NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP

Transcription

1 NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP ZBIGNIEW GALIAS Deprtment of Electricl Engineering, AGH University of Science nd Technology, Mickiewicz 30, Krków, Polnd WARWICK TUCKER Deprtment of Mthemtics, Uppsl University, Box 480, Uppsl, Sweden The question of coexisting ttrctors for the Hénon mp is studied numericlly y performing n exhustive serch in the prmeter spce. As result, severl prmeter vlues for which more thn two ttrctors coexist re found. Using tools from intervl nlysis, we show rigorously tht the ttrctors exist. In the cse of periodic orits we verify tht they re stle, nd thus proper sinks. Regions of existence in prmeter spce of the found sinks re locted using continution method; the sins of ttrction re found numericlly. Keywords: Hénon mp, periodic orit, coexisting ttrctors, intervl rithmetic 1. Introduction The Hénon mp [Hénon, 1976] h, : R 2 R 2 is n invertile mp defined y: h, (x, y) = (1 + y x 2, x), (1) The mp depends on two prmeters nd, which in the clssicl cse hve vlues of = 1.4 nd = 0.3. Originlly introduced s model of Poincré mp for three-dimensionl flow, the mp displys wide rry of dynmicl ehviors s its prmeters re vried, [Zhiping et l., 1991; Picqudio et l., 2002]. In this study, we will concentrte on the dissiptive, orienttion-reversing cse; more precisely, we will consider [0, 2] nd [0, 0.5]. This region contins the clssicl prmeter vlues = 1.4, = 0.3 for which one oserves the so-clled Hénon ttrctor (see Fig. 1). In the setting we re considering, it is known [Newhouse, 1979] tht when sddle point genertes homoclinic intersection, cscde of (periodic) sinks will occur. Furthermore, there will e nery prmeter vlues for which the mp hs infinitely mny coexisting sinks. Nevertheless, coexisting sinks for the Hénon mp pper to e very elusive; due to the dissiptive nture of the mp, the regions in prmeter spce for which sinks pper simultneously re extremely smll. In this work, we report some results of the numericl serch for prmeter vlues for which there exist more thn two ttrctors. It is known [Benedicks & Crleson, 1991] tht there is set of prmeters (ner = 0) with positive Leesgue mesure for which the Hénon mp hs strnge ttrctor. But given specific point (, ) in prmeter spce, it is not possile to prove tht the dynmics of the mp genertes strnge ttrctor. It is, however, possile to prove the existence of (periodic) sink; this is finite computtion, nd ll necessry conditions re open (nd thus roust). Our study shows tht in mny cses where there ppers to e strnge ttrctor, the true underlying dynmics is governed y single (or multiple) periodic sink. 1

2 2 Z. Glis & W. Tucker y x Fig. 1. Trjectory of the Hénon mp (with (, ) = (1.4, 0.3)) composed of points; unstle fixed points (+, + ) 2. Numericl methods 2.1. Serch for coexisting ttrctors The most nturl method to find sink is to follow trjectory nd monitor whether it converges to periodic orit. First, initil itertes re discrded until stedy-stte is oserved. At this stge, we restrt the trjectory y tking the current iterte s initil point. Next, we check if the trjectory periodiclly returns very close to this initil point. The numer of itertions which should e skipped to rech the stedy stte depends on vrious fctors like the eigenvlues of the periodic orit (they define the strength with which the periodic orit ttrcts neighoring trjectories), the size of the sin of ttrction nd its shpe, including the size of the immedite sin. Usully the numer of itertions which hs to e discrded hs to e chosen y tril-nd-error. With chotic ttrctors the sitution is more difficult, since trjectory elonging to chotic ttrctor never exctly comes ck to the initil point. We use n pproch sed on counting the numer of ins visited y the trjectory t stedy-stte. First, the stte spce is covered y finite numer of ins. Agin s in the cse of periodic orits some initil itertes re computed nd discrded to rech the stedy-stte. At this stge, trjectory of fixed (long) length is computed, nd the numer of ins visited y the trjectory is clculted nd recorded. This dt is sved for severl initil points. If, for two different initil points the numers of ins re similr, then most likely the trjectories converge to the sme ttrctor. Note tht this method works for periodic ttrctors too. In this sitution, however, if two trjectories converge to the sme periodic orit, then the numer of ins should e the sme. A more sophisticted (lthough slightly slower) pproch is to lso record the in positions, nd compre whether the trjectories visit the sme ins. This method will distinguish two different periodic orits of the sme period, or two chotic ttrctors oth hving nonempty intersections with pproximtely the sme numer of ins. In order to e le to locte coexisting ttrctors, it is necessry to use mny initil conditions. Since some of the ttrctors my hve very smll sins of ttrction, it is good strtegy to strt the computtions from mny rndomly chosen initil points. In this study, 2000 initil conditions for ech point in prmeter spce were used The existence of ttrctors To prove the existence of ttrctors we use methods from intervl nlysis [Moore, 1966; Neumier, 1990]. These methods llow us to otin rigorous results using computers. Computtions in properly rounded intervl rithmetic produce results which contin oth mchine rithmetic results nd lso true (infinite rithmetic precision) results. Using intervl rithmetic it is possile, for exmple, to prove in finite numer of steps tht n imge of given set is enclosed in other set. Let us ssume tht in simultions periodic or chotic ttrctor is oserved. In this section we descrie generl technique to prove tht there is n ttrctor in neighorhood of the oserved trjectory. In order to prove tht there is n ttrctor for the system, it is sufficient to prove tht there is positively

