Bifurcation structures in maps of Hénon type

Transcription

1 Nonlinerity 11 (1998) Printed in the UK PII: S (98) Bifurction structures in mps of Hénon type Ki T Hnsen nd Predrg Cvitnović NORDITA, Blegdmsvej 17, DK-2100 Copenhgen Ø, Denmrk Center for Chos nd Turulence Studies, Niels Bohr Institute, Blegdmsvej 17, DK-2100 Copenhgen Ø, Denmrk Received 9 My 1997 Recommended y P Grsserger Astrct. We construct series of n-unimodl pproximtions to mps of the Hénon type nd utilize the ssocited symolic dynmics to descrie the possile ifurction structures for such mps. We construct the ifurction surfces of the short periodic orits in the topologicl prmeter spce nd check numericlly tht the Hénon mp prmeter plne (, ) is topologiclly equivlent to two-dimensionl section through the infinite-dimensionl prmeter spce chrcterizing generic mp of the Hénon type. PACS numers: 0320I, 0545B 1. Introduction While the topologicl dynmics of unimodl nd multimodl one-dimensionl mppings is well understood [30, 28], clssifiction of ll possile topologiclly distinct dynmicl systems in two or more dimensions remins n open prolem. The gol of this pper is to develop theory of ifurction digrms which clssify nd order topologiclly distinct ifurction sequences for two-dimensionl invertile mps of the Hénon type [20]. We consider mps which stretch nd fold the phse spce once under one mpping, exemplified y Smle horseshoe [33]. We study here the mps which do not hve complete inry Cntor set repeller such s complete horseshoe mp hs, ut ssume tht the dmissile orits cn still e uniquely identified y suset of the inry symolic dynmics itinerries [12, 13, 7]. This ssumption hs not een proved for the Hénon mp, ut is supported y ll of our numericl results. The Hénon mp [20] is n invertile mpping of two-dimensionl plne into itself: x t+1 = 1 xt 2 + y t y t+1 = x t. Equivlently, the Hénon mp cn e defined y the 2-step recurrence reltion x t+1 = 1 xt 2 + x t 1. (1) The Hénon mp is one of the simplest models of Poincré mp of three-dimensionl invertile flow. Our description of the ifurction digrm for ll mps of the Hénon type Present ddress: ABB Corporte Reserch, PO Box 90, N-1361 Billingstd, Norwy. E-mil: khnsen@nordit.dk Also t: Deprtment of Physics nd Astronomy, Northwestern University, 2145 Sheridn Rod, Evnston, IL , USA. E-mil: predrg@ni.dk /98/ $19.50 c 1998 IOP Pulishing Ltd nd LMS Pulishing Ltd 1233

2 1234 K T Hnsen nd P Cvitnović (once-folding mps) will e generic in the sense tht it will e vlid for ll flows which fold the phse spce t most once etween susequent Poincré sections. Detiled numericl investigtions of such structures for the Hénon mp hve een crried out y Mir nd co-workers [8, 29, 5], s well s mny other uthors [1, 2 4, 12, 13, 15, 20, 27, 29, 32], to cite ut few. Our pproch is different in so fr tht insted of studying the ifurction structure of the Hénon mp or the Lozi mp [22], we offer here topologicl chrcteriztion of the prmeter spce nd the dmissile orits for ll mps of the Hénon type. The pproch is closely relted to the pruning front conjecture [7, 6]. There the phse spce stle unstle mnifolds folitions re replced y strightened-out symol plne street mp pplicle to ny mp of the Hénon type. The totlity of ll turning points of the unstle mnifold of the mp delinetes the pruning front in the symol plne, the order etween the dmissile nd indmissile orits. For unimodl one-dimensionl mppings the pruning front is specified y single prmeter, the kneding invrint [30, 28], ut for two-dimensionl mppings infinitely mny prmeters re required to specify the pruning front, tht is to sy the infinity of the turning points of the unstle mnifold. However, one striking feture of smooth dissiptive once-folding mps is their hierrchic folition; for smll vlues of the modulus of in corsest resolution they look like unimodl mps, under somewht finer resolution two primry folds re discernile, nd so forth. This oservtion is the sis for systemtic pproximtion to two-dimensionl once-folding mps y sequences of n-unimodl one-dimensionl mps tht we shll develop here; we shll construct nested sequences of prmeter topologicl street mps of ll dmissile (ut strongly hierrchiclly ordered) prmetriztions of once-folding mps. A symol plne together with pruning front specifies symolic dynmics of given once-folding mp; our n-unimodl pproximtion to descries ll dmissile once-folding mps, with point in the topologicl prmeter spce corresponding to prticulr topologicl prmeter. The ifurction theory presented here is sed on [16], nd the ifurction structures in multi-unimodl one-dimensionl mps re discussed in the spirit of the work in [18]. 2. Unimodl pproximtion In the 0 limit the unstle mnifold of the Hénon mp shrinks to one-dimensionl rc, folds of the stle mnifold stretch off to infinity, nd the Hénon mp (1) reduces to the one-dimensionl qudrtic mp x t+1 = 1 x 2 t (2) with one criticl point x c = 0. The symolic description for unimodl mp with criticl point x c is defined y s t = { 1 if xt >x c 0 if x t <x c. (3) The infinite symol sequence S(x) = s 1 s 2 s 3... is the (future) itinerry of the point x = x 0. The dynmics cts on this sequence s shift: S(f t (x)) = σ t S(x) = s 1+t s 2+t s 3+t... (4) Symols L nd R re often used [26] insted of 0 nd 1, indicting tht the point x t lies either to the left or right of the criticl point. The criticl point x c my e denoted y s t = C.

