A route to chaos in the Boros-Moll map
AA route to chaos in the Boros–Moll map
Laura Gardini
Department of Economics, Society, Politics (DESP), University of Urbino, ItalyE-mail: [email protected]
V´ıctor Ma˜nosa
Department of Mathematics, Universitat Polit`ecnica de Catalunya, SpainE-mail: [email protected]
Iryna Sushko
Institute of Mathematics, National Academy of Sciences of Ukraine, UkraineE-mail: [email protected]
Abstract
The Boros-Moll map appears as a subsystem of a Landen transformation associated to certain rationalintegrals and its dynamics is related to the convergence of them. In the paper, we study the dynamicsof a one-parameter family of maps which unfolds the Boros-Moll one, showing that the existence of anunbounded invariant chaotic region in the Boros-Moll map is a peculiar feature within the family. We relatethis 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 devotedto explain the main contact/homoclinic bifurcations that occur in the family. We also report some otherbifurcation phenomena that appear in the considered unfolding.
In this paper we study some dynamical properties of the one-parameter family of planar maps G h : R → R given by G h ( x, y ) := (cid:18) h ( x + y ) + xy + 9( x + y + 2) / , x + y + 6( x + y + 2) / (cid:19) . (1)For h = 5, the map G appears as a subsystem on an uncoupled Landen transformation defined in R introduced by Boros and Moll in [3], see also [4]. Roughly speaking, given a definite integral depending onseveral 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 a n +1 = 5 a n + 5 b n + a n b n + 9( a n + b n + 2) / , b n +1 = a n + b n + 6( a n + b n + 2) / ,c n +1 = d n + e n + c n ( a n + b n + 2) / , d n +1 = ( b n + 3) c n + ( a n + 3) e n + 2 d n a n + b n + 2 ,e n +1 = c n + e n ( a n + b n + 2) / , (2)is a Landen transformation of the integral I ( a, b, c, d, e ) = (cid:90) ∞ cx + dx + ex + ax + bx + 1 d x, which means that I ( a n +1 , b n +1 , c n +1 , d n +1 , e n +1 ) = I ( a n , b n , c n , d n , e n ). The map G is, therefore, theone associated with the subsystem of the denominator’s parameters of the integral. This map has been1 a r X i v : . [ n li n . C D ] A p r nvestigated from a dynamical view point by Chamberland and Moll in [5], proving that the set of valuesof 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 G .The study of the global dynamics of map G is a challenging task. The known facts can be summarizedas follows: There is a connected open set which is the basin of attraction of the fixed point ( x, y ) = (3 , The boundary between thesesets is given by one of the two connected components of an algebraic curve (given in Eq. (6), below) that ispart, 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 G given by the one-parameter family (1) and we describe theroute that leads to the appearance of the chaotic set for h = 5. In particular we analyze the maps G h for h ≥
5, say h ∈ [5 , h (cid:38)
5. We relate the existence of the chaotic set for h = 5with 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 thatappear in the family: the one that leads to the appearance of an unbounded chaotic repellor that exists forall the values of the parameter h ∈ (5 , (cid:101) h ], where (cid:101) h (cid:39) . G . 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 G h . In Section2.1 we introduce the notation concerning the so called critical lines , which is one of the main tools in ouranalysis, and we recall some of the main features of the phase portrait of the Boros-Moll map. In Section2.2 we prove the existence of three fixed points for all the values of the parameter h , and we determine thevalue h NS at which a Neimark-Sacker bifurcation of one of the fixed points occurs, that starts the sequenceof bifurcations studied in the paper (see Fig.2 and Fig.9). In Section 2.3 we show that the dynamics of thefamily of maps for values of h greater than h NS is regular.Section 3 is devoted to describe some sequences of bifurcations, the main ones that appear in the selectedroute to chaos in the Boros-Moll map, with special emphasis in the creation and contact bifurcations ofchaotic areas, and homoclinic bifurcations related to snap-back repellors.An Appendix is included with some considerations regarding the partition of the phase space in zonesrelated to the number of rank-1 preimages of the family of maps. G h and main final bifurcations G As mentioned in the Introduction, our aim is to describe how the chaotic set existing for the map G iscreated. In Fig.1 we recall the structure in the phase space.The determinant of the Jacobian matrix of any map G h isdet( J ( G h )) = −
13 ( x + y −
6) ( x − y )( x + y + 2) (3)and therefore, for any fixed h ∈ R the critical curve LC − of a map G h , is given by the two straight lines LC j − : y = x, and LC jj − : x + y − . (4)Since LC j − 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 ) = ( − , − of LC j − consists of two unbounded arcs LC = G h ( LC j − ) = LC j ∪ LC jr which, in the case h = 5, are the It cannot be called a chaotic attractor, since the closure of a generic trajectory looks like the full invariant set, and attractsno other point. See [2] for details related to some properties of maps with a vanishing denominator. P = (3 ,
