Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

This conjecture is true when the Newhouse regions with C r -topology, where 2 ≤ r ≤ ∞, are considered [7]. However, in the real analytic case, it has been proved now only for Newhouse regions existing near reversible maps with nontransversal heteroclinic cycles of two specific types presented in Fig. 1. In the first case the cycle is contracting-expanding and it contains a symmetric couple of saddle periodic orbits (with Jacobians less and greater than 1) and a pair of symmetric transverse and nontransversal heteroclinic orbits, see Fig. 1(a). This case was studied by Lamb and Stenkin [8]. The second case relates again to a nontransversal heteroclinic cycle but of another type: here a symmetric couple of non-transversal heteroclinic orbits to symmetric saddle points exists, see Fig. 1(b). This case was studied in [9]. Note that, both in [8] and [9], the RMD-conjecture was proved for one parameter general unfoldings, i.e. the existence of Newhouse intervals with mixed dynamics was proved. However, probably the most interesting cases are those where the initial reversible map has a homoclinic tangency associated to a symmetric saddle point. If the point is fixed, then three main cases of homoclinic tangencies can be selected, see Fig. 2. The cases (a) and (b) are connected with the a priori presence of only two symmetric orbits, periodic (the fixed point) and homoclinic ones. Therefore, we call these cases, as well as the heteroclinic case from Fig. 1(b), "a priori conservative". In these cases we do not known "a priori" bifurcation mechanisms of symmetry breaking, i.e. (global) bifurcations in one parameter unfoldings leading to the appearance a pair "sinksource" of periodic orbits. Note that for the heteroclinic a priori conservative case (as in Fig. 1(b)) the corresponding symmetry breaking bifurcations were found and studied in [9]. However, for a priori conservative homoclinic cases (as in Fig. 2 (a) and (b)) the problem is still open and we are going to study it in the nearest time. At the same time, bifurcations leading here to the appearance of symmetric elliptic periodic orbits should be well known, since they are the same as in the corresponding conservative case, see [10,11,12,9].
The case (c) of Fig. 2 is connected with the a priori presence of both symmetric orbits, periodic one (the fixed point), and a symmetric couple of orbits, the homoclinic ones. Evidently, the orbit behavior near every homoclinic is nonconservative, typically. Thus, in general, bifurcations of individual homoclinic tangency can lead to attractive (repelling) periodic orbits and, hence, the appearance of pairs "sink-source" of periodic orbits should be an a priori anticipated effect. Therefore, we call this cases, as well as the heteroclinic case from Fig. 1(a), "a priori nonconservative". Moreover, in these cases the main problem becomes a clarification of bifurcation mechanisms of appearance of symmetric periodic orbits and, first of all, elliptic ones.
We will assume, from now on, that the involution R is not trivial and leaves fixed a curve, that is, it satisfies . We say that an object Λ is symmetric when R(Λ) = Λ. To put more emphasis, the notation self-symmetric may be used. By a symmetric couple of objects Λ 1 , Λ 2 , we mean two different objects which are symmetric one to each other, i.e., R(Λ 1 ) = Λ 2 . Note that Symmetric homoclinic (heteroclinic) tangencies can be divided into two main types, namely, when: (i) there is a non-transversal symmetric heteroclinic orbit to a symmetric couple of saddle points, or (ii) there is a symmetric homoclinic tangency or a symmetric couple of non-transversal homo/heteroclinic orbits to symmetric saddle points. Thus, in this paper we consider the case of a symmetric couple of non-transversal homoclinic orbits to a symmetric saddle point. We assume that the multipliers λ and λ −1 are such that 0 < λ < 1. Then the point O divides its the stable and unstable manifolds into two connected parts called, resp., stable and unstable separatrices. If the homoclinic orbits belongs to the intersections of different pairs of the separatrices, as in Fig. 3(a), we will also say about a figure-8 homoclinic tangencies. If If the homoclinic orbits belongs to the intersections of the same pair of the separatrices, as in Fig. 4, we will also say about a figure-fish homoclinic tangencies.
