Computers Math. Applic. Vol. 34, No. 2-4, pp. 195-227, 1997 Pergamon Copyright@1997Copyright©1997 Elsevier Science Ltd e Printed in Great Britain. All rights reserved Printed in Great Britain. All rights reserved 08981221/970898-1221/97 $17.00 + 0.060.00 PII: SOS9S·1221(97)00124·7SO898-1221(97)00124-7
Quasiattractors and Homoclinic Tangencies
S. V. GONCHENKO, L. P. SHIL'NIKOVSHIL’NIKOV AND D. V. TURAEV Research Institute for Applied Mathematics and Cybernetics 10 Ul'janovaUl’janova St.,st., Nizhny Novgorod, 603005, Russia
Abstract-Recent results describing nontrivial dynamical phenomena inin systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction, with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors. many coexisting strange attractors. Keywords-Attractor, Homoclinic bifurcations, Dynamical models. Keywords-Attractor, Homoclinic bifurcations, Dynamical models. 1. INTRODUCTION 1. INTRODUCTION The discovery of dynamical chaos is one of the main achievements in the modern science. At the The discovery of dynamical chaos is one of the main achievements in the modern science. At the aftermath, various phenomena in natural sciences and engineering have obtained an adequate aftermath, various phenomena in natural sciences and engineering have obtained an adequate mathematical description within the framework of differential equations. From the mathematical mathematical description within the framework of differential equations. From the mathematical point of view, dynamical chaos is commonly associated with the notion of a strange attractor-all point of view, dynamical chaos is commonly associated with the notion of a strange attractor-an attractive limit set with the complicated structure of orbit behavior. attractive limit set with the complicated structure of orbit behavior. By now, there does not exist a commonly accepted definition for a strange attractor describing By now, there does not exist a commonly accepted definition for a strange attractor describing dynamical chaos in real systems. Frequently, a strange attractor is regarded as a nontrivial dynamical chaos in real systems. Frequently, a strange attractor is regarded as a nontrivial attractive set which is composed by unstable orbits and which is transitive. There exist two types attractive set which is composed by unstable orbits and which is transitive. There exist two types of attractors which correspond completely to this definition: these are hyperbolic attractors and Lorenzof attractors attractors which (the correspond latters are completely also called to quasihyperbolicthis definition: theseattractors). are hyperbolic Hyperbolic attractors attractors and areLorenz structurally attractors stable (the latters(they satisfyare also to calledSmale’s quasihyperbolic “axiom A”). attractors).Lorenz attractors Hyperbolic are structurahy attractors unstableare structurally and, moreover, stable (theythey composesatisfy toopen Smale's sets in "axiom the space A"). of dynamicalLorenz attractors systems. areThe structurally structural instabilityunstable and,of Lorenzmoreover, attractors they composeis connected open setswith inthe the fact space that of suchdynamical an attractor systems. contains The structural a saddle equilibriuminstability ofstate Lorenz together attractors with is connectedits unstable with separatrices the fact that which such can an attractorform homoclinic contains loopsa saddle of differenteqUilibrium types state when together parameters with vary.its unstableNevertheless, separatrices the property which ofcan transitivity form homoclinic and the propertyloops of differentof instability types of whenindividual parameters orbits arevary. preserved Nevertheless, by perturbations, the property and of transitivityLorenz attractors and theare property similar ofto instabilityhyperbolic ofattractors individual with orbits this are point preserved of view. by perturbations, and Lorenz attractors are similar to Thehyperbolic “transitivity” attractors and with“instability” this point properties of view. give a possibility of rigorous description of dy- namicalThe "transitivity"chaos in hyperbolic and "instability"and quasihyperbolic properties systems give a bypossibility tools of theof rigorousergodic theory.description Therefore, of dy suchnamical attractors chaos inwere hyperbolic called andstochastic quasihyperbolic attractors. systems by tools of the ergodic theory. Therefore, such attractors were called stochastic attractors. This work was supported by EC-Russia Collaborative Project ESPRIT-P9282-ACTCS, by the INTAS Grant No.This 93-0570 work wasext. supportedand by the byRussian EC-Russia Foundation Collaborative of Fundamental Project ESPRIT-P9282-ACTCS,Researches, Grant No. 96-01-01-135. by the INTAS Grant No. 93-0570 ext. and by the Russian Foundation of Fundamental Researches, Grant No. 96-01-01-135.
Typeset by AMS-1EX 195 195 196 S.V.S. V. GONCHENKO et al.
However, it isis necessary to remark that there are no examples (known to thethe authors) of dynamical models from applications where nontrivial hyperbolic attractors are found; and to the present time,time, the study of such attractors is the subject of thethe pure mathematics ratherrather thanthan of thethe nonlinear dynamics. Quasihyperbolic attractors do occur in applications but inin a limited class of problems. At thethe same time, for most of known dynamical systems of natural origination that demon-demon strate chaotic behaviour nontrivial attractive sets have quite different nature. We mention, for instance, spiral attractors [1-3J[l-3] associated with a homoclinic loop to a saddle-focus [4,5];[4,5J; atat- tractors that arise through breakdown of an invariant torus [6-9J;[6--g]; screw-like attractors in thethe Chua circuit [lO,ll];[10,11]; attractors in the H&onHenon map [12-14J;[12-141; attractors forming throughthrough the period-period doubling cascade in strongly dissipative maps; attractors in the Lorenz model
zix=O"(y-x),= a(y - x), e=rx-y-xz,y = rx -y-xz, .i=-bz+xy,z = -bx +xy, at large values of T (for instance, at c = 10, b = 8/3, T > 31) [15-171; attractors in periodically at large values of r (for instance, at 0" 10, b 8/3, r > 31) [15-17J; attractors in periodically forced self-oscillatory systems with one degree of freedom [18-211,[18-21], etc. Strange attractors of such systems are well known to contain not only nontrivial hyperbolic sets but also attractive periodic orbits, and thereby, not being stochastic rigorously speaking. Due to thisthis reason, we will adhere to the definition given in [6,22]:[6,22J: a strange attractor (a qumiattructorquasiattmctor in terms of [6,22]) is an attractive limit set which contains nontrivial hyperbolic subsets anda.nd which may contain attractive periodic orbits of extremely longlong periods. Since neither thethe transitivitytransitivity property nor the property of individual instability of orbits may not be fulfilled in this case1 we property nor the property of individual instability of orbits may not be fulfilled in this casel we will use the term a quasistochastic attractor. will use the term a quasistochastic attractor. We notice that the principal reason of distinguishing the class of quasistochastic attractors is that, in contrast with the genuine stochastic attractors, for them there is no rigorous mathematical that, in contrast with the genuine stochastic attractors, for them there is no rigorous mathematical base for the main notions by using of which chaotic dynamics is analyzed: Lyapunov exponents, base for the main notions by using of which chaotic dynamics is analyzed: Lyapunov exponents, entropy, decay of correlations, sensitive dependence on initial data, etc. Thus, for a large variety entropy, decay of correlations, sensitive dependence on initial data, etc. Thus, for a large variety of dynamical systems of natural origination, the question of the nature of chaos remains open so of dynamical systems of natural origination, the question of the nature of chaos remains open so far. far. The following speculation indicates possible direction for the study of this question. Note The following speculation indicates possible direction for the study of this question. Note that if a system has an attractor which is structurally stable, then according to conventional that if a system has an attractor which is structurally stable, then according to conventional hypothesis (the structural stability theorem [23,24]), the attractor is hyperbolic: either trivial hypothesis (the structural stability theorem [23,24]), the attractor is hyperbolic: either trivial (i.e., a stable periodic orbit) or nontrivial. As we mentioned, no one has ever seen nontrivial (Le., a stable periodic orbit) or nontrivial. As we mentioned, no one has ever seen nontrivial hyperbolic attractors in natural applications. It follows that if any attractor could be made hyperbolic attractors in natural applications. It follows that if any attractor could be made structurally stable by a small perturbation of the system, then in principle, the study of chaotic structurally stable by a small perturbation of the system, then in principle, the study of chaotic dynamics in real systems would be reduced to the study of stable periodic regimes, but t:his would dynamics in real systems would be reduced to the study of stable periodic regimes, but this would be quite strange. be Anquite alternative strange. is to try to find nontrivial attractors for systems which lie in the open regions of Anstructural alternative instability is to tryin tothe find space nontrivial of dynamical attractors systems. for systems At the which present lie intime the twoopen types regions of suchof structural regions areinstability known. in Thethe regionsspace of of dynamical the first typesystems. are filledAt theby presentthe systems time twowith typesLorenz of (quasihyperbolic)such regions are known.attractors. The The regions second of arethe thefirst so-called type are Newhouse filled by regionsthe systems with thewith study Lorenz of which(quasihyperbolic) the present attractors.paper deals. The second are the so-called Newhouse regions with the study of whichThe thescope present of this paper paper deals. is to represent recent results of the authors which show that quasi- attractorsThe scope of systemsof this paperin the isNewhouse to represent regions recent may results exhibit of therather authors nontrivial which features show thatwhich quasi are inattractors a sharp ofdistinction systems inwith the thatNewhouse one could regions expect may in exhibit analogy rather with nontrivialstochastic featuresattractors. which Thus, are wein ashow sharp that distinction the quasistochastic with that oneattractors could expectmay containin analogy structurally with stochastic unstable attractors. and, moreover, Thus, infinitelywe show thatdegenerate the quasistochastic periodic orbits attractors are what maymakes contain the complete structurally description unstable of and,dynamics moreover, and bifurcationsinfinitely degenerate of such attractors periodic orbitsimpossible are whatin any makes finite-parameter the complete family. description of dynamics and bifurcationsWe also establish of such attractorsthat the quasistochastic impossible in anyattractors, finite-parameter in contrast family. with hyperbolic ones, may notWe possess also establishthe property that of the self-similarity, quasistochastic Namely, attractors, there mayin contrastexist infinitely with hyperbolicmany time ones,scales may on not possess the property of self-similarity. Namely, there may exist infinitely many time scales on ‘Even if these properties may hold, they are not preserved under small perturbations. 1 Even if these properties may hold, they are not preserved under small perturbations. Quasiattractors 19.'197 which behavior of the system is qualitatively different. Besides, we show that the quasiattractors may simultaneously contain saddle periodic orbits with different topological indices or, what is the same, with different numbers of positive Lyapunov exponents. The last is also impossible for hyperbolic attractors. Different quasistochastic attractors possess rather different properties (such as “the"the form”form" of attractor, the form of power spectrum, fractal dimension, etc.) Nevertheless, it seems to us that the most important property common for them is the presence of structurally unstable PoincartlPoincar«~ homoclinic orbit either in the system itself or in a nearby system. Recall that a Poincare homoclinic orbit is an orbit of intersection of the stable and unstable manifolds of a saddle periodic orbit. A homoclinichomo clinic orbit is called structurally stable if the in-in·· tersectiontersection is transverse, and it is called structurally unstable (or a homoclinic tangency) if the invariant manifolds are tangent along it (Figure 1).I).