3 Numericl study of coexisting sinks for the Hénon mp 3 invrint set ( trpping region) enclosing the oserved ttrctor. A set A is clled positively invrint for the mp f if f (x) A for ll x A. This cn e estlished y constructing cndidte for trpping region enclosing the region of interest, covering it y set of intervl vectors, nd verifying rigorously tht the imge of ech intervl vector is enclosed y the set A. Usully one uses covering y so clled ε oxes, i.e., intervl vectors with corners lying on rectngulr grid [Glis, 2001]. Ech ε ox is n intervl vector of the form v = ([k 1 ε 1, (k 1 + 1)ε 1 ], [k 2 ε 2, (k 2 + 1)ε 2 ],..., [k m ε m, (k m + 1)ε m ]), where m is the dimension of the stte spce, k i re integer numers, nd ε = (ε 1, ε 2,..., ε m ). This type of representtion is very well suited for rigorous computtions. First, y chnging ε, it is possile to represent given set y ε oxes with ritrry precision. Second, when ε is fixed, n ε ox is defined uniquely y m integer numers. In this work, we use nother method. It hs the dvntge tht no cndidte set hs to e defined priori, nd the trpping region is constructed y the procedure. In this method one strts with single ε ox enclosing n ritrry point from the oserved trjectory or covering of the numericlly oserved ttrctor y ε oxes. This defines the set S of oxes. In ech step, for ech ox from S its imge under f is computed, nd set of ε oxes contining the imge re found. New oxes re dded to the set S. The procedure is continued until no new oxes pper in the process. On the exit, the set S is trpping region for the mp f. To prove tht there is t lest n ttrctors for the system it is sufficient to prove tht there re n trpping regions, nd tht the trpping regions re pirwise disjoint. This is prticulrly esy if the trpping regions re represented s sets of ε oxes of the sme size. It is very importnt to choose ε in correct wy. If ε is too lrge, then either we will not e le to finish the serch procedure or the resulting trpping regions for different ttrctors will overlp. If ε is too smll, then the computtions will generte covering consisting of prohiitively mny oxes The existence of periodic orits In the previous section, we hve descried generl technique to prove the existence of n ttrctor. It involves finding imges of mny ε oxes, nd even for simple ttrctors the numer of oxes which hs to e used my e very lrge. When the ttrctor is periodic, one my use fster technique. Intervl methods provide simple computtionl tests for uniqueness, existence, nd nonexistence of zeros of mp within given intervl vector. Let F : R n R n e continuous mp. In order to investigte the existence of zeros of F in the intervl vector v one evlutes certin intervl opertor over v. In this work, we use the Newton opertor [Neumier, 1990]: N(v) = ˆv F (v) 1 F(ˆv), (2) where F (v) is n intervl mtrix contining the Jcoin mtrices F (v) for ll v v, nd ˆv v. In our implementtion, we choose ˆv to e the center of v. The most importnt property of the Newton opertor cn e used to prove the existence nd uniqueness of zeros. It sttes tht if N(v) v, then F hs exctly one zero in v. Note tht ll opertions in (2), including evlution of F(ˆv) nd F (v), hve to e implemented rigorously in intervl rithmetic for this property to e vlid. In order to study the existence of period p orits of mp h, we construct the mp F defined y [F(v)] k = z (k+1) mod p h(z k ), for 0 k < p, (3) where v = (z 0, z 1,..., z p 1 ). Zeros of F correspond to period p points of h, i.e., F(v) = 0 if nd only if h p (z 0 ) = z 0. In order to prove tht there is single period-p orit enclosed in the intervl vector v it is sufficient to evlute the intervl Newton opertor on v, nd verify tht N(v) v. This pproch comined with the generlized isection technique hs een successfully used to find ll low period cycles for the Hénon mp nd the Iked mp [Glis, 2001]. Appliction to continuous systems ws presented in [Glis, 2006] nd [Glis & Tucker, 2011]. Another choice is to pply the intervl opertor to the mp f = id h p. This pproch involves the computtion of severl itertes of the mp, nd my especilly for longer orits led to huge overestimtions. The former method hs n dvntge tht the prolem of the existence of periodic orits is trnslted into the prolem of the existence of zeros of higher-dimensionl function. This helps to reduce the overestimtion cused y the wrpping effect inherent in ll set-vlued computtions. For detils see [Glis, 2001].

