This is a preprint of: “A Route to Chaos in the Boros–Moll Map”, Laura Gardini, Vı́ctor Mañosa, Iryna Sushko, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 29(4), 1930009:1–21, 2019. DOI: [10.1142/S021812741930009X] A route to chaos in the Boros–Moll map Laura Gardini Department of Economics, Society, Politics (DESP), University of Urbino, Italy E-mail: laura.gardini@uniurb.it Vı́ctor Mañosa Department of Mathematics, Universitat Politècnica de Catalunya, Spain E-mail: victor.manosa@upc.edu Iryna Sushko Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine arXiv:1804.06587v1 [nlin.CD] 18 Apr 2018 E-mail: sushko@imath.kiev.ua Abstract The Boros-Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to the convergence of them. In the paper, we study the dynamics of a one-parameter family of maps which unfolds the Boros-Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros-Moll map is a peculiar feature within the family. We relate this singularity with a specific property of the critical lines that occurs only for this special case. In particular, we explain how the unbounded chaotic region in the Boros-Moll map appears. Special attention is devoted to explain the main contact/homoclinic bifurcations that occur in the family. We also report some other bifurcation phenomena that appear in the considered unfolding. 1 Introduction In this paper we study some dynamical properties of the one-parameter family of planar maps Gh : R2 → R2 given by   h(x + y) + xy + 9 x+y+6 Gh (x, y) := , . (1) (x + y + 2)4/3 (x + y + 2)2/3 For h = 5, the map G5 appears as a subsystem on an uncoupled Landen transformation defined in R5 introduced by Boros and Moll in [3], see also [4]. Roughly speaking, given a definite integral depending on several parameters, a Landen transformation is a map on these parameters that leaves invariant the integral, see [20, p. 412] for a more precise definition, and [1] for a historical account of Landen transformations. Indeed, in the above references it is shown that the dynamical system defined by   5an + 5bn + an bn + 9 an + bn + 6   an+1 = , bn+1 = ,   (an + bn + 2) 4/3 (an + bn + 2)2/3      d n + e n + cn (bn + 3)cn + (an + 3)en + 2dn cn+1 = , dn+1 = , (2)   (an + b n + 2) 2/3 an + bn + 2       cn + en   en+1 = , (an + bn + 2)1/3 is a Landen transformation of the integral Z ∞ cx4 + dx2 + e I(a, b, c, d, e) = dx, 0 x6 + ax4 + bx2 + 1 which means that I(an+1 , bn+1 , cn+1 , dn+1 , en+1 ) = I(an , bn , cn , dn , en ). The map G5 is, therefore, the one associated with the subsystem of the denominator’s parameters of the integral. This map has been 1 investigated from a dynamical view point by Chamberland and Moll in [5], proving that the set of values of the parameters for which the integral associated to the 5-dimensional Landen transformation converges, coincides with the basin of attraction of one of the fixed points of the map G5 . The study of the global dynamics of map G5 is a challenging task. The known facts can be summarized as follows: There is a connected open set which is the basin of attraction of the fixed point (x, y) = (3, 3). Coexisting with this basin, it appears an invariant set with chaotic dynamics.1 The boundary between these sets is given by one of the two connected components of an algebraic curve (given in Eq. (6), below) that is part, together with its preimages, of the stable set of a saddle point that is also contained in the curve. In this work we consider the unfolding of G5 given by the one-parameter family (1) and we describe the route that leads to the appearance of the chaotic set for h = 5. In particular we analyze the maps Gh for h ≥ 5, say h ∈ [5, 6], pursuing the appearance of areas with chaotic dynamics and their bifurcations, and showing that there is a repelling chaotic set for h & 5. We relate the existence of the chaotic set for h = 5 with the merging of a part of the critical line with the mentioned algebraic curve. The paper is structured as follows. Section 2 is devoted to study the two main final bifurcations that appear in the family: the one that leads to the appearance of an unbounded chaotic repellor that exists for all the values of the parameter h ∈ (5, e h], where e h ' 5.032; and the final creation of the invariant unbounded chaotic set of G5 . These bifurcations are studied in Sections 2.4 and 2.5, respectively. The previous subsections are devoted to explain some general issues of the family of maps Gh . In Section 2.1 we introduce the notation concerning the so called critical lines, which is one of the main tools in our analysis, and we recall some of the main features of the phase portrait of the Boros-Moll map. In Section 2.2 we prove the existence of three fixed points for all the values of the parameter h, and we determine the value hN S at which a Neimark-Sacker bifurcation of one of the fixed points occurs, that starts the sequence of bifurcations studied in the paper (see Fig.2 and Fig.9). In Section 2.3 we show that the dynamics of the family of maps for values of h greater than hN S is regular. Section 3 is devoted to describe some sequences of bifurcations, the main ones that appear in the selected route to chaos in the Boros-Moll map, with special emphasis in the creation and contact bifurcations of chaotic areas, and homoclinic bifurcations related to snap-back repellors. An Appendix is included with some considerations regarding the partition of the phase space in zones related to the number of rank-1 preimages of the family of maps. 2 General properties of map Gh and main final bifurcations 2.1 Structure of the phase space of G5 As mentioned in the Introduction, our aim is to describe how the chaotic set existing for the map G5 is created. In Fig.1 we recall the structure in the phase space. The determinant of the Jacobian matrix of any map Gh is 1 (x + y − 6) (x − y) det(J(Gh )) = − 3 (3) 3 (x + y + 2) and therefore, for any fixed h ∈ R the critical curve LC−1 of a map Gh , is given by the two straight lines j LC−1 : y = x, and jj (4) LC−1 : x + y − 6 = 0. j Since LC−1 intersects the line (d) (d) : x + y + 2 = 0, (5) where the denominator of the components of the map vanishes, at the point (x, y) = (−1, −1), the image2 j j of LC−1 consists of two unbounded arcs LC = Gh (LC−1 ) = LC j ∪ LCrj which, in the case h = 5, are the 1 It cannot be called a chaotic attractor, since the closure of a generic trajectory looks like the full invariant set, and attracts no other point. 