Our main goal in this paper is to prove the RMD-Conjecture for general oneparameter reversible unfoldings of a couple homoclinic tangencies. For this goal, we consider a one-parameter family {f µ } of R-reversible maps such that the map f 0 has a couple of symmetric non-transversal homoclinic orbits Γ 1 and Γ 2 to a symmetric saddle fixed point O. Thus, we have that R(O) = O and R(Γ 1 ) = Γ 2 , R(Γ 2 ) = Γ 1 . Since O ∈ Fix R, the condition J(O) = 1 holds always. Therefore, our genericity condition, the condition [C] from Section 1, should be related to nonconservative properties of orbit behavior near homoclinics Γ 1 and, symmetrically, Γ 2 .
We consider a small fixed neighbourhood U of the contour O ∪ Γ 1 ∪ Γ 2 . Note that U is represented as a union of a small neighbourhood U 0 of the point O and a number of small neighbourhoods U j 1 and U j 2 , j = 1, ..., n, of those points of the orbits Γ 1 and  Γ 2 which do not lie in U 0 , see Fig. 3(b). We can assume that U is self-symmetric, i.e. R(U 0 ) = U 0 and R(U j 1 ) = U j 2 , R(U j 2 ) = U j 1 .
Note that we can also consider a homoclinic "figure-fish" configurations, where only one pair of separatrices of the saddle forms a couple of symmetric homoclinic tangencies, see Fig. 4. The proofs are the same, therefore, we do not consider this case separately and illustrate only certain geometric constructions in comparing with the figure-8 case (see Figs. 5 and 6).
In the family {f µ } we study bifurcations of single round periodic orbits, that is, orbits which pass just one time along U . We select three types of such orbits: the 1-orbits that pass trough the neighbourhoods U 0 and U j 1 ; the 2-orbits that pass trough the neighbourhoods U 0 and U j 2 ; and the 12-orbits that pass trough the neighbourhoods U 0 , U j 1 and U j 2 , j = 1, ..., n. Note that the orbits of the first two types are not symmetric but they compose always (when exist) a symmetric couple; while the orbits of the third type can be self-symmetric.
In this paper we show that, in the family {f µ } bifurcations of 1-and 2-orbits lead to the appearance of a couple "sink-source" periodic orbits, while bifurcations of 12-orbits can lead to the appearance of symmetric elliptic periodic orbits. These results imply, almost automatically, the existence of Newhouse regions (intervals) with reversible mixed dynamics for close reversible maps.
For such orbits we construct the corresponding first return (Poincaré) maps and study bifurcations of their fixed points. In Theorem 1 it is stated that two series (cascades) of infinitely many bifurcations of birth of single-round periodic orbits occur when varying µ near 0. The first series includes bifurcations of birth a pair "sinksource" of 1-and 2-orbits, the second series includes bifurcations of birth of symmetric elliptic 12-orbits. In the first case, in the corresponding Poincaré maps (R-symmetrical each other) the fixed attracting and repelling points appear under non-degenerate saddle-node bifurcations, exist for some intervals of values of µ and undergo nondegenerate period doubling bifurcations. In the second case, in the corresponding Poincaré map (which is R-self-symmetric), a fixed elliptic point is born as result of a nondegenerate reversible fold bifurcation.
The main result of this paper ....

Symmetry breaking bifurcations in the case of reversible maps with non-transversal homoclinic figure-8.
Let f 0 be a C r -smooth, r ≥ 4, two-dimensional map, reversible with respect to an involution R with dim Fix(R) = 1. Let us assume that f 0 satisfies the following two conditions: [A] f 0 has a saddle fixed point O belonging to the line Fix(R) and O has multipliers λ, λ −1 with 0 < λ < 1.
[B] f 0 has a symmetric couple of homoclinic orbits Γ 1 and Γ 2 such that Γ 2 = R(Γ 1 ) and the invariant manifolds W u (O) and W s (O) have quadratic tangencies at the points of Γ 1 and Γ 2 .
define reversible maps with symmetric non-transversal homoclinic figure-8 like in Figure 3(a) or homoclinic figure-fish shown in Figure 4(b). We ask them to satisfy one more condition. Namely, consider two points M − 1 ∈ W u loc (O) and M + 1 ∈ W s loc (O) belonging to the same homoclinic orbit Γ 1 and suppose f q 0 (M − 1 ) = M + 1 for a suitable integer q. Let T 1 denote the restriction of the map f q 0 onto a small neighbourhood of the homoclinic point M − 1 . Then, we assume that [C] the Jacobian J 1 of the map T 1 at the point M − 1 is positive and not equal +1. Without loss of generality, we assume that 0 < J 1 < 1. We consider also two points implies that the global maps T 1 and T 2 defined near the corresponding homoclinic points are not conservative maps.