4 4 Z. Glis & W. Tucker Note tht the Newton opertor enles us to prove the existence of unstle periodic orits s well. To show tht the periodic orit is n ttrctor we must show tht it is symptoticlly stle. This prolem is hndled in the following section Properties of periodic orits In this study, the most importnt property of periodic orit is its stility. To study the stility of the orit v = (z 0, z 1,..., z p 1 ) it is sufficient to compute the Jcoin mtrix J long the orit J = (h p ) (z 0 ) = h (z p 1 ) h (z 1 ) h (z 0 ), (4) nd find its eigenvlues. If oth eigenvlues re within the unit circle, then the orit is symptoticlly stle. Sometimes, especilly when eigenvlues re close to the unit circle, it my e esier to use the Jury criterion, which sttes tht the second-order polynomil λ λ+ 0 hs ll zeros within the unit circle if nd only if 0 < 1, > 0, nd > 0. Another importnt question concerning periodic orits is the minimum period. Even if the mp (3) is shown to hve single zero in given intervl vector, it does not follow tht the minimum period of the corresponding periodic orit of h is p. To e sure tht p is the minimum period it is sufficient to verify tht the minimum distnce d self (v) = min z k z l 2 (5) k l etween points elonging to the periodic orit v = (z 0, z 1,..., z p 1 ) is positive. If the intervl vectors z 0, z 1,..., z p 1 contin points elonging to the periodic orit it is sufficient to verify tht the intervl k l z k z l 2 does not contin Distnce etween ttrctors Another prolem is to verify tht two ttrctors re different. This cn e solved y computing the distnce etween them. The distnce etween two sets A nd B is defined s d(a, B) = mx{sup inf z w 2, sup inf z w 2}. (6) z A w B w B z A Two sets re different if their distnce is positive. However, note tht usully we do not know the exct positions of ttrctors; we only hve the regions enclosing them. From the fct tht the distnce etween enclosures of ttrctors is positive it does not follow tht the ttrctors re different. Therefore, in generl, we hve to verify tht the enclosures re not overlpping or, in other words, we hve to compute the infimum distnce etween the two sets, which is defined s d inf (A, B) = inf z w 2, (7) z A,w B nd verify tht it is positive. Formlly, to show tht the ttrctors re different, we use the following property: If sets of oxes k z k nd l w l re enclosures of ttrctors A nd B, respectively, nd the intervl k,l z k w l 2 does not contin zero, then the ttrctors re different. The infimum distnce etween two ttrctors is lso interesting from the dynmicl point of view. It tells us how close two trjectories elonging to different ttrctors could e. Sometimes, the enclosures of some ttrctors re composed of severl components, nd we know tht the ttrctors hve non-empty intersection with ech component; this is usully the cse for periodic ttrctors. Then, y computing distnces etween components we cn otin lower ound of the distnce etween the ttrctors The continution method to find the existence region of sink When point in the prmeter spce with sink is found one my use the continution method to find connected region in the prmeter spce for which this periodic orit exists. The serch is sed on the ssumption tht the position of the orit chnges continuously with the prmeters. Usully, the most effective method to find the region of existence is to continue long the order of the region using for exmple the simplex continution method [Allgower & Georg, 1980]. Unfortuntely, in our cse this procedure did not produce correct results: some prts of the existence regions were not found (for exmple one

5 Numericl study of coexisting sinks for the Hénon mp 5 rnch of the swllowtil structure shown in the next section ws missing). This filure ws cused y the fct tht in some cses two sinks with the sme period coexist, nd continution long the order fils t points where the second sink emerges. Therefore, we use the following method: the prmeter spce is covered y rectngulr grid of points, nd continution in ll directions is crried out. For ech grid point, sinks which were otined y the continution procedure in previous steps re rememered, nd continution is performed in ech direction for ech sink Numericl study of sins of ttrction When we detect more thn one ttrctor for the system, n interesting chllenge is to determine which trjectories converge to which ttrctor. For n ttrctor A we define its sin of ttrction B(A) with respect to the mp f s the set of points for which trjectory strting in this set converges to A: B(A) = {x: d( f n (x), A) 0 for n }. (8) Bsins of ttrction cn e studied numericlly y computing trjectories emnting from different initil conditions, nd checking which ttrctor ech trjectory converges to. The initil points cn e selected ritrrily, ut nturl choice is to select them from rectngulr grid. To visulize the sins of ttrction the positions of initil points whose trjectories tend to different ttrctors re plotted using different colors. An importnt notion in the context of sins of ttrction is the so-clled immedite sin size. It is defined s the lrgest numer r ε such tht ll trjectories strting closer thn r ε from the ttrctor do not escpe further thn ε from the ttrctor, nd converge to the ttrctor: r ε (A) = rg mx{d( f n (x), A) ε n 0 nd d( f n (x), A) n 0 for ll x such tht d(x, A) r}. (9) r The choice of ε is somewht ritrry. Usully, choosing ε couple of times lrger thn the expected immedite sin size works well. 3. Results In order to locte multiple ttrctors, we hve performed the following serch in prmeter spce points were chosen in the rectngle (, ) [0, 2] [0, 0.5]. For ech point in prmeter spce, 2000 initil points in phse spce were rndomly selected. For ech such initil point, trjectory of length ws computed. The first itertions were discrded (to void trnsient ehviour), nd the finl itertions were used to fill ins. For inning only the x-vrile ws used. Divergent trjectories were detected nd discrded. All initil points with less thn 20% ins non-empty were recorded. Similr computtions hve een done for points in the rectngle of prmeter vlues (, ) [1.3, 1.5] [, ]. The results of these computtion re reported in Tle 1, where some exmples of prmeters for which three ttrctors were found re given, long with short descriptions of oserved ttrctors. In ll cses, we hve proved the existence of three ttrctors, with the exception of cse 14, where we were unle to prove the existence of the third ttrctor due to its very smll immedite sin of ttrction (further explntions re given lter). In ll cses where we report sink, we hve proved the existence of sink using the intervl Newton method, with the exception of the period-72 orit for cse 5. In this cse the existence of the orit ws proved, ut we could not verify from the Jcoin mtrix tht the orit is stle. Nevertheless, the existence of n ttrctor enclosing this prticulr orit ws estlished using the generl method descried in Section 2.2. In wht follows, severl exmples re discussed in more detils The cse = 1.338, = Using the intervl Newton opertor, we prove tht for = 1.338, = there exist three stle periodic orits with periods 7, 15, nd 30, respectively. The properties of the sinks re collected in Tle 2. For ech sink we report its period p, rigorous ounds of its self distnce d self s defined in (5), rigorous ounds for the eigenvlue λ 1 with the lrgest solute vlue, the lrgest Lypunov exponent l 1 computed s l 1 = p 1 log λ 1 (l 1 is computed non-rigorously), the immedite sin size r ε, nd the percentge p conv of trjectories converging to this sink, when finding sins of ttrction.

