On local and global aspects of the 1:4 resonance in the conservative cubic H\'enon maps

We study the 1:4 resonance for the conservative cubic H\'enon maps $\mathbf{C}_\pm$ with positive and negative cubic term. These maps show up different bifurcation structures both for fixed points with eigenvalues $\pm i$ and for 4-periodic orbits. While for $\mathbf{C}_-$ the 1:4 resonance unfolding has the so-called Arnold degeneracy (the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient), the map $\mathbf{C}_+$ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by $\pi/4$. For both maps several bifurcations are detected and illustrated.

For area-preserving maps, the basis of the 1:4 resonance phenomenon consists in a local bifurcation of a fixed (or periodic) point with eigenvalues e ±iπ/2 = ±i. However, this can be only the simplest, standard part of the general picture of the resonance. As we show, the 1:4 resonance in the case of maps (1) and (2) can be nontrivial, i.e. it can include not only bifurcations of fixed points with eigenvalues ±i themselves (local aspects) but also a series of accompanying bifurcations (global aspects) of 4-periodic orbits which are initially born from the central fixed point with eigenvalues ±i.
It is well-known that the strong resonances, i.e. the bifurcation phenomena connected with the existence of fixed (periodic) points with eigenvalues e ±2πi/q with q = 1, 2, 3, 4 (that is, the 1:1, 1:2, 1:3 and 1:4 resonances), play a very important role in the dynamics of area-preserving maps. Among them, the 1:4 resonance (with q = 4) is, as a rule, the most complicated and least studied. For example, the resonances with q = 1, 2, 3 are nondegenerate for the standard conservative Hénon mapx = y,ȳ = M − x − y 2 .
The simplest degeneracies occur when only one of the conditions in (5) does not hold.
The local bifurcations at the 1:4 resonance, in the general (not necessarily conservative) setting, were first studied by V. Arnold in the 70's, see [Arn88], for the flow normal formż = εz +Ãz|z| 2 + (z * ) 3 , whereÃ is a coefficient and ε is a small complex parameter. 1 He showed that the structure of such resonance essentially depends on the relation between ReÃ and ImÃ and he studied several cases, e.g. when |Ã| < 1 or |ReÃ| > 1. Numerous other cases (when |ReÃ| < 1 and |Ã| > 1) were studied in many papers, see e.g. [Kra94,Kuz95]. Concerning the conservative case, where ReÃ ≡ 0, the Arnold normal form can be represented aṡ where ε and b = Im(Ã) are real. This normal form is nondegenerate in the case b = 1 and its bifurcations are well-known, see Fig. 1. We see that cases b < −1, Fig. 1a, and |b| < 1, Fig. 1b, are very different. In particular, the equilibrium z = 0 is always stable in the first case, whereas, it can be unstable (a saddle with 8 separatrices, at ε = 0) in the second case. Figure 1: Bifurcations of zero equilibrium in the family (6) for |b| = 1 in the cases (a) b < −1, here only one equilibrium z = 0 (a nonlinear center) exists at ε ≤ 0 and 8 equilibria appear surrounding z = 0 at ε > 0; and (b) |b| < 1, here 5 equilibria (4 saddles and the center z = 0) exist at ε < 0; at ε = 0 all these equilibria merge to the point z = 0 which becomes the nonhyperbolic saddle with 8 separatrices; at ε > 0 the point z = 0 becomes again a center and the 4 saddles appear to be rotated by π/4 with respect to the ones at ε < 0. Note that, in case b > 1, one needs to change both ε by −ε and the time direction in the (a) row of the plot.
