Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

0.69 MB·English

by  A. Delshams

Abundance of attracting, repelling and elliptic
periodic orbits in two-dimensional reversible maps.
2
1 Delshams A.† , Gonchenko S.V.‡, Gonchenko V.S.‡, L´azaro
0
J.T.† and Sten’kin O.‡
2
†UniversitatPolit`ecnicadeCatalunya,Barcelona,Spain
n ‡InstituteforAppliedMathematics &Cybernetics,N.Novgorod,Russia
a
J
Abstract. We study dynamics and bifurcations of two-dimensional reversible
5
maps having non-transversal heteroclinic cycles containing symmetric saddle
2
periodicpoints. Weconsiderone-parameterfamiliesofreversiblemapsunfolding

generally the initial heteroclinic tangency and prove that there are infinitely
]
S sequences(cascades)ofbifurcationsofbirthofasymptoticallystableandunstable
aswellasellipticperiodicorbits.
D
.
h
t
a
1. Introduction
m
[ Reversible systems have a very special status inside the realm of dynamical systems.

Usually, they are positioned “between” dissipative and conservative systems. In the
1
v context of continuous dynamical systems, a reversibility means that the system is
7 invariant under the change of time-direction, t t, and a transformation in the
5 spatial variables. In the discrete context, reversib7→ilit−y of a map f (a diffeomorphism)
3 meansthatf andf 1possessthesamedynamics. Noticethattheterm“thesame”can
5 −
haveratherdifferentmeaning. Iff andf 1aresmoothlyconjugate,i.e.,f h=h f 1
. − −
1 ◦ ◦
and h is a just a diffeomorphism, then f is called weekly reversible. However, much
0
more interesting types of reversibility appear when h possesses some structures. For
2
1 example, if h is an involution, i.e., h2 = Id. In this case, the map f is called
: strongly reversible. Since this last case is the most frequent one in the literature
v
(probably, beginning with Birkhoff), nowadays strongly reversible maps are simply
i
X called reversible maps.
r In contrast to conservative and dissipative systems, the study of homoclinic
a
bifurcations in reversible systems is not so popular. Even for two-dimensional
maps, only few results are known and most of them relate to “conservative and
reversible”mapswhichformacertaincodimension- subclassintheclassofreversible
∞
maps. This situation is probably due to the “common belief” that conservative
and dissipative phenomena of dynamics only exist separately and, thus, there is no
necessity to study them “all together”.
However,they actually canappeartogether inadynamicalsystem,giving riseto
the so-called phenomenon of mixed dynamics, which was recently discovered in [12]
(see also [16, 18, 19]). The essence of this phenomenon consists in the fact that
(i) adynamicalsystemhassimultaneouslyinfinitelymanyhyperbolicperiodicorbits
of all possible types (stable, completely unstable and saddle), and
Attracting, repelling and elliptic orbits in reversible maps 2
(ii) these orbits are not separated as a whole, i.e., the closures of sets of orbits of
different types have nonempty intersections.
It was shown in [12] that the property of mixed dynamics can be generic, i.e.,
it holds for residual subsets of open regions of systems. In particular, it was also
proved that such regions (Newhouse regions, in fact) exist near two-dimensional
diffeomorphismswithnon-transversalheterocliniccyclescontainingatleasttwosaddle
periodic points O and O such that J(O ) > 1 and J(O ) < 1, where J(O ) is
1 2 1 2 i
| | | |
the Jacobian of the Poincar´emap (the diffeomorphism iterated as many times as the
period of O ) at the point O , i=1,2.
i i
Let us recall that a heteroclinic cycle (contour) is a set consisting of saddle
hyperbolic periodic orbits O ,...,O as well as heteroclinic orbits Γ Wu(O )
1 n i,j i
⊂ ∩
Ws(O )), where at least the orbits Γ and Γ for i = 1,...,n 1, are included.
j i,i+1 n,1
−
In general, cycles can include also homoclinic orbits Γ Wu(O ) Ws(O ). An
i,i i i
⊂ ∩
heterocliniccycleiscallednon-transversal (ornon-rough)ifatleastoneofthepointed
out intersections Wu(O ) Ws(O ) is not transverse.
i j
∩
Ifa heterocliniccycle(orahomoclinic orbit)is transverse,then, asiswell-known
afterShilnikov[36],thesetoforbitsentirelylyinginasmallneighbourhoodisalocally
maximal uniformly hyperbolic set. The situation becomes drastically different in the
non-transversal case. One can say even that the corresponding system is infinitely
degenerate,since its bifurcations can produce homoclinic tangencies of arbitraryhigh
orders and, as a consequence, arbitrary degenerate periodic orbits [10, 13, 20].
