Volume 104A, number 6,7 PHYSICS LETTERS 10 September 1984

BIFURCATION SEQUENCES IN HORSESHOE MAPS: INFINITELY MANY ROUTES TO CHAOS '~

Philip HOLMES Department of Theoretical and Applied Mechanics and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA

Received 13 June 1984

We report results obtained using homoclinic bifurcation theory and knot theory for suspensions (flows) of two-dimen- sional Hdnon-like maps Fu, e with two parameters. If one parameter, e, is the j acobian determinant of F/~, e and the other, ~, represents "nonlinearity", we assume there are values s l(e) < h l(e) for each e ~ [0, 1] such that, for ~ < sl(e) all orbits ofFg, e escape, and for u > hl(e), Fu, e has a Smale horseshoe (chaos). Increasing t~ for different e we find infinitely many distinct routes to this chaos.

In this brief note we announce preliminary results which will be reported more fully elsewhere [1,2].

We consider the sequences o f bifurcations occurring as/a increases for fixed e E [0, 1 ] for the two-parame- ter family of planar diffeomorphisms

Fu,,: ( x , y ) ~ y , - e x + ~ _ y 2 ) . (1)

Other than the fact that det(DFu,~) = e is constant, the exact form ofFu, e seems not to be crucial, pro- vided that, as e ~ 0, i tsy-component collapses to a one-dimensional map with negative schwarzian deriva- tive and a unique, nondegenerate maximum [3 -6 ] . For e = 0 the map

fu: y ~ I.t - y2 (2)

n n+l 0- completely describes the orbits { f (Yo , f~ (Y0))n=l of (1)and this fact can be exploited to obtain results for Fu, e with l e I small. The work reported at length in refs. [1,2] and summarized here shows that, while many bifurcation subsequences (M.S.S. sequences [7]) of (2) persist for (1) with e > 0, small, many others are completely disrupted, and that as e ~ 1, the area preserving limit, countably many bifurcation subsequences are precisely reversed.

The results reported here are relevant to the use o f

Work supported by the National Science Foundation under MEA 80-17570

0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

nonlinear dynamical models of physical processes in several ways. It has recently been suggested [3,8] that low dimensional flows and maps might capture many features of the dynamics of temporally chaotic, but spatially ordered, phenomena. Examples include the preturbulent fluid flows in Taylor-Couet te and Rayleigh-Benard experiments [9], oscillating elec- trical circuits [10], and vibrating structures [11,12]. In almost all such models the chaotic solutions corre. spond to the existence o f a subshift o f finite type [13], or, loosely speaking, a Smale horseshoe [14]. As a parameter such as Taylor or Rayleigh number or forcing amplitude increases, a sequence o f bifurca- tions occurs in which simple motions are replaced by successively more complex periodic and ultimately chaotic motions. It has been argued that "universal" features, such as accumulating sequences of period- doubling bifurcations, provide strong evidence that the underlying dynamical models are essentially one- dimensional unimodal maps like (2) [3,10]. The pres- ent results show that a modest extension to even a two-dimensionai map can lead to completely different bifurcation sequences. For example, while period doubling continues, it can be interrupted by the ap- pearance of odd-period motions in other regions o f the phase space [15]. The moral is that there are no universal routes to chaos. Our methods also show, in the two-dimensional case, the one-dimensional limit

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(2) connects with the area preserving hmit of (1) with e = 1. Such area preserving maps occur widely in models for conservative hamiltonian systems with two degrees of freedom [16], and are often used in plasma physics [17].

We note that, for e = -0 .3 and/l = 1.4, (1) can be transformed into the (orientation reversing) map studied by H6non [18]. Here we concentrate on the orientation preserving case, since our main interest in such maps is that they provide good models for cer- tain aspects of the Polncar6 maps of periodically forced single degree of freedom oscillators [ 11,19]. For maps like (1) it is possible to prove [20] that there are two curves tt = s l(e) < / l = h l(e ) such that, for tt < Sl, all orbits ofFu, e escape to ~, while for /1 > hl(e) , Fu, e has an invariant Cantor set ~2, a Smale horseshoe [13,14], on which the map is conjugate to a shift on two symbols, and hence is chaotic. Evident- ly, as tt increases, infinitely many bifurcations occur for example, S2 has ~2n/n orbits of each period n.

For e = 0 ~2 is created forfu m a well understood fashion, the periodic orbits appearing in the "universal" MSS sequence [7] in saddle-node or tangent bifurca- tions at tt = s[ and period doubh.ngs at f~2t. Countably many homoclinic bifurcations hg / 2l also occur at which some iterate f~(0) of the critical point lands on an un- stable periodic orbit. In these symbols k or k2 l de- notes the period of the bifurcating orbit and ] denotes its order of occurrence with increasing/~. The kneading theory of Milnor and Thurston [3., 13,21 ] enables us to uniquely order the s /k' f k 2 and h lk 2 I.