W'(O)
Figure 1. The saddle fixed point 0 whose the stable Ws and the unstable W” manifoldsFigure 1. haveThe asaddle quadratic fixed tangencypoint 0 whoseat the thepoints stable of a W'homoclinic and the orbitunstable I+ (bold W'" pointsmanifolds in the have figure). a quadratic tangency at the points of a homoclinic orbit r (bold points in the figure). As it is well known [25,26], in any neighborhood of a structurally stable Poincare homoclinic or- bitAs there it is existwell knownnontrivial [25,26]' hyperbolic in any neighborhoodsets containing ofa a countable structurally number stable of Poincare saddle periodic homoclinic orbits, or continuumbit there existof nonperiodicnontrivial hyperbolicPoisson stable sets containingorbits, etc. a countableThus, the numberpresence of of saddle a structurally periodic orbits,stable Poincarecontinuum homoclinic of non periodic orbit canPoisson be considered stable orbits, as the etc. universal Thus, thecriterium presence of ofcomplex a structurally dynamics. stable PoincareBifurcations homoclinic of systems orbit withcan behomoclinic considered tangencies as the universalwere studied criterium in a seriesof complex of papers dynamics. beginnin.g withBifurcations [27,28]. An of systemsimportant with result homo was clinic established tangencies by were Newhouse studied in[29], a seriesthat ofin papersthe space beginning of dy- namicalwith [27,28]. systems An there important exist regions result (Newhowewas established regions) by whereNewhouse systems [29], with that structurally in the space unstable of dy Poincarenamical systemshomoclinic there orbits exist regionsare dense. (Newhouse Moreover, regions) as it wherewas found systems in [29-311,with structurally Newhouse unstableregions existPoincare in any homo neighbourhood clinic orbits ofare any dense. system Moreover, with homoclinic as it was tangency. found inNamely, [29-31], the Newhouse following regionsresu1.t isexist valid. in any neighbourhood of any system with homoclinic tangency. Namely, the following result is valid. THEOREM 1. Let fe be a general2 finite parameter family of dynamicai systems which has a saddleTHEOREM periodic 1. Letorbit IF!: L,.be aSuppose generaJ2 that finite at parameterE = 0 there familyexists ofa structurallydynamical systemsunstable whichhomoclimic has a saddleorbit l? periodic of the orbit orbit LO.Le. Then,Suppose values that ofat E€ for= 0which there L,exists has a anstructurally orbit of quadratic unstable homoc1iruichomoclinic tangencyorbit r ofaxe the dense orbit in Lo. some Then, open valuesregions of Ai € forof thewhich parameter Le has space,an orbit accumulating of quadratic at homoc1in;icE = 0. tangencyThe one-parameter are dense inversion some openof this regions theorem Ai ofwas the established parameter by space, Newhouse accumulating in [29], for at the€ = caseo. of two-dimensionalThe one-parameter diffeomorphisms version of thisand theorem it was extendedwas established onto the by general Newhouse multidimensional in [29], for thecase case blyof two-dimensionalus in [30] (the case diffeomorphisms with an arbitrary andnumber it was ofextended parameters onto follows the general immediately multidimensional from [29,30]). case The by us in [30] (the case with an arbitrary number of parameters follows immediately from [29,30]). The 2The exact conditions of general position have been formulated in [30]. In particular, it is required of fo that for the2The tangency exact conditionsto be quadratic, of general the positionorbit I’0 havenot liebeen in theformulated strong stable in [30]. and In strongparticular, unstable it is submanifoldsrequired of /0Wad that and for theWuu, tangencyetc. to be quadratic, the orbit ro not lie in the strong stable and strong unstable submanifolds WU and W"'''', etc. 198198 S.S. V.V. GONCHENKOGONCHENKO etet al.al. multidimensionalmultidimensional casecase waswas alsoalso consideredconsidered partlypartly inin [31].[31]. ThisThis theoremtheorem showsshows thatthat althoughalthough anyany givengiven homoclinichomo clinic tangencytangency cancan bebe removedremoved byby aa smallsmall perturbationperturbation ofof thethe system,system, thethe presencepresence ofof homoclinichomoclinic tangenciestangencies is,is, nevertheless,nevertheless, aa persistentpersistent phenomenon.phenomenon. InIn ourour opinion,opinion, thethe presencepresence ofof structurallystructurally unstableunstable PoincarePoincare homoclinichomoclinic orbitsorbits eithereither inin thethe systemsystem itselfitself oror inin aa nearbynearby systemsystem isis oneone ofof thethe mainmain peculiaritiespeculiarities of quasistochasticquasistochastic systems.systems. AsAs wewe cmcan judge,judge, thethe presencepresence ofof homoclinichomoclinic tangenciestangencies forfor somesome valuesvalues ofof parametersparameters waswas eithereither theoreticallytheoretically provedproved oror foundfound byby computer simulationssimulations inin allall dynamicaldynamical modelsmodels withwith quasiattrac-quasiattrac torstors (see(see thethe listlist above)above) forfor whichwhich thethe problemproblem ofof findingfinding suchsuch parameterparameter valuesvalues waswas explicitly.explicitly set.set. ByBy TheoremTheorem 1, thethe closureclosure of thesethese parameterparameter valuesvalues containscontains openopen regions.regions. NoteNote thatthat thethe sizesize of thesethese regionsregions may bebe ratherrather largelarge inin concrete examplesexamples (see,(see, forfor instance,instance, [14]),[14]), thioughthough thethe theoreticaltheoretical estimatesestimates for thethe size of thethe regionsregions Ai~i thatthat can bebe extractedextracted fromfrom thethe known proofproof of Theorem 1 give us extremely smallsmall values.values. We will call asas thethe NevrhovseNewhouse regions,regions, suchsuch regionsregions inin thethe space ofof dynamicaldynamical systems (or(or inin thethe parameter space while speakingspeaking on a finite-parameter family) where systems with homoclinichomo clinic tangenciestangencies are dense. In thethe case where bifurcations ofof some systemsystem having a saddle periodic orbit withwith aa homoclinichomo clinic tangencytangency are considered, we reservereserve thethe term “Newhouse"Newhouse regions”regions" specifically forfor thosethose inin a small neighborhood of thethe initial system, where systems are dense whichwhich have homoclinic tangenciestangencies of thethe givengiven periodic orbit. As we see, thethe problem of studying dynamical phenomena in the':;he Newhouse regionsregions is an importantimportant part of the global problem of studying thethe nature of chaos in realreal dynamics models. Besides, this problem is of itsits own interest from thethe point of view of the qualitative th.eorytheory and thethe theorytheory of bifurcations of dynamical systems. In the present paper, we describe dynamical phenomena in the Newhouse regionsregions for both thethe two-dimensional and the multidimensional cases. In Sections 2 and 3, we discuss mainmaLn results (Theorems 2-10). In Section 4, we collect geometrical constructions which determine dynamics (Theorems 2-10). In Section 4, we collect geometrical constructions which determine dynamics near homoclinic tangencies. We restrict ourself by the case of diffeomorphisms: the case of flows near homoclinic tangencies. We restrict ourself by the case of diffeomorphisms: the case of flows can be similarly considered by means of the Poincare map. can be similarly considered by means of the Poincare map.