Weremindalsothatsystemswithhomoclinictangenciesaredenseinopenregions
(theso-calledNewhouseregions)inthespaceofsmoothdynamicalsystems[26,27,28].
Moreover,these regions exist near any system with a homoclinic tangency (or a non-
transversal heteroclinic cycle). Importantly, these regions are present in parameter
familiesunfoldinggenerallytheinitialhomoclinic(orheteroclinic)tangencyincertain
opendomainsoftheparameterspaceinwhichtherearedensevaluesoftheparameters
corresponding to the existence of homoclinic tangencies. Certainly, such domains are
called again Newhouse (parameter) regions or Newhouse intervals for one-parameter
families. In general,it should be clear from the context the kind of Newhouse regions
considered.
The existence of Newhouse regions near systems with homoclinic tangencies
was established in [28] for two-dimensional diffeomorphisms, in [11, 29, 35] for the
general multidimensional case (including parameter families [28, 11]) and in [5] for
area-preserving maps. The existence of Newhouse regions near systems with non-
transversal heteroclinic cycles follows immediately from these results, since in such
case homoclinic tangencies appear under arbitrary small perturbations. Moreover,in
this case also the so-called Newhouse regions with heteroclinic tangencies can exist.
It was proved in [12] that if the non-transversal heteroclinic cycle is simple, i.e., it
containsonlyonenon-transversalheteroclinicorbitandthecorrespondingtangencyof
invariantmanifoldsisquadratic,thenNewhouseintervalswithheteroclinictangencies
exist in any general one-parameter unfolding.
The above-mentioned mixed dynamics takes place as a generic phenomenon.
This is the case, for instance, when the initial heteroclinic cycle is contracting-
expanding [12], that is, when it contains contracting and expanding periodic points
(i.e. with the absolute value of its Jacobian being greater or less than 1). It is worth
mentioning that contracting-expanding heteroclinic cycles are rather usual among
reversible maps. An example of such a cycle is shown in Figure 1(a). In this example
the reversible map has two saddle fixed points O and O and two heteroclinic orbits
1 2
Attracting, repelling and elliptic orbits in reversible maps 3
Figure 1. Two examples of planar reversible maps with symmetric non-
transversal(quadratictangency)heterocliniccycles. Onecanthink,forsimplicity,
that the involution R in both cases is linear of form x 7→ x,y 7→ −y and, thus,
FixR={y=0}.
Γ Wu(O ) Ws(O ) and Γ Wu(O ) Ws(O ) such that R(O ) = O and
12 1 2 21 2 1 1 2
⊂ ∩ ⊂ ∩
R(Γ ) = Γ , R(Γ ) = Γ . Besides, the orbit Γ is non-transversal, so that
21 21 12 12 12
the manifolds Wu(O ) and Ws(O ) have a quadratic tangency along Γ . Since
1 2 12
R(O )=O ,itturnsoutthattheirJacobiansverifyJ(O )=J 1(O ). IfJ(O )= 1,
1 2 1 − 2 i
6 ±
i=1,2,then the heteroclinic cycleis contracting-expanding. This conditionis robust
and is perfectly compatible with reversibility.
Certainly, results of [12] can be applied to reversible maps with such heteroclinic
cyclesandsothephenomenonofmixeddynamicsbecomesveryimportantandgeneric.
However, reversible systems are sharply different from general ones by the fact that
they can possess robust non-hyperbolic symmetric periodic orbits, more precisely,
elliptic symmetric periodic points. Thus, one realizes that the phenomenon of mixed
dynamicsinthecaseoftwo-dimensionalreversiblemapsshouldbeconnectedwiththe
coexistence of infinitely many attracting, repelling, saddle and elliptic periodic orbits.