6 6 Z. Glis & W. Tucker Tle 1. Prmeter vlues with multiple ttrctors ttrctors period-7, period-15, period period-7, period-18, period period-7, period-18, period period-7, period-18, period period-22, period-22, period period-8, period-12, period period-8, period-14, period period-8, period-16, period period-6, period-8, chotic? period-6, period-8, chotic? period-14, period-36, chotic? period-12, period-16, chotic? period-8, period-12, chotic? period-14, period-32, chotic? period-8, chotic?, chotic? Tle 2. = 1.338, = ; properties of the sinks, p period of the sink, d self the self distnce of the sink, λ 1 rigorous ounds for the eigenvlue with the lrgest solute vlue, l 1 the lrgest Lypunov exponent, r ε the immedite sin size, p conv the percentge of trjectories converging to the sink k p d self λ 1 l 1 r ε p conv % % % 0.6 y 0 x Fig. 2. = 1.338, = , () period-7 sink (+), period-15 sink ( ), period-30 sink ( ), () sins of ttrction of the three sinks, x [ 2, 2], y [ 1, 1]

7 Numericl study of coexisting sinks for the Hénon mp 7 Tle 3. = 1.338, = , distnces etween sinks k l d inf (o k, o l ) d(o k, o l ) The sinks re shown in Fig. 2(). One cn see tht the period-7 sink is locted fr from the other two sinks, nd tht the two longer sinks lmost coincide. A very low self distnce of the third sink (d self (o 3 ) ) indictes tht the sink might e the result of the period-douling ifurction. Looking t the picture lone, it is difficult to sy whether the two longer periodic orits re different. The distnces etween the sinks computed using the formuls (6) nd (7) re reported in Tle 3. The results confirm tht the shortest sink is fr wy from the other sinks (the orit distnce is lrger thn 0.66, nd the infimum distnce is lrger thn 0.07), nd tht the period-15 nd period-30 orits re locted close to ech other (their distnce is pproximtely nd their infimum distnce is close to ). It is cler tht the sum of immedite sin sizes of two sinks hs to e smller thn the infimum distnce etween the sinks ( < ). Bsins of ttrction of the three sinks were found numericlly initil points re selected uniformly in the ox (x, y) [ 2, 2] [ 1, 1]. The results re shown in Fig. 2(). Initil points for which trjectories converge to different sinks re plotted using different colors. Initil points with diverging trjectories re left white. Thin structures corresponding to immedite sins of ttrction re visile, however in most regions in the stte spce the sins look rndom %, 22.36%, nd 18.56% of trjectories converge to the period-7, period-15, nd period-30 sinks, respectively. The remining 43.96% of trjectories diverge. Using the continution procedure, we found regions in the prmeter plne for which the three sinks oserved for = 1.338, = exist. The results re shown in Fig. 3(,,c). In ll cses the regions hve swllowtil structure (compre [Hnsen & Cvitnović, 1998]). The intersection of the existence regions is shown in Fig. 3(d). This lso illustrtes tht the existence region of the period-7 orit is the lrgest, nd encloses the intersection of the two other regions. A more detiled picture is shown in Fig. 3(e). In order to explin cretion of the region with 3 sinks, let us denote swllowtils corresponding to period-7, period-15 nd period-30 sinks y letters α, β, nd γ, respectively. The rnches in the swllowtil structures re numered in the clockwise mnner, in such wy tht the outer rnches hve numers 1 nd 4 (compre Fig. 3). One cn see tht some of the rnches of the period-15 nd period-30 swllowtils hve common edges. Nmely, β 3 hs common edge with γ 4 nd β 1 hs common edge with γ 1. At these oundries, the period-15 orit looses stility, nd the period-30 orit is creted vi the period-douling ifurction. For us, it is importnt tht β 3 one of the inner rnches of the period-15 swllowtil hs common edge with γ 4. As consequence, β 2 the second inner rnch of the period-15 swllowtil intersects the γ 4 rnch. The intersection hs prllelogrm shpe. In this region of intersection there exist three sinks the third one is the period-7 sink, which exists in the region enclosing the whole picture (compre Fig. 3(e)). Summrizing, the explntion of the coexistence of three sinks with different periods is the following: prt of the swllowtil structure (of the period-15 sink in the cse oserved) is enclosed in the region of existence of nother sink (the period-7 sink in this exmple). Within certin region in the swllowtil structure, two sinks with the sme period (period-15 in this exmple) coexist. When this region is left into one of the swllowtil inner rnches, one of the sinks looses stility nd stle periodic orit is orn through the period-douling ifurction ( period-30 orit in the cse oserved). In neighorhood of the swllowtil, there re mny other regions (perhps infinitely mny) where three sinks of vrious periods exist. Severl exmples of points elonging to these regions re plotted in Fig. 3(e) nd collected in Tle 4. First, note tht in prt of the swllowtil structure two sinks with the sme period coexist. An exmple is point (, ) = (1.388, ) plotted with the + symol in Fig. 3(e). Second, note tht there is symmetric period-30 region locted on the other side of the period-15 swllowtil. This region intersects the β 3 rnch of the period-15 swllowtil, nd s consequence symmetric prllelogrm shped region with three sinks of period 7, 15, nd 30 exists. For exmple, the point (, ) = ( , ) (plotted using the symol) elongs to this

