Stable heteroclinic cycles and symbolic dynamics

In bifurcation theory, we often have to deal with a critical vector field in R"', m '" 2, which satisfies the following properties: there exists a finite sequence of singularities, Po , PI "",PIl-I , of saddle type with only one unstable direction, and a sequence of 2n heteroclinic or homoclinic orbits, r 0,0 , fO,l , ... ,fi,o, f i,l , ... ,fll 1,0, f ll 1,1 ,joining one of the p/s to another. Such a configuration is called a heteroclinic cycle (see Fig. 1). It turns out that, in many situations, the set r= U;'~d(ri.OU r i• l ) is an attractor. In such a case, we call it a stable heteroclinic cycle. Stable heteroclinic cycles are obviously not structurally stable, and consequently the following question arises:


I. INTRODUCTION
In bifurcation theory, we often have to deal with a critical vector field in R"', m '" 2, which satisfies the following properties: there exists a finite sequence of singularities, Po , PI "",PIl-I , of saddle type with only one unstable direction, and a sequence of 2n heteroclinic or homoclinic orbits, r 0,0 , fO,l , ... ,fi,o, f i ,l , ... ,f ll -1 ,0, f ll -1 ,1 ,joining one of the p/s to another.Such a configuration is called a heteroclinic cycle (see Fig. 1).It turns out that, in many situations, the set r= U;'~d(ri.OU r i • l ) is an attractor.In such a case, we call it a stable heteroclinic cycle.
Stable heteroclinic cycles are obviously not structurally stable, and consequently the following question arises: Question: Consider a $)"1 vector field in Rill, m ~ 2, with a stable heteroclinic cycle r.What happens in a neighborhood of r when we perturb the vector field in the fj' topology?
This question has been completely answered, in the particular case called "gluing bifurcation," of a configuration involving only one singularity.1-3However, so far, there is no answer'in the general case, which remains, very interesting for the following two reasons: First, it is not an academic generalization of the gluing bifurcation: there exist some extra difficulties related to the increasing richness of the possible dynamical behavior.
Second, stable heteroclinic cycles (with more than one singularity) occur in problems of bifurcation coming from the PDEs world (for instance hydrodynamiCs 4 ).
In this paper we focus our attention to a particular class of (stable) heteroclinic cycle.These cycles will be called rotating (stable) heteroclinic cycles (see Fig. 2) and correspond to the case when the two heteroclinic or homoclinic orbits rj,o and ri,1 , emanating from a singularity Pi' end at the same singularity Pi.
Consider now a Z"l vector field X 0 in RfII with a rotating stable heteroclinic cycle r and assume that the linearized vector field DXo(P,), at each singularity Pi' is such that its dominating stable eigenvalue is real.Then, generically, there exists a rigorous way to reduce the dynamics of any vector CHAOS 4 (2), 1994 1054-1500/94/4(2)/407113/$6.00 field X, fjl-close to X 0 in a neighborhood of r, to the dynamics of a map fixi on the interval with a finite number of discontinuities (see Fig. 3).Furthermore, this map is monotone and is a contraction of each interval of continuity.This reduction to a one-dimensional dynamics is the subject of Sec.II.
Some simple extra conditions on the vector field Xo ~i!lds maps fix) from the union of n intervals into itself, mapping one interval into another with a single discontinuity.
in each interval.
For a vector field X, fjl-close to Xo , there is a natural way of coding the invariant curves of X which remain in a neighborhood of r.
Roughly speaking, we can code an invariant curve with a periodic sequence of 2n symbols corresponding to the fact that the curve has to follow successively some of the 2n heteroclinic or homoclinic orbits of Xo .In Sec.III, we show how this coding corresponds, on the interval, to the classical Milnor-Thurston coding for the periodic orbits of the map fix)• We show also that each code of a periodic orbit of fix) is also the code of the periodic orbit of a map fix) which also• has one discontinuity in each interval.the same monotonicity type of fix) but which is an interval exchange.When this interval exchange preserves the orientation it can be seen as a map from the union of n-circles into itself, which maps each circle onto another by a rotation.We call this type of map a composition of rotations in n circles (see Fig. 4 for an example of these maps).Sections IV-VIII are devoted to' the study of the symbolic dynamics of these maps.