3 Bifurction structures in mps of Hénon type 1235 To ny given itinerry S we ssocite the point γ(s) [0, 1] constructed s follows { wt if s t+1 = 0 w t+1 = w 1 = s 1 1 w t if s t+1 = 1 (5) γ(s)=0.w 1 w 2 w 3...= w t /2 t. t=1 The numer γ(s) is independent of detils of prticulr unimodl mp nd preserves the ordering of x in the sense tht if ˆx >xthen γ(s(ˆx)) > γ (S(x)) for ny unimodl mp. We shll refer to γ(s)s the (future) topologicl coordinte or the (future) symolic coordinte Kneding vlues If the prmeter in the qudrtic mp (2) is >2then the itertes of the criticl point x c diverge for t. As long s 2, ny sequence S composed of letters s i ={0,1} is dmissile, nd ny vlue of 0 γ < 1 corresponds to n dmissile orit in the non-wndering set of the mp. The corresponding repeller is complete inry Cntor set. For <2only suset of the points in the intervl γ [0, 1] corresponds to dmissile orits. The foridden symolic vlues re determined y oserving tht the lrgest x t vlue in n orit x 1 x 2 x 3... hs to e smller thn or equl to the imge of the criticl point, the criticl vlue f(x c ). Let K = S(x c ) e the itinerry of the criticl point x 0 = x c, denoting the kneding sequence of the mp. The corresponding topologicl coordinte is clled the kneding vlue [28] = γ(k)=γ(s(x c )). (6) If γ(s)>γ(k), the point x whose itinerry is S would hve x>f(x c )nd cnnot e n dmissile orit. Let ˆγ(S)=sup γ(σ m (S)) (7) m e the mximl vlue, the highest topologicl coordinte reched y the orit x 1 x 2 x 3... Theorem 1 ([31, 26, 14, 30, 28]). Let e the kneding vlue of the criticl point, nd ˆγ(S) e the mximl vlue of the orit S. Then the orit S is dmissile if nd only if ˆγ(S). We shll cll the intervl (, 1] the primry pruned intervl. The orit S is indmissile if γ of ny shifted sequence of S flls into this intervl. While unimodl mp my depend on mny ritrrily chosen prmeters, its dynmics determines nd is determined y unique kneding vlue. There exists mp from the prmeter of specific unimodl mp to the -line, nd thus we cn use to prmetrize ny unimodl mp. We shll cll the topologicl prmeter of the mp. The jumps in s function of correspond to indmissile vlues of the topologicl prmeter. Ech jump in corresponds to stility window ssocited with stle cycle of smooth unimodl mp. For the qudrtic mp (2) increses monotoniclly with the prmeter, ut in generl such monotonicity need not e the cse Periodic orits A periodic point (or cycle point) x i elonging to cycle of period n is rel solution of f n (x i ) = x i, i = 0,1,...,n 1, f r (x i ) x i for r<n. (8)

4 1236 K T Hnsen nd P Cvitnović Tle 1. The mximl vlues of unimodl mp cycles up to length 5. S ˆγ(S) = = = = = = = = = = = = = = The nth iterte of unimodl mp crosses the digonl t most 2 n times. Similrly, the ckwrd nd forwrd Smle horseshoes intersect t most 2 n times, nd therefore there will e 2 n or fewer periodic points of length n. A cycle of length n corresponds to n infinite repetition of length n symol string, customrily indicted y line over the string: S = (s 1 s 2 s 3...s n ) =s 1 s 2 s 3...s n. If s 1 s 2...s n is the symol string ssocited with x 0, its cyclic permuttion s k s k+1...s n s 1...s k 1 corresponds to the point x k 1 in the sme cycle. A cycle p is clled prime if its itinerry S cnnot e written s repetition of shorter lock S. A cycle of differentile one-dimensionl mp is stle if d dx f n (x 1 ) = f (x n )f (x n 1 )...f (x 2 )f (x 1 ) < 1. A cycle is superstle if the ove product vnishes, i.e. if the orit includes criticl point. The intervl of prmeter vlues for which cycle p is stle is clled the stility window of p. Ech cycle yields n rtionl vlues of γ. It follows from (5) tht if the repeting string s 1,s 2,...s n contins n odd numer of 1 s, the string of well-ordered symols w 1 w 2...w n hs to e of the doule length efore it repets itself. The vlue γ is geometricl sum which we cn write s the finite sum 22n 2n γ(s 1 s 2...s n )= w 2 2n t /2 t. 1 t=1 Using this we cn clculte the ˆγ(S) for ll short cycles. For orits up to length 5 this is done in tle Bifurctions Periodic orits in smooth unimodl mps re genericlly creted either s pir with one stle nd one unstle length n orit in sddle-node ifurction point, or s period 2n orit in period-douling ifurction where period n orit ecomes unstle.

5 Bifurction structures in mps of Hénon type ε 100ε 1001ε 10ε 1011ε Figure 1. Bifurction points from tle 1 plotted s function of the topologicl prmeter. Grey res re indmissile intervls of corresponding to stle windows in smooth unimodl mp. As shorthnd nottion for pirs of orits we use the letter ɛ to denote either 0 or 1. The line over the symol strings is omitted. Immeditely fter sddle-node ifurction the two creted orits oth hve the sme itinerry s 1 s 2...s n with n even numer of 1 s nd with the topologicl prmeter vlue (s 1 s 2...s n ) = ˆγ(s 1 s 2...s n ). Orits with this itinerry exist for ll unimodl mps with ˆγ(s 1 s 2...s n ). As the prmeter in the smooth unimodl mp increses the stle orit psses superstle point nd chnges its symolic dynmics. If we now ssume tht the symol string s 1 s 2...s n is the cyclic permuttion giving the mximum γ vlue, then the itinerry of the stle orit fter the superstle point is s 1 s 2...s n 1 (1 s n ), since the point closest to the criticl point psses through the criticl point. The topologicl prmeter vlue of the mp is then (s 1 s 2...s n 1 (1 s n )). The indmissile topologicl prmeter intervl ((s 1 s 2...s n ), (s 1 s 2...s n 1 (1 s n ))) is then uniquely relted to the prmeter intervl in etween the sddle-point ifurction nd the superstle point, or more loosely speking; to the intervl where the orit s 1 s 2...s n 1 (1 s n ) is stle. In the sme wy there will e n intervl ((s 1 s 2...s n 1 (1 s n )), (s 1 s 2...s n 1 (1 s n )s 1 s 2...s n )) corresponding to the intervl in from where the orit s 1 s 2...s n 1 (1 s n )is superstle to the point where the orit s 1 s 2...s n 1 (1 s n )s 1 s 2...s n is superstle. This intervl includes the period-douling ifurction where the 2n orit s 1 s 2...s n 1 (1 s n )s 1 s 2...s n is creted. From tle 1 we cn find some of the lrgest intervls in corresponding to the stility windows in smooth unimodl mp. The stle period 3 orit window on the prmeter -xis corresponds to the intervl ( 6 7, 8 ) on the line nd so on, see figure Bi-unimodl pproximtion The unimodl pproximtion is n exct description for the Hénon mp for 0, ut not very ccurte for 0. We therefore continue to the next order of refinement nd pproximte the unstle mnifold in figure 2() with two unimodl mps, one ove the other, s sketched in figure 2(). It is importnt to note tht the points in the orit re forced to e on one of the two functions in figure 2() depending on one symol in the pst itinerry: if n orit hs point on the right-hnd side of the horseshoe (symol 1) then its imge is on the upper function nd if n orit hs point on the left-hnd side of the horseshoe (symol 0) then its imge is on the lower function. This is illustrted in figure 2 where we hve drwn the unstle mnifold of the Hénon mp ( = 1.4, = 0.3) nd one period 7 orit. For ech point in the orit we hve written the future itinerry of the point (omitting the line over the