Once stated the general conditions for f 0 , let us embed it into a one-parameter family {f µ } of reversible maps that unfolds generally at µ = 0 the initial homoclinic tangencies at the points of Γ 1 and Γ 2 . Then, without loss of generality, we can take µ as the corresponding splitting parameter. By reversibility, the invariant manifolds W u 1 (O) and W s 1 (O) split as W u 2 (O) and W s 2 (O) do when µ varies. Therefore, since these homoclinic tangencies are quadratic, only one governing parameter is needed to control this splitting.
We study first bifurcations of single-round 1-and 2-orbits as well as symmetric 12-orbits. Any point of such an orbit is a fixed point of the corresponding first-return map T 1k or T 2k , that is constructed by orbits of f µ with k iterations (of f µ ) staying entirely in U 0 ∪ U j 1 and U 0 ∪ U j 2 , j = 1, ..., n, respectively. The first main result is as follows: Theorem 1 Let {f µ } be a one-parameter family of reversible diffeomorphisms that unfolds, generally, at µ = 0 the initial homoclinic tangencies. Assume that f 0 satisfies conditions [A]- [C]. Then, in any segment [−ǫ, ǫ] with ǫ > 0 small, there are infinitely many intervals δ k = µ k sn , µ k pd and δ c k = µ k f , µ k pdC accumulating at µ = 0 as k → ∞ and such that the following holds.
(1) The value µ = µ k sn and µ = µ k pd correspond to simultaneous non-degenerate saddle-node and period doubling bifurcations of single-round 1-and 2-orbits of period k. Thus, the first-return map T 1k and T 2k has at µ ∈ δ k by two fixed points: sink and saddle for T 1k and source and saddle for T 2k . At µ = µ k pd the sink and source undergo simultaneously a non-degenerate (soft) period doubling bifurcations.
(2) The value µ = µ k f and µ = µ k pdC correspond to symmetric and conservative nondegenerate fold and period doubling bifurcations of single-round 12-orbit of period 2k. Thus, the corresponding first-return map T 12k has at µ ∈ δ c k two symmetric fixed points, elliptic and saddle ones. At µ = µ k pdC the elliptic point undergoes a symmetric period doubling bifurcation. ‡ However, the property of a symmetric saddle periodic point to be a priori area-preserving is more delicate. It is well-known (see, for instance, [13]) that a symmetric reversible saddle map is "almost conservative" since its (local) analytical normal form and its C ∞ formal normal form (up to "flat terms") are conservative.)

Preliminary geometric and analytic constructions
Consider the map f µ . Denote T 0 ≡ f µ U0 . The µ-dependent map T 0 is called the local map. We introduce also the so-called global maps T 1 and T 2 by the following relations: Then the first-return maps T 1k : Π + 1 → Π + 1 , T 2k : Π + 2 → Π + 2 and T 12km : Π + 1 → Π + 1 are defined by the following composition of maps and neighbourhoods: (1.1) see Fig. 5 and 6 . Accordingly, As usual, we need such local coordinates on U 0 in which the map T 0 has the simplest form. We can not assume the map T 0i is linear, since by condition [A], only C 1 -linearization is possible here. Therefore, we consider such C r−1 -coordinates in which the local maps have the so-called main normal form or first order normal form. This form is given by the following lemma.