2. MAIN RESULTS: THE TWO-DIMENSIONAL CASE Before studying the general multidimensional case, we consider the case of two-dimensional Before studying the general multidimensional case, we consider the case of two-dimensional maps. Let f be a two-dimensional diffeomorphism having a saddle fixed point 0 with multipliers X maps. Let I be a two-dimensional diffeomorphism having a saddle fixed point 0 with mulltipliers ), and 7, where IX] < 1, ]y] > 1. Let Ws and W” be, respectively, the stable and unstable manifolds and ,,(, where 1)'1 < 1, 11'1 > 1. Let WS and WU be, respectively, the stable and unstable manifolds of 0. Suppose they have a quadratic tangency at the points of some homoclinic orbit P (Figure 1). of O. Suppose they have a quadratic tangency at the points of some homoclinic orbit r (Figure 1). According to the traditional approach going back to Andronov, to study the bifurcations of a Accordinggiven system to theis to traditional embed it approachin an appropriate going back finite-parameter to Andronov, tofamily, study thenthe bifurcationsto divide the of aparameter given system space isinto to theembed regions it inof anstructural appropriate stability, finite-parameter to determine family,the bifurcation then to set,divide and theto parametersplit the bifurcation space into set the into regions connected of structural components stability, corresponding to determine to identical the bifurcation phase portraits set, and (into splitthe sense the bifurcation of topological set equivalence).into connected Accordingly, components a corresponding good model must to identicalpossess aphase sufficient portraits number (in theof parameters sense of topological allowing oneequivalence). to analyze Accordingly,bifurcations aof good each modelperiodic, must homoclinic, possess a sufficientheteroclinic number orbit ofoccurred. parameters allowing one to analyze bifurcations of each periodic, homoclinic, heteroclinic orbit occurred.In a general finite-parameter family containing f, the splitting parameter p must clearly be one of Inthe a maingeneral parameters. finite-parameter We define family the containingsplitting parameter I, the splitting as follows. parameter Take J1,a mustpoint clearlyof homoclinic be one oftangency the main on parameters.Ws (the point We M define + in theFigures splitting 2 and parameter 3). The manifold as follows. W” Takehas a pointparabola-like of homo shape clinic tangencynear this onpoint WS for (the all pointmaps M+close in to Figures f. We 2denote and 3).as Thep the manifold distance WUbetween has a W3parabola-like and th.e bottom shape nearof the this parabola. point forThe all mapssign of close p is to chosen I. We such denote that as f,,J1, thehas distanceno homoclinic between orbits WS andat p the:a 0bottom which ofare the close parabola. to P, and The there sign are of twoJ1, is structurallychosen such stable that I,..such has orbits no homo at p clinic< 0 (Figureorbits at2). J1, :> 0 which areAs close we noticed,to r, and values there of are p twofor whichstructurally the map stable f,, has such “secondary” orbits at J1, homoclinic< 0 (Figure tangencies 2). accu- mulatesAs we noticed,at /.J = 0.values Indeed, of J.ttake for whicha pair theof pointsmap I,..belonging has "secondary" to P and homocliniclying near tangencies0 : M+ E accu Wi, mulatesand M- atE W&J1, = O.(see Indeed, Figure take3). Takea pair ,U of a bitpoints smaller belonging than zero. to r Takeand alying piece near C of 0 the : M+part E ofWI~c the andunstable M- Emanifold Wl~c (see that Figure lies near 3). TakeM+ andJ1, a beginbit smaller to iterate than it. zero. After Take some a piecenumber C ofof theiterations part of (thethe unstable manifold that lies near M+ and begin to iterate it. After some number of iterations (the QuasiattractorsQUasiattractors 199199
W” W” s 1, I, lw W” Ww Ofo o Ofo
(4(a) PJ.I. < 0.O. (b) /.L = 0. (4(c) PJ.I. > 0.O. Figure 2.2. The splittingsplitting parameter pJ.I. isis chosen suchsuch thatthat WUW" has aa tangencytangency withwith W3W· at a homoclinic point M+ at p = 0, there is no homoclinic intersection near M+ at at a homoclinic point M+ at J.L == 0, there is no homoclinic intersection near M+ at ~1 > 0 and there are two points of intersection at p < 0. J.L > 0 and there are two points of intersection at J.t < O.
a seccfndaryhomoclinic a secondarytangency homoclinic tangency Figure 3. The figure shows how a secondary homoclinic tangency of the manifolds Ws andFigure W” 3. mayThe figurebe obtained. shows howTake a secondarya pair of homoclinicpoints which tangency belong of tothe r manifoldsand lie near W· and0 : M+W" Emay WC, be andobtained. M- E W&.Take aTake pair pof a pointsbit smaller which than belong zero. to Taker and a liepiece near C of0: the M+ part E WI~cof W” andthat M- liesE nearWI~c' M+Take and p, abegin bit smallerto iterate than it. zero.After Takesome a piecenumber C of theiterations part of(the W" closer that liesC is nearto WP,,,M+ andthe beginlarger tothe iteratenumber), it. Afterit may someapproach number a ofsmall iterations neighborhood (the closerof M-. C is Since,to WI~c' at pthe = larger0, the thepoint number),M- goes it mayat M+ approachby some a smallfinite degreeneighborhood of f, it followsof M-. that Since, a small at p,neighborhood = 0, the pointof M-M- isgoes mapped at M+into bya smallsome neighborhoodfinite degree ofof f, M+it followsby the thatsame a smalldegree neighborhoodof fp at all small of M-/.L isThus, mapped completing into a smallone neighborhoodround along the of M+initial byorbit the sameof homoclinic degree of tangency,II' at all smallthe curvep,. Thus,C may completing return to one a roundneighborhood along theof M+.initial While orbit doingof homoclinic that, the tangency,curve C is theexpanded curve Cand mayfolded, return thereby to a formingneighborhood a “parabola” of M+. Whilef!(C) doingwhich that,may theclearly curve have C isa expandedtangency andwith folded,WC, atthereby some pointforming near a "parabola"M+, if fi and f!(C) C are which appropriately may clearlychosen. have a tangency with Wl~c at some point near M+, if p, and C are appropriately chosen. closer C is to the stable manifold, the larger the number), it may approach a small neighborhoold closerof M-. C isSince, to the at stable,LL= 0, manifold, the point the M- larger goes the at number),M+ by someit may finite approach degree a ofsmall f, itneighborhood follows that ofa smallM-. neighborhoodSince, at J.L = of0, M-the pointis mapped M- goesinto ata smallM+ byneighborhood some finite ofdegree M+ ofby f,the it samefollows degree that aof smallffi at neighborhoodall small p. Thus, of M-the iscurve mapped C may into return a small to aneighborhood neighborhood ofof M+ M+ byfor thesome same number degree Ic of f,."iterations at all smallof fiL J.L. (weThus, will thespeak curve that C mayC makes return a tosingle a neighborhood round along ofI?). M+ While for somedoing numberthat, the k of iterations of f", (we will speak that C makes a single round along r). While doing that, the 200200 S.S. V.V. GONCHENKO etet al.al.
o
the cuCll c ic tangency Figure 4. Take p a bit greater than in Figure 2. Then, after one more round along theFigure initial 4. Takehomoclinic jJ. a bitorbit, greater the thanimage in ofFigure C takes 2. Then,a distorted after oneform morewhich roundallows alongone tothe obtain initial ahomoclinic cubic tangency orbit, ofthe Ws imageand W”.of C takes a distorted form which allows one to obtain a cubic tangency of W' and W". curve C is expanded and folded, thereby forming “parabola” f,(C). Fitting 1-1and C, one can curve C is expanded and folded, thereby forming "parabola" f!(C). Fitting J..L and C, one can clearly obtain a secondary homoclinic tangency. clearly obtain a secondary homo clinic tangency. Making more rounds, other homoclinic tangencies can be obtained with an appropriate variation of Making p. According more rounds,to Theorem other 1, homoclinic values of p tangencies corresponding can beto obtainedthe multi-round with anhomoclinic appropriate ixmgencies variation offill J..L.densely According intervals to Theorem accumulating 1, values at ofp J..L= corresponding0. to the multi~round homo clinic tangencies fill Wedensely note also,intervals that accumulatinga small perturbation at J..L O.of f may imply cubic homoclinic tangencies. Figure 4 showsWe notehow also,it can that be achieved.a small perturbation Consider a ofsystem f may with imply the cubicsecondary homoclinic homoclinic tangencies. tangency Figure (Fig- 4 showsure 3). howWe ittake can thebe achieved.parabola Considerf,k(C) and a systemchange withp a littlethe secondarybit, so that homo the clinicparabola tangency lies above(Fig ureWs. 3).By Wesome take number the parabola k’ of iterations, f!(C) andthe change parabola J..L a carrieslittle bit,out soone that more the round parabola along lies I?. above The S curveW • Byf k+k’(C) some numberis a “distorted k' of iterations,parabola” the(Figure parabola 4) which carries can out be onemade more cubically round tangentalong r.to TheW” B curveby a small f!+k' perturbation (C) is a "distorted (for this, parabola" two control (Figure parameters 4) which arecan necessary). be made cubically tangent to W byIncreasing a small perturbation the number (forof rounds this, twoalong control I, homoclinic parameters tangencies are necessary). of higher and higher orders can be Increasingobtained inthe a numberneighborhood of rounds of alongthe initial r, homoclinic quadratic tangenciestangency. ofSince higher systems and higher with ordersquadratic can betangencies obtained are in densea neighborhood in the Newhouse of the regions,initial quadraticwe arrive tangency.at the following Since systemsresult. with quadratiC tangencies are dense in the Newhouse regions, we arrive at the following result. THEOREM 2. (See [32,33].) Systems with homociinic tangencies of any order (definite or indefi- nite)THEOREM are dense 2. (Seein the [32,33}.) Newhouse Systems regions. with homoc1inic tangencies of any order (definite or indefi nite) are dense in the Newhouse regions. Recall the definition of the order of tangency of two Cr-smooth curves yr and ys on a plane. LetRecall the curvethe definition yr be given of theby orderthe equation of tangency y = of0 two and cr“yz -smooth be given curves by the '/'1 equationand '/'2 ony a= plane.q(s), Let the curve '/'1 be given by the equation y = 0 and '/'2 be given by the equation 'Ii = cp(x), QuasiattractorsQuasiattractors 201201