II. ROTATING STABLE HETEROCLINIC CYCLES
We note that most of the techniques used in this section are now quite standard.We refer to Refs. 1 and 2 for the details of quite similar proofs.
Hypothesis H 4 .We know that, with the above conditions, both orbits [;,0 and f;,l arrive to P7(i) tangent to the line Xl =X3 = ... =Xm =0.In addition to this generic condition we assume that these two orbits arrive to P ,(i) at the same side: x,>O.
It is very easy to check that, thanks to the conditions we have given on the eigenvalues of the linearized vector field at Furthermore, this neighborhood is also invariant by the "time t" (t~O) map of the flow of any vector field /iI-close toX o • Thanks to the hyperbolicity of the singularities p, , and up to a smooth change of coordinates, it is not restrictive to assume that for any vector field IO'-close to X 0 , the points Po, PI,•••, P 11-1 remain hyperbolic singularities, the local stable manifold is given by the equation x] =0 and the local unstable manifold by the equations X2=X3= •"=x",=O.
For hand r positive and small enough consider, near each singularity Pi' the rectangle R i defined by x2 = hand IXj I<r for j=1,3, ... ,m (seeFig.5).
By following the flow induced by X 0, the rectangles R"o=R,n{x, <O} and R,,] =R,n{x]>O} are mapped in RT(j) on two singular triangles whose singular extremities lie on XI =0 (see Fig, 6), These points correspond to the intersection off"o and f,,1 withR~i)' For a perturbed vector field X, f?1-close to X o , we have the same situation except that the two branches of the unstable manifold at p, do not need to cross R~,) on the line x,=O (see Fig, 7), In other words, we may not have a heteroclinic or a homoclinic connection.
The two maps we have constructed from R"o to R~i) and froI?R"I to R~,) can be extende~ by continuity to maps flam Rj,o to R7(j) an_d from Ri,l to RT(j) , where R"o=R,n{xI<>O} and R"I =R,n{xI~O}, This yields a map T lx ]" :R,-'>R~,) which is bivalued on the line XI =0, From Hypothes}s H 2 , it}s easy to check that the maps T[X),i restricted to Ri,o and Ri,l are contractions.
We set The dynamics on the leaves is a contraction and, on the quotient space (which is one dimensional) we are reduced to a map fiX) on the union of n intervals 1,=[ -r,r] such that fIX)(I,)CI T(i) and   (0,1) (1,0) (0,0) (1,0).

III. SYMBOLIC DYNAMICS
Consider a vector field X 0 with a stable heteroclinic cycle f, and let U be a small tubular neighborhood of f.Any oriented simple closed curve in U yields, by retraction, an oriented loop in f.This loop is homotopic to a loop consisting on a succession of arcs fi,o or fi,I followed in a positive (time increasing) or negative direction.Consequently, to any oriented simple closed curve in U we can associate a periodic sequence ( .. .xOXI... XIXO"') of symbols in {-I,l}X{O,I, ... ,n -I }X{O,I}.If this simple closed curve is an invariant curve of a vector field f?1-close to X o , then the coding is simpler because there is a natural orientation of the orbits in U.It follows that the corresponding sequence will be in {I} X{O,I, ... ,n -l}X{O,I}.Consequently, we can forget about the symbol {I} which means the time increasing direction and associate, to each invariant curve Y in U of a vector field f?1-close to X o , a sequence of symbols I(Y)={YI} where the y/s belong to {O,I, ... ,n-I}X{O,I} (see Fig. 8 for an example), Assume now that the vector field X satisfies the assumptions from HI to Hs.A simple oriented closed curve in U of a vector field X, f?1-close to Xo , corresponds to a periodic orbit of T IX ) and i[X) .