2 See [2] for details related to some properties of maps with a vanishing denominator. 2 Figure 1: Chaotic set (in black) and basin of attraction of P1 = (3, 3) (in red) for the Boros-Moll map G5 . In (a) in the phase plane (x, y). In (b) in the plane scaled as (arctan(x), arctan(y)). two connected components of the resolvent curve given by R(x, y) = −x2 y 2 + 4 x3 + 4 y 3 − 18 xy + 27 = 0. (6) j For h = 5, the image of the portion of straight line LC−1 from infinity (−∞) to (x, y) = (−1, −1) gives j LC , the arc bounding the chaotic area in Fig.1, and belongs to the stable set of the saddle fixed point j P2 ' (−4.20557, 3.95774), [12, Proposition 8]. The second branch of G5 (LC−1 ), which is LCrj , is the image j of the half line of LC−1 taken from (−1, −1) to infinity (+∞), and it is also shown in Fig.1a. It includes the fixed point P1 = (3, 3) which is an attracting node whose basin is also bounded by LC j . The third fixed point P3 ' (−5.30914, 0.83118) is an unstable focus belonging to the chaotic area. As noticed in [5], the fact that (x − y)2 R(G5 (x, y)) = R(x, y), (7) (x + y + 2)4 implies that the curve LC is invariant, i.e. Gn5 (LC) = LC for any n > 0. By using the fact that LC j is part of the stable set of the saddle point P2 we also have that the arc LC j is invariant. From this fact, and from the characterization of the basin of attraction of P1 given in [5], the (seemingly chaotic) area bounded by the curve LC j is also invariant. In fact, in [19] and [20, p. 442–444] it seems suggested that such invariant area is exactly a true chaotic area (with dense periodic points, which are all homoclinic), but this is still to be proved rigorously. In order to understand the appearance of the chaotic area we consider the 2-parametric family of maps Gg,h : R2 → R2 given by   gx + hy + xy + 9 x+y+6 Gg,h (x, y) := , , (x + y + 2)4/3 (x + y + 2)2/3 for which a numerical exploration evidences that unfolding the Boros-Moll map there are maps with different qualitative behavior. Indeed, one can check that in the parameter plane (g, h) depicted in Fig. 2 there appear several regions indicating qualitative differences. The different colors in this figure represent regions in the parameter space related to the existence of attracting cycles of different periods. The yellow region in Fig. 2 represents the values of the parameters at which the dynamics is regular, namely there exist two attracting fixed points, P1 in the positive side and P3 in the negative side, whose basins of attraction are separated by the stable set of a saddle fixed point P2 . Differently, the brown region represents the values of the parameters at which the only attracting set is the positive fixed point P1 . The existence of these fixed points, as a function of the parameters, is commented below. The white points denote either existence of a cycle 3 Figure 2: In (a) two-dimensional bifurcation diagram related to an i.c. close to the fixed point P3 . In (b) enlargement of the rectangle marked in (a). of period higher than 45 or the existence of a chaotic attractor. Since the positive fixed point P1 is always attracting, in all these regions the attractor P1 coexists with some other attracting set. To simplify our analysis, which is oriented to understand the dynamics of the Boros-Moll map, we consider g = h, which gives the one-parametric family (1), and we will focus mainly in the interval h ∈ [5, 6] of the parameter values. Thus, in the two-dimensional bifurcation diagram the straight line of equation g = h represents the path that we shall follow in order to describe some of the bifurcations occurring from a regular regime existing for h = 6 to the chaotic one existing for h = 5. In fact, changing the parameter h along the path in Fig.2(b), decreasing h, the fixed point P3 becomes an attracting focus and then it becomes a repelling focus via a Neimark-Sacker (NS from now on, for short) bifurcation, as commented below, giving rise to a sequence of bifurcations that we will summarize in its main features. Decreasing h we can comment the appearance/disappearance of attracting k-cycles for k > 1 of different periods or attracting closed invariant curves as well as the existence of chaotic attractors/repellors. 2.2 Fixed points of Gh and first Neimark-Sacker bifurcation We notice that for any h ∈ R, map Gh has exactly three different fixed points. In order to characterize the fixed points of map Gh , and following [5], we introduce the auxiliary variable m3 = x + y + 2, so that the fixed points are characterized by   c1 (x, y, m) := m3 − x − y − 2 = 0, c2 (x, y, m) := −xm4 + xy + hx + hy + 9 = 0, (8)  c3 (x, y, m) := −ym2 + x + y + 6 = 0. Isolating x and y from the first and third equations we get: m5 − m3 − 2 m2 − 4 m3 + 4 x= 2 and y = . (9) m m2 Substituting these expressions in the second equation we obtain that the variable m must satisfy c4 (m) = m11 − m9 − 3m8 − hm7 − 3m6 − 2m5 + (2h − 9) m4 + 8m3 + 8m2 + 16 = 0. 