As it was shown in [Bir87,SV09], in the case of map (3), the degeneracy A = 1 (here B 21 = −B 03 ) takes place. Note that the Hénon map (3) has a fixed point with eigenvalues ±i at M = 0. As it was shown in [Bir87], the familyx = y,ȳ = ε 1 − x − y 2 + ε 2 y 3 can be considered as a two-parameter general unfolding for the study of bifurcations of this point. This result is quite important, since such maps (conservative Hénon maps with small cubic term) appear naturally as rescaled first return maps near homoclinic orbits to saddle-focus equilibria of divergence-free three-dimensional flows [BS89] or near quadratic homoclinic tangencies of area-preserving maps [GG09,DGG15]. 2 In the present paper we show that degeneracy A = 1 can take place only in the case of cubic map (1) (here B 03 < 0 always). This occurs for M 1 = ±16/27 and M 2 = 1/3. For those parameter values one has B 21 = 0 and B 03 = 0, see Section 3 for the concrete expressions of these coefficients as a function of M 2 . Then, according to [Bir87], case A = 1 is generic whenever the coefficient Ω = Re(B 32 − B 50 − B 14 ) is non-zero, which guarantees a non-vanishing twist for the 4th power of the local normal form (4). We show in Fig. 2 the graph of Ω for C − and C + as a function of M 2 . In particular, we see that Ω ≈ −0.25 when degeneracy A = 1 takes place for C − . A description of the local bifurcations in normal form (4) with A = 1, Ω = 0 was also given in [Bir87]. However, we do not restrict ourselves only to the study of the local structure of this resonance -the analysis of the corresponding normal form is quite standard. We also study the global effect of this resonance on the dynamics of map (1) as a whole. To this end, we find (analytically and/or numerically) bifurcation curves relevant to describe the bifurcation diagrams related to both bifurcations of the fixed point with eigenvalues ±i as well as to accompanying bifurcations of saddle and elliptic 4-periodic orbits belonging to the corresponding resonant garland surrounding the fixed point. We collect the corresponding results in Section 3.
We also show that degeneracy B 03 = 0 can take place only in the case of the cubic map (2), when M 1 = ±20/27 and M 2 = −1/3 (for this map B 21 = 0 always). Moreover, for map C + , the value of A is always greater than 1. As in the case of map (1), we study both the local and global aspects of this resonance, see Section 4.
As far as we know, this type of the conservative 1:4 resonance (with B 03 = 0) has not been studied before, 3 therefore we describe the main elements of the local bifurcations at such resonance in the Appendix B. In this case we assume B * 14 = 5B 50 = 0. Note that the previous equality of the coefficients guarantees the Hamiltonian character of the corresponding flow normal form (see Eq.(9) in Section 2), while the nonvanishing of those coefficients is an additional nondegeneracy condition. In Fig. 3 we show the graphs of B 14 and B 50 , for the case of map (2), as a function of M 2 . In particular, when degeneracy B 03 = 0 takes place in map (2) (M 1 = ±20/27, M 2 = −1/3), one has that B 14 ≈ −5/64. We note that the cubic Hénon maps (1)-(2) have an important meaning for the theory of dynamical systems. For example, they appear as truncated normal forms of first return maps near cubic homoclinic tangencies. In Fig. 4 we illustrate the geometric idea how such maps can be obtained. Let a two-dimensional map f have a fixed saddle point O and a homoclinic orbit Γ at whose points the manifolds W u (O) and W s (O) have a cubic tangency. Let M + ∈ W s loc 2 The main bifurcations of area-preserving maps with quadratic homoclinic tangencies were studied in [MR97,GG09,DGG15], and with cubic ones in [GGO17]. In all these papers, the main technical tool was the so-called rescaling method by which it was possible to represent the first return map in the form of a map being asymptotically close to the quadratic or to the cubic Hénon map.