8 8 Z. Glis & W. Tucker 0.35 () α () β 1 β α α 3 α (c) γ 1 β β (d) γ 2 γ 3 γ (e) α 2 γ 2 β 2 γ 3 γ 4 β Fig. 3. Existence regions of the three sinks, () period-7 sink, () period-15 sink, (c) period-30 sink, (d) intersection of the existence regions, (e) enlrgement of (d) nd positions of points elonging to other regions of coexistence of three sinks region. Another region is creted t the intersection of two period-30 swllowtils. This hppens in the corner etween rnches β 2 nd β 3 outside the period-15 swllowtil. In the rhomus shped region two sinks of period-30 coexist. For exmple, the point (, ) = ( , ) (plotted with the symol) elongs to this region. Yet nother region exists in the β 2 rnch of the period-15 swllowtil, where period-30 sink looses stility, nd period-60 sink is orn vi the period-douling ifurction. For exmple, the point (, ) = ( , ) plotted with the symol is in this region. One my stte the hypotheses tht there is sequence of period-douling ifurctions locted within the β 2 rnch of the period-15 swllowtil. If this is the cse, then there re infinitely mny regions of coexistence of three sinks with periods 7, 15, nd 15 2 n, for n = 0, 1, 2,.... We hve confirmed this result for

9 Numericl study of coexisting sinks for the Hénon mp 9 Tle 4. Fig. 3(e). Prmeter vlues with multiple ttrctors close to = 1.338, = nd symol used to plot the corresponding point in symol sinks periods , 15, , 15, , 15, , 30, , 15, , 15, 30 n = 0, 1, 2. The finl interesting region which we descrie here is creted y the intersection of the γ 3 rnch of the period- 30 swllowtil nd the β 2 rnch of the period-15 swllowtil. The point (, ) = ( , ) plotted with the symol elongs to this region. One cn expect tht similrly the period-60 structure intersects the period-15 swllowtil, nd nother region is creted. Using intervl methods, we hve verified tht ll periodic orits listed in Tle 4 exist nd re stle, with the exception of the period-60 orit, for which only the existence hs een proved. Here, the Jcoin mtrix otined hs very wide entries, nd rigorous computtion of the eigenvlues is not fesile. Nturlly, it would e possile to use higher precision intervl rithmetic to prove stility in this cse The cse = , = 586 We prove tht for = , = 586 there exist three sinks with periods 7, 18, nd 36, respectively. Properties of the sinks re collected in Tle 5. Tle 5. = , = 586; properties of the sinks p d self λ 1 l 1 r ε p conv % % % 5 The sinks nd their sins of ttrction re plotted in Fig. 4. The structure of sins of ttrction is similr to the previous cse (compre Figure 2()). This time much fewer trjectories (only 5.38%) converge to the period-7 sink. The existence regions of the period-7, period-18, nd period-36 stle periodic orits otined using the continution method re shown in Fig. 5. The sitution is similr to the cse = 1.338, = The min difference here is tht the size of the period-7 region in the re of interest is much smller. Actully, this is nother rnch of the period-7 swllowtil structure discussed erlier (see Fig. 3()). Another difference is tht the swllowtil structure is less symmetric. But the explntion for the emergence of three sinks is the sme. Prt of the swllowtil structure of the period-18 sink is enclosed in the region of existence of the period-7 sink. One of the period-18 sinks undergoes the period-douling ifurction, nd period-36 sink is orn. As in the previous cse, other regions with three sinks in the neighorhood of the swllowtil structure cn lso e oserved The cse = 0.98, = 415 Using the intervl Newton opertor, we estlish tht for = 0.98, = 415 there exist three stle periodic orits with periods 8, 12, nd 20, respectively. Properties of the sinks re collected in Tle 6. The sinks nd their sins re shown in Fig. 6(). Note tht the stripes in sin structures re much wider thn in the previous exmples. This is relted to considerly lrger immedite sin size of the period-8 sink. The existence regions of the period-8, period-12, nd period-20 stle periodic orits otined using the continution method re shown in Fig. 7(,,c). Note tht the first structure is rther complex. The intersection of the