6 1238 K T Hnsen nd P Cvitnović x t xt () x t-1 () xt Figure 2. () The strnge ttrctor (unstle mnifold) nd period 7 orit in the Hénon mp ( = 1.4, = 0.3).() A sketch of i-unimodl pproximtion with the sme periodic orit. symols). The choice etween the upper nd lower hlf of the unstle mnifold depends on the preceding point in the orit, nd hence on the next to lst symol in the symol string lelling the point. We stress tht this mp, constructed from two unimodl mps, is not multivlued mp, since ech point is ssigned unique vlue. We denote this one-dimensionl mp iunimodl insted of imodl not to confuse it with other imodl mps frequently studied in the literture, such s the cuic mp. A point in n orit with itinerry S =...s t 2 s t 1 s t s t+1 s t+2... is mpped in the i-unimodl pproximtion y the one-dimensionl mp { f0 (x t ) if s t 1 = 0 x t+1 = f st 1 (x t ) = f 1 (x t ) if s t 1 = 1. (9) The two criticl points of the functions f 0 nd f 1 yield the two kneding sequences K 0 nd K 1, with the corresponding topologicl prmeter vlues 0 nd 1. The i-unimodl mp f s is descried y the point ( 0, 1 ) in the two-dimensionl topologicl prmeter plne. For n order-reversing two-dimensionl mp which flips, stretches, nd folds the phse spce, the criticl vlue of f 1 is lrger thn tht of f 0, 1 > 0. This is the cse for the Hénon mp with >0. For n order-preserving mpping which stretches nd folds without flipping the criticl vlue of f 1 is smller thn f 0 nd 1 < 0. This is the cse for the Hénon mp with <0. The line = 0 is mpped into the line 1 = 0, the unimodl mp discussed ove. We shll now trce out some of the chrcteristic ifurction structures for the iunimodl pproximtion in this two-dimensionl topologicl prmeter plne. Ech orit (except the two fixed points 0 nd 1) hs two mximl vlues ˆγ 0 nd ˆγ 1 defined s for the unimodl mp (7), ut with the restriction tht the symol s m 1 is equl to the index of ˆγ. If the orit is given y the itinerry S =...s 2 s 1 s 0 s 1 s 2... we hve ˆγ s (S) = sup γ(σ m (S)) with s m 1 = s (10) m

7 Bifurction structures in mps of Hénon type 1239 where σ is the shift (4). An orit S is dmissile if nd only if ˆγ 0 (S) 0 (11) ˆγ 1 (S) 1 so the orit S exists within rectngle in the ( 0, 1 ) plne. The prmeter point point 0 = 1, 1 = 1 corresponds to complete Smle horseshoe for which ll orits exist. In order to hve i-unimodl mp, we hve to require tht the imges of the criticl points re not ove the smllest criticl point. In terms of kneding sequences this constrins s to s = γ(k s ) γ(σ(k s )) (12) which is true if s 0.10 = 2 3. (13) This requirement is less constrining in higher-order multi-unimodl pproximtions Mximl vlues of short cycles We cn now proceed to determine ll cycles up to given length nd determine the topologicl vlues ˆγ s (S) of ll their cycle points. The fixed point 0 hs s 1 = 0, with the only mximl vlue ˆγ 0 (0) = 0. This fixed point exists for 0 ˆγ 0 (0) = 0. In other words, if there is nything in the non-wndering set, the fixed point 0 exists. The fixed point 1 hs s 1 = 1, with the corresponding topologicl coordinte ˆγ 1 (1) = It exists for topologicl prmeter plne vlues 1 ˆγ 1 (1) = 0.10 = 2 3. The 2-cycle 10 hs two cyclic permuttions. Cycle point x 10 with itinerry s 1 s 2 = 10 hs s 0 = s 2 = 0 nd s 1 = s 1 = 1 giving the mximl vlue ˆγ 1 (10) = The second point in the period 2 orit, x 01,isonmpf 0 since s 1 = 0 nd the mximl vlue is ˆγ 0 (01) = Thus this orit exists for the topologicl prmeter vlues 0 ˆγ 0 (10) = = ˆγ 1 (10) = = 4 5. (14) There re two 3-cycles, 100 nd 101 with s 1 = s 2 determining the fold to which cycle point elongs. The 100-cycle cycle points hve the following topologicl coordintes: γ 0 (100) = , γ 1 (010) = , γ 0 (001) = The two mximl vlues re ˆγ 0 (100) nd ˆγ 1 (100), so the region in the topologicl prmeter plne for which 100 exists is given y 0 ˆγ 0 (100) = = ˆγ 1 (100) = = 4 9. (15) The topologicl coordintes of the other 3-cycle cycle points re γ 0 (101) = 0.110, γ 1 (110) = 0.100, γ 1 (011) = 0.010, so the cycle exists for 0 ˆγ 0 (101) = = 6 (16) 7 1 ˆγ 1 (101) = = 4 7. (17)

8 1240 K T Hnsen nd P Cvitnović Tle 2. The mximl vlues of short cycles of the i-unimodl mp. S ˆγ 0 (S) S ˆγ 1 (S) = = = = = = = = = = = = = = = = = = = = = = = = = = Figure 3. Bifurction lines of the period 5 cycles yielding i-unimodl swllowtil ( crossrod re [29, 5]) in the topologicl prmeter plne ( 0, 1 ). We cn continue these clcultions for longer cycles; the ˆγ s vlues for cycles up to length 5 re summrized in tle 2. These vlues yield the ifurction lines for the cycles in the topologicl prmeter plne spce ( 0, 1 ) Bifurction lines in the prmeter plne The ifurction lines given y tle 2 re esier to understnd if we drw the lines in the ( 0, 1 ) plne. The period 1, 2, 3, nd 4 cycles yield single stle cycle nds. For ech cycle only one mximum vlue is lrger thn 2. We note tht the 1, 10, nd 1011 cycles 3 ifurcte long constnt 1 vlues, while 10ɛ nd 100ɛ yield windows long constnt 0. We cn find similr structure for the Hénon mp close to the = 0 line. However, the i-unimodl pproximtion descries lso more interesting higher codimension structures. The simplest exmple is given y the four period 5 cycles 10ɛ 1 1ɛ 2, ɛ i {0,1}. The ifurction lines for these cycles re drwn in figure 3. Ech cycle exists