This curve is a stable periodic orbit of X if its corresponding periodic orbit of T IX ) (resp. of fiX) avoids the line To a periodic point x of T IX ) (resp, of fiX) we can associate a periodic sequence (Y,) in the symbols {O,I, ... ,n-l} X{O,I} by setting yl=(m[o8,) if and only if T'(x) ER m "" (resp.i'(x) El, and i'(X)81~0).
It is straightforward to check that the code I(Y) of an oriented closed invariant curve in U of a vector field X, f?1close to X ° , coincides with the code of the corresponding periodic orbits of T IX ) and fiX) defined above, In the sequel, we are going to describe the different codes we can get for periodic orbits of the maps fiX)' In order to do it, let us consider a periodic orbit e of a map fiX) .
By changing the position of the points of e in the interval without changing their mutual order nor their positions with respect to zero, it is easy to see that there exists a map .
where Ii are intervals which contain zero and satisfy the following conditions: (iii) it is an isometry on each interval of continuity (with the same monotonicity as fix); (iv) it possesses a periodic orbit with the same code (with respect to the new natural partition) as O.
By construction, this map fix) is an interval exchange transformation.
To understand the symbolic dynamics of the invariant closed curves in U of a vector field VI-close to X o , it is thus necessary to understand the symbolic dynamics of some classes of interval exchanges transformations.This is a subject too vast and too rich.In the next section we are going to restrict our attention to the description of the symbolic dynamics associated to a special class of interval exchange transformations: Hypothesis H6 .Assume that on each interval of monotonicity the functions fix) and fix) are increasing and that f ['.,.).i(O+) < 0 < fix)., (O-).Assume also that T is a bijection and that the intervals J, =[f(:r1.r-1(i)(O+),fi~-I,r -1(i) (O-)] have all the same length and satisfy Notice that Hypothesis H6 is nothing more than conditions on the heteroclinic cycle r (T bijective) and on the type of perturbed vector fields allowed.More precisely, one can think on the perturbations on the vector field in the following way.We can unfold the bifurcation diagram around the stable heteroclinic cycle in a 2n-parameter space.Each parameter corresponds to the breaking of a heteroclinic connection.In this setting, Hypothesis H6 restricts the allowed perturbations to a submanifold with dimension n in the parameter space.With all these restrictions, fix) maps each interval J, onto J ~i) as follows (see Fig. 9): (i) It possesses one discontinuity at 0; (ii) it is increasing with slope 1 on each interval of continuity; (iii) it is surjective; (iv) it is injective in the interior of J, but the two end points of J i have the same image.By identifying the end points of the intervals J, we get a map F[xJ from the union of n circles sA, sL""S}'_l into itself (each of these circles has an origin 0) with the follow_ ing properties.There exists a bijection r.{O,I, ... ,n-l} -,,{O,l, ... ,n -l} and a map a:{O,I, ... ,n -l}-'>R such that FIX)(S!)=S~(i) and p~,)oFIXllsloPil is a rotation with an angle a(i), where Pi is an identification of the circle sf to a reference circle SI, We call these maps compositions of rotation in n circles.They will also be denoted by F r, •.
Remark 3.1.The composition of two compositions of rotations in n circles is again a composition of rotations in n circles.More precisely, we have F r,aoF r',a' =F 'fOT' ,a' +aOT' .o Remark 3.2.If in the definition of a composition of rotations in n circles we replace the assumption that 'T is a permutation by the weaker assumption that 'T is just a map, then it is easy to reduce this problem to the case in which T is bijective.Moreover, since each permutation can be decomposed in a product of independent cyclic permutations, the case in which T is a cyclic permutation is the one which is going to keep our attention in the sequeL However, the study of certain compositions of rotations in n circles with T not being a cyclic permutation (mainly the case r=id) is still important and will be done later.0 In the rest of the paper we are going to study the symbolic dynamics of this class of maps.In fact we shall look for the characterization of the kneading sequences associated to a natural coding and we shall describe some of their properties.