3) (in red) for the Boros-Moll map G .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 ) = − x y + 4 x + 4 y − xy + 27 = 0 . (6)For h = 5, the image of the portion of straight line LC j − from infinity ( −∞ ) to ( x, y ) = ( − , −
1) gives LC j , the arc bounding the chaotic area in Fig.1, and belongs to the stable set of the saddle fixed point P (cid:39) ( − . , . G ( LC j − ), which is LC jr , is the imageof the half line of LC j − taken from ( − , −
1) to infinity (+ ∞ ), and it is also shown in Fig.1a. It includesthe fixed point P = (3 ,
3) which is an attracting node whose basin is also bounded by LC j . The third fixedpoint P (cid:39) ( − . , . R ( G ( x, y )) = ( x − y ) ( x + y + 2) R ( x, y ) , (7)implies that the curve LC is invariant, i.e. G n ( LC ) = LC for any n >
0. By using the fact that LC j is partof the stable set of the saddle point P we also have that the arc LC j is invariant. From this fact, and fromthe characterization of the basin of attraction of P given in [5], the (seemingly chaotic) area bounded bythe curve LC j is also invariant. In fact, in [19] and [20, p. 442–444] it seems suggested that such invariantarea is exactly a true chaotic area (with dense periodic points, which are all homoclinic), but this is still tobe proved rigorously.In order to understand the appearance of the chaotic area we consider the 2-parametric family of maps G g,h : R → R given by G g,h ( x, y ) := (cid:18) gx + hy + xy + 9( x + y + 2) / , x + y + 6( x + y + 2) / (cid:19) , for which a numerical exploration evidences that unfolding the Boros-Moll map there are maps with differentqualitative behavior. Indeed, one can check that in the parameter plane ( g, h ) depicted in Fig. 2 there appearseveral regions indicating qualitative differences. The different colors in this figure represent regions in theparameter space related to the existence of attracting cycles of different periods. The yellow region in Fig. 2represents the values of the parameters at which the dynamics is regular, namely there exist two attractingfixed points, P in the positive side and P in the negative side, whose basins of attraction are separatedby the stable set of a saddle fixed point P . Differently, the brown region represents the values of theparameters at which the only attracting set is the positive fixed point P . The existence of these fixed points,as a function of the parameters, is commented below. The white points denote either existence of a cycle3igure 2: In (a) two-dimensional bifurcation diagram related to an i.c. close to the fixed point P . 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 P is alwaysattracting, in all these regions the attractor P coexists with some other attracting set. To simplify ouranalysis, which is oriented to understand the dynamics of the Boros-Moll map, we consider g = h , whichgives the one-parametric family (1), and we will focus mainly in the interval h ∈ [5 ,
6] of the parametervalues. Thus, in the two-dimensional bifurcation diagram the straight line of equation g = h representsthe path that we shall follow in order to describe some of the bifurcations occurring from a regular regimeexisting for h = 6 to the chaotic one existing for h = 5. In fact, changing the parameter h along the pathin Fig.2(b), decreasing h , the fixed point P becomes an attracting focus and then it becomes a repellingfocus via a Neimark-Sacker (NS from now on, for short) bifurcation, as commented below, giving rise toa sequence of bifurcations that we will summarize in its main features. Decreasing h we can comment theappearance/disappearance of attracting k -cycles for k > G h and first Neimark-Sacker bifurcation We notice that for any h ∈ R , map G h has exactly three different fixed points. In order to characterize thefixed points of map G h , and following [5], we introduce the auxiliary variable m = x + y + 2, so that thefixed points are characterized by c ( x, y, m ) := m − x − y − ,c ( x, y, m ) := − xm + xy + hx + hy + 9 = 0 ,c ( x, y, m ) := − ym + x + y + 6 = 0 . (8)Isolating x and y from the first and third equations we get: x = m − m − m − m and y = m + 4 m . (9)Substituting these expressions in the second equation we obtain that the variable m must satisfy c ( m ) = m − m − m − hm − m − m + (2 h − m + 8 m + 8 m + 16 = 0 . m of c ( m ; h ) does not vanish for any h ∈ R , hence no rootenters from infinity. On the other hand, the discriminant of c is given by∆ m ( c ( m ; h )) = 927712935936 h − h + 211982405861376 h + 2739735613145088 h + 118044964423729152 h − h − h − h + 2232890826264743510016 h − h + 39121043805342246371328 h − h + 8262772575788649550970880 h − h + 35844037650190215269056512 . By using the Sturm method we obtain that ∆ m ( c ( m )) > h ∈ R , hence c ( m ) has no multiple roots,so that no roots of c ( m ) appear from C and there is no value of h such that the roots collide. Finally, usingagain the Sturm method we have that for h = 5, c ( m ) has only three distinct real roots. Hence for all h ∈ R the polynomial c ( m ) has three different real roots, which give rise to three different fixed points viaEq. (9).As already noticed in [5] and recalled above, G has the fixed points P , P and P . Since the locationof the fixed points vary continuously with the parameter h , we will denote the fixed points of the maps G h as P ( h ) , P ( h ), and P ( h ), not indicating the dependence on h when it is not necessary.Since the fixed points P and P persist for h ∈ [5 ,