9 Bifurction structures in mps of Hénon type Figure 4. The topologicl prmeter plne ( 0, 1 ) ifurction lines of the period 5, 6 nd 7 swllowtils. in rectngle in the topologicl prmeter plne. The inccessile topologicl prmeter vlues re shded grey. The 0 = 1 line necessrily crosses the sme stle windows 1011ɛ nd 1001ɛ s the unimodl mp, figure 1, ut long the 1 = 1 line the cycles pir differently, s 1110ɛ nd 1010ɛ. We find in the ( 0, 1 ) plne topologicl structure which we shll refer to s swllowtil, prmeter region within which the two pirs of cycles exchnge prtners. This structure is denoted crossrod re in [29, 5]. This swllowtil crossing is the distinctive feture of i-unimodl mps; we shll illustrte it y finding ll swllowtils for the short cycles up to length 9. If f 0 nd f 1 re smooth functions then the function f (5) (x) x will hve the norml form g = x 3 + ux + v. Solving g = 0 for x, u, nd v close to zero will depend on the two prmeters u nd v, nd the dimensionlity of the norml form prmeter spce (u, v) is clled the codimension of the ifurction [11]. Hence, the swllowtil such s the one illustrted in figure 3 is codimension-2 ifurction structure. The ifurction digrm for the period 6 cycles yields one swllowtil similr to the period 5 swllowtil with the symolic dynmics given y 100ɛ 0 1ɛ 1. In the i-unimodl pproximtion the other period 6 cycles yield simple windows with stle cycles. The period 7 cycles yield three different swllowtils in the topologicl prmeter plne. The swllowtils for period 5, 6, nd 7 cycles re drwn together in figure 4. Longer cycles comine into incresing numers of swllowtils. In figures 5() nd () we disply ll swllowtil crossings for cycles of periods 8 nd 9. The swllowtils re given y the following itinerries. Period 5; 10ɛ 0 1ɛ 1. Period 6; 100ɛ 0 1ɛ 1. Period 7; 1000ɛ 0 1ɛ 1, 10ɛ 0 111ɛ 1 nd 10ɛ 0 101ɛ 1. Period 8; 10000ɛ 0 1ɛ 1, 100ɛ 0 101ɛ 1, 100ɛ 0 111ɛ 1, 10ɛ ɛ 1 nd 1001ɛ 0 10ɛ 1, where the lst swllowtil lies elow the digonl nd occurs for the orienttion-preserving mps ( <0 for the Hénon mp). Period 9; ɛ 0 1ɛ 1, 1000ɛ 0 101ɛ 1, 100ɛ ɛ 1, 100ɛ ɛ 1, 10ɛ ɛ 1, 10ɛ ɛ 1, 10ɛ ɛ 1, 10ɛ ɛ 1, 10011ɛ 0 10ɛ 1 nd 10001ɛ 0 10ɛ 1 where the lst two swllowtils exist for orienttion-preserving mps. Note tht the figures descrie oth the numer of swllowtils of different lengths nd

10 1242 K T Hnsen nd P Cvitnović () () Figure 5. The ifurction lines of the period () 8 nd () 9 swllow tils in the topologicl prmeter plne ( 0, 1 ). their reltive positions in the prmeter plne. We oserve tht numer of swllowtils re ordered simply y rows nd columns. For exmple, ll swllowtils with the symol strings 10 k ɛ 0 1ɛ 1 with k {1,2,...} re plced ove ech other in the ( 0, 1 ) plne, with ech swllowtil nested in etween the two tils of the swllowtil of the preceding shorter cycle, see figure 4. The symolic description for generic swllowtil in the i-unimodl pproximtion is given y the following proposition. Proposition 1. The four cycles tht form i-unimodl swllowtil of n once-folding mp hve following itinerries: S = s 1 s 2 s m 0ɛ 0 s m+3 s m+4...s n 2 1ɛ 1 (18) with the kneding vlues ˆγ 1 (S) = γ(s 1 s 2 s m 0ɛ 0 s m+3 s m+4...s n 2 1ɛ 1 ) (19) ˆγ 0 (S) = γ(s m+3 s m+4...s n 2 1ɛ 1 s 1 s 2 s m 0ɛ 0 ). The swllowtil crossing elongs to the orienttion-reversing mp ( >0for the Hénon mp) if ˆγ(S) = ˆγ 1 (S), nd the orienttion-preserving mp ( <0for the Hénon mp) if ˆγ(S) =ˆγ 0 (S).

11 Bifurction structures in mps of Hénon type ε1001 1ε101 0 Figure 6. Bifurction lines for the homoclinic orits 1ɛ 0 10ɛ 1 1 in the topologicl prmeter plne Aperiodic orits The periodic orits hve ifurctions structures in the i-unimodl prmeter plne similr to those discussed ove, ut the ifurction structure of periodic orits in one-dimensionl i-unimodl mps, discussed in [17], is more complicted thn the ifurction of periodic orits. We will descrie here riefly the ifurction structures of some homoclinic orits. The ifurction lines of the four homoclinic orits 1ɛ 0 10ɛ 1 1, with ɛ 0,ɛ 1 {0,1}re drwn in the topologicl prmeter plne in figure 6. All four orits hve ˆγ 0 = 0.10, the two orits 1ɛ hve ˆγ 1 = 0.110, nd the two orits 1ɛ hve ˆγ 1 = As shown in [17], there exists complicted we of ifurctions connecting these ifurction lines to other ifurction lines in the prmeter plne. The lines of crisis ifurctions nd nd merging re of this type. 4. Four-unimodl pproximtion The i-unimodl pproximtion developed ove cn explin most ut not ll of the ifurction structures oserved in the Hénon mp (, ) prmeter plne discussed elow. To explin further oserved structures we hve to refine the pproximtion nd pproximte the unstle mnifold in figure 2() with four unimodl functions insted of just two s in figure 2(). This four-unimodl reproduces ll ifurctions of the unimodl nd i-unimodl pproximtions, nd yields in ddition more complicted ifurction structures. The choice of the rnch t ech itertion is now determined y the symols of the two preceding points in the orit s 2 s 1, so we lel the four functions y the four symol strings f 10, f 00, f 01 nd f 11. The reltive ordering of the four rnches is given y the wy the horseshoe mp cts on the phse spce, with the functions nested s f 10 <f 00 <f 01 <f 11 for orienttion-reversing mps (the Hénon mp with >0) nd s f 01 <f 11 <f 10 <f 00 for the orienttion-preserving mps (the Hénon mp with <0). Ech mp hs criticl point with n ssocited topologicl prmeter s s determined y the kneding sequence of its criticl point. The reltive ordering of s s is the sme s of the functions themselves. For orienttion-reversing mps , (20) nd for orienttion-preserving mps (21)