3 However, it was noted in [GLRT14,GT17] that such type resonances can provoke symmetry-breaking bifurcations (of pitchfork type) which, in the case of reversible maps, lead to the appearance of nonconservative periodic orbits (e.g. periodic sinks and sources). and M − ∈ W u loc be a pair of points of Γ and σ k be a small (horizontal) strip near M + . Under some number m of iterations of f the strip σ k is mapped into a vertical stripσ k located near the point M − . The (local) map from σ k intoσ k can be represented, for simplicity, as the linear mapx = λ m x,ȳ = λ −m y, where (x, y) are coordinates of points in σ k and (x,ȳ) are those inσ k . Let q be an integer such that f q (M − ) = M + . Then, since the curve f q (W u loc ) have a cubic tangency with W s loc at the point M + , the image f q (σ k ) ofσ k will have the form of a cubic horseshoe. Thus, the geometry of the first return map f k : σ k → σ k , where k = m + q, is like the one of a cubic horseshoe map. If one rescales the initial coordinates (x, y) and the initial parameters µ 1 and µ 2 (that are the usual parameters that unfold the initial cubic homoclinic tangency between the curves f q (W u loc ) and W s loc at the point M + ) in the appropriate way, then one can rewrite the map f k : σ k → σ k in the form of a cubic Hénon maps with some terms that are asymptotically small as k → ∞. Note that if λ > 0, then there are two different types of cubic homoclinic tangencies: the tangency "incoming from above", see Fig. 4a, and the tangency "incoming from below", see Fig. 4b. In the first case, the truncated rescaled map is map (1), while in the second case it is map (2). For more details, see [GGO17] for the area-preserving case and [GST96,GSV13] for the dissipative case. The main bifurcations of the dissipative cubic Hénon maps (when the absolute value of the Jacobian is less than 1) were studied in [Gon85,GK88]. In the paper [DM00] some bifurcations of conservative maps (1) and (2) were studied. Note that one of the main goals of [DM00] was to describe the bifurcation structure of the 1:1, 1:2 and 1:3 strong resonances. The present work as well as the papers [Gon05,GGO17] are devoted mainly to the study of the local and global aspects of the 1:4 resonances, which complements the research of [DM00] on the dynamics and the bifurcations of the conservative cubic Hénon maps.
The paper is organised as follows. In Section 2 we review the 1:4 resonance for general area-preserving maps and comment about the degenerate cases. In Section 3 we consider map C − and describe the local and global aspects of the 1:4 resonance in this concrete case. The degeneracy A = 1 occurs here and we study its influence on the global bifurcation diagram. The same type of local and global bifurcation analysis is performed in Section 4 for the 1:4 resonance of the map C + . We show that degeneracy B 03 = 0 takes place in this case. As far as we know, this degeneracy has not been studied before, therefore we include the normal form analysis of this bifurcation in Appendix B. For both maps, some of the bifurcation curves have been explicitly obtained and their equations are derived in Appendix A. Finally, in Section 5 we comment on related topics where the results obtained in this paper could be relevant.
2 Local aspects of the 1:4 resonance in area-preserving maps.
The unfolding of the non-degenerate 1:4 resonance leads to the one-parameter family of area-preserving maps beingβ a real parameter characterizing the deviation of the angle argument ϕ of the eigenvalues of the fixed point from π/2 (ϕ =β + π/2), the coefficients B 21 := B 21 (β) and B 03 := B 03 (β) are real and depend smoothly onβ. If B 03 = 0, the fourth iteration of map (7) can be locally embedded into the one-parameter family (6) of Hamiltonian flows, being b(β) = B 21 (β)/B 03 (β) and ε = 4β/B 03 . Local bifurcations of this Hamiltonian system are shown in Fig. 1.
The case A = 1 appears in the bifurcation diagram of C − , see Section 3. The unfolding of this case leads to a two-parameter family of area-preserving maps. The fourth iteration of such a family is close-to-identity and can be approximated by the flow of the Hamiltonian systeṁ where β = 4β/B 03 , µ is the parameter responsible for the deviation from A = 1. The coefficientsB ij are related to those in (4), see [Bir87,GG14,SV09]. Namely, one has The vector field has zero divergence provided that 5B 50 =B 14 .
The bifurcation diagram for (8), when considering (β, µ) in a small neighbourhood of the origin (case A = 1), is displayed in Fig. 5. There are three bifurcation curves L 1 , L 2 and L 3 dividing the (β, µ)-parameter plane into 3 domains. Curves L 1 : {β = 0, µ > 0} and L 2 : {β = 0, µ < 0} correspond to the passage from I to II and II to III, respectively, and reconstructions of nonzero saddle equilibria occur. Curve L 3 : {µ = √ −βΩ + O(|β|)} corresponds to a π/2-equivariant parabolic bifurcation: 4 saddles and 4 elliptic equilibria appear when crossing from I to III. This bifurcation takes place far away from the origin of coordinates and it is a codimension one bifurcation for (8) since the flow is invariant under the rotation of angle π/2. Note that the origin is a non-degenerate conservative center for β = 0; it is a degenerate saddle with 8 separatrices for (β, µ) ∈ L 1 and a degenerate conservative center for (β, µ) ∈ L 2 . The origin in the (β, µ)-plane, which corresponds to the case A = 1, is the endpoint of all the three bifurcation curves. Note that, in Fig. 5, we represent the bifurcation diagram for the case Ω < 0 because for map C − , as we have computed, one has Ω ≈ −0.25, see Fig. 2.
