � @ � �� � 3A@and % 37 � � ��% ! � ��� ��% � � - � � ��% � % � !�! � �� � � � 3 @ EFor each integers � � under consideration, we have

� � � � � @ � � !�! � 1���� � � @ � � ! � � � � � @ � � !�!

� -�� ��� � � @ � � ! � > � � @ � � ! � � � � @ � � !�! C�� �� � � � � @ � � !�! � ���

�� � � @ � � ! � � - � � � � � @ � � ! � > � � @ � � ! � � � � @ � � !�! C� E

With (57) and knowing that

� � ��% � %�� ��% !�! � � � � 6 % � � &2&2& � @ � �� � 3A@ �� � ��% � %�� ��% !�! � � � � 6 % � � � � ��% ! � &2&2& � �� � � � 3 @ �

37

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it follows that

� � � � � @ � � !�! � 1 � � �� � @ � � ! � > � � � ! � @ � -��

�� �'� -%! � � �� � @ � � ! � -�� � � ��� -+!DC

� - � � � �'� - ��� ��� � � � !�! � �4@ � E E E � � 3 @ � (58)

� � � � � @ � � !�! � ��� � �� � @ � � ! � � -�� � � �'� - ��� ��� � � � !�! � � � � � E E E � � 3 @ � (59)

where � � and� � are smooth in � and

-, polynomial in � and

� � � � ! and satisfy

� � � S � � 6 � &2&2& �� � � � � � 6 � � � � � � � ! � &2&2& E

Write< � : � � � @ � � !�! � � � � � � � ! . We get

� � � � � � ! � � 6 � �� � � E E E � � � � : � � � � : � � � � � : � � � � : � � ) � � � � ! E

From (58) and (59), there exist matrices�� : � � �

�:� � � ! � � 6 ������� � � � ��� 6 ������� � � � �� : � � �� :� � � ! � � 6 ������� � � � ��� � ������� � �

�� : � -%! � � �0 :� � � � -+!�! � � 6 ������� � � � ��� � ������� � � � �� � �� � � � ! � � � � � ������� � � � � ��� � ������� � � �and

�with entries which are polynomial in � and

� � � � ! and analytic in-

suchthat

�:� � � �

�:� � � � &2&2& � �0 :� � � � 0 :� � � � &2&2& � �� :� � � � � :� � � � &2&2& �

�� � � � � � � � � � &2&2& � �� � � � � � � � � � &2&2& �and such that��

�� 6...� � : �

���� � �� :

�����

� 6� � �...

� � : � ��� : ������� �� � � ! �� :

�����

1 � �...1 � : � � � : �� ���

������

� �� :�����

1 � �...1 � : � ��� : �� � �

������ �

�����

�� 6 � � � -+!...

�� � : � � � � -%!�� � : � � � � -%!������ �

where for each integer � � � � E E E � � 3 @,

�� � � � � -+!F� � � /� � � � � -+! � � � � /� � � � � -%! � &2&2& E38

Page 39: Catastrophe Theory in Dulac Unfoldingsbroer/pdf/bnr.pdf · 6 using Standard Catastrophe Theory [6, 30, 44, 54], we recover the generic bifurcation theory of limit cycles as it now

Define also the two following matrices�� � � �

� � � � ! � � � ������� � � � : � � ��� � ������� � � � �� � � � � �� � � � � � , � � �

� � � � � �� :� � � ��� � � �� � � � ! � � � ������� � � � : � � ��� � ������� � � : � � �� � � � �

�� � � � �

��O� � �< � � � ! � � � � � ������� � � � : � � � � � ������� � � : � � �< � � � � �� � � � EFrom Lemma 1, these 3 matrices remain invertible for

-sufficiently small. We get����

�� �...� � � : �� � � : �

������ � > �� � � �� � � C

�����

1 � �...1 � : � � � : �� ���

������ �

�����

� � � � � -+!...� � � : � � � � -%!� � � : � � � � -%!������ �

where for each integer � � � � E E E � #�� 3 @,� � � � � -+!F� � � /� � � � � -%! � � � � /� � � � � -%! � &2&2& � E

We follow the same strategy as in the previous section. We claim that for each in-teger � � @ � E E E � � 3 @

, there exist maps�� � : � � � � � � -%! � �� � � � � � � -%! and coefficients

�� � � � � -+! ,�5 � � � � -+! which are smooth in

-and � � � � � ! , (

�� � is linear in�

), such that

�� � � � � -%! � � � � &2&2& � �5 � � � � -+!F� 5 � � &2&2& ��� � : � � � � � � -%! � /� � : � � � � &2&2& � �� � � � � � � -+!F� � � � &2&2&

and such that by putting

1 � � � � : � � � � ! /1 � � � � � � -+! � ��� � /� � � � � � � -%! � � : � � � � � ! �/1�� � � � � � -+! � 3 ���� � � � � -%! 3 � : � � � ! �� � : � � � � � � -%! � �� � � � � � � -%!

and

/� � : � � � � � � -+! � 3 � : ��

� � �� �� � : � � � � � � -+! � : � � � ! � �0 � : � � � � � � -+!�! /1�� � � � � � -%!3 � � �� � : � � � � � � -+! � �

� � : � � � � -+!�! � : � � � !� �5 � : � � � � -+! � : � � � ! ��� � : � � : � � � ! �/��� � � � � � -+! � 3 � : ��

� � � � ��

�� � � � � � -%! /� � � � � � � -%!

3 � : ��

��� �� �� � � � � � � -+! � : � � � ! � �0 � � � � � � -+!�! /1�� � � � � � -+!

3 � � �� � � � � � � -%! � �� � � � � -+!�! � : � � � ! � �5 � � � � -+! � : � � � !� � � � : � � � ! � � � � � E E E � � 3 # �39

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we gets

��� )� � � + � � ! � � � � � � ! ��� 6 � ��� &2&2& � � � � : 1 � � � : 1 ����� � � � � -%! � � � : � � ) � � � � ! �

where��� � � � � � !#"� � .

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44