Lemma 1 [9]
Let a C r -smooth map T 0 be reversible with dim Fix T 0 = 1. Suppose that T 0 has a saddle fixed (periodic) point O belonging to the line Fix T 0 and having multipliers λ and λ −1 , with |λ| < 1. Then there exist C r−1 -smooth local coordinates near O in which the map T 0 (or T n 0 , where n is the period of O) can be written in the following form:

2)
where h 1 (0) = −h 2 (0). The map (1.2) is reversible with respect to the standard linear involution (x, y) → (y, x). In fact, it can be expressed in the so-called cross-form: In the case that T 0 is linear, i.e. of the formx = λx,ȳ = λ −1 y, its j-th iterates (x j , y j ) = T j 0 (x 0 , y 0 ) are given simply by x j = λ j x 0 , y j = λ −j y 0 in standard explicit form or by x j = λ j x 0 , y 0 = λ j y j in cross-form. If T 0 is nonlinear such cross-form expression for T j 0 exists too. Precisely, the following result holds: Lemma 2 [9] Let T 0 be a saddle map written in the main normal form (1.2) (or (1.3)) in a small neighbourhood V of O. Let us consider points (x 0 , y 0 ), . . . , (x j , y j ) from V such that (x l+1 , y l+1 ) = T 0 (x l , y l ), l = 0, . . . , j − 1. Then one has where the functions h j (y j , x 0 ) are uniformly bounded with respect to j as well as all their derivatives up to order r − 2.
Remark 1 (a) Both lemmas 1 and 2 are true if T 0 depends on parameters. Moreover, if the initial T 0 is C r with respect to coordinates and parameters, then the normal form (1.2) is C r−1 with respect to coordinates and C r−2 with respect to parameters (see [14], Lemmas 6 and 7).
(b) Bochner Theorem (see [15]) ensures that any involution R with dim Fix R = 1 is locally smoothly conjugated to its linear part around a symmetric point. It is not a loss of generality to assume that maps f µ are reversible under an involution R with linear part given by L(x, y) = (y, x). As it will be shown, this fact will be very convenient in the construction of the local maps T 01 and T 02 . (c) Similar results related to finite-smooth normal forms of saddle maps were established in [16,17,18,19] for general, near-conservative and conservative maps. The proof of lemmas 1 and 2 is just an adapted version to the reversible setting.

Construction of the local and global maps
We choose in U 0 local coordinates (x, y) given by Lemma 1. In these coordinates, the local stable and unstable invariant manifolds of the point O are straightened: x = 0 is the equation of W u loc (O) and y = 0 is the equation of W s loc (O). Then, we can write the (x, y)-coordinates of the chosen homoclinic points as follows: , then we have that y − 2 = x + 1 and x + 2 = y − 1 . Taking into account the homoclinic geometry of the figure-8 case according to Fig. 5 we can assume that In the homoclinic figure-fish case we obtain that Fig. 6.
We assume that Then the domains of definition of the successor map from Π + i into Π − j , i, j = 1, 2, under iterations of T 0 consists of infinitely many non-intersecting strips σ 0ij k which belong to Π + i and accumulate at W s loc (O)∩Π + i as k → ∞. In turn, the range of the succesor map consists of infinitely many strips σ 1ij k = T k 0 (σ 0ij k ) belonging to Π − i and accumulating at W u loc (O) ∩ Π − i as k → ∞ (see Figure 7). Thus, the first return maps under consideration are defined as follows: (1.6) By Lemma 2 the map T k 0 : σ 0ij k → σ 1ij k can be written in the following form (for large enough values of k) T k 0 : where (x 0 , y 0 ) ∈ σ 0ij k , (x 1 , y 1 ) ∈ σ 1ij k , i, j = 1, 2. We write now the global map T 1 : Π − 1 → Π + 1 in the following form The Jacobian J(T 1 ) has, obviously, the following form: J(T 1 ) = −bc + O |x 11 | + |y 11 − y − 1 | . Thus, J 1 = −bc and we assume (condition C) that 0 < J 1 < 1.
Concerning the global map T 2 , we cannot write it now in an arbitrary form. The point is that after written a formula for the map T 1 it is necessary to use the reversibility relations to get the one associated to it: for constructing T 2 . Then, by (1.8), we obtain that the map T −1 y 12 )} must be written as follows Relation (1.9) allows to define the map T 2 : Π − 2 {(x 12 , y 12 )} → Π + 2 {(x 02 , y 02 )}, but in the implicit form:

Proof of item 1 of Theorem 1
Proposition 1 Let f ε redbe the family under consideration redsatisfying conditions A-D. Then, for every sufficiently large k, the first return map T 1k : σ 0 k → σ 0 k can be brought, by a linear transformation of coordinates and parameters, to the following formX Proof. In virtue of (1.3) and (1.8) the first return maps T 1k is written in the form (1.11) After a series of changes of coordinates it can be brought to the required form (1.10).