10 10 Z. Glis & W. Tucker y () () x Fig. 4. = , = 586, () period-7 sink (+), period-18 sink ( ), period-36 sink ( ), () sins of ttrction of the three sinks, x [ 2, 2], y [ 1, 1] 65 () 6 () Fig. 5. () Existence regions of the three sinks round = , = 586, () enlrgement of () Tle 6. = 0.98, = 415; properties of the sinks p d self λ 1 l 1 r ε p conv % % % existence regions is shown in Fig. 7(d). The regions of existence of the sinks re not relted s in the previous exmples. They do not hve common order nd do not form common structure like the swllowtil in the previous exmples. Hence, the explntion of the coexistence of three sinks in this exmple is different; one my sy tht the sinks re independent.

11 Numericl study of coexisting sinks for the Hénon mp y () () x Fig. 6. = 0.98, = 415, () period-8 sink (+), period-12 sink ( ), period-20 sink ( ), () sins of ttrction, x [ 2, 2], y [ 1, 1] Tle 7. = 0.95, = 795; properties of the sinks p d self λ 1 l 1 r ε p conv % % % 4 Note tht the intersection of the period-12 nd period-20 regions is not entirely enclosed in the period-8 region. Hence, one cn expect tht there is nother region with three sinks locted within the intersection of the period-12 nd period-20 regions just ove the period-8 region. Indeed, we hve verified tht for = , = 45 there re three sinks with periods 12, 16, nd 20, respectively (compre lso Fig. 7(d)). The second sink is creted vi the period-douling ifurction from the period-8 sink. One my expect sequence of period-douling ifurctions nd corresponding period-(8 2 n ) regions. This my crete finite or infinite sequence of regions with three sinks depending on how wide the period-(8 2 n ) regions re The cse = 0.95, = 795 Using the intervl Newton opertor, we prove tht for = 0.95, = 795 there exist three stle periodic orits with periods 8, 14, nd 16, respectively. Properties of the sinks re collected in Tle 7. The distnces etween sinks re collected in Tle 8. Note tht the sinks re locted further wy from ech other compred to the previous exmples. The sinks nd their sins of ttrction re plotted in Fig. 8. Note tht only 0.69% of trjectories converge to the period-14 sink. Its immedite sin size is very smll (r ε ). Most orits converge to the period-16 sink with the lrgest immedite sin size of r ε The results on the existence of the sinks otined using the continution method re shown in Fig. 9. First, note tht the region of existence of the period-8 sink is the sme s the one in Fig. 7(). The region of existence of the period-14 sink is very nrrow. Finding point elonging to this region using rute-force computtions requires very fine smpling of the prmeter spce. Also note tht the regions intersect mny times nd s in the previous exmple

12 12 Z. Glis & W. Tucker 0.55 () 0.9 () (c) (d) Fig. 7. Existence regions of the three sinks round = 0.98, = 415, () period-8 orit, () period-12 orit, (c) period-20 orit, (d) orders of the existence regions t the intersection Tle 8. = 0.95, = 795, distnces etween sinks k l d inf (o k, o l ) d(o k, o l ) do not hve common order The cse = 0.972, = 575 In ll previous exmples, we found three coexisting sinks. For = 0.972, = 575, however, we oserve two stle periodic orits nd third ttrctor which ppers to e chotic. Using the intervl Newton opertor we prove tht in this cse there exist two stle periodic orits with periods 12, nd 16, respectively. Properties of the ttrctors re collected in Tle 9. The three numericlly oserved ttrctors re shown in Fig. 10. The period-16 ttrctor looks like period-8 orit; its self distnce is only d self = The period-12 orit is locted very close to wht ppers to e chotic ttrctor. Using the technique descried in Sec. 2.2, we prove tht for = 0.972, = 575 there re three nonoverlpping trpping regions. For this we used ε oxes with ε = (2 10 5, ). The trpping regions found for the period-12 sink, period-16 sink nd the third ttrctor re composed of 9527, 20425, nd ε oxes,

13 Numericl study of coexisting sinks for the Hénon mp y () () x Fig. 8. = 0.95, = 795, () period-8 sink (+), period-14 sink ( ), period-16 sink ( ), () sins of ttrction, x [ 2, 2], y [ 1, 1] 0.55 () 0.5 () Fig. 9. () Existence regions of three sinks round = 0.95, = 795, () enlrgement of () Tle 9. = 0.972, = 575; properties of the ttrctors p d self λ 1 l 1 r ε p conv % % 3? 0? 62.44% respectively. The trpping region for the chotic ttrctor is composed of four components. For lrger oxes the method fils; when ε = (10 4, 10 4 ) the trpping region found for the period-12 orit encloses oth the period-12 orit nd the third ttrctor.