6] as an attracting node and a saddle, respectively,we are interested in the fixed point P which may be attracting or repelling, as we have seen in Fig. 2. Sowe restrict the scope of our analysis to the proof of a NS bifurcation that occurs at h (cid:39) . DG h ( x, y )) = 1,obtaining c ( x, y, m ) := 3 ( x + y + 2) + ( x + y −
6) ( x − y ) = 0 . Using the polynomials c , c and c that characterize the fixed points, and taking successive resultants weget that if a NS bifurcation takes place then the parameter h must be a root of the polynomial P NS ( h ) = 582371795533824 h − h + 41393275596177408 h − h + 10296285821493313536 h − h + 970090676944581427200 h − h + 8531720267711624773632 h − h + 6956611779457796014080 h − (cid:20) , (cid:21) . Hence h NS (cid:39) . P ( h ) is a hyperbolic stable focus for h (cid:38) h NS , a hyperbolic unstable one for h (cid:46) h NS and non-hyperbolic for h = h NS . In the next subsectionwe show that for h > h NS there is coexistence of the two attracting fixed points P and P and their basinsof attraction are separated by the stable set of the saddle P . G h As remarked above, for large values of h the dynamics of a map G h is quite regular. In fact, the map hastwo attracting fixed points, P and P . An example of the phase plane in such a situation is shown in Fig.3,for h = 6, where P (cid:39) (3 . , . , P (cid:39) ( − . , . P is shown in red, B ( P ) , while that of P is shown in yellow, B ( P ) . The related basins ofattraction are separated by the stable set of the saddle fixed point P . The main difference with respect to the case h = 5 is that the main branch of the stable set of the saddle P is not a critical curve and that the basin of the fixed point P is not connected (for example, there areopen sets of points with both negative coordinates that have a trajectory convergent to P ). In Fig.3 it5igure 3: Basins of attraction of map G h at h = 6.is shown the straight line LC j − up to the point ( − , −
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 ( P ) in one point Q (see the enlargement in Fig.3) . The region denoted by H in Fig.3, which is the portion of phase planebounded above by the critical curve LC j and below by the stable set W s ( P ) on the right side of the point Q, belonging to B ( P ) , is responsible of further preimages belonging to the basin B ( P ).In the Appendix, it is reported how the critical lines split the plane in different zones named Z i wherethe sub-index i denotes the number of different rank-1 preimages. In what follows, we will use this notation.In fact, the arc of LC j on the boundary of H has two merging rank-1 preimages in the half line LC j − upto the point Q − , while the arc of stable set W s ( P ) on the right side of the point Q has two distinct rank-1preimages issuing from the point Q − ∈ LC j − , one at the right side of LC − (on the boundary of the set H − , ) and the other at the left side (on the boundary of the set H − , ), forming the boundary of the set H − = H − , ∪ H − , . Since H − belongs to the zone Z whose points have two distinct rank-1 preimages,we have two more preimages of H − and so on, ad infinitum. Thus, the basin B ( P ) includes also the infinitesequence of preimages ∪ n ≥ G − nh ( H − ) which consists of infinitely many portions of the plane (in fact, due tothe properties of the regions Z k described in Appendix, even if a preimage may happen to belong to Z (sohaving no further preimages), another one above the diagonal exists, and thus it belongs to Z k with k ≥ . P The existence of non-connected regions of the phase plane belonging to the basin B ( P ) are relevant to provethat close to the value h = 5 a chaotic attractor becomes a chaotic repellor which persists for h ∈ (5 , (cid:101) h )for a certain value (cid:101) h , although almost all the points of the phase plane have a trajectory convergent to thefixed point P . Similarly, a chaotic repellor exists for h ∈ (5 − ε,
5) with ε (cid:38)
0, and almost all the pointsof the plane have a trajectory convergent to the fixed point P . Thus, the occurrence of an invariant set ofpositive measure (which seems chaotic) for h = 5 as shown in Fig.1 is a very peculiar phenomenon in theframework of dynamical systems theory. The peculiarity may be related to the basin B ( P ) which is an opensimply connected set only for h = 5 , and thus related to a peculiar behavior of the critical curves. In fact,for h ∈ (5 − ε, (cid:101) h ) \ { } the basin B ( P ) is an open set but not simply connected (while for h ≥ (cid:101) h it is nonconnected).We show that there exists a particular value (cid:101) h (cid:39) .