4 Observe that the term with highest degree in m of c4 (m; h) does not vanish for any h ∈ R, hence no root enters from infinity. On the other hand, the discriminant of c4 is given by ∆m (c4 (m; h)) = 927712935936 h14 − 29686813949952 h13 + 211982405861376 h12 + 2739735613145088 h11 + 118044964423729152 h10 − 1417849672260648960 h9 − 3159536320125075456 h8 − 89793115515877588992 h7 + 2232890826264743510016 h6 − 8668279665625562873856 h5 + 39121043805342246371328 h4 − 958082327734127689728000 h3 + 8262772575788649550970880 h2 − 28385116733510952733900800 h + 35844037650190215269056512. By using the Sturm method we obtain that ∆m (c4 (m)) > 0 for all h ∈ R, hence c4 (m) has no multiple roots, so that no roots of c4 (m) appear from C and there is no value of h such that the roots collide. Finally, using again the Sturm method we have that for h = 5, c4 (m) has only three distinct real roots. Hence for all h ∈ R the polynomial c4 (m) has three different real roots, which give rise to three different fixed points via Eq. (9). As already noticed in [5] and recalled above, G5 has the fixed points P1 , P2 and P3 . Since the location of the fixed points vary continuously with the parameter h, we will denote the fixed points of the maps Gh as P1 (h), P2 (h), and P3 (h), not indicating the dependence on h when it is not necessary. Since the fixed points P1 and P2 persist for h ∈ [5, 6] as an attracting node and a saddle, respectively, we are interested in the fixed point P3 which may be attracting or repelling, as we have seen in Fig. 2. So we restrict the scope of our analysis to the proof of a NS bifurcation that occurs at h ' 5.6105116077. To find the parameter value where the NS bifurcation takes place we consider the equation det(DGh (x, y)) = 1, obtaining 3 c5 (x, y, m) := 3 (x + y + 2) + (x + y − 6) (x − y) = 0. Using the polynomials c1 , c2 and c3 that characterize the fixed points, and taking successive resultants we get that if a NS bifurcation takes place then the parameter h must be a root of the polynomial PN S (h) = 582371795533824h11 − 7007247733358592h10 + 41393275596177408h9 − 557755817005154304h8 + 10296285821493313536h7 − 129039497006709473280h6 + 970090676944581427200h5 − 4072496100968654438400h4 + 8531720267711624773632h3 − 7149146865159609778176h2 + 6956611779457796014080h − 20284715289100324700160 By using the Sturm method we get that this polynomial has a unique real root in   808559935549815353 404279967774907677 , . 144115188075855872 72057594037927936 Hence hN S ' 5.6105116077302352. Now we check that the point P3 (h) is a hyperbolic stable focus for h & hN S , a hyperbolic unstable one for h . hN S and non-hyperbolic for h = hN S . In the next subsection we show that for h > hN S there is coexistence of the two attracting fixed points P1 and P3 and their basins of attraction are separated by the stable set of the saddle P2 . 2.3 Regular regime of the map Gh As remarked above, for large values of h the dynamics of a map Gh is quite regular. In fact, the map has two attracting fixed points, P1 and P3 . An example of the phase plane in such a situation is shown in Fig.3, for h = 6, where P1 ' (3.37349, 3.00069), P3 ' (−5.47727, 0.48573). In that figure the basin of attraction of the fixed point P1 is shown in red, B(P1 ), while that of P3 is shown in yellow, B(P3 ). The related basins of attraction are separated by the stable set of the saddle fixed point P2 . The main difference with respect to the case h = 5 is that the main branch of the stable set of the saddle P2 is not a critical curve and that the basin of the fixed point P1 is not connected (for example, there are open sets of points with both negative coordinates that have a trajectory convergent to P1 ). In Fig.3 it 5 Figure 3: Basins of attraction of map Gh at h = 6. j is shown the straight line LC−1 up to the point (−1, −1) and its image is the curve LC j , shown in black, which intersects the line (d) (see (5)) in one point and also intersects the stable set W s (P2 ) in one point Q (see the enlargement in Fig.3)3 . The region denoted by H0 in Fig.3, which is the portion of phase plane bounded above by the critical curve LC j and below by the stable set W s (P2 ) on the right side of the point Q, belonging to B(P1 ), is responsible of further preimages belonging to the basin B(P1 ). In the Appendix, it is reported how the critical lines split the plane in different zones named Zi where the sub-index i denotes the number of different rank-1 preimages. In what follows, we will use this notation. j In fact, the arc of LC j on the boundary of H0 has two merging rank-1 preimages in the half line LC−1 up s to the point Q−1 , while the arc of stable set W (P2 ) on the right side of the point Q has two distinct rank-1 j preimages issuing from the point Q−1 ∈ LC−1 , one at the right side of LC−1 (on the boundary of the set H−1,1 ) and the other at the left side (on the boundary of the set H−1,2 ), forming the boundary of the set H−1 = H−1,1 ∪ H−1,2 . Since H−1 belongs to the zone Z2 whose points have two distinct rank-1 preimages, we have two more preimages of H−1 and so on, ad infinitum. Thus, the basin B(P1 ) includes also the infinite sequence of preimages ∪n≥1 G−nh (H−1 ) which consists of infinitely many portions of the plane (in fact, due to the properties of the regions Zk described in Appendix, even if a preimage may happen to belong to Z0 (so having no further preimages), another one above the diagonal exists, and thus it belongs to Zk with k ≥ 2. 2.4 Homoclinic bifurcation of P2 The existence of non-connected regions of the phase plane belonging to the basin B(P1 ) are relevant to prove that close to the value h = 5 a chaotic attractor becomes a chaotic repellor which persists for h ∈ (5, e h) e for a certain value h, although almost all the points of the phase plane have a trajectory convergent to the fixed point P1 . Similarly, a chaotic repellor exists for h ∈ (5 − ε, 5) with ε & 0, and almost all the points of the plane have a trajectory convergent to the fixed point P1 . Thus, the occurrence of an invariant set of positive measure (which seems chaotic) for h = 5 as shown in Fig.1 is a very peculiar phenomenon in the framework of dynamical systems theory. The peculiarity may be related to the basin B(P1 ) which is an open simply connected set only for h = 5, and thus related to a peculiar behavior of the critical curves. In fact, for h ∈ (5 − ε, e h)\ {5} the basin B(P1 ) is an open set but not simply connected (while for h ≥ e h it is non connected). We show that there exists a particular value e h ' 5.032 such that for h > e h, there exists an invariant unbounded absorbing region A, which is bounded by segments of images of finite rank of a generating segment j g belonging to the critical curve LC−1 for x < −1 , that is, below the line (d) of vanishing denominator. For 3 To simplify the notation, in the figures we write only LC in place of LC j , since these are the only arcs of interest for the dynamics of the map, so there cannot be any confusion. 