The first result one obtains for Fu, c concerns the extension of these bifurcation points into curves in the (/a, e)-plane [22]. In fact for generic families Fu, e every point s~, ¢/2t extends to a bifurcation curve which crosses the plane to meet e = 1 and every homoclmic point h~2 t extends and opens into an uncountable "Cantor fan" o f homoclinlc bifurcation curves crossing to e = 1 [1]. Extension for small (nonuniform) e IS easily obtained by the implicit function theorem and global methods involving transversality arguments are used to continue the bifurcation curves. However, al- though all the periodic curves extend (and new ones do not necessarily occur), their order need not be preserved. In fact we have.

Theorem 1. For any integer N < ~ there exists e N > 0 such that for 0 < e ~< ear, the periodic bifurcations

of period ~< N of Fu, e occur in the same sequence as those ofj 'u In contrast, for any e > O, Infinitely many periodic bifurcations of b~,~ occur in sequences differ- ing from those of J~.

The first part of this result was obtained earlier by Guckenhelmer [22] The second part IS obtained by studying families of saddle-node bifurcations which accumulate on the homochnic tangencles occurring as /~ increases for e C (0, 1), fixed [23]. For example, one constructs periodic orbits with "one-dimensional'" itineraries containing arbitrarily long blocks of the symbol "L", denoting that the orbit spends a long time near the unstable fixed point at the left-hand end of the invariant interval before "following" the homo- chnlc orbit. Such Itineraries can be constructed for each homoclinlc bifurcation with a "non trivial" homoclmic piece and a long L-sequence in each period- lC block. Upon ordering the kneading invarlants corre- sponding to such orbits, one finds that orbits of period k ~> M which appear directly before the last homochnlc bifurcation for e > e(M) are the penultimate ones of period k for e = 0, while those appearing directly before the first homoclmic hifurcation are the last to appear for e = 0. This argument can be extended to homo- clinic bifurcations associated with other periodic points in the interval [ 1 ]

We note that the orbits constructed above do not settle the question of whether the two-dimensional map can complete its first period-doubling sequence without the appearance of (stable) periodic orbits of period j v a 21. In fact Whltley [24] has constructed a C 1 planar dIffeomorphism which does complete its penod-doubhng sequence in such a manner. However, that example cannot be made C ~, or even C 2 The reason seems to be that, to prevent "'extra" saddle- nodes, one must make the unstable manifold of each saddle of period 2 l wrap very tightly around the saddle of period 2 l+1. For e close to 1 it is easy to see that this does not occur [1], and period k saddle-nodes occur before even the first period doubling

Finally we give a result obtamed using knot theory for the "natural" suspension of F u e defined on a disc D C Iq 2. This suspension yields a flow in a "penodic" 3-manifold D X S 1 a solid torus. Using methods of Birman and Wtlliams [25], one finds that infinitely many torus knots arise in this flow. A (p, q)- toms knot IS obtained by wrapping an orbit p times

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longitudinally and q times meridionaUy around a toms. If the period of the orbit is q, we call the orbit a resonant toms knot. We prove that there are precise- ly two resonant toms knots for each relatively prime integer pair (p, q) with p <~ q/2. Moreover, we have:

ly prime integers with Pi <~ qi/2 and p 1/q 1 > P 2/q 2. Let Si(e), e E [0, 1) denote the/a-value for which the unique pair of resonant (Pt, qi) toms knots first appear in a saddle-node bifurcation for the map F#,c. Then

lim~_~l Sl(e ) > l ime~ 1 S2(e ) and SI (0 ) < $2(0 ).

Theorem 2. Let (Pi, qi), i = 1, 2 be pairs of relative- Corollary. Infinitely many saddle-node (and period-

(t,7)H,6)(4,SJ(t,4) 12,7) (2,5) (3,7) S 9 ~4 ©3 ¢ S 4 c3

O5

t o

~ o 5

0 -1 0 t h 2 2 h t 3 4 5 (b) ,u.

Fig. 1. (a) Periodic bifurcation curves and (b) homoclinic bifurcation curves for the H6non map (x, y ) ~ (y, - e x + p - y2) In (a) the notations s j and f J2 indicate saddle-node and period-doubhng bifurcations respectively. The subscripts denote the period of g bifurcating orbit(s) and the superscripts the order of occurrence within that penodic class for e = 0 Absence of a superscript im- phes that there is only one bifurcation of that type and period. Bracketed integer pairs denote resonant torus knot types.