~(0) y = &i-J+ El2 + . . . + Es-&-l + x8+l + o(z8+l). (2.1)(2.1) TheThe values ciCi areare parametersparameters whichwhich control thethe bifurcationsbifurcations ofof thethe intersectionsintersections ofof IV”WU and IV9WS (the(the lastlast hashas thethe formform yy == 0). We see thatthat thethe bifurcationbifurcation analysis requiresrequires at leastleast anan s-parameter familyfamily inin thisthis case.case. According toto Theorem 2, one can obtainobtain tangenciestangencies of arbitrarily high order byby a smallsmall per-per turbationturbation ofof thethe initialinitial mapmap ff with thethe orbit of homoclinichomoclinic tangencytangency of order 1.1. Therefore, wewe have toto conclude that nono finite number of control parameters isis sufficient forfor thethe complete studystudy ofof thethe bifurcations in a small neighborhood of a homoclinic tangency,tangency, independentlyindependently of the order of it. The impossibilityimpossibility of giving thethe complete description of thethe bifurcations of systems with struc-struc turallyturally unstable Poincare homoclinic orbits appearsappears also asas thethe presencepresence of systems with arbi-arbi trarily degenerate periodic orbits inin thethe Newhouse regions. It is well known that if, forfor some C-smoothCr -smooth map, an orbit of period jj has one multiplier equal to v = fl and all the other multipliers do not lie on the unit circle, then in the case Y = 1, the to v = ±1 and all the other multipliers do not lie on the unit circle, then in the case v = 1, the restriction of thethe Jlh‘th degree of thethe map onto the center manifold can be written either in thethe form g = y + L,y8+1 + 0 (ys+l) 1 l (a) s = 1. (a) 8 == 1. (b)(b) s8::: = 2. Figure 5. The firstfirst returnreturn map forfor thethe cases oEof: (a)(a) quadraticquadra.tic tangency;tangency; (b)(b) cubic tangency.tangency. Let [Xyl1'>'')'1 < 1 (the case IXyl1'>'')'1 > 1 isis reducedreduced toto thethe case IA+yl1'>'')'1 < 1 byby transitiontransition fromfrom f toto its inverseinverse map). Then map (2.5)(2.5) isis closeclose toto thethe well-known one-dimensional parabola map &=M-y2, (2.6)(2.6) for k large enough; the resealed splitting parameter M may be take arbitrary finite values (the for k large enough; the rescaled splitting parameter M may be take arbitrary finite values (the larger k, the larger the interval of allowed values of M). larger k, the larger the interval of allowed values of M). In the case of sthth order tangency, the resealed map T, is close to the one-dimensional map (see In the case of 8 order tangency, the rescaled map Tk is close to the one-dimensional map (see Lemma 2 in Section 4) g=Eo+E1y+ ' * * + E,-lys-' + ys+l + 0 (ys+l) ) (2.7) where Ee, El,. . . , ES-i are resealed parameters EO,EI,. . . ,es-i from (2.1) and they may take where Eo,E1, ... ,Es-l are rescaled parameters co,eb ... ,es-l from (2.1) and they may take arbitrary finite values. Particularly, if Es = Ez = a.. = Es-1 = 0, El = fl, then map (2.7) arbitrary finite values. Particularly, if Eo E2 E - 0, El ±1, then map (2.7) has a fixed point with the multiplier fl and= with= the... order s of1 degeneracy= which can be made has a fixed point with the multiplier ±1 and with the order of degeneracy which can be made arbitrarily high by increasing the value of s. Since Tk is close to map (2.7) it also has a highly arbitrarily high by increasing the value of s. Since Tk is close to map (2.7) it also has a highly degenerate fixed point for Eo, E2, . . . , Es-1 close to zero and El close to fl. degenerate fixed point for Eo, E , . .. , E - close to zero and El close to ±l. Thus, by a small perturbation 2of a systems 1 with a homoclinic tangency of a large order, one can Thus, by a small perturbation of a system with a homoclinic tangency of a large order, one can achieve a periodic orbit of a high order of degeneracy to arise. Since systems with homoclinic achieve a periodic orbit of a high order of degeneracy to arise. Since systems with homoclinic tangencies of any order are dense in the Newhouse regions (Theorem 2), it follows that systems tangencies of any order are dense in the Newhouse regions (Theorem 2), it follows th;a.t systems with arbitrarily degenerate periodic orbits are also dense there. withWe arbitrarily see again thatdegenerate no finite periodic number orbitsof control are alsoparameters dense there. is sufficient for the complete study of theWe Newhouse see again regions: that no now,finite fornumber the study of control of the parameters bifurcations is sufficientof periodic for orbits. the complete In other studywords, of thefrom Newhousethe point regions:of view ofnow, the approachfor the study traditional of the bifurcationsto the bifurcation of periodic theory, orbits. any dynamical In other words,model from(a finite-parameter the point of viewfamily of the of approach dynamical traditional systems) tois, the in bifurcationterms of [32,33], theory, badany indynam.ical the Newhouse model regions.(a finite-parameter Apparently, familyhere itof is dynamical necessary systems)to give up is, the in ideologyterms of of[32,33], complete bad description in the Newhouse and to regions.restrict oneselfApparently, to the herecalculation it is necessary of some toaverage give up quantities the ideology and ofto completethe study descriptionof certain generaland to restrictproperties. oneself to the calculation of some average quantities and to the study of certain general properties.In particular, such a general property is that in the Newhouse regions there exist nontrivial hyperbolicIn particular, sets; i.e.,such there a general is always property a countable is that number in the ofNewhouse saddle periodic regions orbits there andexist structurally nontrivial hyperbolicstable Poincare sets; i.e.,homoclinic there is orbits.always a countable number of saddle periodic orbits and structurally stableAnother Poincare important homoclinic feature orbits. of systems in the Newhouse regions is the absence of complete self-similarity.Another important Notice thatfeature the ofhomoclinic systems inorbits the ofNewhouse high orders regions of tangency is the absencethat we ofo’btained complete by self-similarity.perturbations ofNotice the map that f themake homoclinic quite a large orbits number of high of orders rounds of along tangency P (for that instance, we obtained the cubic by perturbationstangency can beof theformed map after f make three quite rounds). a large It numberis clear ofthat rounds the higher along ther (fororder instance, of tangency, the cubic the tangency can be formed after three rounds). It is clear that the higher the order of tangency, the Quasiattractors 203 more rounds is required to get it. Near the homoclinichomo clinic tangencies of high orders there appear the maps described by formula (2.7). Since the first return map near such a tangency corr~pon&,corresponds, at the same time,time, to many rounds along the initial homoclinic orbit I’,r, maps close to the map 1f exhibit dynamics which is described on large time scales by maps (2.7): the larger the number of rounds, thethe larger the value of s. The maps given by formula (2.7) are completely &f&rentdifferent for different values of s. Thus, systems belonging to the Newhouse regions may show completely different qualitative behavior on different time scales. Note that nothing similar happens in hyperbolic systems where thethe number of essential scales is always finite. One more important feature is the coexistence of orbits of different topological types. If we consider a structurally stable Poincare homoclinic orbit, then we see that all periodic orbits lying in a small neighborhood of it have a saddle-type [25,26]. On the contrary, near a structurally unstable homoclinic orbit there may well known exist both structurally unstable and attractive periodic orbits in addition to saddle ones. Namely, the following theorem is valid. THEOREM 4. (See [27,28]')127,281.) Let the product of the multipliers of 0 be less than unity in absolute value: 1..\,1]Xy] < 1. Then for a general one-parameter familyfamily 1/1fM there exists a sequence of intervals 6i accumulating at p = 0, such that at p E &i the map fp possesses an attractive periodic orbit in Di accumulating at J.t = 0, such that at J.t E Di the map IJ1. possesses an attractive periodic orbit in a small neighborhood of F and, at p belonging to the boundary of &, the map has a structurally a small neighborhood ofr and, at J.t belonging to the boundary of Di, the map has a structurally unstable periodic orbit. If ]Xy] > 1, then the analogous result is also valid: the map has here a repelling periodic orbit If 1..\')'1 > 1, then the analogous result is also valid: the map has here a repelling periodic orbit (a source) for p E 6i. Theorems 4 and 1 imply the following result. (a source) for J.t E Di. Theorems 4 and 1 imply the following result. THEOREM 5. If ]Xr] < 1, then, for a general one-parameter family fP, in the Newhouse re- THEOREM 5. If 1..\,1 < 1, then, for a general one-parameter family IJ1.' in the Newhouse re g-ions Ai, parameter values are dense for which the map, in addition to a countable number of gions D.i' parameter values are dense for which the map, in addition to a countable number of saddle periodic orbits, also possesses a countable number of attractive3 periodic orbits. saddle periodic orbits, also possesses a countable number of attractive3 periodic orbits. In its initial weaker formulation (not for intervals in one-parameter families but for regions in In its initial weaker formulation (not for intervals in one-parameter families but for regions in the space of dynamical systems) this theorem was proved in [35]. The proof of the one-parameter the space of dynamical systems) this theorem was proved in [35]. The proof of the one-parameter version can be obtained, for instance, in the following way. Let ]Xy] < 1 and ~0 E A,. By version can be obtained, for instance, in the following way. Let 1..