The existence of Newhouse regions (intervals) in which this property is generic was
already established in [23] for the case of reversible two-dimensional maps close to a
map having a heteroclinic cycle of the type depicted in Figure 1(a).
However, it appears to be true that the phenomenon of mixed dynamics is
universal for reversible (two-dimensional) maps with complicated dynamics when
symmetric structures (symmetric periodic, homoclinic and heteroclinic orbits) are
involved. This universality can be formulated as the following
Reversible Mixed Dynamics Conjecture Two-dimensional reversible maps with
mixed dynamics are generic (compose residual subsets) in Newhouse regions in which
there are dense maps with symmetric homoclinic or/and heteroclinic tangencies.
Wewillassume,inwhatfollows,thattheinvolutionRisnottrivial,i.e. itsatisfies
R2 =Id, dimFixR=1. (1.1)
WewillsaythatanobjectΛissymmetricwhenR(Λ)=Λ. Toputmoreemphasis,
sometimesthenotationself-symmetricmaybeused. Byasymmetric couple of objects
Attracting, repelling and elliptic orbits in reversible maps 4
Λ ,Λ , we will mean two different objects that are symmetric to each other, i.e.,
1 2
R(Λ )=Λ .
1 2
Then the symmetric homoclinic (heteroclinic) tangencies from the RMD-
Conjecturecanbedividedintotwomaintypes: 1)thereisanon-transversalsymmetric
heteroclinic orbit to a symmetric couple of saddle points, or 2) there is a symmetric
couple of non-transversalhomo/heteroclinic orbits to symmetric saddle points.
Notice that the heteroclinic quadratic tangency shown in Fig. 1(a) relates to the
type 1), whereas an example of reversible map having a heteroclinic cycle of type 2)
is shown in Figure 1(b). The latter map has two symmetric saddle fixed points O
1
and O (R(O ) = O , R(O ) = O ) and a symmetric couple of heteroclinic orbits
2 1 1 2 2
Γ Wu(O ) Ws(O ) and Γ Wu(O ) Ws(O ) (R(Γ )=Γ ).
12 1 2 21 2 1 12 21
⊂ ∩ ⊂ ∩
Aswementionedabove,thecaseofnon-transversalheterocliniccyclesoftype1),
as in Fig.1(a), wasstudied in the paper [23]where, infact, the RMD-Conjecture was
proved for general one-parameter (reversible) unfoldings, under the generic condition
J(O )=J 1(O )=1.
1 − 2
6
In the first part of this paper, Sections 2 and 3, we state and prove the RMD-
Conjecture for one-parameter families which unfold generally heteroclinic tangencies
of type 2), as in Fig. 1(b). We will call this type as reversible maps with a
symmetric couple of heteroclinic tangencies. We notice that in this case the condition
J(O )=J(O )=1holdsalwayssinceO FixR,showingthatgenericconditionsare
1 2 i
∈
different in systems with different types. The generic condition that we are going to
assumeforsystemsoftype2)isdenotedbycondition[C]inTheorem1. Thiscondition
(see (2.1) and comments to it) amounts to say that the (global) map defined near a
heteroclinicpointisneitherauniformcontraction(expansion)noraconservativemap.
Moreprecisely,ithastohaveanon-constantJacobianinthoselocalcoordinates(near
O andO )inwhichthesaddlemapsare,apriori,area-preserving. Inparticular,such
1 1
local coordinates are given by Lemma 2 in which the normal form of the first order
for a saddle map is derived.
‡
Moreover,as we show,symmetry breakingbifurcationshave alsoanothernature,
incomparisonwith[23]. Wefindatwo-step“fold pitch-fork”scenarioofbifurcations
⇒
in the first-return maps leading to the appearance of non-conservative fixed points
which can be either attracting and repelling or saddle with Jacobian greater and less
than 1 (see Theorem 1 and Figure 4). Notice that in the case of heteroclinic cycles
of type 1), such non-conservative points appear just under fold bifurcations [23] in
a symmetric couple of first-return maps, whereas elliptic points appear in symmetric
first-return maps.