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doubling) bifurcation curves cross each other in the 01, e) parameter plane between e = 0 and e = 1. In

particular, each resonant bifurcation sequence of period q for the area preserving map (e = 1) is precise-

ly reversed for the one-dimensional map (e = 0) there are infinitely many routes to chaos.

This theorem xs proved using an algorithm which

permits one to generate the itinerary, and hence the kneading invariant, from the integers (p,, q,) [ 1 ]. Tins

allows us to locate the bifurcation points S 1 (0) <

$2(0 ) for e = 0 using the Mflnor-Thurston theory [3,21]. In contrast, for e = 1, the well-known theory of resonant hamiltonian bifurcations imphes that

(Pi, qi) torus knots are born m the order 2, 1 ifPl/ql > P2/q2, since these orbits bifurcate out from the eUiptic fLxed point b as the eigenvalues of the lineari-

zation DFu, l (b ) pass through exp(+i27rPi/qi ). For/1 = $1(1 ) the exgenvalues are exp(+i. 0) = 1 and for the subsequent period doubling they are exp( + 1r r) = - 1 '

as/a increases from $1(1), the eigenvalues exp(+i0) pass through every 0 E (0, rr). The fact that the knot type is an invariant, and the uniqueness of each reso- nant (Pi, qi) torus knot pair, then permits one to con- nect up the e = 1 and e = 0 bifurcation points.

To illustrate our results, we show numerically com-

puted bifurcation curves for orbits of periods k ~< 7 m fig. 1, together with some homoclimc bifurcation curves. For additional numerical results, see ref. [26].

Also see ref. [15] for related remarks on the non- uniqueness of bifurcation sequences for two-dimen- sional maps.

All aspects of the reordering of the bifurcation se-

quences for e near 1 shown in fig. 1 can be predicted by the methods outlined here [1,2]. We stress that these methods - homoclinic bifurcation and knot theory - yield many more results for (1) and are generally applicable to two-dimensional diffeomor- phisms arising as Poincar6 maps for three-dimensional flows.

References

[1] PJ Holmes andDC Whltley, Phil Trans R. Soc London Ser A, to be pubhshed

[2] P J Holmes and R F Wllhams, to be pubhshed [3] P Collet and J -P Eckmann, Iterated maps on the

interval as dynamical systems (Barkhauser, Basel, 1980) 14] M. Mlsiurewlcz, Inst HautesEtudesSca Publ Math 53

(1981) 17 [5] J Guckenheimer, Invent Math 39 (1977) 165, Commun

Math Phys 70(1979)133 [6] D Singer, SIAM J Appl Math 35 (1978) 260 [7] M, Metropolis, M L Stem and P R Stem, J Combm

Thy 15 (1973) 25 [8] D Ruelle and F Takens, Commun Math Phys 20

(1971) 167 [9] H L Swmney and J P Gollub, eds, Hydrodynamic

mstabahtles and the transition to turbulence (Springer, Berlin, 1981).

[10] J Testa, J PerezandC Jeffrles, Phys Rev Lett 48 (1982) 714

[11] PJ Holmes, Ph:l Trans R Soc London Ser A292 (1979) 419

[12] FC Moon andPJ Holmes, J SoundVlb 65 (1979) 285

[13] J. Guckenhelmer and P.J Holmes, Nonhnear oscillations, dynamical systems and bifurcations of vector fields (Springer, Berlin, 1983)

[141 S. Smale, Bull Am Math Soc 73 (1967) 747 [15] A Arneodo, P Coullet, C Tresser, A Llbchaber,

J Maurer and D d'Humi~res, Physlca 6D (1983) 385 [16] C W Horton, L E Relchl and A G Szebehely, eds,

Long tune pred:ctlon m dynamics (Wiley, New York, 1983)

[17] A J Llchtenberg and M.A. Lleberman, Regular and stochastic mot:on (Springer, Berlin, 1982)

[18] M. H6non, Commun Math Phys 50 (1976) 69 [19] P.J Holmes and D C Whltley, Physlca 7D (1983) 111. [20] R. Devaney and Z Niteckl, Commun Math. Phys 67

(1979) 137 [21] J Mtlnor and W. Thurston, unpubhshed note~ [22] J Guckenheuner, Ann. N Y Acad So 316 (1979) 78 [231 N K Gavnlov and L P Sxlmkov, Math USSR Sbornik

88 (1972) 465; 90 (1973) 139 [24] D.C Whltley, to be pubhshed [25] J. Barman and R F Wilhams, Topology 22 (1983) 47,

Contemp. Math 20 (1983) 1 [26] H E1HamoulyandC Mara, C R Acad Sci 293 (19811

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