032 such that for h > (cid:101) h , there exists an invariantunbounded absorbing region A , which is bounded by segments of images of finite rank of a generating segment g belonging to the critical curve LC j − for x < − d ) of vanishing denominator. For 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 thedynamics of the map, so there cannot be any confusion. < (cid:101) h such an invariant region does not exist . Moreover, for h > (cid:101) h (resp. h < (cid:101) h ) the saddle fixed point P is not homoclinic (resp. is homoclinic). For h > (cid:101) h (when the unbounded absorbing region A exists) the stable set of the saddle separates two basinsof attraction: the basin B ( P ) of the attracting fixed point P 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 ( P ) = ∂ B ( P ) = ∂ B ( A ) . Figure 4: Phase portrait of map G . . The white area belongs to the basin of the invariant chaotic area.The red area belongs to the basin B ( P ): In green, the images of the generating segment g of the critical line LC j − . The region H − , which is part of the basin B ( P ), is close to contact with the chaotic area boundedthe arcs given by the images of the generating segment.In Fig.4 we illustrate the unbounded chaotic area at h = 5 . g which is the arc of LC j − belonging to the invariant area itself(see [18]). It can be seen that the boundary is very close to contact the point Q − ∈ LC j − on the boundaryof the set H − which belongs to the basin B ( P ) . Recall that this boundary belongs to the stable set of thesaddle, W s ( P ) , while the left branch of the unstable set of the saddle, W u,l ( P ) , entering the chaotic area, isdense in that area (differently, the other branch W u,r ( P ) issuing from the saddle P is a branch connectingthe saddle with the attracting fixed point P ). When such a contact occurs, say at h = (cid:101) h, that is, when theboundary of H − has a contact with the critical curve bounding the chaotic area, then this contact leads (for h < (cid:101) h ) to the destruction of the invariant chaotic area and simultaneously to the appearance of homoclinicpoints of the saddle P (but on the left side only). The role played by contact bifurcations of this kind havebeen described in several works (see e.g. [7] and [18]), see also the bifurcations called “crisis” in [13, 14]).The contact bifurcation occurring at h = (cid:101) h is qualitatively shown in Fig.5. The contact point betweenthe region H − (whose boundary belongs to the stable set W s ( P )) and the chaotic area bounded by arcs ofcritical curves, is denoted by p k ∈ LC jk for a suitable integer k , and it is the image of a point p − ∈ g ⊂ LC j − . If the area is truly chaotic then the closure of the unstable set W u,l ( P ) is exactly the chaotic area (i.e. W u,l ( P ) is dense in the area), which means that the arcs of critical curves are limit sets of the unstableset (arcs shown in blue in Fig.5). The image of p k is the point p k +1 ∈ LC jk +1 belonging to the boundaryof H on the arc of the stable set W s ( P ), so that the same property holds for the forward images which For h = 5 there exist an invariant area, but it is not absorbing. h = (cid:101) h. are converging to the saddle fixed point P : that is, they are contact points between critical arcs on theboundary of the invariant area A and the main branch of the stable set W s ( P ). For h < (cid:101) h when the critical curves intersect the regions H − and H also the unstable set W u,l ( P ) is crossingthese regions, and thus is crossing the stable set W s ( P ) leading to homoclinic points of the saddle, andsince an invariant absorbing area no longer exists the generic trajectory converges to the attracting fixedpoint P .Recall that for noninvertible maps we cannot refer to a general theorem stating that when a homoclinicorbit of a saddle cycle exists then close to the homoclinic orbit we can find an invariant set which is chaotic (inthe sense of Devaney [6] and of Li and