The second part of this work, Section 4, has a more applied character. We show
in Subsection4.1 thatreversibletwo-dimensionalmaps with apriorinon-conservative
orbit behaviour can be obtained as certain periodic perturbations of two-dimensional
conservative flows of form x˙ = y, y˙ = F(y) . We require that these perturbations
include explicitly the “friction term” x˙ and preserve only reversible properties of the
initial flow (for example, they keep the perturbed systems to be invariant under the
change x x,y y,t t). In this way, in particular, we can obtain the
7→ 7→ − 7→ −
reversible maps of type 2), see Figure 2. As a concrete example, we consider, in
Subsection 4.1, the periodically perturbed Duffing equation. In Subsection 4.2 we
‡ However, the property of a symmetric saddle periodic point to be a priori area-preserving is
more delicate. It is well-known, see e.g. [4], that a symmetric reversible saddle map is “almost
conservative”,i.e. itsanalyticalnormalformisexactlyconservative,anditsC∞ formalnormalform
(upto“flatterms”)isconservative.)
Attracting, repelling and elliptic orbits in reversible maps 5
consider an example of a reversible map from [30] defined on the two-dimensional
torus which shows a visible non-conservative orbit behaviour. This map describes
the dynamics of three coupled simple rotators with small symmetric couplings. The
couplings are chosen here in such a way that they preserve the reversibility of the
initial uncoupled three simple rotators.
In the third part of the paper, at Section 5, we consider a series of problems
relatedtotherepresentationofreversibletwo-dimensionalmapsintheso-calledcross-
form. It is well-known that such type of cross-forms of maps are very convenient for
studyinghyperbolicpropertiesofsystemswithhomoclinicorbits,bothtransversal[36]
and non-transversal [6, 7, 21]. We show that the cross-forms and the corresponding
§
cross-coordinates are natural for reversible maps, since they allow expressing many
reversible structures explicitly and simplify analytical treatment. In particular, in
the correspondinglocalcross-coordinatesnear a symmetric saddle periodic point, the
normalformofthesaddlemapbecomesverysimple. Andlast(butnotleast),Section6
contains the proofs of the lemmas needed in the proof of the main theorem 1.
Figure2. Examplesofreversiblemapswithnon-transversalheterocliniccyclesin
the cases of (a) “inner tangency”; (b) “outer tangency”. These diffeomorphisms
canbeconstructed asthePoincar´emaps forperiodicallyperturbedconservative
planar systems (left) when the corresponding perturbations preserve the
reversibility(forexample, withrespecttothe change x→x,y→−y,t→−t, as
in the system x˙ =y, y˙ =x−x3+εx˙cost). Notice that the resulting reversible
mapcannotbearea-preserving,ingeneral
§ Since L.P. Shilnikov was the first author introducing such forms and coordinates into dynamical
systems,theyareoftenreferredtoas“Shilnikovcross-form”and“Shilnikovcoordinates”.
Attracting, repelling and elliptic orbits in reversible maps 6
1.1. Out of the general rule: a collection of reversible maps with codimension one
homoclinic and heteroclinic tangencies
It is important to notice that there are many other cases of reversible maps with
homoclinic and heteroclinic tangencies for which one needs to prove the RMD-
Conjectureinthe frameworkofone-parametergeneralfamilies. InFigure3wecollect
some simple examples of such maps. They differ by the type of fixed points and
tangencies: homoclinicorheteroclinic,quadraticorcubic ,etc. However,itturnsout
k
to be more important that all this kind of maps could be separated into two groups:
a firstone including those maps which are a priorinon-conservativeanda secondone
with those maps where this non-conservativity is, in some sense, hidden.
Figure 3. Some examples of reversible maps with homoclinic and heteroclinic
tangencies
A map as in Figure 1(a) (the first one in Figure 3) belongs to the first group,
since the condition J(O )=1 destroys certainly the conservativecharacter. Indeed,
1,2
6
k The existence of symmetric cubic homoclinic or heteroclinic tangencies is a codimension one
bifurcation phenomenon in the class of reversible maps. Therefore, these cubic tangencies should
bealsoconsideredasthemainonesjointlywiththepointedoutquadraticones.