\')'1 < 1 and J.to E D.i. By Theorem 1, arbitrarily close to 110 there exists ~1 E Ai such that 0 has a quadratic homoclinic Theorem 1, arbitrarily close to J.to there exists J.tl E D.i such that 0 has a quadratic homoclinic tangency at ~1 = ,ui. By Theorem 4, near p = ~1 there exists a small interval di c Ai such tangency at J.t = J.tl' By Theorem 4, near J.t = J.tl there exists a small interval d C D.i such that f,, has an attractive periodic orbit at p E di. Again, since dr C Ai there exists1 a value that has an attractive periodic orbit at J.t E d . Again, since d C D.i there exists a value p2 E IJ1.dl such that 0 has a quadratic tangency at 1p = ~2 and some1 new interval dz c dl such thatJ.L2 E f,d 1 hassuch one that more 0 hasattractive a quadratic periodic tangency orbit atat pJ.L E= da. J.t2 Repeatingand some newthe arguments,interval d2 weC dobtain,1 such thatin arbitrary IJ1. has closenessone more ofattractive the given periodic value ~0, orbit the atsystem J.L E d2.of embeddedRepeating intervals the arguments, dr > dz >we . .obtain, . such thatin arbitrary fp has atcloseness least j attractiveof the given periodic value J.Lo,orbits the atsystem p E dj. of embeddedThe intersection intervals of alld1 :)dj d2is :)nonempty. ••• such thatIt contains IJ1. has atat leastleast onej attractive point p* periodicand the orbitsmap fpat hasJ.L E ad jcountable• The intersection number of allattractive dj is nonempty.periodic orbitsIt contains at ,u =at p*.least one point J.t* and the map fJ1. has a countable number of attractive periodic orbitsTheorems at J.L = 4J.t* and • 5 provide a theoretical basis for the fact that most presently known strange attractorsTheorems contain 4 and attractive5 provide aperiodic theoretical orbits basis within. for theAs fact a rule,that mostthe attractive presently periodicknown strangeorbits inattractors a quasiattractor contain attractivehave very periodiclong periods orbits and within. narrow Asbasins a rule, of attraction,the attractive and periodicthey are orbitshard toin aobserve quasiattractor in applied have problems very longbecause periods of theand presencenarrow basinsof noise. of attraction,However, inand the they space are of hard the parametersto observe inof theapplied model problems there can because exist regions of the where presence individual, of noise. relatively However, short-period in the space attractive of the periodicparameters orbits of thecan model be seen; there these can existregions regions are called where windowsindividual, of stability.relatively short-period attractive periodic orbits can be seen; these regions are called windows of stability. 3. MAIN RESULTS: THE MULTIDIMENSIONAL CASE 3. MAIN RESULTS: THE MULTIDIMENSIONAL CASE Theorems 2 and 3 can be extended onto the general multidimensional case [36]. Thus, the conclusionTheorems on 2 impossibilityand 3 can beof extendeda finite-parameter onto the generalcomplete multidimensionaldescription is also case valid [36]. for Thus,this case. the conclusionHowever, the on situationimpossibility connected of a finite-parameterwith the coexistence complete of periodic description orbits is alsoof different valid fortopological this case. typesHowever, is considerably the situation more connected complicated. with the Here,coexistence the windows of periodic of stabilityorbits ofmay different contain topological narrow types is considerably more complicated. Here, the windows of stability may contain narrow 3Repelling if IA71> 1. 3Repelling if IA11 > 1. 204 S. V. GONCHENKO et al.at. invariant tori and even chaotic attractors. Moreover, not only saddle and attractive (or saddle and repelling) periodic orbits may exist simultaneously, but saddle periodic orbits with the different numbers of positive Lyapunov exponents may also coexist. These statements are based on the results represented below. Let f be a multidimensional diffeomorphism with a structurally unstable homoclinichomo clinic orbit r of some saddle fixed point 0.O. We are interested in the structure of the set N of all orbits which lie entirely in a small neighborhood U of the set 0 uU I’.r. Suppose the map satisfies some conditions of general position [36] (the tangency is quadratic, l?r does not lie in W8SW8* and W”“,WUu, etc.). Let Xi,.AI, .... . , Am, 11.71,...... ,3;1, In be the rnultipliersmultipliers of 0, lin(+yn] I 12:: ..•**- 22:: lyll1111 > 1 > 1x11IA11 2::> ***••. ~2 IX,].IAml· We use the notation XA = 1x11,IA11, yI = 1111.In]. The multipliers Xi,yjAi"j nearest to the unit circle (i.e.,(Le., those for which IAi]Xi] I = X,A, ]-yj]IIj I = y)I) we call Eeadingleading and the rest we call nonleading. The coordinates in a neighborhood of 0 that correspond to the characteristic directions of these multipliers we call, respectively, leading and nonleading. We assume that the leading multipliers are simple. We designate the number of leading stable multipliers by p,Ps and the number of leading unstable multipliers by p,.PU' Accordingly, we assign the type (pS,pu)(Pa, Pu) to thethe system. The four following cases are possible here: (1,l).(1,1). AlAI and yiII are real and XA > I&],IA21, yI < I"nl,]TZ], (2,l).(2,1). XiAl = 'x2X.2 = XeiV,Aei Figure 6. An example of the “center” manifold MC (the union of M&, with the dashedFigure 6.regions An exampleoutside M&,of the in "center"the figure) manifoldfor the Methree-dimensional (the union of Mfoccase where with the dashedmultipliers regionsX2, X1,outside and Mfocy of the in thefixed figure)point forare thesuch three-dimensionalthat 0 < AZ < A1 case< 1 where< 7. the multipliers A2, AI, and 'Y of the fixed point are such that 0 < A2 < Al < 1 < 'Y. Figure 6 represents an example of the manifold MC for the three-dimensional case where the multipliersFigure 6 representsof 0 are such an examplethat 0 WU n (a) The orbit of tangency belongs to Wad.B (b) The vector that is tangent to W* and (a) The orbit of tangency belongs to W 8. (b) The vector that is tangent to w· and W” at M+ is parallel to the nonleading W U at M+ is parallel to the nonleading eigendirection. (c) The image of the surface lI; is tangent to Wd at W+ (c) The image ofthe surface TI;;- is tangent to W8 at W+. Figure 7. The exceptional cases where the smooth invariant manifold does not exist. Figure 7. The exceptional cases where the smooth invariant manifold does not exist. this is Case (1,l) where Xi and yi are the leading multipliers and X2 is the nonleading stable multiplier.this is Case The (1,1) point where 0 is Al the and fixed 1'1 pointare theof theleading stable multipliers node-type andfor theA2 isrestriction the nonleading of the mapstable f ontomUltiplier. W”. TheThe nonleading point 0 is themanifold fixed pointWss ofexists the stablein Ws node-typesuch that foriterations the restriction of any pointof the of map Wss f S 8S tendonto toW 0 • likeThe a nonleadinggeometric progressionmanifold Wwithexists the ratio in W8 X2. such The thatorbits iterations lying in ofWs any \ W”” point tend of toW8S 0 alongtend tothe 0 leadinglike a geometriceigendirection progression and the with distance the ratio to 0 A2' decreases The orbits as alying geometric in W8 progression\ was tend withto 0 thealong ratio the Xi. leading eigendirection and the distance to 0 decreases as a geometric progression with theIn ratiothis AI.case, in a small neighborhood of 0 there are well known [37] to exist two-dimensional invariantIn this case,Cl-manifolds in a small each neighborhood of which contains of 0 thereWlu,, are and well intersects known [37]Ws toat exista curve two-dimensional tangential to invariant C 1-manifolds each of which contains Wl~c and intersects W S at a curve tangential to 206206 S.V.S. V. GONCHENKO etet al.al. thethe leadingleading direction.direction. AccordingAccording toto TheoremTheorem 6,6, atat leastleast one ofof them,them, ML,,Mfoc' cancan bebe extended alongalong thethe orbits of f,I, formingforming aa global attractiveattractive invariantinvariant manifold MCMe whichwhich contains r.r. The manifold MCMe isis attractive inin thethe sense,sense, any pointpoint thatthat do not belong toto MCMe leaveleave thethe smallsmall neighborhood U of r with thethe iterationsiterations of thethe inverseinverse map f-1- ‘.1 • ThisThis impliesimplies thatthat bfc.Nt C containscontains thethe whole setset N of orbits lyinglying inin U entirely.entirely. The invarianceinvariance of MCMe means thatthat ifif one takestakes a small area II;TI; cC MfO,Mfoc containingcontaining thethe point M- of r and iterates thisthis area k times,times, thenthen itit returnsreturns inin the neighborhood of 0 for some k, soso thatthat fk(H;)Ik(TI;) cC MTO,Mfoc (Figure 6). This occurs possiblepossible if thethe mapmap fI satisfy some conditionsconditions of general position. The excludedexcluded cases where thethe smooth invariantinvariant manifold does notnot existexist are shown inin Figure 7: thethe homoclinichomoclinic orbit I’r belongsbelongs toto VPW 8S (Figure(Figure 7a); thethe vectorvector tangential to W”WU atat M+M+ isis parallelparallel toto the nonleadingnonleading eigenvector (Figure(Figure 7b); thethe surface f”(H,)Ik(fI;;) isis tangenttangent toto W”WS atat M+ (Figure 7~).7c). The reductionreduction theorem immediatelyimmediately givegive us essential restrictionsrestrictions on possible typestypes of orbits ofof thethe set N forfor thethe map fI itselfitself and forfor all nearby maps. Thus,Thus, since there isis a strong exponentialexponential contraction along the stable nonleading directions and thethe number of such linearly independentindependent directions isis (m - ps),Ps), orbits of N must have at least (m - p,)Ps) negative Lyapunov exponents. Analogously, thethe strong expansions along thethe nonleading unstable directions causes ,thatthat orbits of N must have at leastleast (n - pU)Pu) positive Lyapunov exponents. This means that dimensions of stable and unstable manifolds of any periodic orbit in U may not be less than (m - p,)Ps) and (n - pU),Pu), respectively. InIn particular, if 0 has unstable nonleading multipliers (i.e.,(Le., p,Pu < n), thenthen neither fI nor any nearby map has attractive periodic orbits in U. In general, these restrictions are not final. More precise estimates for the number of positive and negative Lyapunov exponents can be foundfound if one consider thethe (ps(Ps + p,)-dimensionalpu)-dimensional map which is the restrictionrestriction of thethe initialinitial map onto MC.Me. Let us introduce the quantity D which is equal to the absolute value of the product of all Let us introduce the quantity D which is equal to the absolute value of the product of all leading multipliers, i.e., D = XPsyPu. Note that D is the Jacobian of the restriction of 1’ onto MC, leading multipliers, Le., D = )..P''YPu . Note that D is the Jacobian of the restriction of f onto Me, calculated at the point 0. If D < 1, then the map fl~c contracts (p, +p,)-dimensional volumes calculated at the point O. If D < 1, then the map tiMe contracts (Ps + pu)-dimensional volumes exponentially near 0, and if D > 1, then it expands the volumes. Since any orbit that lies in U, exponentially near 0, and if D > 1, then it expands the volumes. Since any orbit that lies in U, entirely spends most of the time in a small neighborhood of 0, the map flm= contracts (pa +pu)- entirely spends most of the time in a small neighborhood of 0, the map liMo contracts (Ps + Pu) dimensional volumes in a neighborhood of the orbit at D < 1 and it expands the volumes at dimensional volumes in a neighborhood of the orbit at D < 1 and it expands the volumes at D > 1. Therefore, any orbit of N has at least one negative Lyapunov exponent at D < 1 and it D > 1. Therefore, any orbit of N has at least one negative Lyapunov exponent at D < 1 and it has at least one positive Lyapunov exponent at D > 1. has at least one positive Lyapunov exponent at D > 1. If to summarize what is said above, we arrive at the following result. If to summarize what is said above, we arrive at the following result. THEOREM 7. (See [36].) Let f b e a map with a homoclinic tangency in general position. If the THEOREM 7. (See (36].) Let I be a map with a homoclinic tangency in general posit;ion. If the saddle fixed point 0 has unstable nonleading multipliers (pu < n) or if D > 1, then neither f saddle fixed point 0 has unstable nonleading multipliers (pu < n) or jf D > 1, then neither I nor maps close to it have attractive periodic orbits in a small neighborhood of 0 n I?. nor maps close to it have attractive periodic orbits in a small neighborhood of 0 n r. A statement that is, in a sense, opposite to this theorem, is also valid. A statement that is, in a sense, opposite to this theorem, is also valid. THEOREM 8. (See [36].) If 0 has no unstable nonleading multipliers (p,, = n) and if D < 1, then systemsTHEOREM with 8. infinitely(See (36J.) many If 0 attractivehas no unstable periodic nonleading orbits are multipliers dense in the(pu Newhouse= n) and ifregions D < 1,Ai. then systems with infinitely many attractive periodic orbits are dense in the Newhouse regions ~i' This theorem does not follow from the reduction theorem. Here, the proof is based on the study of Thisthe first theorem return doesmap not Tk follow of some from small the stripreduction ff close theorem. enough Here,to M+. the Note proof that is basedthe maps on the Tk study may ofbe thedifferent first returnin different map Tksituations. of some smallNamely, strip let(J close0, do enough not have to M+. unstable Note nonleadingthat the maps multipliers Tk may andbe differentlet D < in1. differentThen, insituations. the case (ps,pu)Namely, = let(1, 0,l), dothe not map have Tk unstableis close tononleading the one-dimensional multipliers mapand let(like D in< the1. Then,two-dimensional in the case case; (Ps,Pu) see the= (1,1),previous the section)map Tk is close to the one-dimensional map (like in the two-dimensional case; see the previous section) g=M-y2, (3.1) Y- = M -y,2 (3.1) in some resealed coordinates. The same formula holds in the case (ps,pu) = (2,1) at Xy C 1. In inboth some cases, rescaled only onecoordinates. variable isThe relevant same andformula all the holds others in theare case suppressed (Ps, Pu) by= (2,1)strong atcontraction. >''''1 < 1. In bothIn thecases, case only (ps,pu) one variable= (2,l) isat relevant Xy > 1, and the allresealed the others map areTk issuppressed close to the by Henonstrong mapcontraction. In the case (Ps, Pu) = (2, 1) at ).."'1 > 1, the rescaled map Tk is close to the Henon map 3 = y, g=M-y2-Bx, (3.2) x = Y, fi = M - y2 - Ex, (3.2) at an appropriate choice of u. at an appropriate choice of (J. QuasiattractorsQU88iattractors 207207 2 InIn thethe casescases (ps,(Pa,Pu) pU) == (1,2)(1,2) andand (pa,(Ps,Pu) pU) == (2,2)(2,2) atat Xy2)..'Y << 1, thethe resealedrescaled mapmap TkTk isis close,close, forfor thethe appropriatelyappropriately chosenchosen u,0', toto thethe mapmap zx === y,y, fj=M-x2--y,y=M _x 2 Cy, (3.3)(3.3) and,and, inin thethe case (ps,(Pa,Pu) pU) = (2,2)(2,2) atat X+y2)....,,2 >> 1,1, itit isis close toto thethe mapmap zx=y, = y, gfi == z,z, y=M-y2-Cz-B~,fi = M y2 - Cz Bx, (3.4)(3.4) wherewhere M isis thethe resealedrescaled splittingsplitting parameterparameter CL,j.L, and B andand C areare some trigonometrictrigonometric functionsfunctions ofof kcpkcp and k+,k1{J, respectively.respectively. At kk largelarge enough,enough, parameters M,M, B, and C maymay taketake arbitrary finitefinite values. TheThe lastlast twotwo mapsmaps have not beenbeen studiedstudied sufficiently, unlikeunlike thethe parabolaparabola mapmap (3.1)(3.1) and thethe HenonHenan map (3.2).(3.2). However, the bifurcation analysis of thethe fixed pointspoints of thesethese maps isis comparatively simple. Thus,Thus, for each ofof mapsmaps (3.1)-(3.4)(3.1)-(3.4) one can easily findfind parameterparameter valuesvalues such thatthat there exists anan attractive fixedfixed point. Thus, an analogue of Theorem 4 isis valid: if therethere are no n&ableunstable nonleading multipliers and ifif DD < 1, thenthen a smallsmall perturbation of ff can provideprovide an appearance of anan attractive periodicperiodic orbit. Unlike to the two-dimensional case, inin dependence on the situation, not only thethe splitting parameter p may be required here, but also there may be necessary the perturbation of values cp parameter j.L may be required here, but also there may be necessary the perturbation of values cp and $1{J which control thethe variation of BBand and C, respectively.respectively. By the use of the construction with the system of embedded disks (analogous to that ap By the use of the construction with the system of embedded disks (analogous to that ap plied in the two-dimensional case at the proof of Theorem 5), the theorem on infinitely many plied in the two-dimensional case at the proof of Theorem 5), the theorem on infinitely many attractive periodic orbits (Theorem 8) follows immediately now for the Newhouse regions Ai in attractive periodic orbits (Theorem 8) follows immediately now for the Newhouse regions Ai in corresponding one-, two-, or three-parameter families. corresponding one~, two-, or three~parameter families. Actually, the analysis of fixed points of maps (3.1)-(3.4) allows us to establish much more than Actually, the analysis of fixed points of maps (3.1)-(3.4) allows us to establish much more than the existence of attractive periodic orbits. Thus, for maps (3.2) and (3.3) there exist the values the existence of attractive periodic orbits. Thus, for maps (3.2) and (3.3) there exist the values of M and, respectively, B or C at which the map has a fixed point with a pair of multipliers of M and, respectively, B or C at which the map has a fixed point with a pair of multipliers equal to unity in absolute value, while map (3.4) has a fixed point with three multipliers equal equal to unity in absolute value, while map (3.4) has a fixed point with three multipliers equal to unity in absolute value for some M, B, and C. If we select now the three cases (recall that to unity in absolute value for some M, B, and C. If we select now the three cases (recall that D = PyPU is less than one): D )..P''YP'' is less than one): o+> (Pa,P¶L) = (1,1)1 or (P~,P~) = C&l) and X-Y < 1, (1+) (Ps,Pu) = (1,1), or (Ps,Pu) = (2,1) and)"'Y < 1, (2+) (ps,pu) = (2,l) ad h > 1, or (P~,P,) = (L2) or (P~,P~> = (2,2) and h2 < 1, (2+) (Ps,Pu) = (2,1) and )...." > 1, or (Ps,Pu) = (1,2) or (Ps,Pu) (2,2) and )..'Y2 < 1, (3+) (P~,PJ = C&2) and X-v2 > 1, (3+) (Ps,Pu) = (2,2) and )....,,2> 1, then we arrive at the following result. then we arrive at the following result. THEOREM 9. (See (361.) Suppose that D < 1. Then, in Case (l+), systems having periodic THEOREMorbits with 9.1 multipliers(See (36].) equal Suppose to unity tbat in D absolute < 1. Tben,value inare Casedense (1+), in the systems Newhouse baving regions periodic Ai. orbits witb l multipliers equal to unity in absolute value are dense in tbe Newbouse regions Ai. This theorem has quite nontrivial consequences. Note that an invariant curve can be born fromThis the theorem points withhas quitetwo unit nontrivial multipliers consequences. (an invariant Note torus, that if anwe invariantconsider acurve flow) canand bechaotic born fromattractors the pointscan be with formed two unitin the multipliers case of three (an invariantmultipliers torus, equal if weto unityconsider in absolutea flow) andvalue. chaotic For attractorsinstance, an can attractor be formed similar in theto thecase Lorenz of three attractor multipliers can beequal born to at unity local inbifurcations absolute value.of a fixedFor instance,point with an two attractor multipliers similar equal to theto -1Lorenz and attractor one equal can+l, beand born a spiralat local attractor bifurcations can be of borna fixed in pointthe case with of twothree multipliers multipliers equalequal to to -1 -1 and(see one [38,39], equal where+1, andan analysisa spiral attractorof corresponding can be bornnormal in theforms case is ofcarried three out).multipliers equal to -1 (see [38,39J, where an analysis of corresponding normal formsUsing is carriedthe construction out). with embedded disks again, we find that systems with infinitely many invariantUsing thetori construction and systems with with embedded infinitely disksmany again, coexisting we find chaotic that systemsattractors urith are infinitely dense inmany the invariantNewhouse toriregions and in systems Cases (2+)urith andinfinitely (3+), manyrespectively. coexisting chaotic attractors are dense in the NewhouseTo conclude, regions we inconsider Cases the(2+) question and (3+), on therespectively. coexistence of saddle periodic orbits with different numbersTo conclude, of positive we consider Lyapunov the questionexponents. on the coexistence of saddle periodic orbits with different numbers of positive Lyapunov exponents. 208208 S.S. V.V. GONCHENKOGONCHENKO etet al.al. THEOREMTHEOREM 10.10. (See(See [36].)[36].) LetLet DD << 1, andand letlet 00 havebave nono unstableunstable nonleadingnonleading multipliers4multipliers. 