There exists a shift ξ = x − x + + O(λ k ), η = y − y − + O(λ k ) which cancels the constant term in the first equation and the term linear in y −y − in the second equation of (1.11), and the map takes the form Rescaling the variables corresponding to the existence of a fixed point with a multiplier +1 (saddle-node fixed point) and a fixed point with a multiplier −1 (period doubling bifurcation), respectively. For −1 < M 2 < 0, the has no fixed points below the curve L +1 , has a stable (sink) fixed point in the region between the bifurcation curves L +1 and L −1 , while at L −1 a period doubling bifurcation takes place and a stable 2-periodic orbit appears above the curve L −1 .

Proof of item 2 of Theorem 1
We take in U 0 the local C r−1 -coordinates (x, y) from lemma 1. We consider the homoclinic points M and, moreover, the equalities (1.5) hold. Now we will construct the first return map T 12km = T 2 T m 0 T 1 T k 0 . Briefly, the method we use -based on a rescaling technique § -will allow us to prove that the first-return map T 12km can be written asymptotically close (as k, m → ∞) to the area-preserving (symplectic) map of the form (see also [9]): H : x =M +cx − y 2 , cȳ = −M + y +x 2 , (1.12) in which the coordinates (x, y) and the parameterM can take arbitrary values. Note thatc = − b c is a constant andc < 0, since bc < 0 by condition C. The bifurcation diagram of the map (1.12) is shown in Figure 8. We represent it on the (c,M )-plane, since a character of bifurcations depend on the value ofc. Equations of the bifurcation curves F 0 (fold bifurcation), P D 1 and P D 2 (period doubling), P F (pitch-fork) and P D(asym) (period doubling of a couple of symmetric § The rescaling method [20] appears to be very efficient to study homoclinic bifurcations. fixed points) are as follows: We denote the coordinates (x, y) on Π + i as (x 0i , y 0i ) and on Π − i as (x 1i , y 1i ), i = 1, 2. Then, by Lemma 2, the map T k 0 : Π + 2 → Π − 1 will be defined on the strip σ 021 Analogously, the strips σ 011 Fig. 7. Then the first return map T 12k can be constructed as the following composition We can write these relations by the coordinates as follows: x 11 = λ k x 02 (1 + kλ k h k (x 02 , y 11 )) , y 02 = λ k y 11 (1 + kλ k 1 h k (y 11 , x 02 )), x 12 = λ m x 01 (1 + mλ m h m (x 01 , y 12 )) , y 01 = λ m y 12 (1 + mλ m h m (y 12 , x 01 )), (1.14) In these formulas we use the cross-coordinates (x 0 , y 1 ) on the strips σ 0 k , since, by Lemma 1, the coordinates (x 0 , y 0 ) of any point of σ 0 k are recalculated via the crosscoordinates as (x 0 , λ k y 1 (1 + kλ k h k (y 1 , x 0 )). Accordingly, the coordinates y 0 appear in (1.14) only as auxiliary ones. On the other hand, the coordinates (x 02 , y 11 ) are dynamical ones, since formulas (1.14) define the map (x 02 , y 11 ) → (x 02 ,ȳ 11 ), whereȳ 02 = λ kȳ 11 (1 + kλ k 1 h k (ȳ 11 ,x 02 )). Thus, the coordinates (x 01 , y 12 ) are only intermediate.
Our consideration covers formally also the case of nonorientable reversible maps. Indeed, in this case we assume that f 0 satisfies the conditions A and B and now the condition Note that the necessary calculation for the first return maps are obtained the same as in the orientable case. Only the result in the case of the map T 12km will be different. Here, the first-return map T 12km can be written asymptotically close (as k, m → ∞) to the area-preserving (symplectic) map of the form (1.12) again. However, c = b c is positive here, since bc > 0 by condition C'. The bifurcation diagram of the map (1.12) on the (c,M )-half-plane, wherec > 0 is shown in Figure 9.
Equations of the bifurcation curves F 0 (fold bifurcation), P D 1 , P D 2 (period doubling) and P F (pitch-fork) are as follows: Note that in this case the pitch-fork bifurcation leads to the appearance of a couple of symmetric saddle fixed points that do not bifurcate at further varying values of parameters. Accordingly, the bifurcation curve P D(symp) as in casec < 0 is absent here.