Yorke [15]), because the standard theorem holds for diffeomorphisms.However, also for endomorphisms (having not a unique inverse) it is possible to prove that following thehomoclinic orbit in the proper way, with the suitable inverses applied to the homoclinic points, it is possibleto show that a horseshoe can be rigorously defined, leading to the result. Examples can be found in [10] and[16].Thus, for 5 < h < (cid:101) h the chaotic attractor becomes a chaotic repellor, the only attracting set is the fixedpoint P , and its basin B ( P ) is almost the whole plane. The points of the plane on the left side of the mainbranch of the stable set of the saddle P that are leaving that side converging to the attracting fixed point P are doing so ultimately crossing through the regions called H − and H , where H is always defined asabove: it is the strip between the main branch of the stable set of P and the arc of critical curve LC j aboveit, which exists as long as h > P for5 < h < (cid:101) h has points crossing through these regions. That is: arcs belonging to the unstable set W u,l ( P )as well as arcs belonging to the critical lines LC jn for some n > H − and H and then converge to the attracting fixed point P . h = 5 . Role of the critical lines At h = 5, the scene described in Section 2.4 is no longer possible because the strip H reduces to the arc LC j which merges with the main branch of the stable set of the saddle P , and thus the portion H − reducesto an arc of LC j − . The portion of plane bounded by the arc of curve LC j (belonging to the resolvent curvegiven in Eq.(6)) is invariant: it is the situation shown in Fig.1. Notice that the saddle P is homoclinic,since the whole segment LC j − and all its preimages belong to the stable set of P and intersections betweenthe stable and unstable sets are infinitely many, but only on the left side of the arc LC j . This means that,8ifferently from the case for 5 < h < (cid:101) h , even if the saddle P is homoclinic no one point belonging to theunstable set W u,l ( P ) can have a trajectory convergent to P (since no one point can be mapped to the rightside of LC j ) and similarly no point on the images of the critical line LC j − , of any rank, can converge to P . So we have seen that for 5 ≤ h < (cid:101) h the saddle P is homoclinic only on the left side. The characteristicFigure 6: Images of an arc of LC j − for x ≤ − h = 5 . h = 5 and h = 4 . h = 5 may be related to the critical curves asfollows. For 5 < h < (cid:101) h the image of the critical curve LC j − for x ≤ − LC j whichintersects the line of vanishing denominator ( d ) in two points, so that LC j consists in three unboundedcomponents. In Fig.6 we show a portion of phase plane, an enlargement close to ( − , −
1) at h = 5 .
01, atthis value of the parameter almost all the points have a trajectory converging to P (cid:39) (3 . , . . At h = 5 the image of LC j − for x ≤ − LC j which merges with the main arc of the stable setof P , which is tangent (from above) to ( d ) in the point ( − , − LC j = LC jn for any n ≥ . For h < LC j − for x ≤ − LC j completely above ( d ), and thusits points converge to the attracting fixed point P . Clearly, all the points in the strip H between the mainarc of the stable set of P and the arc LC j are also converging to the attracting fixed point P , and so dothe points of its preimage belonging to a strip including LC j − as well all the further preimages of any rank.For instance, at h = 4 .
99, a chaotic repellor still exists (since the fixed points P and P are homoclinic, aswell as infinitely many repelling cycles exist and are homoclinic), and almost all the points are convergingto P (cid:39) (2 . , . . For h < P , butalso the unstable set W u,l ( P ) has again arcs going to the right side and converging to P . In Fig.7 we illustrate an arc of LC j − for x ≤ − , LC j in red and its first images, at h = 5 .
01, whichare all on the left side of the curve LC j , while at h = 4 .