12 1244 K T Hnsen nd P Cvitnović An orit S now hs four mximum vlues restricted to the four mps ˆγ s s(s) = sup γ(σ m (S)) with s m 2 = s,s m 1 =s. (22) m As in the i-unimodl pproximtion, we cn esily determine the ˆγ s s of ll the short cycles, nd study ll possile ifurctions of orits in the four-dimensionl topologicl prmeter spce ( 10, 00, 01, 11 ). Since we lck the ility to visulize three-dimensionl ifurctions hyperplnes in four-dimensionl prmeter spce, we will drw ifurction lines in the different two-dimensionl topologicl prmeter sections nd some ifurction plnes in three-dimensionl sections of the full prmeter spce. The six projections of the four-dimensionl spce into two-dimensionl suspces re ( 10, 00 ), ( 10, 01 ), ( 10, 11 ), ( 00, 01 ), ( 00, 11 ), nd ( 01, 11 ). These projections will revel the codimension-2 ifurction structures possile in generic once-folding mp in the four-unimodl. We shll recover the unimodl nd i-unimodl structures lredy discussed ove, together with some new ifurction structures. The projections of the four-dimensionl topologicl prmeter spce into different twodimensionl spces re non-trivil ecuse of the ordering constrints (20) nd (21). In simple one-dimensionl tri-unimodl nd four-unimodl mps [18] we cn scn twodimensionl topologicl prmeter plne ( i, j ) while we let ll the other topologicl prmeter vlues hve the extremum vlue tht llows mximum numer of orits. The two-dimensionl plnes give us ll possile codimension-2 ifurction structures in the system. For the four-unimodl mps discussed here we hve drwn two-dimensionl - plnes where the other s s vlues re s lrge s possile ut restricted y (20) nd (21). The six plnes re: ( 10, 00 ) with 01 = 11 = 1 for >0 nd 01 = 11 = 10 for <0, ( 10, 01 ) with 00 = 01, 11 = 1 for >0 nd 11 = 10, 00 = 1 for <0, ( 10, 11 ) with 00 = 01 = 11 for >0 nd 01 = 11, 00 = 1 for <0, ( 00, 01 ) with 10 = 00, 11 = 1 for >0 nd 11 = 10 = 00 for <0, ( 00, 11 ) with 10 = 00, 01 = 11 for >0 nd 01 = 11, 10 = 00 for <0, ( 01, 11 ) with 10 = 00 = 01 for >0 nd 10 = 00 = 1 for <0. Here the Hénon mp prmeter is used to indicte whether the mp is orienttion reversing or preserving. Some of the ssumed prmeter limits, for exmple 00 = 01 = 11, re impossile in ny smooth mp. This introduces some uncceptle structures (see elow), ut ensures tht we cpture ll possile codimension-2 structures existing in the four-dimensionl prmeter spce. Inequlities (20) nd (21) imply tht the prmeter plnes ( 10, 00 ), ( 01, 11 ) only consist of the upper tringles 00 > 10 nd 11 > 01 respectively nd we do not know the sign of in these plnes. In the other four plnes the digonl corresponds to = 0in the Hénon mp. The ifurctions lines re not necessrily continuous cross the digonl ecuse the projections re different for >0nd <0. The four plnes ( 10, 01 ), ( 10, 11 ), ( 00, 01 ), nd ( 00, 11 ) (e.g. figures 8() (e)) cn e regrded s eight tringulr plnes drwn together for convenience Period 4 orit cusp ifurction The shortest orits which exhiit new type of codimension-2 singulrity in the fourunimodl re the period 4 orits 1000, 1001 nd In figure 7 the ifurction lines for these three orits re drwn in the topologicl prmeter plne ( 10, 00 ). The unstle orit 1001 is common in the two tils nd t point where the two tils meet this orit yields cusp ifurction. This cusp is similr to the codimension-2 cusp in the centre of

13 Bifurction structures in mps of Hénon type Figure 7. The ifurction of the orits 1000, 1001 nd 1011 in the topologicl prmeter plne ( 10, 00 ). Tle 3. The four symolic vlues 10, 00, 01 nd 11 of the nine period 6 cycles τ(000010) = τ(100000) = τ(000011) = τ(100001) = τ(100011) = τ(111000) = τ(100010) = τ(101000) = τ(010011) = τ(101001) = τ(110011) = τ(111001) = τ(110010) = τ(011001) = τ(111011) = τ(101011) = τ(000100) = τ(001100) = τ(000110) = τ(011100) = τ(001110) = τ(010100) = τ(110100) = τ(100110) = τ(111100) = τ(100111) = τ(100101) = τ(010110) = τ(111101) = τ(101111) = τ(110101) = τ(101110) = the swllowtils discussed in the i-unimodl cse, ut unlike i-unimodl swllowtil there is no connection to two other tils Period 6 swllowtils It turns out tht the period 5 orits do not yield ny new nd interesting structures in the four-unimodl pproximtion. For the period 6 orits we find four new codimension-2 structures. All six topologicl prmeter plnes of period 6 orits re drwn in figures 8() (f ). The symolic vlues ˆγ(S) used to drw these figures re given in tle 3. We now discuss figure 8 in detil. The period 6 swllowtil structure in figure 8(e) is the i-unimodl swllowtil 100ɛ 1 1ɛ 2 lredy discussed nd drwn in figure 4.

14 1246 K T Hnsen nd P Cvitnović () 10 () (c) 10 (d) (e) 00 (f) 01 Figure 8. The ifurction lines of period 6 orits in the six four-unimodl topologicl prmeter plnes.