Attracting, repelling and elliptic orbits in reversible maps 7
under splitting such heteroclinic cycle, homoclinic tangencies appear both to saddles
with Jacobian greater and less than 1 and, thus, attracting and repelling periodic
pointscanbeborn[6]. Bythisprinciple,alltheothercaseswhereasymmetriccouple
of fixed (periodic) points are involved, can be referred to as the class of a priori non-
conservative maps (e.g., all maps of the first, third and fourth rows of Figure 3). For
mapsofthis type,the problemoffindingsymmetricperiodicorbits(i.e.,elliptic ones)
has to be considered as very important.
The maps at the second row of Figure 3 have only symmetric fixed points.
Evidently, the first, third and fourth maps can be assigned to maps with ”hidden
non-conservativity”, since it is not clear, in advance, the existence of bifurcation
mechanisms leading to the appearance of attracting and repelling periodic orbits.
It is not the case of the third map (at the second row) which has a symmetric couple
of (quadratic) homoclinic tangencies to the same symmetric fixed saddle point. One
can assume, without loss of reversibility, that the map near a homoclinic point is not
conservative (the Jacobian is greater or less than 1). Then, clearly, stable (unstable)
periodic orbits can be born under such homoclinic bifurcations. Thus, the main
problem here is to prove the appearance of elliptic periodic orbits and, as a first
step, to do it in the one-parameter setting.
1.2. A short description of the main results
We describebriefly the centralideas underlyingour mainresults(Theorems1 and2).
Let assume f to be a R-reversible map of type as in Figure 1(b), that is, having
0
two symmetric saddle fixed points O and O (i.e. R(O ) = O , i = 1,2) and
1 2 i i
two asymmetric non-transversal heteroclinic orbits Γ Wu(O ) Ws(O ) and
12 1 2
⊂ ∩
Γ Wu(O ) Ws(O ), satisfying that R(Γ ) = Γ . Let us consider f any
21 2 1 12 21 µ
⊂ ∩ { }
generalone-parameterunfolding off with µ being the parametersplitting the initial
0
heteroclinic tangency. The main goal of this paper is to show that under general
hypotheses, any of these unfoldings undergoes infinitely many (in fact, a cascade) of
symmetry-breaking bifurcations of single-round periodic orbits. Such bifurcations in
these first-return maps (defined near some point of the heteroclinic tangency) follow
from the following scenario:
¶
fold bifurcation pitch-fork bifurcation
⇒
Under fold-bifurcation, a symmetric parabolic fixed point appears which falls
afterwardsintotwosymmetricsaddleandellipticfixedpoints. Concerning(reversible)
pitch-fork bifurcations, they can be of two classes depending essentially on the type
of initial heteroclinic cycle they exhibit (see Figure 2 and 4): in case (a) we say that
f has a heteroclinic cycle of “inner tangency” while in case (b) we say that it is of
0
“outertangency”. Inthefirstcase,“innertangency”,fromthesymmetricellipticfixed
pointthreefixedpointsareborn: asymmetricsaddleandasymmetricsinkandsource.
In the second case, “outer tangency”, under a pitch-fork bifurcation, the symmetric
saddle point becomes a symmetric elliptic point and two fixed asymmetric saddle
points with Jacobian greater and less than 1. Both scenarios are showed in Figure 4.
Inthe bifurcationdiagramofFigure 8, relatedto a conservativeapproximationofthe
rescaledfirst-returnmap,valueofc˜<0correspondstothe “innertangency”caseand
¶ Notice that inthe Lamb-Sten’kin case [23], see Fig. 1(a), foldbifurcations are directlysymmetry
breaking and lead to the appearance of two pairs of asymmetric periodic orbits: (saddle, sink) and
(saddle,source).
Attracting, repelling and elliptic orbits in reversible maps 8
c˜> 0 corresponds to the “outer tangency” one (situation c˜= 0 is singular and not
realisable for our maps; therefore there are no transitions from c˜<0 to c˜>0).
We finish this Introduction by describing the structure of the paper. Section 2
is devoted to the two above-mentioned main results of the paper, Theorems 1 and 2.
Their proofis presentedin Section3 andrelies to five lemmas whoseproofis deferred
to Sections 5, 6 and 6.3. Section 4 contains two concrete examples of applications of
the main results of this paper.