6 h e h (resp. h < e h) the saddle fixed point P2 is not homoclinic (resp. is homoclinic). 2.4.1 Situation before the contact and qualitative description of the contact bifurcation For h > e h (when the unbounded absorbing region A exists) the stable set of the saddle separates two basins of attraction: the basin B(P1 ) of the attracting fixed point P1 and the basin B(A) of the absorbing area A (inside which several attracting sets may exist). That is, the frontier of these two basins is that stable set: W s (P2 ) = ∂B(P1 ) = ∂B(A). Figure 4: Phase portrait of map G5.032 . The white area belongs to the basin of the invariant chaotic area. The red area belongs to the basin B(P1 ): In green, the images of the generating segment g of the critical line j LC−1 . The region H−1 , which is part of the basin B(P1 ), is close to contact with the chaotic area bounded the arcs given by the images of the generating segment. In Fig.4 we illustrate the unbounded chaotic area at h = 5.032, whose boundary consists of a finite j number of images of the generating segment g which is the arc of LC−1 belonging to the invariant area itself j (see [18]). It can be seen that the boundary is very close to contact the point Q−1 ∈ LC−1 on the boundary of the set H−1 which belongs to the basin B(P1 ). Recall that this boundary belongs to the stable set of the saddle, W s (P2 ), while the left branch of the unstable set of the saddle, W u,l (P2 ), entering the chaotic area, is dense in that area (differently, the other branch W u,r (P2 ) issuing from the saddle P2 is a branch connecting the saddle with the attracting fixed point P1 ). When such a contact occurs, say at h = e h, that is, when the boundary of H−1 has a contact with the critical curve bounding the chaotic area, then this contact leads (for h 5 (as shown below). Notice that also the unstable set of the saddle P2 for 5 1 have points which cross through H−1 and H0 and then converge to the attracting fixed point P1 . 2.5 Final bifurcation for h = 5. Role of the critical lines At h = 5, the scene described in Section 2.4 is no longer possible because the strip H0 reduces to the arc LC j which merges with the main branch of the stable set of the saddle P2 , and thus the portion H−1 reduces j to an arc of LC−1 . The portion of plane bounded by the arc of curve LC j (belonging to the resolvent curve given in Eq.(6)) is invariant: it is the situation shown in Fig.1. Notice that the saddle P2 is homoclinic, j since the whole segment LC−1 and all its preimages belong to the stable set of P2 and intersections between the stable and unstable sets are infinitely many, but only on the left side of the arc LC j . This means that, 8 differently from the case for 5 < h < e h, even if the saddle P2 is homoclinic no one point belonging to the unstable set W u,l (P2 ) can have a trajectory convergent to P1 (since no one point can be mapped to the right j side of LC j ) and similarly no point on the images of the critical line LC−1 , of any rank, can converge to P1 . e So we have seen that for 5 ≤ h < h the saddle P2 is homoclinic only on the left side. The characteristic j Figure 6: Images of an arc of LC−1 for x ≤ −1 at h = 5.01, h = 5 and h = 4.99. commented above associated with the Boros-Moll map at h = 5 may be related to the critical curves as follows. For 5 < h < e h the image of the critical curve LC−1j for x ≤ −1 is an unbounded branch LC j which intersects the line of vanishing denominator (d) in two points, so that LC1j consists in three unbounded components. In Fig.6 we show a portion of phase plane, an enlargement close to (−1, −1) at h = 5.01, at this value of the parameter almost all the points have a trajectory converging to P1 ' (3.00375, 3.00000). j At h = 5 the image of LC−1 for x ≤ −1 is the branch LC j which merges with the main arc of the stable set of P2 , which is tangent (from above) to (d) in the point (−1, −1), and it is invariant, so that LC j = LCnj j for any n ≥ 1. For h < 5 the image of LC−1 for x ≤ −1 is the branch LC j completely above (d), and thus its points converge to the attracting fixed point P1 . Clearly, all the points in the strip H0 between the main arc of the stable set of P2 and the arc LC j are also converging to the attracting fixed point P1 , and so do j the points of its preimage belonging to a strip including LC−1 as well all the further preimages of any rank. For instance, at h = 4.99, a chaotic repellor still exists (since the fixed points P2 and P3 are homoclinic, as well as infinitely many repelling cycles exist and are homoclinic), and almost all the points are converging to P1 ' (2.99625, 3.00000). For h < 5 not only the critical curves are on the right side of the main branch of the stable set of P2 , but also the unstable set W u,l (P2 ) has again arcs going to the right side and converging to P1 . j In Fig.7 we illustrate an arc of LC−1 for x ≤ −1, LC j in red and its first images, at h = 5.01, which are all on the left side of the curve LC , while at h = 4.99 the images are all on the right side to the LC j j (clearly for h = 5 this branch of LC j is invariant). The above study evidences that the arc LC j is intersecting the main arc of the stable set W s (P2 ) in one point Q as long as h > 5, they are merging when h = 5, and LC j is above that stable set for h < 5. 