4 Then,Tben, inin CaseCase (l+),(1 +), systemssystems thattbat forfor anyany jj == 0,.0, .... . ,,1 1 havebave aa countablecountable numbernumber ofof periodic periodic orbitsorbits withwitb jj multipliersmultipliers greatergreater thantban unityunity inin absoluteabsolute valuevalue areare densedense inin thetbe NewhouseNewbouse regionsregions &.At. AtAt thetbe samesame time,time, nono mapmap closeclose toto fI cancan have,bave, inin aa smailsmall neighborhoodneigbborbood UU ofof 0 0 Uu F,r, aa periodicperiodic orbitorbit withwitb moremore thantban 1 multipliersmultipliers greatergreater thantban unityunity inin absoluteabsolute value.value. The secondsecond partpart of thethe theoremtheorem followsfollows fromfrom thethe easily verifiedverified factfact that,that, inin CaseCase, (l+),(1+), thethe mapmap fI (and(and anyany nearbynearby map)map) contractscontracts exponentially (1+(1 + l)-dimensionalI)-dimensional volumesvolumes o:non MCMC inin aa smallsmall neighborhoodneighborhood ofof 0,0, and hence,hence, inin aa smallsmall neighborhoodneighborhood ofof anyany orbit lyinglying inin #I;rU entirely.entirely. Therefore,Therefore, anyany suchsuch orbitorbit cannotcannot havehave moremore thanthan 1I positivepositive LyapunovLyapunov exponents.exponents. TheThe firstfirst partpart of thethe theoremtheorem isis provedproved byby thethe linearlinear analysis oror fixed points ofof maps (3.1)-(3.4):(3.1 )-( 3.4): forfor any of thesethese maps, regionsregions ofof parameter values can be easily foundfound wherewhere thethe mapmap has aa fixedfixed pointpoint with jj multipliers greater thanthan unityunity inin absoluteabsolute valuevalue (0(0 5$ jj <$ 1). ThisThis impliesimplies that,that, for anyjany j =o,...,0, ... ,1,1, a periodic orbitorbit withwith jj positivepositive Lyapunov exponentsexponents cancan arise at anan arbitrarilyarbitrarily small perturbation of fI in a corresponding Z-parameterl-parameter family. Using the construction with embedded disksdisks again, wewe findfind that thethe parameterparameter values are dense in thethe NewhouseNewhouse regionsregions Ai atat whichwhich thethe map has now infinitelyinfinitely many such orbitsorbits simultaneously for each jj = 0,.0, .... . ,, 1.l. Theorem 10 hashas aa direct relationrelation toto the problem of hyperchaos. Usually, thosethose attractors areare calledcalled hyperchaotic for whichwhich more thanthan one positive Lyapunov exponent isis found.found. As we see, inin contrast with hyperbolic systems, the number of positive Lyapunov exponents may vary for different orbits ifif thethe system belongs toto a Newhouse region. It isis not clear, therefore, in ,whatwhat sense thethe number of positive Lyapunov exponents can be considered as a characteristics of thethe system as a whole. At the same time, considerations based on estimates of contraction and expansion of volumes are still effective here: the quantity 1 inin Theorem 10 is none other thanthan the integralintegral part of the Lyapunov dimension calculated at the point 0 by the Kaplan-Yorke formula [40] for part of the Lyapunov dimension calculated at the point 0 by the Kaplan-Yorke formula [40] for the restriction of thethe map fI onto the “center”"center" (or “inertial”)"inertial") manifold MC. 4. GEOMETRIC CONSTRUCTIONS AND CALCULATIONS We discuss here in greater detail, the geometric constructions that determine the dynamics We discuss here in greater detail, the geometric constructions that determine the dynamics near homoclinic tangencies. First, we consider the two-dimensional case. Namely, we consider near homoclinic tangencies. First, we consider the two-dimensional case. Namely, we consider a CT-smooth (r 1 3) two-dimensional diffeomorphism f, which has a saddle fixed point 0 with a Or-smooth (r :::: 3) two-dimensional diffeomorphism I, which has a saddle fixed point 0 with multipliers X and y where 0 < IX] < 1, ]y] > 1. We consider the case where ]Xy] < I.. Suppose multipliers ,A and "I where 0 < IAI < 1, 1"11 > 1. We consider the case where IA"II < 1. Suppose the stable and unstable manifolds of 0 have a quadratic tangency at the points of the homoclinic the stable and unstable manifolds of 0 have a quadratic tangency at the points of the homoclinic orbit l?. orbit r. Let U be a small neighborhood of the set 0 U I’. The neighborhood U is the union of a small Let U be a small neighborhood of the set 0 U The neighborhood U is the union of a small disc Uo containing 0, and of a finite number of smallr. disks surrounding the points of I’ which are disc Uo containing 0, and of a finite number of small disks surrounding the points of r which are located outside U,J (Figure 8). We denote by N the set of orbits of the map f that lie entirely located outside Uo (Figure 8). We denote by N the set of orbits of the map f that lie entirely in U. Let To be the restriction of f onto Uo (it is called the local map). Note that the map TO in in U. Let To be the restriction of onto U (it is called the local map). Note that the map To in some C’-’ -coordinates (z, y) can fbe writteno in the form [41,42] some Or-l·coordinates (x, y) can be written in the form [41,42] z = xx + f(x, y)x2y, ?J= YY + dx7 Y)XY2. (4.1) (4.1) By (4.1), the equations of the local stable manifold IV{, and local unstable manifold IV& are y By= 0 (4.1),and z the= 0,equations respectively. of theThe local representation stable manifold (4.1) WI~c is convenient and local inunstable that, in manifoldthese coordinates WI~c are Ythe = 0map and Tt x =for 0, anyrespectively. sufficiently The large representation k is linear (4.1)in the is lowestconvenient order. in that,Specifically, in these wecoordinates have the thefollowing map T~representation for any sufficiently [4l] of the large map k T,kis linear : (20, yo)in thet+ (2k,lowest yk) order. Specifically, we have the following representation [41] of the map T~ : (xo, Yo) t-+ (Xk' Yk) xk = AkZO + i~lkl-fl-ktk(xO,yk), Xk = ,AkXO + IAlkl"ll-kek(xo,Yk), ?/O = Y-“Yk + ld-2k~k(z0,yk),2k (4.2) Yo = "Y-kYk + bl- 'l'/k(xo,Yk), (4.2) 4The contribution of the unstable nonleading multipliers is trivial: instead of “‘j multipliers greater than unity” 4Thewe should contribution write “(n of - thepU unstable+ j) multipliers nonleading . ;”multipliers the case Dis trivial:> 1 is reducedinstead toof the"j multiplierscase D < 1greater by considering than unity"the wemap shouldf-’ insteadwrite "(nof f,- P-uso +everywhere j) multipliers through ... j"the the theorem case D the> 1words is reduced “greater to thethan caseunity” D Figure 8. The neighborhood lJ of the contour 0 U r (bold points in the figure) is a Figureunion of 8. a Thesmall neighborhooddisk We containing U of the0 andcontour of a finite0 u r number(bold pointsof small in theneighborhoods figure) is a. ofunion that ofpoints a. small of diskr which Uo containinglie outside 0Uo. and of a finite number of small neighborhoods of that points of r which lie outside Uo. where & and qk are functions uniformly bounded at all k along with their derivatives up to the where ek and 17k are functions uniformly bounded at all k along with their derivatives up to the order (r - 2). order (r - 2). Let M+(s+,O) and M-(0, y-) be a pair of points of I’ which lie in Ue and belong to I+‘& Let M+(x+,O) and M-(O,y-) be a pair of points of r which lie in Uo and belong to Wl~c and J$:,, respectively. Without loss of generality, we can assume x+ > 0 and y- > 0. Let II+and andlVj~c' II- respectively.be sufficiently Without small lossneighborhoods of generality, of wethe canhomoclinic assume x+points > 0M+ and andy- >M- O. suchLet and be sufficiently small neighborhoods of the homoclinic points and such n+that Te(II+)n- n II+ = 0 and Te(II-) n II- = 0. Evidently, there exists an integerM+ Q suchM- that that To(n+) n n+ 0 and To(n-) n n- 0. Evidently, there exists an integer q such that fQ’(M-) = M+. We denote the map fqq : II- ---) II+ as 2’1 (it is called the global map, see We denote the map --+ as Tl (it is called global map, see Figurefq(M-) 9). = The M+. map Ti can obviously bef written: n- in then+ form the Figure 9). The map Tl can obviously be written in the form 2 - x+ = ax + b(y - y-) + +n. , x - x+ = ax + b(y - y-) + ... , g = cx + d(y - y-)2 + . . . , (4.3) fi = ex + d(y _ y-)2 + ... , (4.3) where bc # 0 since Tl is a diffeomorphism, and d # 0 since the tangency is quadratic. whereNote bethat f; 0the since orbits Tl isof aN diffeomorphism,must intersect theand neighborhoods d f; 0 since theII+ tangencyand II- is(otherwise, quadratic. these orbits wouldNote be that far thefrom orbits I). However,of N must not intersect all orbits the that neighborhoods start in II+ n+arrive and in n- III-.(otherwise, The set of these the pointsorbits wouldwhose beorbits far fromget into r). II-However, form anot countable all orbits number that startof strips in I1+ ui arrive = II+ inn n-. Tt”II- The thatset ofaccumulate the points whoseon IV”. orbitsThe getway intoof constructing n- form a countablethese strips number is obvious of strips from lT2 Figure = n+ n10. To-kn- In turn,that the accumulate images of S onthe Wstrips• The 0: wayunder of theconstructing maps T$ thesegive onstrips II- isa obvioussequence from of verticalFigure 10.strips In C$turn, that the accumulate images of onthe W&strips (Figure lT2 under 11). the maps T~ give on n- a sequence of vertical strips IT! that accumulate onThe WI~c images (Figure of 11).the strips 0: under the map Tl have the shape of horseshoes, accumulating on theThe “parabola” images ofTIN’& the strips (F i glTk ure under 13). theIt ismap clear Tl thathave thethe orbitsshape ofof Nhorseshoes, must intersect accumulating II+ at theon thepoints "parabola" of intersection Tl WI~c of (Figurethe horseshoes 13). It Tl$is clear and that the thestrips orbits oio . ofTherefore, N must intersectthe structure n+ atof the points of intersection of the horseshoes T1lTJ and the strips IT?. Therefore, the structure of the 210 S. V. GONCHENKO et al. • • • • • \\0 ••• • • • W U Figure 9. The local and global maps Ti and Tl. Figure 9. The local and global maps To and Tl' set N depends strongly on the geometric properties of the intersection of the horseshoes and the set N depends strongly on the geometric properties of the intersection of the horseshoes and the strips. strips. To be specific, we shall assume that X > 0 and y > 0. Then, depending on the signs of c and d, "I fourTo different be specific, cases we of shall mutual assume arrangement that A > °ofand the manifolds> 