IV. SYMBOLIC DYNAMICS OF COMPOSITIONS OF ROTATIONS IN n CIRCLES
We shall start by choosing a model to represent the com-pOSitions of rotations in n circles.In what follows we shall denote by E (.) the integer part function.Let a be a map from a subset of Z to R. We shall denote by a the decimal part of a (Le., a = a-Eoa).
We shall model a composition of rotations in n circles  identify the circle sf with the interval [i, i+ 1) for each i= O,I, ... ,n -1, then p 7(ktf7,.oP;; J is a rotation by an angle a(k), The itinerary of a point is then defined by the sequence The n-tuple of itineraries [[(O),[(1), ... ,[(n -1)] will be called the kneading invariant associated to f 7 a' We note that all itineraries of the form [(k) with k EjO,I, ... ,n -I} start with kO ... , That is, they can be written in the form [(k)=kOd j d 2 ... for each k=O,I, ... ,n-l.
We endow the space of itineraries with the following total ordering relation:  • We are interested in giving a full characterization of the kneading invariants of the compositions of rotations in n circles.To do this, in addition to the above two properties, a third condition is necessary.It turns out that this last condition is strongly related to the characterization of the kneading sequences of rotations.Therefore, we shall study first this particular case.This is the subject of the next section.

V. ROTATIONS
The main results in this section are closely related to the ones developed by Morse and Hedlund in Ref. 5 when studying Sturmian series and follow the ones from Gambaudo (see Ref. 1) with few improvements.We are going to characterize the kneading invariants of rotations by means of some structural properties among which the lexicographical ordering of sequences plays a fundamental role.However. in Ref. 5, different motivation can be found.[n fact, they study the relation between rotations and all associated symbolic sequences (not only kneading sequences) through the rotation number.Therefore, because of these differences in motivation and approach with Ref. 5, and also for completeness we shall develop Our own study in full detail.
A rotation can be modeled as a composition of rotations in 11 circles in the trivial case in which 11 = 1.Then, of course, r-::id.'We note that in the case 11> 1 and 7'=-id we have n noncoupled rotations and, hence, the characterization of the kneading invariant in this case follows directly from the corresponding characterization for rotations.
When we consider a rotation as a composition of rotations in One circle the use of T is superfluous and, hence, it will be omitted.Also, a(O) will be denoted simply by a.Moreover, the usc of the first symbol in the itinerary of a point is also superfluous and So it will be removed.Thus, when talking about rotations, the itinerary [(x) of a point will be defined to be the sequence a(x)a(f,,(x) )a(f;,(x))'" .
We extend the notions of kneading invariant, ordering of itineraries, and shift operation to this new framework in the natural way.
From all that was mentioned in the previous section it follows that if [= dod I'" is the kneading invariant of a rotation, then it satisfies the following two conditions: (A) do=O; (B) [';(J"j(l)';ldld, ... for all j'30.
We want to see that each kneading invariant of a rotation satisfies one more condition.This third condition, together with conditions (A) and (B), characterizes the kneading invariants of rotations.This will be shown at the end of this section.To state the property of rotations we are looking for, we need some definitions and technical lemmas.