2. Symmetry breaking bifurcations in the case of reversible maps with
non-transversal heteroclinic cycles
Let f be a Cr-smooth, r 4, two-dimensional map, reversible with respect to an
0
≥
involution R satisfying dim Fix(R)=1. Let us assume that f satisfies the following
0
two conditions:
[A] f has two saddle fixed points O and O belonging to the line Fix(R) and that
0 1 2
any point Oi has multipliers λi,λ−i 1 with 0<λi <1, i=1,2.
[B] The invariant manifolds Wu(O ) and Ws(O ) have quadratic tangencies at the
1 2
pointsofsomeheteroclinicorbitΓ and,therefore,byreversibility,themanifolds
12
Wu(O ) and Ws(O ) have quadratic tangencies at the points of a heteroclinic
2 1
orbit Γ =R(Γ ).
21 12
Hypotheses[A]-[B]definereversiblemapswithnon-transversalsymmetricheteroclinic
cycleslikeinFigure1(b). Weaskthemtosatisfyonemorecondition. Namely,consider
two points M Wu (O ) and M Ws (O ) belonging to the same heteroclinic
orbit Γ and1su∈pposloecfq(1M ) = M2 ∈for aloscuita2ble integer q. Let some smooth local
12 0 1 2
coordinates(x ,y )bechosennearthepointsO insuchawaythatthelocalinvariant
i i i
manifolds are straightened, i.e., Wu (O ) and Ws (O ) have, respectively, equations
loc i loc i
x = 0 and y = 0. Let T denote the restriction of the map fq onto a small
i i 12 0
neighbourhood of the point M . Then, we assume that
1
[C] the Jacobian of T is not constant and, moreover,
12
∂J(T )
12
Q= =0 (2.1)
∂y M16
(cid:12)
Condition J(T12) = const is(cid:12)well defined only when certain restrictions on the
6
local coordinates hold. One possibility is when these coordinates (x ,y ) around O
i i i
are chosenin such a way that Wu (O ) and Ws (O ) are straightened. However,the
loc 1 loc 2
sign of J(T ) depends also on the orientation chosen for the coordinate axes. To be
12
precise,wechoosetheseorientationsinsuchaawaythat: (i)they andx coordinates
−
of the heteroclinic points M Wu (O )and M Ws (O ) arepositive; (ii)for the
1 ∈ loc 1 2 ∈ loc 2
symmetric points, M =R(M ) Ws (O ) and M =R(M ) Wu (O ), the x and
1′ 1 ∈ loc 1 2′ 2 ∈ loc 2
y-coordinates are positive as well.
Two classes of reversible maps satisfying conditions [A]-[C] can be distinguished:
those maps with “inner” (heteroclinic) tangency and those with “outer” tangency,
corresponding to J(T ) > 0 and J(T ) < 0, respectively. Two examples of such
12 12
diffeomorphisms are shown in Figure 2. Notice that in both cases the globalmap T
12
is orientable. In the case (a) the axes x ,y and x ,y have the same orientation,
1 1 2 2
whereas the orientations are different in the case (b).
Once stated the general conditions for f , let us embed it into a one-parameter
0
family f of reversible maps that unfolds generally at µ=0 the initial heteroclinic
µ
{ }
Attracting, repelling and elliptic orbits in reversible maps 9
tangencies at the points of Γ . Then, without loss of generality, we can take µ
12
as the corresponding splitting parameter. By reversibility, the invariant manifolds
Wu(O ) and Ws(O ) split as Wu(O ) and Ws(O ) do when µ varies. Therefore,
1 2 2 1
since these heteroclinic tangencies are quadratic, only one governing parameter is
needed to control this splitting.
Let U be an small enough neighbourhood of the contour C = O ,O ,Γ ,Γ .
1 2 12 21
{ }
It can be represented as the union of two small neighbourhoods (disks) U and U
1 2
of the saddles O and O and a finite number of small disks containing those points
1 2
of Γ and Γ which do not belong to U and U (see Figure 2). We will focus our
12 21 1 2
attentiononthebifurcationsoftheso-calledsingle-roundperiodicorbits,thatis,orbits
lying entirely in U andhaving exactly one intersectionpoint with everydisk fromthe
set U (U U ). Any point of a single-round periodic orbit is a fixed point of the
1 2
\ ∪
corresponding first-return map T , that is constructed by orbits of f with k and m
km µ
iterations (of f ) in U and U , respectively. We will call them single-round periodic
µ 1 2
orbit of type (k,m). The values of k and m will be always prescribed a priori. The
first main result is as follows:
Theorem 1 Let f be a one-parameter family of reversible diffeomorphisms that
{ µ}µ
unfolds, generally, at µ=0 the initial heteroclinic tangencies. Assumethat f verifies
0
the conditions [A]-[C].