The unfolding of case B 03 = 0 reduces to the study of the Hamiltonian floẇ z = iβz + iB 21 z|z| 2 + i z * 3 + iB 32 |z| 4 z + iB 50 z 5 + iB 14 |z| 2 z * 3 + O(|z| 7 ), Note that normal forms (8) and (9) are Hamiltonian and reversible with respect to two linear involutions: involution R : z → z * (in the real coordinates (x, y), it corresponds to involution (x, y) → (x, −y)) and involution R * : z → iz * (it corresponds to involution (x, y) → (y, x)). Since flows (8) and (9) are π/2-equivariant, two additional involutions also exist:R : z → −z * (it corresponds to (x, y) → (−x, y)) andR * : z → −iz * (it corresponds to the involution (x, y) → (−y, −x)). The lines of fixed points of these involutions are the following: Returning to the bifurcation diagram of Fig. 6, we see that at (β, ) ∈ I only one equilibrium exists (the trivial equilibrium z = 0 that is a center), while nontrivial equilibria (centers and saddles) appear in the other regions of the diagram. In regions II and IV there are 8 nontrivial equilibria while in region III there exist 16 nontrivial equilibria. In the case of regions II and IV, all nontrivial equilibria are symmetric, i.e. belong to the lines of fixed points of the involutions: in II two of the four centers belong to the axis y = 0 (Fix(R)) while the other two belong to x = 0 (Fix(R)), and two of the four saddles belong to the bisectrix x = y (Fix(R * )) while the other two belong to x = −y (Fix(R * )). We see in region IV another disposition of these equilibria: the picture seems turned by an angle of π/4. Due to the strong reversibility properties of system (9), such simple rotation of the garland is impossible without bifurcations. The corresponding (providing such a rotation) symmetry breaking bifurcations, at the passage from domain II to domain IV, are schematically shown in Fig. 6. When passing from I to III b the centers undergo (supercritical) pitchfork bifurcations: they all become saddles and four pairs of nonsymmetric centers are born. The curve L hom in domain III corresponds to a global π/2-equivariant bifurcation of creation of heteroclinic connections between all 8 saddles. 4 The passage through L hom from III b to III a is related to reconstructions of the separatrices of the saddles. After this, at passage from III a to IV, we observe a (subcritical) pitchfork bifurcation where asymmetric centers merge with symmetric saddles and the latter become centers.
• The curve L 3 4 , given by is associated with pitchfork bifurcations of elliptic 4-periodic orbits. This is a consequence of the fact that the periodic orbit has a point on the symmetry line y = x, see details in the proof of Lemma 1.
• The curve L 4 4 , with equation corresponds to a parabolic bifurcation curve for 4-periodic (non-symmetric) orbits. These orbits are not of Birkhoff type since they are not ordered orbits surrounding the elliptic fixed point P ± π/2 . This curve has a special property. For parameters on L 4 4 with M 1 > 0 (resp. M 1 < 0) the fixed point Q − = (− (M 2 − 2)/3, − (M 2 − 2)/3) (resp. Q + = −Q − ) of map C − is created at a parabolic bifurcation. That is, both parabolic bifurcations, for the 4-periodic orbit and the fixed point, take place simultaneously. See details below. This peculiarity is due to the simple form of the cubic Hénon map and clearly is not a persistent property under arbitrary small perturbations.
Remark 1. The curves L 1,2 4 together with L − π/2 reflect a peculiarity of local aspects of the 1:4 resonance in the case of cubic map C − . On the other hand, the curves L 3 4 and L 4 4 and the related bifurcations can be considered as peculiarities of the global aspects. Moreover, the curve L 4 4 has no direct relation with the problem of 1:4 resonance since it is a parabolic bifurcation curve for simultaneously two non-symmetric 4-periodic orbits which are symmetric one to other with respect to involution R : (x, y) → (y, x).