• The above proposition motivates the next definition.Let [ be an admissible sequence.We shall say that l is a-nice (respectively, I-nice) if there exists b;;31 such that I=O"HIOh]10 iJ2 1 ... (respectively, I=Ol h o Ol/JIOl iJ2 0 ... ) with b, E {b,b+ I} for all i'30 and b,=b+ 1 for some i.The number b will be called the order of l.A nice sequence will be a sequence either a-nice or I-nice or equals to (Olr or 0" or 01"'"', Remark 5.3.Notice that each sequence of the form (O[kr with k>1 is I-nice of order k-1.Therefore, in view of Proposition 5.2, each extremal sequence is nice.However, as we shall see later, there exist nice sequences which are not extremal.0 Now we define the deflation operation 8 from the space of nice sequences to the space of admissible ones.First we set 8((0I)")=8(0")=0~ and 8( 01 We shall study now the action of 8 on the set of extremal sequences.We want to show that {; preserves the extremality and, therefore, it can be iterated infinitely many times on the set of extremal sequences.This will be done in Proposition 5.6.•Prior to the proof of this fact we shall see that the deflation operation preserves ordering and, in some sense, commutes with (J".[n the rest of the section we shall use freely the notation from the definitions of a nice sequence and 8. Lemma 5.4. Let [<.f.be t-nice sequences of the same order with tE{O,I}.Then 8(l)<8(.f.).
Proof We shall prove the lemma in the case t = O.Proof.If m = 0 then since [ is extremal and t-nice it follows that B,,=O=a and 1=0.So, 1'=0 and there is nothing to prove.Now assume that m>O.We also assume t=O and a = 1.In the other three cases the proof follows similarly.Clearly, U'•(D=O"'IO"'•" 1... with b,E{b,b+l} for all i?;o I. Since a = 1 and t = 0, in view of the definition of 0, it follows that B,=1 and b,=b.We claim that b,=b+ 1 for some i>l Proof.If [E {(Olr,O~,OI~) then there is nothing to prove.Thus, we assume that [",{(Olr,OOO,OC}.By Proposition 5.2 and Remark 5.3, I is t-nice with tE{0,1}.Assume that [ is O-nice.If [ is 1-nice the proof follows analogously. If o(1)=01 x the proposition holds trivially.So, we also assume that oW *01 x.In view of Proposition 5.  (o(l).We have to see that u" '(o([»,;;IB,B, ..  The foHowing lemma shows that the kneading invariants of rotations cannot be any extremal sequence.Lemma 5,8.Let I be the kneading invariant of a rotation.Then, I::foOl~. Proof.Assume that [ is the kneading invariant of the rotation with angle a.Without loss of generality we may assume that aE[O,I).The statement [=Ol~ is equivalent to a(ka)=1 for all k?;ol and, by the definition of address, this is equivalent to the condition ka-E(ka)?;oI-afor all k;;,1.If a is rational, say p/q, then taking k=q we get ka-E(ka)=O<I-a:.If a: is irrational then the sequence {ka-E(ka)hEN is dense in [0,1).Thus, again there exists k;;,1 such that ka-E(ka)<I-a.hence, l*Olx.
• By Proposition 5.7 and Lemma 5.8 we see that if l is the kneading invariant of a rotation, then 8"(l)*01% for all In ~ O. Therefore, it is interesting to characterize the set of extremal sequences having some iterate by 0 equal to 01 x.This is our next step.
Let l be an admissible sequence.We shall say that I is periodic if it exists 1'>0 such that ui>(l)=[.The smallest such I' will be called the period of I (in this case [ will also be called p-periodic).The sequence [ is p-eventually periodic if there exists m;;'O such that u"'([) is periodic of period p.
The p-eventually periodic sequences which are not periodic will be called p-preperiodic (or simply preperiodic).If 1 is a finite sequence then 111 will denote the cardinality of 1.
Moreover, if I is not eventually periodic then o([) is not eventually periodic.
Finally, in order that o(l) be eventually periodic, from the definition of 0, it follows that [ has to be eventually periodic too.This ends the proof of the lemma.

•
The next proposition characterizes the extremal sequences which have some iterate by 0 equal to 01%.Proposition 5.12.An extremal sequence l satisfies 8"([)=0Ix for some m;;'O if and only if l is p-preperiodic for some p~ 1.