3 Route to chaos In order to describe the dynamics of map Gh (leading to the unbounded chaotic set shown in Fig.4) we account some of the phenomena that appear when the parameter h is decreased to h = 5 after the N S bifurcation of the fixed point P3 , that is, decreasing h from hN S ' 5.6105. The N S bifurcation is of supercritical type, as can be seen from the attracting closed curve Γ which appears surrounding the repelling focus P3 , an example is illustrated in Fig.8a. The restriction of the map to the invariant curve in general leads either to an attracting cycle (when the rotation number is rational and the closed curve consists of a saddle-node connection) or to quasiperiodic orbits, dense in the closed curve (when the rotation number is irrational). 9 Figure 7: Critical curves LCkj for 1 ≤ k ≤ 4 at h = 5.01 in (a), at h = 4.99 in (b). It is worth noting that as long as the curve Γ is in a region without intersections with the critical curve j LC−1 then all the points inside the area bounded by Γ cannot be mapped outside the area (i.e. the area bounded by Γ is invariant). At the same time, the points external to that area which are attracted from Γ have the trajectory completely outside Γ (i.e. the points are approaching the closed curve from outside). The dynamics in the phase plane change, as well as the global shape of the curve Γ, after a crossing of the j critical curve LC−1 . j Figure 8: Bifurcation of the closed curve Γ. In (a) h = 5.6, smooth curve not intersecting LC−1 . In (b) j h = 5.44, Γ intersects LC−1 . j Decreasing h a contact between Γ and LC−1 occurs at h ' 5.506, after which there is a portion of Γ j which is below the critical curve and thus the portion of area bounded by Γ and crossing LC−1 is folded j j on LC so that Γ becomes tangent to LC in two points (the rank-1 images of the two intersection points j Γ ∩ LC−1 ). And clearly the contacts between Γ and the critical curves LCnj persist for any n. In Fig.8b it is well evident the tangency between Γ and the critical curve LC1j . This mechanism is a peculiar phenomenon related to noninvertibility (i.e. it cannot occur in maps with a unique inverse). It is related to the folding of the phase plane due to the critical curves, and breaks the invariance of the area bounded by the closed curve Γ. In fact, all the points in the area bounded by Γ and j LC−1 are mapped outside Γ, between Γ and LC j , and its further images between Γ and LC1j , and so on (see 10 Figure 9: One-dimensional bifurcations diagrams. Figure 10: One-dimensional bifurcations diagrams. Enlargements of the rectangles shown in color in Fig.9a. the related properties in [8] and [18]). In particular, this fact allows the existence of cycles of the map with periodic points both inside and outside the area bounded by the closed curve. We shall see an example in the next subsection. Let us first illustrate a sequence of attractors and their bifurcations via one-dimensional bifurcation diagrams as a function of the parameter h. Since we know that unbounded attracting sets exist, in Fig.9 we show the scaled variable (as we have done in Fig.1b) to better emphasize when an attractor is unbounded. In Fig.9 it is evident the range in which decreasing h from hN S the closed invariant curve Γ increases in size, since approaching the straight line (d) the shape becomes wider. In the same figure we see that there are two large windows associated with cycles and interesting dynamics, which are enlarged in Fig.10 and commented in the next subsections. 3.1 Bifurcations occurring in the first enlargement. As noticed above, the oscillations on the closed curve Γ become pronounced as h is decreased. Moreover, other attracting sets appear by saddle-node bifurcation. An example is shown in Fig.11. At h =5.4379 besides P1 only the closed curve Γ is attracting, while at h =5.4378 we can see that a pair of 11-cycles have 11 appeared. One of them is attracting, while the other is a 11-cycle saddle whose stable set bounds the basin Figure 11: Coexistence of the closed curve Γ and an attracting 11-cycle at h =5.4378. of attraction of the attracting 11-cycle, separating its basin from the basin of Γ. In Fig.11a we show in red (resp. yellow) the basin of P1 (resp. the closed curve Γ) and we show with different colors the basins of the 11 fixed points of the 11-th iterate of the map Gh , to emphasize the shape of the basin of the attracting 11-cycle. In Fig.11b and related enlargement it can be seen that some points of the attracting 11-cycle are outside the closed curve Γ and part are inside. One more peculiarity related to the immediate basins of the 11 fixed points of map G11h is that the stable set of the 11 saddles (related to the 11-cycle saddle) leads to a closed curve, which is possible only in noninvertible maps, due to the crossing of the critical curves of the map (other examples and explanation of the mechanism can be found in [18]). From the same Fig.11 we can Figure 12: In (a) h =5.437, attracting 11-cycle and its basin of attraction for the 11-th iterate of the map. In (b) h =5.412, coexistence between the attracting 11-cycle and the attracting 4-cycle (in yellow and azure their respective basin of attraction). see that the curve Γ is quite close to the stable set of the saddle 11-cycle on the boundary of the basin of attraction. A contact between these two invariant sets may lead to the disappearance of the closed curve Γ, and its transition to some repelling set. In any case, at h =5.437 besides P1 only the 11-cycle is attracting, and the stable set of the saddle 11-cycle has now a quite different shape, confirming the disappearance of the 12 closed invariant curve, as shown in Fig.12a, where the crossing of LC j of the stable set of the saddle leads to the disconnected components evidenced in Fig.12a for the 11-th iterate of the map. Decreasing h, one more saddle-node bifurcation occurs, now giving rise to a 4-cycle. As can be seen in Fig.10b this occurs at h ' 5.4125 leading to another coexistence of attracting sets on the left side of the main branch of the stable set of P2 . In Fig.12b at h =5.412 besides P1 (with basin in red) the basins of attraction of the 11-cycle (in yellow) and of the 4-cycle (in azure) are shown. The basins’ structure of the two coexisting attractors is clearly