Then, at any segment [ ǫ,ǫ] with ǫ>0 , there are infinitely many intervals δ
km
−
withborder pointsµ(k,m) andµ(k,m) suchthatδ 0ask,m andthefollowing
fold pf km → →∞
holds:
(i) The value µ = µ(k,m) corresponds to a non-degenerate conservative fold
fold
bifurcation and, thus, the diffeomorphism f has at µ δ two symmetric,
µ km
∈
saddle and elliptic, single-round periodic orbits of type (k,m).
(ii) Thevalueµ=µ(k,m) correspondstoasymmetric(andnon-degenerateifcondition
pf
[C] holds) pitch-fork bifurcation depending on the type of f :
0
(ii) In the case of “inner” tangency, single-round asymmetric attracting and
a
repelling periodic orbits of type (k,m) are born and, moreover, these orbits
undergo simultaneously non-degenerate period doubling bifurcations at the
value µ=µ(k,m) (where µ(k,m) 0 as k,m ).
pd pd → →∞
(ii) For the “outer” tangency, two single-round saddle periodic orbits of type
b
(k,m)withJacobiangreaterandlessthan1,respectively,areborn. Moreover,
they do not bifurcate any more (at least for µ <ǫ).
| |
We refer the reader to Figure 4 for an illustration of this theorem.
Theorem 1 and its counterpart result in [23] show that the appearance of non-
conservative periodic orbits under global bifurcations can be consider as a certain
generic property of two-dimensional reversible maps.
Briefly,themethodweuse-basedonarescalingtechnique-willallowustoprove
that the first-return map T can be written asymptotically close (as k,m ) to
km
→ ∞
an area-preservingmap of the form:
x¯=M˜ +c˜x y2,
−
H : M˜ 1 1 (2.2)
 y¯= + y+ (M˜ +c˜x y2)2,
 − c˜ c˜ c˜ −
in which the coordinates (x,y) and the parameters (M˜,c˜) can take arbitrary values

except c˜=0. The regionc˜<0 will stand for the “inner” tangency case and c˜>0 for
Attracting, repelling and elliptic orbits in reversible maps 10
Figure 4. We mark, by shading, some intervals corresponding to the existence
oftwoasymmetricsingle-roundperiodicorbits.
the “outer” one. Its bifurcation diagram is showed in Figure 8. The map (2.2) is, in
fact,theproductoftwoH´enonmapswithJacobian c˜and c˜ 1,(seeequations(3.4)).
−
− −
Thus, we can state (see also [40]) that map (2.2) has a complicated dynamics in the
corresponding parameter intervals. The latter means, in particular, that all fixed
points become saddles and all of them have homoclinic and heteroclinic intersections
for all values of the parameter µ including (quadratic) tangencies for dense subsets –
Newhouse phenomenon.
An analogous “homoclinic tangle” can be observed for the first-return map T
km
(see Lemma 3). However, although the map (2.2) is reversible and conservative (its
Jacobian is identically 1), the original first-return map T is also reversible but not
km
conservativeingeneral(seeLemma4). Precisely,itwillbeshownthatinsomeregions
of the space of parameters (c˜,M˜) the map T possesses chaotic dynamics and has
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four saddle fixed points, two of them symmetric conservative and a symmetric couple
of fixed points (that is, symmetric one to each other and with Jacobian greater and
less than 1, respectively). According to [12, 23, 20], the following result holds:
Theorem 2 Let f betheone-parameterfamilyofreversiblemapsfromTheorem1.
µ
{ }
Then, in any segment [ ε,ε] of values of µ, there are Newhouse intervals with
−
mixed dynamics connected with an abundance of attracting, repelling and elliptic
periodic orbits. This is, values of parameters corresponding to maps f exhibiting
µ
simultaneously infinitely many periodic orbits of all these types form a residual set (of
second category) in these intervals.