Now suppose that 8"(l) = 0 1 x for some m;;' O. Then the proposition follows directly from the inductive use of Lemma 5.11.• We note that, from Proposition 5.12 and the iterative use of Lemma 5.11 we get that each extremal p-preperiodic sequence is of the form 0(11) ~ with 1 a finite sequence of O's and I's.
From Proposition 4.3 we see that each kneading invariant of a rotation [ is extremal.By Proposition 5.7 we get that 8 111 (l) is a kneading invariant of a ''rotation for each m30.Therefore, by Lemma 5.8, 8"(O*0Ix for each m;;'O.So, [ is not preperiodic by Proposition 5.12.Next we show that indeed the extremality and the non-preperiodicity of a sequence characterize the kneading invariants of rotations.Theorem 5.13.An admissible sequellce l is the kneading invariant of a rotation if and only if is extremal and nol preperiodic.
Proof.The "only if" part follows from Propositions 5.7 and 5.12 and Lemma 5.8.Now we prove that if [ is extremal and not preperiodic then it is the kneading invariant of a rotation.Let us split the study into two cases.
FIG.7.The standard situation for X, r.:;1 close to Xo.
FIG. 9.The map fix] when the hypotheses from HI to H6 are fulfilled.
FIG. 10.Some compositions of rotations in two circles.
dk<d",where k",O is such that d;= d; for i= 0, I, ... ,k-I and < is the usual ordering of real numbers.Clearly, x"'y implies [(x)g(y),Finally we define the shift operation (Y in the space of itineraries bydsdod J ••• ) = r(s )d j d 2 ....We note that the shift operation depends on r and, therefore, on the map under consideration.In what follows we shall denote the composition uouo"'O(y (n times) by <7".As it is usual, for each x E [0, n) we have (Y([(x»=[( f7,.(x».The fOllowing proposition gives us an algorithm to compute the kneading invariant associated to f T,a • Proposition 4.2.Let f7,.be a composition of rotations in n circles.Let k E {O,I, ... ,n -I} and [(k)=kdod j ... '.Then, =0,1, .... Proof.From Remarks 4.1 and 3.1 we obtain by induction that for 1"'0.Thus, Hence, d{=1 if and only if Therefore, • A first characterization of the kneading invariant of a composition of rotations in n circles is given by the following proposition.Proposition 4.3.Let [[(O), ... ,[(n -1)] be a kneading invariant of a composition of rotations in n circles f7,.' Let kE{0,I,2, ... ,n-I} and j"'O.Then we have Then there exists b'31 such that [ is either OlbOl b I Ol b 2 0 ... or Ob+'10 iJ qOb 2 1... with b,E{b,b+l} for all i'3l.Proof.Let [=Odld, ... .By using Lemma 5.1 with j= I we get d'_I=d l for all k>1 such that d,*d l .That is, if d l =0 then in the sequence [ there always is a 0 before any I and if d I = I then there is a 1 before any O.Then, [ is either Ol" II 01 iJ IOl b 20 ... or Oh ll lO"QOIJ2 ... where bi~l for all i'30.Assume that [ is of the form 0"01 O"iJ 0"' ....The proof in the other case follows similarly.

( 1 )
If a< 1/2 then I is O-nice and oeD is the kneading invariant of the rotation by angle f3 1'-1-£(1'-1)+1' (2) If a> 1/2 then [ is I-nice and oW is the kneading invariant of the rotatiol/ by angle y-E( If a=1/2 then 1=(01)~ and oW=O~ is the kneading invariant of the rotation by angle f3=0.Conversely, if [=aoa,,,,*OOO is the kneading invariant of the rotation by angle f3 and kEN then the new sequence obtained by applying to [ the rule { OHI, if i=O,aj---'110k, if i>O and ai= 1, 0, if i>O and a;=O, or, respectively, is the kneading invariant of the rotation by angle a where a satisfies the relation give in (1) and k=£[(l-a)/a] (respectively, the relation given in(2) and k=£[a!(I-a)J).