14 14 Z. Glis & W. Tucker 0.8 y x Fig. 10. = 0.972, = 575, () period-12 sink (+), period-16 sink ( ), nd chotic (?) ttrctor (dots), () sins of ttrction, x [ 2, 2], y [ 1, 1] Tle 10. = 0.984, = 33; properties of the ttrctors p d self λ 1 l 1 r ε p conv % % 6 3? 0? 66.60% 3.6. The cse = 0.984, = 33 In this cse, we lso oserve two sinks nd (possily) chotic ttrctor. The difference is tht the chotic ttrctor now occupies much smller re in the stte spce. Using the intervl Newton opertor we show tht for = 0.984, = 33 there exist two stle periodic orits with periods 8, nd 12, respectively. Properties of the ttrctors re collected in Tle 10. The ttrctors re shown in Fig. 11. Agin, we cn prove tht there re three non-overlpping trpping regions in this cse. For this ε oxes with ε = (2 10 6, ) were used. The trpping regions found for the period-8 sink, period-12 sink nd the third ttrctor re composed of 1539, 6527, nd ε oxes, respectively. The trpping region for the chotic ttrctor is composed 20 components. Therefore, one my expect tht the ttrctor is creted y sequence of period-douling ifurctions originting from period-20 sink The cse = 1.044, = 43 In this cse, s in two previous cses we lso oserve two sinks nd (possily) chotic ttrctor. This cse is interesting ecuse we were le to locte mny regions with three sinks in its neighorhood. The ttrctors nd their sins of ttrction re plotted in Fig. 12. Not tht the ttrctors re more seprted from ech other thn in the previous exmple. Using the intervl Newton opertor we prove tht for = 1.044, = 43 there exist two stle periodic orits with periods 14 nd 32, respectively. Properties of the ttrctors re collected in Tle 11. For the period-32 orit we were not le to compute the eigenvlues. Nevertheless, the orit ws proved to e stle using Jury s criterion.

15 Numericl study of coexisting sinks for the Hénon mp y x Fig. 11. = 0.984, = 33, () period-8 sink (+), period-12 sink ( ), nd chotic (?) ttrctor ( ), () sins of ttrction, x [ 2, 2], y [ 1, 1] y () () x Fig. 12. = 1.044, = 43, period-12 sink ( ), period-32 sink (+) nd chotic(?) ttrctor ( ), () sins of ttrction, x [ 2, 2], y [ 1, 1] Despite our ttempts to use vrious sizes of ε oxes, we were not le to prove the existence of trpping region enclosing the third ttrctor. This is perhps due to its very smll sin of ttrction, only 0.52% of the computed trjectories converge to this ttrctor.

16 16 Z. Glis & W. Tucker Tle 11. = 1.044, = 43; properties of the ttrctors p d self λ 1 l 1 r ε p conv % ? % 3? 0.52% One cn expect tht the (seemingly) chotic ttrctor is creted y sequence of period douling ifurctions. If this is the cse there could e prmeter region close to = 1.044, = 43 with three sinks. To confirm this hypothesis we hve crried out locl serch for multiple ttrctors. A numer of points in the prmeter spce with three coexisting sinks were locted. The results re collected in Tle 12. For ech region with three sinks only one exmple is given. Tle 12. Prmeter vlues with three sinks close to = 1.044, = 43 sinks periods , 16, , 16, , 16, , 16, , 16, , 16, , 22, , 22, , 32, , 32, , 32, , 32, , 14, 32, , 42, 64 We hve proved tht for = nd = 43 there exist three sinks with periods 14, 16, nd 22, respectively. It is interesting to know tht their immedite sin sizes re , , nd , respectively. The extremely smll vlue of the immedite sin size for the third sink might e the reson why it ws not possile to find trpping region enclosing the chotic ttrctor for = 1.044, = 43. Note tht finding trpping region for the period-22 orit for = , = 43 lso did not work. One cn see from Tle 12 tht in neighorhood of the point = 1.044, = 43 there re mny different regions with three coexisting sinks. In order to get etter understnding of the reltions etween these regions, we hve found existence regions of vrious sinks. The serch ws performed using the continution method strted t points from Tle 12. The results re shown in Fig. 13. The lrgest region is the region of existence of period-14 sink. Aprt from it there re three structures of prllel stripes. Two of them re very nrrow. The widest structure contins regions of existence of seven sinks. The first three of them re period-16, period-32, nd period-64 sinks. The corresponding existence regions hve common orders, which mens tht period-32 nd period-64 sinks re orn y the period-douling ifurction. The widest structure contins lso four nrrow prllel stripes, which re the regions of existence of period-80, period-48, period-80 nd period-64 sinks. Note tht these regions re seprted from ech other. The second set of prllel stripes (lelled 22 ) is lmost horizontl. Although it is very nrrow (in fct one cn see only single stripe in the picture), it contins four existence regions. The regions of existence of period-22, period-44, nd period-88 sinks hve common orders. The lst stripe in this structure is the region of existence of period-66 sink, nd is seprted from other stripes in this structure. The lst structure (lelled 42 ) contins regions of existence of period-42 nd period-84 sinks. The intersection of ech of the two stripes elonging to the prllel structures descried ove cretes region of existence of three sinks. It follows tht there re t lest 50 ( ) distinct regions of coexistence of three sinks shown in Fig. 13. Some exmples were lredy given in Tle 12.