15 Bifurction structures in mps of Hénon type 1247 The swllowtil in figure 8(c) is legl swllowtil of the once-folding mp ut one which we did not find in the i-unimodl topologicl prmeter plne. The symolic description of the orits in this swllowtil is 10ɛ 1 11ɛ 2. Figure 8() illustrtes some interesting new ifurction structures. Here we find two cusp structures involving three orits ech. One cusp ifurction involves the three orits , , nd while the other cusp involves the three orits , , nd The til 11100ɛ ifurcting long the 00 direction in figure 8() is lso til ifurcting long the 00 direction in figure 8(e) strting t the i-unimodl swllowtil. We will focus on these structures ecuse, s we shll see elow, they re oserved in the Hénon mp. In figure 8(f ) we find new cusp ifurction involving the three orits , , nd Figure 8() shows the topologicl prmeter plne ( 10, 01 ) nd yields some unimodl structures nd the cusp ifurction lso drwn in figure 8(f ). Figure 8(d) ppers to e slightly more complicted ut it contins no structures not lredy descried ove. A discontinuity t the digonl 00 = 01 is clerly visile here. At the digonl the different folds switch ordering, so some ifurction lines re discontinuous t this line in the symol plne. This does not imply tht there re ny discontinuities in the Hénon mp. The prt of swllowtil in the upper tringle here is the sme i-unimodl swllowtil s in figure 8(e) nd not new structure. The cusp in the lower tringle is the sme cusp s in figure 8(f ) nd only new imge of this. There re in ddition some other topologicl ifurction structures in figure 8 which cnnot e interpreted s ifurctions. These do not give topologicl lines in pirs s required for ifurction in dynmicl system. The interesting ifurction plnes cn lso e drwn in three-dimensionl prmeter spce. It turn out tht the structure we get y comining the two swllowtils in figures 8(e) nd 8(c) nd the two cusps in figure 8() is of the sme type s the ifurction structure for the period 8 orits discussed elow in section 4.4, see figure Cusp ifurctions The cusp ifurction discussed ove clerly shows the min prolem tht we fce in defining symolic dynmics for the Hénon-type mps. We illustrte this here in some detil y discussing one of the cusps. A conjecture of universlly vlid definition of symols in Hénon mp is stted elsewhere [16]. Figure 8() shows the cusp with the period 6 orits , , nd , nd figure 9 shows the stle nd the unstle mnifolds t the cusp point for the Hénon mp. Our four-unimodl mp cnnot e good description when one of the folds no longer hve turning point corresponding to one-dimensionl criticl point. Figure 2() seems to justify four-unimodl pproximtion, ut figure 9 shows tht we lose tngency point close to the period 6 orit for these prmeter vlues. Closer exmintion of figure 9 shows tht ctully two of the folds lose their turning points corresponding to the criticl points of the one-dimensionl mps t the cusp nd proper pproximtion is then the i-unimodl pproximtion with two unimodl mps. The prolem is how to choose the symol for point on the fold tht does not hve primry turning point. Moving in prmeter spce such tht turning point is creted shows tht there is not one unique point yielding the symol prtition, ut the position of the point will depend on the pth we choose in the prmeter plne. This implies tht orits my chnge symolic dynmics moving round the cusp in the prmeter spce s shown in [15] (see lso [9]). The ifurction lines for

16 1248 K T Hnsen nd P Cvitnović 2.0 x t x t Figure 9. The stle nd unstle mnifolds t the cusp point for the period 6 orit nd ehind the cusp in figure 8() should therefore not e understood s ifurction lines where n orit is creted in dynmicl system, ut n indiction of where the description of the orit using the four-unimodl symolic dynmics is correct. The orit will lso exist etween the lines ut without the sme symolic description in this pproximtion. The short digonl lines on the cusps in figure 8 indicte the chnge from four-unimodl to i-unimodl pproximtion. The chnge of symolic dynmics t cusp lso hs consequences for the method proposed y Bihm nd Wenzel [3, 4] to find cycles in the Hénon mp. As discussed in [16], there will e region ehind the cusp where the method does not converge. The chnge in modlity tkes plce when the criticl point on the lower-most unimodl mp itertes directly into the criticl point of the second lower-most unimodl mp. We cn stte this with symolic dynmics using the kneding sequences of the mps. Proposition 2. In the four-unimodl there is ifurction from four-unimodl to iunimodl pproximtion of the once-folding mp t prmeter vlues where the kneding sequences of the mps stisfy the following condition; for order-reversing mpping ( > 0) K 00 = σ(k 10 ), (23) for order-preserving mpping ( < 0) K 11 = σ(k 01 ). (24) Assuming s 1 = 1 for the kneding sequences we get the conditions on the topologicl prmeter vlues for the ifurction: for >0 00 = , (25)

17 Bifurction structures in mps of Hénon type () 10 () (c) 10 Figure 10. Bifurction lines of some period 8 orits in the two-dimensionl projections of the topologicl prmeter spce () ( 10, 00 ),()( 00, 11 ),(c)( 10, 11 ). nd for <0 11 = (26) This requirement is stisfied for ll the cusp points of the periodic orits discussed ove. One exmple is the cusp in figure 8() with the orit giving 2 2 ˆγ 10 (100111) = 2 2γ(111100) = = = = γ(111001) =ˆγ 00 (100111). Depending on whether the numer of 1 s in the repeting string is odd or even, the stle orit is either inside cusp or it surrounds the cusp point of n unstle orit Bifurction of period 8 orits Another more complicted exmple of ifurctions oserved in the Hénon mp explinle y the four-unimodl pproximtion is the ifurction structure of period 8 orits. We therefore investigte the ifurction structures in the topologicl prmeter plne for the period 8 orits. In the sme wy s for the period 6 orits we cn construct two-dimensionl topologicl prmeter plnes for ll period 8 orits. This will yield very complicted picture; in

18 1250 K T Hnsen nd P Cvitnović ε 1 111ε 2 10ε ε Figure 11. Bifurction lines of some period 8 orits in the three-dimensionl topologicl prmeter ( 10, 00, 11 ). figure 10 the ifurction lines for some period 8 orits re sketched in the three plnes ( 10, 00 ), ( 00, 11 ), nd ( 10, 11 ). These drwings show tht there re two cusp structures nd two swllowtils in these topologicl prmeter spces quite similr to the period 6 orits discussed ove. One of the swllowtils is the swllowtil 100ɛ 0 111ɛ 1 from the i-unimodl pproximtion (figure 5()), while the other structures ppers only in the four-unimodl pproximtion. We cn comine the three pictures in figure 10 to drw three-dimensionl projection of the full four-dimensionl topologicl prmeter spce. This will descrie how the codimension-2 ifurction structures re connected with stle windows in the prmeter spce. In figure 11 the exct ifurction plnes re drwn in the topologicl prmeter spce ( 10, 00, 11 ). The rnges of the xes re 0.64 < 10 < 0.69, 0.67 < 00 < 0.71, nd 0.82 < 11 < The line 00 = yielding cusp structures eqution (25) is lso drwn. In this three-dimensionl spce the ifurctions tke plce t plnes nd n orit exist inside three-dimensionl ox with one corner t (1, 1, 1). A scn of the (, ) plne of the Hénon mp corresponds to two-dimensionl hypersurfce cutting through the ifurction plnes in this three-dimensionl topologicl prmeter spce, yielding ifurction line whenever the (, ) prmeter hypersurfce intersects the ifurction plne of n orit Are-preserving mps The lines =1 in the Hénon mp correspond to re-conserving mps. This limit is not specil line in the topologicl prmeter spce, ut we cn show tht certin codimension-2 ifurctions require tht the mp is re conserving. To show this we hve to use the symmetry etween the stle nd unstle mnifolds.