fractal, which means that the saddle 11-cycle on the immediate basin of the 11-cycle is now homoclinic on one side (that external to the immediate basin which is clearly visible in yellow via bounded closed pieces). As evidenced in Fig.10b the attracting 4-cycle undergoes a supercritical N S Figure 13: In (a) h = 5.41, N S bifurcation of the 4-cycle. In (b) h = 5.40676, chaotic piece showing that the 4-cycle is a snap-back repellor. bifurcation at h ' 5.411 which leads to 4-cyclic closed invariant curves (see an example in the enlargement of Fig.13a at h = 5.41). The closed curves undergo bifurcations to annular chaotic regions and to 4-cyclic chaotic pieces (see the portion of phase plane in Fig.13b at h = 5.40676). This sequence, as described in [9] (see also [18]), leads to the homoclinic bifurcation of the expanding 4-cycle, repelling focus, which becomes a snap-back repellor (following Marotto [17]). The chaotic piece of the 4-cyclic set shown in Fig.13b is very close to the boundary of its basin of attraction, and a contact with the basin leads to the disappearance of the 4-cyclic chaotic pieces, which become a chaotic repellor. Already at h = 5.40675 a chaotic transient can be observed and the only attractor left on the left side of the main arc of the stable set W s (P2 ) is the 11-cycle (with a chaotic repellor on its basin boundary). The attracting 11-cycle disappears with the companion saddle cycle in a (reversed) saddle-node bifurca- tion. After this saddle-node bifurcation we cannot observe a simple attracting cycle, but a closed invariant absorbing area exists, inside which the dynamics seem chaotic. At h = 5.4 the area shown in Fig.14a in- j tersects the critical line LC−1 in two segments, forming together the generating segments whose images bound the absorbing area via segments of critical curves. The area is unbounded as in fact the image of the shortest segment (evidenced in dark green in Fig.14a) is mapped in an arc of LC j which crosses the line (d) of vanishing denominator (evidenced with a dark green arrow in Fig.14a), thus its image consists in two unbounded arcs of LC1j on the boundary of the area. In Fig.14b it is shown a wider portion of phase space and in green the arcs of critical curves on the boundary of the invariant area, while in Fig.14c it is represented the unbounded attracting area in the whole plane, related to the scaled variables. 3.2 Bifurcations occurring in the second enlargement. In the previous subsection we have described a few bifurcations which occur decreasing h and evidenced in Fig.10b, leading to an unbounded invariant absorbing area, with complex dynamics. Clearly, inside this invariant area it is possible that stable cycles of high period exist, not detectable in our numerical simulations. 13 Figure 14: Unbounded absorbing area, with chaotic dynamics, at h = 5.4. In (a) the generating segments j on LC−1 are evidenced. In (b) the attracting set is shown in a larger portion of the phase plane. In (c) the attracting set is shown the plane scaled as (arctan(x), arctan(y)). However, cycles with homoclinic orbits also exist, both saddles and expanding cycles, as the 4-cycle described above which is a snap-back repellor (in fact, although an invariant 4-cyclic attracting chaotic area no longer exists, as evidenced by the complex dynamics, the homoclinic 4-cycle still exists). From Fig.9 we can see that the unbounded invariant absorbing area persists for a wide interval of values for the parameter h, and the appearance of an attracting 4-cycle leads to another window with interesting dynamics, enlarged in Fig.10a. The pair of 4-cycles appears by saddle-node bifurcation inside the invariant absorbing area. In Fig.15a (at h = 5.128) we show the unbounded invariant area confined by portions of critical curves, j and the generating segment is now a wider connected segment of LC−1 , in red are indicated the points where a pair of 4-cycles is going to appear. In Fig.15b (at h = 5.127) an attracting 4-cycle exists inside the invariant area, and its basin of attraction is evidenced with respect to the 4 points (in black) which are fixed for the 4-th iterate of the map Gh . The complex structure of the basins evidences a chaotic repellor on the basin boundary, while the immediate basins are (as in previous examples) bounded by the stable set of the saddle 4-cycle forming closed curves, denoting intersection with the critical curves. From Fig.10a we can see that Figure 15: Appearance of an attracting 4-cycle. In (a) at h = 5.128 the cycles do not exsist. In (b) at h = 5.127 a pair of 4-cycles exist inside the unbounded invariant area. the 4-cycle undergoes a N S bifurcation (at h ' 5.11), leading to 4-cyclic closed invariant curves which, from 14 j smooth (for h close to the N S bifurcation) become nonsmooth after the crossing of the critical line LC−1 . We also see in Fig.10a that an attracting 12-cycle appears, which is related to the closed curves. In fact, at h ' 5.10299 the attracting set is no longer a smooth curve, since it is an invariant set with self intersections, as reported in Fig.16 (for the mechanism leading to the formation of loops and self intersections in closed curves see [8] and [18]). Figure 16: Enlargement showing one piece of the 4-cyclic invariant set at h = 5.10299 and appearance of the 12-cycle at h = 5.10298 represented by the red point in the further enlargement. From Fig.10a we also see that, in its turn, the attracting 12-cycle undergoes a N S bifurcation, and after a sequence of bifurcations 12-cyclic chaotic pieces are created (see Fig.17a at h = 5.097). Other bifurcations are illustrated in Fig.17b at h = 5.096 (enlarged portion of phase plane, evidencing one piece of the 12-cyclic set). This, in its turn, is followed by a contact bifurcation at h = 5.095, leading to an expansion of the invariant absorbing area with chaotic dynamics which has now 4-cyclic chaotic pieces, one of them shown in Fig.17c. This also denotes that the repelling 4-cycle inside that area is now homoclinic, i.e. a snap-back repellor. It is worth noting that all these attracting sets belong to the unbounded invariant absorbing area, gener- ated by the segment as shown in Fig.15a. However, inside this region, we can find other invariant absorbing j areas as the 4-cyclic areas which are all bounded by few images of the generating segment intersecting LC−1 and shown in Fig.17c. The 4-cyclic chaotic areas are shown at a lower value of h in Fig.18a, together with Figure 17: Enlargements showing a part of a chaotic attractor. Three pieces of an attractor of 12 connected components for h = 5.097 in (a); six pieces belonging to an invariant set of 24 connected components for h = 5.096 in (b); and one of 4 connected components for h = 5.095 in (c). 