99 the images are all on the right side to the LC 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 ( P ) in onepoint Q as long as h >
5, they are merging when h = 5, and LC j is above that stable set for h < . In order to describe the dynamics of map G h (leading to the unbounded chaotic set shown in Fig.4) we accountsome of the phenomena that appear when the parameter h is decreased to h = 5 after the N S bifurcationof the fixed point P , that is, decreasing h from h NS (cid:39) . . The
N S bifurcation is of supercritical type,as can be seen from the attracting closed curve Γ which appears surrounding the repelling focus P , anexample is illustrated in Fig.8a. The restriction of the map to the invariant curve in general leads eitherto an attracting cycle (when the rotation number is rational and the closed curve consists of a saddle-nodeconnection) or to quasiperiodic orbits, dense in the closed curve (when the rotation number is irrational).9igure 7: Critical curves LC jk for 1 ≤ k ≤ 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 LC j − then all the points inside the area bounded by Γ cannot be mapped outside the area (i.e. the areabounded by Γ is invariant). At the same time, the points external to that area which are attracted fromΓ 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 thecritical curve LC j − . Figure 8: Bifurcation of the closed curve Γ. In (a) h = 5 . , smooth curve not intersecting LC j − . In (b) h = 5 . , Γ intersects LC j − . Decreasing h a contact between Γ and LC j − occurs at h (cid:39) . LC j − is foldedon LC j so that Γ becomes tangent to LC j in two points (the rank-1 images of the two intersection pointsΓ ∩ LC j − ). And clearly the contacts between Γ and the critical curves LC jn persist for any n . In Fig.8b itis well evident the tangency between Γ and the critical curve LC j . This mechanism is a peculiar phenomenon related to noninvertibility (i.e. it cannot occur in maps witha unique inverse). It is related to the folding of the phase plane due to the critical curves, and breaks theinvariance of the area bounded by the closed curve Γ . In fact, all the points in the area bounded by Γ and LC j − are mapped outside Γ , between Γ and LC j , and its further images between Γ and LC j , and so on (see10igure 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 withperiodic points both inside and outside the area bounded by the closed curve. We shall see an example inthe next subsection.Let us first illustrate a sequence of attractors and their bifurcations via one-dimensional bifurcationdiagrams as a function of the parameter h . Since we know that unbounded attracting sets exist, in Fig.9 weshow 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 h NS the closed invariant curve Γ increases insize, since approaching the straight line (d) the shape becomes wider. In the same figure we see that thereare two large windows associated with cycles and interesting dynamics, which are enlarged in Fig.10 andcommented in the next subsections. 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 . P only the closed curve Γ is attracting, while at h =5 . h =5 . . 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 P (resp. the closed curve Γ) and we show with different colors the basins of the11 fixed points of the 11-th iterate of the map G h , to emphasize the shape of the basin of the attracting11-cycle. In Fig.11b and related enlargement it can be seen that some points of the attracting 11-cycle areoutside the closed curve Γ and part are inside. One more peculiarity related to the immediate basins of the11 fixed points of map G h is that the stable set of the 11 saddles (related to the 11-cycle saddle) leads toa closed curve, which is possible only in noninvertible maps, due to the crossing of the critical curves of themap (other examples and explanation of the mechanism can be found in [18]). From the same Fig.11 we canFigure 12: In (a) h =5 . , attracting 11-cycle and its basin of attraction for the 11-th iterate of the map.In (b) h =5 . , coexistence between the attracting 11-cycle and the attracting 4-cycle (in yellow and azuretheir 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 ofattraction. 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 P 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 the12losed invariant curve, as shown in Fig.12a, where the crossing of LC j of the stable set of the saddle leadsto 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 inFig.10b this occurs at h (cid:39) . P . In Fig.12b at h =5 .
412 besides P (with basin in red) the basins ofattraction of the 11-cycle (in yellow) and of the 4-cycle (in azure) are shown. The basins’ structure of the twocoexisting attractors is clearly fractal, which means that the saddle 11-cycle on the immediate basin of the11-cycle is now homoclinic on one side (that external to the immediate basin which is clearly visible in yellowvia bounded closed pieces). As evidenced in Fig.10b the attracting 4-cycle undergoes a supercritical N S
Figure 13: In (a) h = 5 . , N S bifurcation of the 4-cycle. In (b) h = 5 . , chaotic piece showing thatthe 4-cycle is a snap-back repellor.bifurcation at h (cid:39) .