19 Bifurction structures in mps of Hénon type 1251 To discuss this symmetry we first hve to define quntity for the stle mnifold equivlent to γ(s). This is given in [7, 6] s for >0 { 1 wt if s t 1 = 0 w t 1 = w t if s t 1 = 1 δ(x) = 0.w 0 w 1 w 2...= w 1 t /2 t, t=1 w 0 = s 0 (27) nd for <0 { wt if s t 1 = 0 w t 1 = 1 w t if s t 1 = 1 δ(x) = 0.w 0 w 1 w 2...= w 1 t /2 t. t=1 w 0 = s 0 (28) From the pruning front conjecture [7, 6] it follows tht since for re-preserving mps this is symmetry etween the unstle nd stle mnifolds it will lso e symmetry etween the pruning fronts in γ nd in δ. We cn use this symmetry to discuss the cusp ifurction. At cusp point singulrity n orit hs two points on the pruning front corresponding to two cyclic permuttions of the periodic symol string; S nd S = σ k (S). The represerving mp symmetry implies tht the γ -vlues of these strings re symmetric to the δ-vlues of one ckwrd shift of the sme symol string, s the re-preserving pruning front is symmetric to the ckwrd itertion of the pruning front. At cusp in two-dimensionl prmeter plne ( ss, s s ) the symol string S yields the vlue ss nd shifted string S yields the vlue s s. The cusp cn only exist for the order-reversing re conserving mp ( = 1) line if γ(s) = 1 δ(σ 1 (S )) δ(s) = 1 γ(σ 1 (S )) γ(s )=1 δ(σ 1 (S)) δ(s ) = 1 γ(σ 1 (S)) (29) nd for the order-preserving re conserving mp ( = 1) if γ(s) = δ(σ 1 (S )) δ(s) = γ(σ 1 (S )) γ(s )=δ(σ 1 (S)) δ(s ) = γ(σ 1 (S)) (30) where σ 1 is the inverse shift opertion of the symol string, corresponding to n itertion once ckwrd in time. This implies tht the two periodic points on the pruning front in the symol plne re symmetric to ech other with respect to symmetry line. The cyclic permuttions of the period 4 orit 1001; S = 1001 nd S = 1100 yields 00 = γ(1001) = nd 10 = γ(1100) = , figure 7. Using the definitions (5)

20 1252 K T Hnsen nd P Cvitnović nd (27) we find the following reltions etween the symolic vlues of the symol string: γ(1001) = = = 1 δ(0110) = 1 δ(σ 1 (1100)) δ(1001) = = = 1 γ(0110) = 1 γ(σ 1 (1100)) γ(1100) = = = 1 δ(1100) = 1 δ(σ 1 (1001)) δ(1100) = = = 1 γ(1100) = 1 γ(σ 1 (1001)). This is the symmetry reltion (29) corresponding to the Hénon mp with = 1. The cusp of the period 6 orits with the orits , , nd hs the orit common in the two tils with the cyclic permuttions S = nd S = giving the symolic vlues 00 nd 10 t the singulr point. Direct clcultion using definitions (5) nd (27) yields γ(101001) = 1 δ(σ 1 (110100)) δ(101001) = 1 γ(σ 1 (110100)) γ(110100) = 1 δ(σ 1 (101001)) δ(110100) = 1 γ(σ 1 (101001)), the symmetry in (29) which restricts the cusp to the = 1 line. There is cusp structure for period 6 orits involving the three orits , , nd The common orit in the two tils re nd the cyclic permuttions S = nd S = gives the symolic vlues 11 nd 01 t the singulr point. Using the definitions (5) nd (28) for <0wefind γ(100111) = δ(σ 1 (110011)) δ(100111) = γ(σ 1 (110011)) γ(110011) = δ(σ 1 (100111)) δ(110011) = γ(σ 1 (100111)) which is the symmetry eqution (30) for the Hénon mp t = 1. (31) 5. Hénon mp ifurctions We shll now try to verify the ifurction structure descried ove for generic topologicl prmeter spce in the specific prmeter spce (, ) of the Hénon mp. The different ifurction lines nd mny of the swllowtils in the i-unimodl pproximtion cn e found numericlly in this (, ) plne. Mny of these ifurction structures hve een drwn in e.g. [8, 29]. The ifurction curves for the cycles with period 1, 2, 3 nd 4 for < 1 give only simple windows similr to the ifurction lines otined in the topologicl prmeter spce. In figure 12 we hve drwn the ifurction lines for the period 4 orits in the prmeter plne (, ) close to the = 1 line. We find here the cusp predicted in the topologicl prmeter plne. In greement with the rguments ove, we do find tht the cusp point is exctly on the = 1 line. A scn of the (, ) plne for the Hénon mp, serching for stle period 5 orits revels the swllowtil ifurction s drwn in figure 13. We notice tht in figure 3 the swllowtil crossing in the symol plne tkes plce for 1 > 0, corresponding to n orienttionreversing horseshoe, tht is >0 for the Hénon mp. The period douling to two period 10 swllowtil crossings, four period 20 crossings etc, is found for the Hénon mp exctly

21 Bifurction structures in mps of Hénon type e e Figure 12. The ifurction curves of the period 4 orits in the Hénon mp Figure 13. The swllowtil of period 5 orits in the prmeter plne (, ) for the Hénon mp: res with stle period 5 orit. s constructed in the i-unimodl mp symol plne [16]. This i-unimodl ifurction structure is the sme s the well studied one-dimensionl imodl mps in [23 25, 10, 27]. The reltive position etween two swllowtils in the topologicl prmeter plne is topologicl feture which is vlid lso in ny 2-prmeter plne (, ) for once-folding mp. If one swllowtil crossing is etween two other tils in the topologicl prmeter plne or if til from one swllowtil crosses til from different swllowtil, then this will e true lso in (, ) prmeter plne.