15 the related four basins of attraction for the 4-th iterate of the map. Besides a fractal structure with a chaotic repellor on the boundary, this figure shows that the areas are very close to the boundary of their immediate basins, denoting that the parameter is very close to a contact bifurcation value, leading to the disappearance of the 4-cyclic areas, and the appearance of a unique unbounded absorbing area inside which the dynamics seem chaotic, see Fig.18b. Figure 18: Contact bifurcation leading to expansion of the invariant area. In (a) h = 5.09397 4-cyclic absorbing areas exist. In (b) h = 5.09396 the attracting set is an unbounded absorbing area. The unbounded absorbing area previously obtained, persists for lower values of h, as shown in Fig.9, up to the bifurcation described in Sec.2.4.1 occurring at h = e h. Decreasing h up to the final bifurcation, it seems that the dynamics inside the unbounded absorbing area become more complex. That is, more cycles appear and undergo homoclinic bifurcations. For example the fixed point P3 which is a repelling focus may become a snap-back repellor. From Fig.18b it is not clear whether it is homoclinic or not. However, we have evidence of this, at the same value h = 5.09396 used in Fig.18b. ∗ Figure 19: Images G21h∗ (γ) at h = 5.09396 of different arcs γ close to P3 corresponding to different values of the radius s. In (a) two different arcs are shown. In (b) a third arc is added to the two shown in (a). The values of s are given in the text. In fact, let us consider the fixed point P3 and its rank-1 preimage different from itself, P3−1 , which is the point symmetric to P3 with respect to the main diagonal. In order to show that P3 is a snap-back repellor 16 it is enough to show that there is a point q0 in a neighborhood of P3 which is mapped into P3 in a finite −1 number of iterations, say Gm h (q0 ) = P3 , and such that the local inverse which satisfies Gh (P3 ) = P3 (which −n is a contraction), leads to limn→∞ Gh (q0 ) = P3 . Let us consider an arc γ in such a neighborhood of P3 (i.e. such that the preimages of its points by the local inverse G−1 h converge to P3 ), given by a half circle of equation (x − x(P3 ))2 + (y − y(P3 ))2 = s2 with the radius s sufficiently small. In Fig.19a we show such an arc γ in black corresponding to the radius ∗ s = 0.3, and the image G21 h∗ (γ) (with h = 5.09396 as used in Fig.18b) in black which is close to the preimage P3−1 , below it. Differently an arc γ in red corresponding to the radius s = 0.25, has the image G21h∗ (γ) in red which is close to the preimage P3−1 , but above it. This means that there exists a value s∗ ∈ (0.25, 0.3) such −1 that the related arc γ ∗ has the image G21 ∗ h∗ (γ ) which crosses through P3 . This implies the existence of a ∗ 22 point q0 ∈ γ such that Gh∗ (q0 ) = P3 . As an example, considering s = 0.29 in Fig.19b we show in pink the related arc γ and, also in pink, the −1 image G21 h∗ (γ) which is above the black arc and closer to P3 . A similar construction can be used also at any smaller value of h (in particular also at h = 5 or smaller), showing that P3 persists in being a snap-back repellor. 4 Conclusions We have considered a one-parameter family of maps which unfolds a Landen-type map obtained by Boros and Moll, and we have described the main bifurcations that appear in the route towards the parameter value that corresponds to this map. We have focused on those homoclinic and contact bifurcations that can give information about the unbounded invariant chaotic region that appear in the Boros-Moll map. Our study confirms that the dynamics of this map is singular in the considered family, and we give evidences that this singularity is related with a specific property of the critical lines: the fact that one of the critical lines merges with one separatrix of the saddle point only for the parameter value corresponding with the map introduced by Boros and Moll. Acknowledgements V.M. is supported by Ministry of Economy, Industry and Competitiveness–State Research Agency of the Spanish Government through grant DPI2016-77407-P (MINECO/AEI/FEDER, UE) and acknowledges the group research recognition 2017-SGR-388 from AGAUR, Generalitat de Catalunya. He also acknowledges Prof. A. Gasull for helpful discussions. V.M. and I.S. as visiting professor are grateful to the University of Urbino Carlo Bo for the hospitality during a stage at the University in February 2018. References [1] G. Almkvist, B. Gert. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies diary. Amer. Math. Monthly 95 (1988), 585–608. [2] G.I. Bischi, L. Gardini, C. Mira. Maps with denominator. Part 1: some generic properties. Int. J. Bifurcation & Chaos 9 (1999), 119–153. [3] G. Boros, V. H. Moll. A rational Landen transformation. The case of degree six. In M. Knopp et al. eds, “Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis”, Contemporary Mathematics 251. AMS, Providence RI 2000, 83–91. [4] G. Boros, V. H. Moll. Landen transformations and the integration of rational functions. Mathematics of Computation 71 (2001), 649–668. 