411 which leads to 4-cyclic closed invariant curves (see an example in the enlargementof Fig.13a at h = 5 . h = 5 . h = 5 . W s ( P ) is the11-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 invariantabsorbing area exists, inside which the dynamics seem chaotic. At h = 5 . LC j − in two segments, forming together the generating segments whose imagesbound the absorbing area via segments of critical curves. The area is unbounded as in fact the image ofthe shortest segment (evidenced in dark green in Fig.14a) is mapped in an arc of LC j which crosses theline (d) of vanishing denominator (evidenced with a dark green arrow in Fig.14a), thus its image consists intwo unbounded arcs of LC j on the boundary of the area. In Fig.14b it is shown a wider portion of phasespace and in green the arcs of critical curves on the boundary of the invariant area, while in Fig.14c it isrepresented the unbounded attracting area in the whole plane, related to the scaled variables. In the previous subsection we have described a few bifurcations which occur decreasing h and evidencedin Fig.10b, leading to an unbounded invariant absorbing area, with complex dynamics. Clearly, inside thisinvariant area it is possible that stable cycles of high period exist, not detectable in our numerical simulations.13igure 14: Unbounded absorbing area, with chaotic dynamics, at h = 5 . . In (a) the generating segmentson LC j − are evidenced. In (b) the attracting set is shown in a larger portion of the phase plane. In (c) theattracting 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 describedabove which is a snap-back repellor (in fact, although an invariant 4-cyclic attracting chaotic area no longerexists, as evidenced by the complex dynamics, the homoclinic 4-cycle still exists). From Fig.9 we can see thatthe unbounded invariant absorbing area persists for a wide interval of values for the parameter h , and theappearance 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 . LC j − , in red are indicated the points wherea pair of 4-cycles is going to appear. In Fig.15b (at h = 5 . G h . The complex structure of the basins evidences a chaotic repellor on the basinboundary, while the immediate basins are (as in previous examples) bounded by the stable set of the saddle4-cycle forming closed curves, denoting intersection with the critical curves. From Fig.10a we can see thatFigure 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 (cid:39) . h close to the N S bifurcation) become nonsmooth after the crossing of the critical line LC j − .We also see in Fig.10a that an attracting 12-cycle appears, which is related to the closed curves. In fact, at h (cid:39) . h = 5 . h = 5 . N S bifurcation, and aftera sequence of bifurcations 12-cyclic chaotic pieces are created (see Fig.17a at h = 5 . . Other bifurcationsare illustrated in Fig.17b at h = 5 .
096 (enlarged portion of phase plane, evidencing one piece of the 12-cyclicset). This, in its turn, is followed by a contact bifurcation at h = 5 . , leading to an expansion of theinvariant absorbing area with chaotic dynamics which has now 4-cyclic chaotic pieces, one of them shownin Fig.17c. This also denotes that the repelling 4-cycle inside that area is now homoclinic, i.e. a snap-backrepellor.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 absorbingareas as the 4-cyclic areas which are all bounded by few images of the generating segment intersecting LC j − and shown in Fig.17c. The 4-cyclic chaotic areas are shown at a lower value of h in Fig.18a, together withFigure 17: Enlargements showing a part of a chaotic attractor. Three pieces of an attractor of 12 connectedcomponents 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).15he related four basins of attraction for the 4-th iterate of the map. Besides a fractal structure with a chaoticrepellor on the boundary, this figure shows that the areas are very close to the boundary of their immediatebasins, denoting that the parameter is very close to a contact bifurcation value, leading to the disappearanceof the 4-cyclic areas, and the appearance of a unique unbounded absorbing area inside which the dynamicsseem chaotic, see Fig.18b.Figure 18: Contact bifurcation leading to expansion of the invariant area. In (a) h = 5 . h = 5 . h , as shown in Fig.9,up to the bifurcation described in Sec.2.4.1 occurring at h = (cid:101) h. Decreasing h up to the final bifurcation, itseems that the dynamics inside the unbounded absorbing area become more complex. That is, more cyclesappear and undergo homoclinic bifurcations. For example the fixed point P which is a repelling focus maybecome a snap-back repellor. From Fig.18b it is not clear whether it is homoclinic or not. However, we haveevidence of this, at the same value h = 5 . G h ∗ ( γ ) at h ∗ = 5 . γ close to P corresponding to different values ofthe radius s . In (a) two different arcs are shown. In (b) a third arc is added to the two shown in (a). Thevalues of s are given in the text.In fact, let us consider the fixed point P and its rank-1 preimage different from itself, P − , which is thepoint symmetric to P with respect to the main diagonal. In order to show that P is a snap-back repellor16t is enough to show that there is a point q in a neighborhood of P which is mapped into P in a finitenumber of iterations, say G mh ( q ) = P , and such that the local inverse which satisfies G − h ( P ) = P (whichis a contraction), leads to lim n →∞ G − nh ( q ) = P . Let us consider an arc γ in such a neighborhood of P (i.e. such that the preimages of its points by thelocal inverse G − h converge to P ), given by a half circle of equation( x − x ( P )) + ( y − y ( P )) = s 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 G h ∗ ( γ ) (with h ∗ = 5 . P − , below it. Differently an arc γ in red corresponding to the radius s = 0 .
25, has the image G h ∗ ( γ ) in redwhich is close to the preimage P − , but above it. This means that there exists a value s ∗ ∈ (0 . , .
3) suchthat the related arc γ ∗ has the image G h ∗ ( γ ∗ ) which crosses through P − . This implies the existence of apoint q ∈ γ ∗ such that G h ∗ ( q ) = P . As an example, considering s = 0 .