Let us give further details on how bifurcation curves organize the parameter space. Fix a vertical line M 1 = C with |C| > 4/3 √ 3. For parameters on this line, as we change M 2 from bottom to top, one has the following sequence of bifurcations: • For C > 4/3 √ 3 (resp. C < −4/3 √ 3) the elliptic 4-periodic orbit created on L 1 4 (resp. L 2 4 ) becomes hyperbolic when crossingL 1 4 (resp.L 2 4 ) and, at the crossing of this period-doubling bifurcation curve, an elliptic 8-periodic orbit is born.   • For larger M 2 , there appears an elliptic 4-periodic orbit at the inverse period-doubling bifurcation that takes place onL 2 4 (resp.L 1 4 ).
• This elliptic 4-periodic orbit undergoes a pitchfork bifurcation when crossing L 3 4 and two elliptic 4-periodic orbits persist. The location of the latter elliptic orbits is symmetric with respect to involution R : (x, y) → (y, x). See more details in the proof of Lemma 1.
In Fig. 8 we show a sequence of phase space plots where these bifurcations can be observed. The illustrations are for parameters on the vertical line M 1 = 0.8. The Fig. 8 top left corresponds to M 2 = 0.65, which is located between L 1 4 andL 1 4 . We see the main island around the fixed point and the 4-periodic satellite islands. There are an elliptic 4periodic orbit and a hyperbolic 4-periodic one (this is located near the peaks of the main stability island, the invariant manifolds of each one of this saddle 4-periodic points surround the corresponding satellite island). In Fig. 8 top center we see a magnification of the 4-periodic satellite island of stability around the elliptic 4-periodic orbit. This elliptic orbit undergoes a period-doubling bifurcation when crossingL 1 4 , the two stability islands can be seen in Fig. 8 top right for M 2 = 0.748. Finally the Fig. 8 bottom left and right, for M 2 = 1.36 and M 2 = 1.38 respectively, illustrate the pitchfork bifurcation taking place when crossing L 3 4 . Curve L 4 4 is the curve of parabolic 4-periodic orbits with double eigenvalue 1 which is a parabolic bifurcation curve for the 4-periodic orbits. As already said, this curve also coincides with the curve which corresponds to a parabolic bifurcation of the fixed point (or, when M 1 = 0, M 2 = 2, to a pitchfork bifurcation of the fixed point). Let us give further details on the sequence of bifurcations when crossing the linesL 4 4 and L 4 4 in Fig. 7 right. To this end consider the vertical line M 1 = 0.0003. When moving M 2 from bottom to top in Fig. 7 right one has the following sequence of bifurcations: • First we have the crossing ofL 4 4 . At this crossing two non-symmetric elliptic 4-periodic orbits and two nonsymmetric hyperbolic 4-periodic orbits are created as a result of an inverse period-doubling bifurcation. The position of the two elliptic 4-periodic orbits is shown in Fig. 9 top left. A magnification of one of the satellite islands is shown in Fig. 9 top center.
• When crossing L 4 4 there is a parabolic bifurcation and two new 4-periodic orbits, one elliptic and the other of saddle type, are created. This can be seen in Fig. 9 top right.
For parameters (M 1 , M 2 ) above the curve L + π/2 the local phase space is topologically equivalent to that of region I in Fig. 5, that is the fixed point P ± π/2 is elliptic and there are no 4-periodic orbits surrounding it.
Crossing the curve L + π/2 through (M 1 , M 2 ) ∈ L + π/2 \ {P r,l 4 } is analogous to cross the line L 2 in Fig. 5 from region I to II. In particular, around P ± π/2 there appear a saddle 4-periodic orbit whose invariant manifolds bound a 4-periodic island of stability with an elliptic 4-periodic orbit inside.