22 1254 K T Hnsen nd P Cvitnović Figure 14. Swllowtils in the Hénon mp: res in the (, ) prmeter plne corresponding to stle period 5, 6 or 7 orit re mrked in lck. We now compre the i-unimodl dmissile swllowtils of the short orits with the swllowtils relized y the Hénon mp. In figure 4 the swllowtils for period 5, 6 nd 7 re drwn together in the topologicl prmeter plne. Oserve the topologicl structure; which tils tht cross other tils nd which swllowtils re nested within other swllowtils. There is one horizontl row of period 7, 5 nd 7 swllowtils nd there is one verticl column with period 5, 6 nd 7 swllowtils. Figure 14 is scn of the (, ) plne of the Hénon mp, with the res corresponding to stle period 5, 6 nd 7 orits re mrked in lck. The swllowtils re rrnged topologiclly s in figure 4, with only few differences in the structure. One of the tils from the period 6 swllowtil crosses til of the period 5 swllowtil; ccording to the i-unimodl topologicl prmeter plne this should not occur. As we will discuss elow, this rises from the four-unimodl pproximtion. Also the period 7 swllowtil ove the period 6 swllowtil hs one til crossing period 5 til. This period 7 swllowtil 1000ɛ 0 1ɛ 1 is not complete swllowtil ut is roken up into cusp nd n isolted til. The ifurction lines re correctly descried y the i-unimodl topologicl prmeter plne ut ecuse the tils ifurcte on different folds with finite distnce the orit is not stle in the whole region where the i-unimodl mp is stle. In figure 15 we find tht one of the tils from the swllowtil 100ɛ 0 1ɛ 1 is connected to cusp ifurction. This is the ifurction predicted y figures 8() nd (e). In oth figures 8() nd (e) the til 11100ɛ ifurctes t 00 vlue. The til is connected to the swllowtil 100ɛ 0 1ɛ 1 in figure 8(e) nd to the cusp with the orits , , nd in figure 8(). This is the til connecting the two codimension-2 structures in the (, ) plne in figure 15. Another connection etween codimension-2 structures predicted from figures 8() nd (e) is the til 10100ɛ which connect the swllowtil 100ɛ 0 1ɛ 1 with the cusp consisting of the orits , , nd We hve found ove tht this cusp hs the symmetry

23 Bifurction structures in mps of Hénon type Figure 15. The (, ) prmeter plne regions with stle period 6 orit in the Hénon mp. restricting it to the = 1 line. Numericlly the cusp is found t 2.75, = 1. The third cusp in figure 8(f ) with the orits , , nd is predicted to e connected to the i-unimodl swllowtil with the til 10011ɛ nd exist t the = 1 line. Numericlly this cusp is found t 3.0, = 1. The swllowtil in figure 8(c) is not found for the Hénon mp. It uses some of the sme orits s the other codimension-2 structures nd it will therefore e difficult to hve this together with the other structures in the sme (, ) plne. This cusp is relized y other once-folding mps; it hs een found in the two-dimensionl Lozi mp [22, 16]. The ifurctions of the period 8 orits turns out to e the most complicted of the short cycles. The period 8 swllowtils in figure 5() with symolic description 100ɛ 0 111ɛ 1 do not exist for the Hénon mp ut cn exist for slightly pertured Hénon mp. In the fourunimodl pproximtion this swllowtil is in figure 11 connected to one other swllowtil nd two cusps. To show tht the rther strnge-looking ifurction we find for the Hénon mp is descried y the ifurction plnes in figure 11 we study vrition of the Hénon mp where we dd x 4 term with third prmeter c: x t+1 = 1 x 2 t cx 4 t + x t 1. (32) This mp is once-folding for c>0. For c<0 the mp is in principle thrice-folding, ut close to c = 0 the mp ehves like once-folding mp for smll vlues of x. Figure 16 shows the prmeter vlues with stle period 8 orit for the pertured Hénon mp (32). For c = 0, figure 16(c), this is the Hénon mp. We find in figure 16 tht the ifurction structures chnges smoothly with the new prmeter c nd for c = 0.08 nd for c = 0.06 we find comintions of fmilir codimension-2 structures, swllowtils nd cusps, while for c = 0 more complicted structure emerges. The (, ) plne in figure 16() corresponds to plne tht cuts through the two cusps on the top nd through the swllowtil on the left-hnd side of figure 11. This is the structure

24 1256 K T Hnsen nd P Cvitnović () () (c) (d) (e) (f) Figure 16. The prmeter vlues giving stle period 8 orits in the pertured Hénon mp (32) in the prmeter spce (, ) with different vlues of c. ()c=0.08, () c = 0.02, (c) c = 0 (the Hénon mp), (d) c = 0.013, (e) c = 0.02, (f ) c = drwn in figures 10() nd (c). The (, ) plne in figure 16(f ) corresponds to plne tht only cuts through the swllowtil on the right-hnd side of figure 11 (figure 10()). The Hénon mp in figure 16(c) is plne cutting through the structure in the middle of figure 11 where the swllowtils nd the cusps merge together. This illustrtes true codimension-3 ifurction for mps of the Hénon type. The reder is refered to works of Mir [29] nd Crcssés [5] for detiled study giving

25 Bifurction structures in mps of Hénon type ε1 10ε1 (d) (c) ε1001 () 0.3 1ε101 () Figure 17. The ifurction lines of some homoclinic orits of the Hénon mp in the prmeter plne (, ). The lels indicte the prmeter vlues in figure 18. more exmples of ifurction structures in the Hénon mp. We hve shown here how the ifurctions in the Hénon mp cn e understood if we extend the mp with third prmeter nd consider the ifurctions s structure in threedimensionl (,,c) prmeter spce. With this procedure we find complete greement etween the predictions of the topologicl prmeter spce nd the numerics. Hence the proper wy to study ifurctions of cycles in the Hénon mp is to extend the investigtion to n infinite-dimensionl topologicl prmeter spce of ll Hénon-like mps Aperiodic orits The ifurction of homoclinic orits in smooth i-unimodl mp would give ifurction lines similr to the ifurction lines in the symol plne, figure 6, s discussed in [17]. The ifurction lines in the Hénon mp for the homoclinic orits with symolic description 1ɛ 0 10ɛ 1 1 re drwn in figure 17. The ifurction line 1ɛ101 is where the ttrctor merges from two prts into one connected ttrctor. This is nlogous to the nd-merging ifurctions in unimodl mp. This ifurction tkes plce long the curve 1ɛ101 (figure 18()) until the cusp re nd from the cusp re long the line 1010ɛ1 until the mrker in figure 17. Aove this point there is different homoclinic tngency, the line 1ɛ10001, which is the order etween two or one connected chotic ttrctor. The other ifurction curves re other homoclinic ifurctions s illustrted in figures 18() (d). The ifurction curves hve similr shpes s in the topologicl prmeter plne (figure 6), ut the ifurction curve corresponding to the 0 = 0.10 line is split into two curves nd one of the curves hs cusp. The cusp is not s nrrow s the homoclinic orit cusp one finds in i-unimodl mps [17]. Numericlly it seems to e the sme type of cusp s we hve in the centre of the swllowtil where the width of the cusp increses s the distnce to the power 3. The second smooth curve in figure 17 seems to lck the singulrity in the 2 derivtive found for i-unimodl mps [17]. The homoclinic orits re chnging the symolic description in the neighourhood of the cusp point. We find tht the homoclinic orit 101 ifurctes t point () nd (c) in