29 in Fig.19b we show in pink the related arc γ and, also in pink, theimage G h ∗ ( γ ) which is above the black arc and closer to P − . A similar construction can be used also at any smaller value of h (in particular also at h = 5 or smaller),showing that P persists in being a snap-back repellor. We have considered a one-parameter family of maps which unfolds a Landen-type map obtained by Borosand Moll, and we have described the main bifurcations that appear in the route towards the parameter valuethat corresponds to this map. We have focused on those homoclinic and contact bifurcations that can giveinformation about the unbounded invariant chaotic region that appear in the Boros-Moll map. Our studyconfirms that the dynamics of this map is singular in the considered family, and we give evidences that thissingularity is related with a specific property of the critical lines: the fact that one of the critical lines mergeswith one separatrix of the saddle point only for the parameter value corresponding with the map introducedby Boros and Moll.
Acknowledgements
V.M. is supported by Ministry of Economy, Industry and Competitiveness–State Research Agency of theSpanish Government through grant DPI2016-77407-P (MINECO/AEI/FEDER, UE) and acknowledges thegroup research recognition 2017-SGR-388 from AGAUR, Generalitat de Catalunya. He also acknowledgesProf. A. Gasull for helpful discussions. V.M. and I.S. as visiting professor are grateful to the University ofUrbino Carlo Bo for the hospitality during a stage at the University in February 2018.
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The map G h has a non unique inverse, and the number of distinct rank-1 preimages of a point ( x, y ) differsdepending on the region of the phase plane which the point ( x, y ) belongs to. These regions, or zones Z i (related to i distinct preimages of rank-1) following the notation introduced in [18], are usually bounded bycritical curves LC of the map, which are the images of curves of merging preimages, denoted LC − , whichbelong to the set where the Jacobian determinant of the map vanishes.In our case, the Jacobian determinant vanishes either at the points LC j − (defined by y = x ) or the pointsin LC jj − (defined by x + y − LC j − intersects ( d ), its image LC j consists of two unbounded curves of the plane, LC j ∪ LC jr (noticethat if an arc γ crosses the line ( d ), where the denominator vanishes in a point in which the numerators ofthe components of the map do not vanish, which is always the case for the considered map, then its image G h ( γ ) consists of two unbounded arcs). 18igure 20: Critical curves for the map G h at h = 5 . . In (a) LC − ; in (b) LC .The curve LC jj − does not intersects ( d ), its image LC jj = G h ( LC jj − ) consists of a single arc, and it iseasy to see that it is a segment of straight line y = 3 (up to x = 3). Indeed, by setting x n = t and y n = − t +6(a point on LC jj − ), and taking ( x n +1 , y n +1 ) = G h ( x n , y n ) we get x n +1 = (6 t − t + 39) /
16 (which takes itsmaximum value in x = 3) and y n +1 = 3, that is, the image of the line LC jj − 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 . LC = LC j ∪ LC jr ∪ LC jj split the plane in 5 distinctopen regions. We can figure out the number of preimages by considering a sort of “folding of the plane”, asqualitatively shown in Fig.21. By computing how many preimages have a particular point in each of theseregions, we obtain the different zones Z i . In Fig.21 they are displayed for the particular case h = 5 .
5. Noticealso that a point belonging to LC has two merging rank-1 preimages in a point belonging to LC − . For eachpoint in a zone Z i it is also possible to show where the preimages are located (on opposite side with respectto the related branch of LC − ).We report how to obtain the number of preimages in each of the zones limited by the curves LC . We fixa point ( u, v ) and we set G h ( x, y ) = ( u, v ). By introducing the notation z = x + y + 2 and from the equationof the second component of the map, we have that vz / = z + 4, hence v = ( z + 4) /z . (10)By studying the function f ( z ) = ( z + 4) /z we easily get that for v > v = 3 there is one negative solution and two merging solutions given by z = 8; and for −∞ < v < P = xy and S = x + y , and by using the equation of the first component, weobtain 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 ) / − h ( z − − S = z − . Since any preimage ( x, y ) associated to ( u, v ) satisfies y − Sy + P = 0 and x = S − y , we have x = S ± √ S − P y = S ∓ √ S − P . The final number of preimages depends on the sign of S − P .For the original Boros-Moll map, the case h = 5, the curve LC j belongs to the curve R ( x, y ) = − x y +4 x + 4 y − xy + 27 = 0, it is part of the stable set of the saddle P , and the zones Z i are displayed inFigure 22. 19igure 21: Qualitative shape of the foliation of the plane. Critical curves and Zones Z i for the map G h at h = 5 . . Figure 22: Critical curves and Zones for the Boros-Moll map ( hh