In Fig. 11 we display the sequence of bifurcations taking place when getting inside/outside the region bounded by curvesB r 4 and B r 4 (by symmetry, bifurcations through B l 4 andB l 4 are analogous). For illustrations we choose M 2 = −0.5 and change M 1 . For M 1 = 0.7 (top left plot in Fig. 11) we see the 4-periodic island having the elliptic point on the symmetry line y = x. For parameters onB r 4 this point undergoes a pitchfork bifurcation: the elliptic 4-periodic orbit becomes a saddle 4-periodic orbit and a pair of elliptic 4-periodic orbits appear nearby, see Fig. 11 top center. Then the separatrices of the saddle 4-periodic orbits become larger, see Fig. 11 top right, and at some moment between M 1 = 0.718 and M 2 = 0.719 the separatrices merge 6 with the exterior separatrices of the other saddle 4-periodic orbits and a sequence of bifurcations related to the reconstruction of homo/heteroclinic connections takes place, see Fig. 11 bottom left. After that, an inverse pitchfork bifurcation occurs, the two elliptic 4-periodic collide into the saddle 4-periodic orbit which becomes elliptic, see Fig. 11 bottom center and right. Note that for M 1 = 0.73 the saddle 4-periodic orbit is on the symmetry line y = x. This sequence of bifurcations (pitchfork, reconnection of the invariant manifolds and inverse pitchfork) happens generically at the unfolding of the degenerate case B 03 = 0, see details in Appendix B.

Other bifurcation curves related to 4-periodic orbits
We have considered some of the local bifurcations of 4-periodic orbits emanating from the 1:4 resonance. The evolution in phase space of the associated 4-periodic orbits leads to other bifurcation curves of other 4-periodic orbits which interact with the obtained ones. In this section we aim to illustrate some aspects of their configuration in parameter space.
Remark 2. The curves B l,r 4 andB l,r 4 are related to the local aspects of the 1:4 resonance. On the other hand, the bifurcations curves considered in this section are not related to the (local) 1:4 resonance problem.
Some of the bifurcation curves to be considered already appear in Fig. 10. The curves C r,l 4 correspond to trace= 2 (double eigenvalue 1), given by C r,l 4 : and the curvesC r,l 4 to trace= −2 (double eigenvalue −1). To explain the bifurcations that take place let us consider a horizontal line M 2 = C, with C 1 < C < C 2 , where C 1 ≈ −1.6220 is the M 2 -coordinate of the intersection point between C r 4 andC r 4 (there is only one intersection point, shown in Fig. 10 left) and C 2 ≈ −1.0647 is the M 2 -coordinate of the bottom intersection point betweenC r 4 andC l 4 shown in the Fig. 10 right. When varying M 1 from right to left in Fig. 10 one has the following bifurcation sequence: • For M 1 in the right hand side ofB r 4 there are saddle 4-periodic orbits along with a pair of elliptic 4-periodic orbits. Recall that curveB r 4 corresponds to a pitchfork bifurcation, hence to the left of this curve we get a 4-periodic island of stability creating a garland containing saddle and elliptic 4-periodic orbits.
• When decreasing M 1 , an inverse period-doubling bifurcation occurs at the first crossing with curveC r 4 in Fig. 10 left. The bifurcation is as follows: for parameters to the right ofC r 4 , in a neighborhood of the elliptic 4-periodic orbit, there appears a saddle 8-periodic orbit. This 8-periodic orbit bifurcates from the saddle 4-periodic orbit that remains to the left ofC r 4 .  Figure 12: Other bifurcation curves of 4-periodic orbits in map C + . In the left plot we see the two (green) curves D l,r 4 that correspond to parabolic 4-periodic orbits with double eigenvalue −1. The curves D l,r 4 correspond to perioddoubling of 2-periodic orbits. The other curves are better seen in the right plot, which is a magnification of the left one. In the right plot we see that curvesD l,r 4 , corresponding to parabolic 4-periodic with double eigenvalue 1 respectively, become tangent to D l,r 4 . Finally, curvesD l,r 4 , which correspond to a pitchfork bifurcation of 4-periodic orbits, end up at (M 1 , M 2 ) = (0, −3).
• For parameters on the curve C r 4 , a parabolic bifurcation for a 4-periodic orbit takes place. At this bifurcation a saddle and an elliptic 4-periodic orbits are created. One of the pairs of the elliptic and hyperbolic 4-periodic orbits that bifurcate lie on the symmetry line y = x. The elliptic orbit undergoes a period-doubling bifurcation when crossing the curveC r 4 . See Fig. 10 right.