17 [5] M. Chamberland, V. H. Moll. Dynamics of the degree six Landen transformation. Discrete and Contin- uous Dynamical Systems A 15 (2006), 905–919. [6] R. Devaney. “An introduction to chaotic dynamical systems”, 2nd ed. Westview Press, Boulder CO 2003. [7] D. Fournier-Prunaret, C. Mira, L. Gardini. Some contact bifurcations in two-dimensional examples. In “Grazer Mathematishe Berichte” (special issue Proceedings ECIT94) (1997), 77–96. [8] C.F. Frouzakis, L. Gardini, Y.G. Kevrekidis, G. Millerioux, C. Mira. On some properties of invariant sets of two-dimensional noninvertible maps. Int. J. Bifurcation & Chaos 7 (1997), 1167–1194. [9] L. Gardini. Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding periodic points Nonlinear Analysis: Theory, Methods & Applications 23 (1994), 1039–1089. [10] L. Gardini. Homoclinic orbits of saddles in two-dimensional endomorphisms In W. Förg-Rob et al. (editors) “Iteration Theory. Proceedings of the European Conference”. World Scientific, Singapore 1996. [11] L. Gardini, I. Sushko, V. Avrutin, M. Schanz. Critical homoclinic orbits lead to snap-back repellers. Chaos, Solitons & Fractals 44 (2011), 433–449. [12] A. Gasull, M. Llorens, V. Mañosa. Periodic points of a Landen transformation. Preprint, arXiv:1801.04195 [math.DS] [13] C. Grebogi, E. Ott, J. A. Yorke. Crisis: sudden changes in chaotic attractors and transient chaos. Physica D 7 (1983), 181–200 [14] C. Grebogi, E. Ott, F. Romeiras, J. A. Yorke. Critical exponents for crisis-induced intermittency. Physical Review A 36, 11 (1987), 5365–5380. [15] T. Li, J.A. Yorke. Period three implies chaos. Amer. Math. Month. 82 (1975) 985–992. [16] Y. Maistrenko, I. Sushko, L. Gardini. About two mechanisms of reunion of chaotic attractors. Chaos, Solitons & Fractals 9 (1998), 1373–1390. [17] F. R. Marotto. Snap-back repellers imply chaos in Rn . J. Math. Anal. Appl. 63 (1978) 199–223. [18] C. Mira, L. Gardini, A. Barugola, J.C. Cathala. “Chaotic Dynamics in Two-Dimensional Noninvertible Maps”, World Scientific, Nonlinear Sciences, Series A, Singapore 1996. [19] V. H. Moll. The evaluation of integrals: a personal story. Notices Amer. Math. Soc. 49 (2002), 311–317 [20] V. H. Moll. “Numbers and functions. From a classical-experimental mathematician’s point of view”. Student Mathematical Library, 65. American Mathematical Society, Providence RI 2012. 5 Appendix The map Gh has a non unique inverse, and the number of distinct rank-1 preimages of a point (x, y) differs depending on the region of the phase plane which the point (x, y) belongs to. These regions, or zones Zi (related to i distinct preimages of rank-1) following the notation introduced in [18], are usually bounded by critical curves LC of the map, which are the images of curves of merging preimages, denoted LC−1 , which belong to the set where the Jacobian determinant of the map vanishes. j In our case, the Jacobian determinant vanishes either at the points LC−1 (defined by y = x) or the points jj in LC−1 (defined by x + y − 6 = 0), see Eqs. (3) and (4). j Since LC−1 intersects (d), its image LC j consists of two unbounded curves of the plane, LC j ∪LCrj (notice that if an arc γ crosses the line (d), where the denominator vanishes in a point in which the numerators of the components of the map do not vanish, which is always the case for the considered map, then its image Gh (γ) consists of two unbounded arcs). 18 Figure 20: Critical curves for the map Gh at h = 5.5. In (a) LC−1 ; in (b) LC. jj jj The curve LC−1 does not intersects (d), its image LC jj = Gh (LC−1 ) consists of a single arc, and it is easy to see that it is a segment of straight line y = 3 (up to x = 3). Indeed, by setting xn = t and yn = −t+6 jj (a point on LC−1 ), and taking (xn+1 , yn+1 ) = Gh (xn , yn ) we get xn+1 = (6t − t2 + 39)/16 (which takes its jj maximum value in x = 3) and yn+1 = 3, that is, the image of the line LC−1 is folded on the half-line y = 3, as t increases from −∞ to +∞ and x = t = 3 is the folding point. In Fig.20 the critical curves for h = 5.5 are displayed. In summary, the components of the critical curve LC = LC j ∪ LCrj ∪ LC jj split the plane in 5 distinct open regions. We can figure out the number of preimages by considering a sort of “folding of the plane”, as qualitatively shown in Fig.21. By computing how many preimages have a particular point in each of these regions, we obtain the different zones Zi . In Fig.21 they are displayed for the particular case h = 5.5. Notice also that a point belonging to LC has two merging rank-1 preimages in a point belonging to LC−1 . For each point in a zone Zi it is also possible to show where the preimages are located (on opposite side with respect to the related branch of LC−1 ). We report how to obtain the number of preimages in each of the zones limited by the curves LC. We fix a point (u, v) and we set Gh (x, y) = (u, v). By introducing the notation z = x + y + 2 and from the equation of the second component of the map, we have that vz 2/3 = z + 4, hence v 3 = (z + 4)3 /z 2 . (10) By studying the function f (z) = (z + 4)3 /z 2 we easily get that for v > 3 there are three solutions of Eq. (10) (one negative and two positive); for v = 3 there is one negative solution and two merging solutions given by z = 8; and for −∞ < v < 3 there is only one negative solution. Introducing the notation P = xy and S = x + y, and by using the equation of the first component, we obtain that given (u, v) and a value of z satisfying Eq. (10), there is a unique value for P and S given by P = u(z 4 )1/3 − h(z − 2) − 9 and S = z − 2 . Since any preimage (x, y) associated to (u, v) satisfies y 2 − Sy + P = 0 and x = S − y, we have √ √ S ± S 2 − 4P S ∓ S 2 − 4P x= and y = . 2 2 The final number of preimages depends on the sign of S 2 − 4P . For the original Boros-Moll map, the case h = 5, the curve LC j belongs to the curve R(x, y) = −x2 y 2 + 4 x3 + 4 y 3 − 18 xy + 27 = 0, it is part of the stable set of the saddle P2 , and the zones Zi are displayed in Figure 22. 19 Figure 21: Qualitative shape of the foliation of the plane. Critical curves and Zones Zi for the map Gh at h = 5.5. Figure 22: Critical curves and Zones for the Boros-Moll map (h = 5). 20