17 Numericl study of coexisting sinks for the Hénon mp Fig. 13. Existence regions of three sinks round = 1.044, = 43 Tle 13. = 0.97, = 66; properties of the ttrctors p d self λ 1 l 1 r ε p conv % 2? 0? 55.27% 3? 0? 8.17% 3.8. The cse = 0.97, = 66 In the previous three cses, two sinks nd one (possily) chotic ttrctor were oserved. In the present cse we hve only one periodic ttrctor, nd two potentilly chotic ttrctors. The existence of the sink is estlished using the intervl Newton opertor. Properties of the ttrctors re collected in Tle 13. The three ttrctors nd their sins of ttrction re shown in Fig. 14. Using the technique descried in Sec. 2.2, we prove tht there re three non-overlpping trpping regions. For ε = (10 4, 10 4 ) the trpping regions found for the two chotic ttrctors overlp. For ε = (10 5, 10 5 ), the three trpping regions do not overlp. They re composed of 8385, , nd ε oxes, respectively. These trpping regions re composed of 8, 12, nd 4 connected regions respectively. Using ε = (2 10 6, ) gives trpping regions composed of 9369, , nd oxes. The lst trpping region is decomposed into 20 connected components, which is much finer enclosure of the ttrctor. 4. Conclusions In this pper, we hve explored lrge prmeter domin for the Hénon mp, nd locted regions for which there re (t lest) three coexisting ttrctors. In the strongly dissiptive cse ( [0, 0.5]) tht we hve considered, it is hrd to find sinks due to their smll immedite sins of ttrction. The ttrctors were locted y n ensemle of rute-force computtions, nd lter verified y intervl nlysis nd rigorous numerics. This ensures tht the reported sinks re not numericl rtifcts, ut true mthemticl ojects. In order to understnd the cretion (nd destruction) of coexisting sinks, we computed their locl existence regions. This provided cler picture of the locl dynmics, nd lso suggested other prmeter regions where coexisting sinks reside. Acknowledgments This work ws supported in prt y the AGH University of Science nd Technology, grnt no , nd the Swedish Reserch Council Grnt no

18 18 REFERENCES 0.8 y () () x Fig. 14. = 0.97, = 66, () period-8 sink (+), first chotic(?) ttrctor ( ), second chotic(?) ttrctor ( ), () sins of ttrction, x [ 2, 2], y [ 1, 1] References Allgower, E. & Georg, K. [1980] Simplicil nd continution methods for pproximting fixed points nd solutions to systems of equtions, SIAM Review 22, Benedicks, M. & Crleson, L. [1991] The dynmics of the Hénon mp, Annls of Mthemtics 133, Glis, Z. [2001] Intervl methods for rigorous investigtions of periodic orits, Int. J. Bifurction nd Chos 11, Glis, Z. [2006] Counting low-period cycles for flows, Int. J. Bifurction nd Chos 16, Glis, Z. & Tucker, W. [2011] Vlidted study of the existence of short cycles for chotic systems using symolic dynmics nd intervl tools, Int. J. Bifurction nd Chos 21, Hnsen, K. & Cvitnović, P. [1998] Bifurction structures in mps of Hénon type, Nonlinerity 11, Hénon, M. [1976] A two dimensionl mp with strnge ttrctor, Commun. Mth. Phys. 50, Moore, R. [1966] Intervl Anlysis (Prentice Hll, Englewood Cliffs, NJ). Neumier, A. [1990] Intervl methods for systems of equtions (Cmridge University Press). Newhouse, S. [1979] The undnce of wild hyperolic sets, Pul. Mth. Inst. Hutes Etudes Sei. 50, Picqudio, M., Hnsen, R. & Pont, F. [2002] On the Mendès Frnce frctl dimension of strnge ttrctors: the Hénon cse, Int. J. Bifurction nd Chos 12, Zhiping, Y., Kostelich, E. & Yorke, J. [1991] Clculting stle nd unstle mnifolds, Int. J. Bifurction nd Chos 1,