We have found other bifurcation curves related to 4-periodic orbits. We note that the corresponding 4-periodic orbits do not lie on the symmetry line y = x of map C + . The bifurcation curves are shown in Fig. 12 left. Note that, in the left plot, curves D l,r curves. To start with we consider M 1 = −0.047, which is between the curves D r 4 andD r 4 . For parameter values (M 1 , M 2 ) = (−0.047, −3.1) the phase space shows up a 2-periodic island and a 4-periodic island (which is a 2periodic satellite of the 2-periodic island). Denote by e 2 (resp. by e 4 ) the elliptic points around which the 2-periodic (resp. the 4-periodic) islands of stability are organized. The 2-periodic elliptic island, for M 2 = −3.1, persists 7 for −0.047 < M 1 ≤ 0. When changing M 1 on the line M 2 = −3.1, the following bifurcations are observed: • At the crossing ofD r 4 from left to right, the point e 4 undergoes a period-doubling bifurcation, and an 8-periodic elliptic point is created.
• When crossing D r 4 from right to left, the point e 2 undergoes a period-doubling. Consequently, a 4-periodic elliptic orbit is born, denote it byẽ 4 .
• At the crossing ofD l 4 from right to left, there is an inverse pitchfork bifurcation at which the two 4-periodic elliptic points e 4 andẽ 4 collide and give rise to a 4-periodic elliptic orbit. This 4-periodic elliptic orbit persists until the crossing ofD l 4 where disappears at a parabolic bifurcation.
Other sequences of bifurcations can be observed for other lines M 2 = C (specially when considering M 2 in the range shown in Fig. 12 right). For example, for M 2 = −2.9 and moving M 1 from right to left starting from M 1 = 0, one has that the 2-periodic elliptic orbit undergoes a period-doubling bifurcation at D l 4 (roughly for M 1 ≈ −0.1). The 4-periodic elliptic orbit that bifurcates from the previous bifurcation undergoes a period-doubling atD l 4 (which happens for M 1 < −0.2).
As a final comment, we note that the bifurcation curves considered before allow us to explain the sequences of bifurcations of 4-periodic orbits that we have observed when plotting the islands of stability for different values of (M 1 , M 2 ). Hence, we believe that the bifurcation diagram for the 4-periodic orbits in the (M 1 , M 2 ) regions shown is complete (although we have no proof of this fact).

Conclusions and related topics
We have obtained a detailed picture of the bifurcation diagrams near the 1:4 resonance of maps C ± in (1)-(2). A description of the bifurcations taking place when crossing the main bifurcation curves derived (either analytically or numerically) has been provided. Special emphasis has been put to the clarification of the scenarios related to the degeneracies of the 1:4 normal form. We have shown that degeneracy A = 1 happens for C − while degeneracy B 03 = 0 happens for C + , and we have analysed the dynamical consequences of these degeneracies in these concrete cases.
We believe that the results presented in this work are relevant for related studies. Namely, we want to emphasize that the study of the cubic Hénon maps (1) and (2) (and naturally the quadratic one (3)) is important because • these maps are the simplest nonlinear symplectic maps of the plane, and therefore the understanding of the basic properties of the dynamics and the bifurcations in such maps will be very useful in more general contexts.
• as pointed out in the introduction, these maps are, in fact, normal forms of the first return maps near the quadratic (map (3)) and cubic (maps (1) and (2)) homoclinic tangencies: it is easy to relate the structure of the bifurcations of these maps with the global bifurcations happening at the homoclinic tangency.
However, there is another important reason why the 1:4 resonance in the cubic Hénon maps is of interest. It is connected with mixed dynamics -a new third type of dynamical chaos characterized by the principal inseparability of attractors from repellers 8 and from the conservative elements of dynamics (for example, periodic sinks, sources and elliptic points), see e.g. [GST97,LS04,DGGLS13,Gon16,GT17]. It is worth noting that the mixed dynamics can be an open property of reversible non-conservative chaotic systems in which self-symmetric orbits are conservative (e.g., symmetric elliptic trajectories), while asymmetric ones appear in pairs and have the opposite type of stability (e.g., "sink-source" pairs). Such symmetric/asymmetric orbits emerge usually as a result of various homoclinic bifurcations, see more details in [DGGLS13], including local symmetry breaking bifurcations like reversible pitchfork ones [LT12].
However, the structure of such bifurcations in many cases is not known, as happens for example in the case of symmetric cubic homoclinic tangencies. It can be deduced from [GGO17] that the first return map near a symmetric cubic homoclinic tangency must coincide in the main order with the cubic Hénon map either of form (1) or (2) which are reversible maps. When studying the problem of 1:4 resonance in these maps, we have shown that pitchfork bifurcations appear accompanying the resonance local bifurcation. These bifurcations should lead to the birth of a "sink-source" pair of periodic orbits in general reversible contexts.
We believe that these topics certainly deserve future devoted studies and we hope that the results presented here will contribute to facilitate them.
where P (y) = M 1 + M 2 y − δy 3 , being δ = 1 in case of map (1) and δ = −1 in case of map (2). The point Q is a parabolic 4-periodic orbit with trace 2 for C ± if the following conditions are satisfied: where D stands for the Jacobi matrix.
The last equations show that pairs (y, P (x)) and (x, P (y)) are the coordinates of points on the ellipse given by equation X 2 − XY + Y 2 = δM 2 , and that (19) has solutions for M 2 ≥ 0 in case of C − and for M 2 ≤ 0 in case of C + . Thus, we introduce the parametrization with parameters t 1 and t 2 along this ellipse in such a way that for some −π ≤ t 1 , t 2 ≤ π we have: where one can see that variables cos t 1 and cos t 2 satisfy the equation of a hyperbola 4X 2 + 16XY + 4Y 2 + 3 = 0. Thus, they can be parametrized in the following way: The natural conditions | cos t 1,2 | ≤ 1 are fulfilled only for t − ≤ |t| ≤ t + , where t ± = √ 3 ± 1 √ 2 . Substituting this parametrization into (24) and excluding t gives the curve L 4 4 in the case C − . Note that for this parametrization, we have M 2 = − 4t 2 t 4 − 4t 2 + 1 which takes only positive values for t − ≤ |t| ≤ t + , and this case does not provide any bifurcation curve for C + .
The other cases (analogous to Cases 1.2, 2.1 and 2.2 in the proof of Lemma 1) lead to cumbersome equations that we omit. B Local analysis of the 1:4 resonance with degeneracy B 03 = 0 Consider a one-parameter family F δ : R 2 → R 2 of area-preserving maps which admit the reversibility (x, y) → (y, x). Let us consider δ ∈ R, small enough, being a parameter that unfolds a 1:4 resonant fixed point. Without loss of generality, we assume F δ (0) = 0 and SpecDF 0 (0) = ±i. There exists a (formal) change of coordinates such that it reduces F δ to its Takens normal form N δ (see [Tak74,Bro90]) which commutes with the linearized map Λ 0 of F 0 at 0, that is, The normal form is easily expressed in terms of complex conjugated variables z = x + iy, z * = x − iy. The map Λ −1 0 N δ is near-the-identity and can be (formally) interpolated by a Hamiltonian flow, that is, where the Hamiltonian function H δ is Λ 0 -invariant and is the sum of resonant terms H δ (z, z * ) = j−k∈Γ h j,k z j (z * ) k , being Γ = {s ∈ Z, s = 0 (mod 4)} the set of resonant monomials.
We are interested in the analysis of the unfolding of the degenerate case (i.e. the situation with B 03 = 0). This means that b 1 = µ = 0 at the exact resonance (when δ = 0). Note that it is natural to adjust the angle variable so that ϕ 1 = 0. Taking into account that b 1 = δ one has H(I, ϕ) = δI + b 2 I 2 + b 3 I 3 + µI 2 cos(4ϕ) + BI 3 cos(4ϕ) + O(I 4 ). Our goal is to describe the bifurcations when δ, µ = 0 but small. Note that this degenerate 1:4 resonance case leads to a codimension two bifurcation problem.