0. Then, IS’,& depending and TrW,& on theare signspossible of c [27,28]and d, (Figurefour different 12). Ifcases Z’rW’,& of mutual is tangent arrangement to W’,& offrom the below manifolds (d < Wl~c0) (Figures and Tl Wl~c12a,b), are thenpossible the [27,28]set N has(Figure a trivial 12). structure:If Tl Wl~c Nis tangent= (0, I’} to[27,28]. Wl~c fromThis belowis related (d To-‘(II-) rT1 • • • • • o Figure 10. This figure illustrates the construction of the strips a:, lying on lI+, such Figure 10. This figure illustrates the construction of the strips 0'2, lying on rr+, such that ui is the domain of definition of the map T$ : Tf+ 4 II-. The points on II+ that 0'2 is the domain of definition of the map T~ : rr+ ..... rr-. The points on rr+ that lie in II- after k iterations of the map To, belong to the set Tck(FI-) n fI+. that lie in rr- after k iterations of the map To, belong to the set To-k(rr-) n rr+. The neighborhood II- is contracted in the vertical direction by a factor of y-’ and The neighborhood n- is contracted in the vertical direction by a factor of ,,(-1 and expanded in the horizontal direction by a factor of X-’ under the action of the map expanded in the horizontal direction by a factor of >. - 1 under the action of the map Tr’, and moreover, T,’ (II-) n II- = 0. Correspondingly, the set Tck(II-) in a To-I, and moreover, TO-1 (rr-) n rr- 0. Correspondingly, the set To-k(n-) in a narrow rectangular are expanded along the x axis and displaced from it by a distznce narrow rectangular are expanded along the x axis and displaced from it by a distence of the order of y-lr. Moreover, the rectangles Z’G”(TI-) and Z’i’k+l’(II-) do not of the order of ,,(-k. Moreover, the rectangles To"(rr-) and To-(k+1)(rr-) do not intersect. For sufficiently large k, the intersection of To-(“)(If-) with II+ is a strip UE intersect. For sufficiently large k, the intersection of To-(k) with is a strip as in the figure. As k -+ 00, the strips oz accumulate on the(rr-) segment rr+W* f~ II+. 0'2 as in the figure. As k ..... 00, the strips 0'2 accumulate on the segment W· n n+. Figure 11. The range of the map T,k : fI+ + ll- is the vertical strip o:. Figure 11. The range of the map T~ : rr+ ..... rr- is the vertical strip CT~. 212 S.s. V. GQNCHENKOGONCHENKO et al. o (a) c c: 0, d < 0. (a) c < 0, d < 0. (b) cc> > 0, d < 0.O. o o (c) c < 0, d > 0. (d) c > 0, d > 0. (c) c < 0, d> O. (d) c > 0, d> 0. Figure 12. The four different cases of homoclinic tangencies. These cases differ not onlyFigure in 12.the Themutual fourarrangement different casesof theof homoc1inicstable and tangencies.unstable manifolds These cases(tangent differ from not onlybelow: in Figuresthe mutual (a) andarrangement (b); tangent of thefrom stableabove: andFigures unstable (c) manifoldsand (d), but (tangent also in fromthat howbelow: the Figuresshaded (a)semineighborhood and (b)j tangent of fromthe pointabove: M- Figuresis mapped (c) andinto (d), the butneighborhood also in that ofhow the the point shaded M+ semineighborhoodunder the action ofof thethe pointglobal M-mapis mapped‘71. If X into> 0 the and neighborhoody > 0, these fourof thecases point are M+distinguished under the actionby the ofcombinations the global mapof signs TI. ofIf the>. > parameters0 and '"Y > 0,c andthese d offour the casesmap areTl. distinguished by the combinations of signs of the parameters c and d of the map TI. Below, to be specific, we consider only the case c > 0, d > 0. To describe maps close to f we mustBelow, introduce to be specific,the splitting we considerparameter only p: the when case p c <> 0,0, thed > parabola0. To describe TlW’& mapsintersects close toWiC f weat twomust points; introduce when the p splitting= 0, the parameterparabola TlW,S,, 11: when is tangent11 < 0, theto Wi,parabola at one Tl point,Wl~c intersectsand w:hen lVj~c Jo > at0 theretwo points;is no intersection.when 11 = 0, Itthe is parabolaclear that Tlif Wl~c the isbottom tangent of tothe Wl~c parabola at one descends point, and sufficiently when J1. >low ° there(large isand no negativeintersection. p), thenIt is eachclear horseshoe that if the intersects bottom ofeach the strip. parabola In this descends case, thesufficiently set NP islow a hyperbolic(large and negativeset similar 11), to thenthe eachinvariant horseshoe set in intersectsthe Smale eachhorseshoe. strip. InHowever, this case, if pthe is Sl3tsufficiently Np. is a largehyperbolic and positive, set similar then tothe the horseshoes invariant andset inthe the strips Smale do nothorseshoe. intersect However,at all, and if 11all isof sufficiently the orbits exceptlarge and0 willpositive, escape then from the U. horseshoes and the strips do not intersect at all, and all of the orbits exceptThe main0 will question escape fromis what U. happens when the parameter p varies from the large negative to theThe large main positive question values. is what First happens of all, itwhen is necessary the parameter to study 11 variesthe structure from the of large the negativebifurcation to theset correspondinglarge positive tovalues. one strip,First that of all,is, toit isstudy necessary the bifurcations to study thein thestructure family of the firstbifurcation return mapsset corresponding Tk(p) E TlT,k to : one0: +strip, 0:. thatThe is,following to study result the bifurcationsis valid. in the family of the first return maps Tk(J.L) == T1T~ : 0"2 -+ O"k' The following result is valid. LEMMA 1. The map Tk(p) can be brought to the form LEMMA 1. The map Tk(l1) can be brought to the form z = y + 0 (Xkyk + yk) ( k x = Y + 0 ().k"l + 'Y- ) , (4.4) g = M - y2 + 0 (Xkyk + 7-k) , (4.4) f} = M y2 + 0 ().k'Yk + 'Y-k) , QuasiattractorsQuasiattractors 213213 , ... 1 /'l;~ T1(Jjl (a) cc < 0, d < 0. (b) c > 0, d < 0. T1(Jjl M’M+ (c) c < 0, d > 0. (d) c > 0, d > 0. (c) c < 0, d> 0. (d) c > 0, d> O. Figure 13. Basic elements of the geometry of the intersection of a strip up and Figure 13. Basic elements of the geometry of the intersection of a strip u? and a horseshoe Tr(ujl) for the case 1x71 < 1. In the case of tangency from below a horseshoe Tl (uJ) for the case 1>'1'1 < 1. In the case of tangency from below (Figures (a) and (b)), the horseshoe Tr(a~) lies below “its” strip up. In this case, Tl(U{) ur. either(Figures Z’r(oi) (a) andintersects (b), thethe horseshoestrips uy only if j lies>> ibelow (the case"its" c strip> 0, d < 0)In orthis it case,does either Tl (ui) intersects the strips uJ only if j ;$ i (the case c 0, d < 0) or it does not intersect any strips at all (the case c < 0, d < 0). For this >reason, the structure not intersect any strips at all (the case c 0, d 0). For this reason, the structure of the set N is trivial in this case: N = 0 the 0 structureand c > 0, of d the> 0set is thatN is thenontrivial intersection here. ofThe any differencehorseshoe inwith the anycases strip c < is0, regulard > ° inand the c >first 0, dcase, > °whileis that in thethe intersectionlatter case thereof any can horseshoe be nonregular with anyas stripwell asis regular intersections.in the first case,As whilea result, in theall thelatter orbits case of therethe set can N beexcept nonregular r can beas shownwell as toregular be of theintersections. saddle-type Asin a theresult, case allc Smale saddle- CHAOS Smale saddle horse-shoe node ill '~~07e ...... ------ period homoclinic doubling bifurcations 4 Figure 14. The bifurcation interval bt’ , @] that corresponds to the sequence of Figure 14. The bifurcation interval [Ptl, 1L~'1 that corresponds to the sequence of bifurcations in the development of the Smale horseshoe on the strip ug, beginning bifurcations in the development of the Smale horseshoe on the strip 0"£, beginning with the first bifurcation of the generation of a saddle-node fixed point at p = ,ut’ with the first bifurcation of the generation of a saddle-node fixed point at 1L :=; 1Ltl and ending with the last one corresponding to a homoclinic tangency for p = @, and ending with the last one corresponding to a homoc1inic tangency for 1L = lL~s, after which the horseshoe appears. after which the horseshoe appears. Now, resealing the variables: Now, rescaling the variables: x + --$ykx,b -k Y + +-‘Y,1 -k x -t --'"d I x , Y -t -d"( y, brings equations (4.6) to form (4.4) where M = -dy2kMr. This completes the proof of the brings equations (4.6) to form (4.4) where M _d"(2k Ml. This completes the proof of the lemma. lemma.Map (4.4) is close to the one-dimensional parabola map Map (4.4) is close to the one-dimensional parabola map fj=M-y2, (4.8) (4.8) whose bifurcations have been well studied, so that it is possible to recover the bifurcation picture whosefor the bifurcationsinitial map haveTk. beenFor the well parabola studied, map,so that the it isbifurcation possible toset recover is contained the bifurcation in the intervalpicture Tk. for[-(l/4),2] the initialof valuesmap of MFor : atthe M parabola= -l/4 map,there appearsthe bifurcation a fixed setpoint is withcontained the multiplier in the intervalequal to[-(1/4),2] +l, this offixed values point of isM attractive: at M ::::.:: at-1/4 M Ethere (-(l/4), appears (3/4)) a fixedand pointit undergoes with the a multiplierperiod-doubling equal tobifurcation +1, this fixedat M point= 314; is attractivethe cascade at ofM period-doublingE (-(1/4), (3/4)) bifurcations and it undergoes lead to achaotic period-doubling dynamics whichbifurcation alternates at M with= 3/4;stability the cascadewindows ofand period-doubling the bifurcations bifurcations stop at M lead= 2, towhen chaotic: the restrictiondynamics ofwhich the alternatesmap onto withthe nonwandering stability windows set becomesand the bifurcationsconjugate to stopthe atBernoulli M = 2, shiftwhen of the two restriction symbols andof the it mapno longer onto bifurcatesthe nonwandering as M increases. set becomes conjugate to the Bernoulli shift of two symbols andBy itLemma no longer 1, similarbifurcates bifurcations as M increases. take place for the map Tk (see Figure 14). The map has anBy attractive Lemma fixed1, similar point bifurcationsok at fi E take(cl:‘, place&‘) forwhich the arisesmap Tkat (seethe saddle-nodeFigure 14). bifurcationThe map hasat pan = attractivepi1 and losesfixed stabilitypoint Ok(at at ,u 11 = EPC’) (I1t1, at 11kthe I) period-doublingwhich arises at bifurcation.the saddle-node Here bifurcation at 11 = I1tl and loses stability (at 11 = I1kl) at the period-doubling bifurcation. Here ’ -m +. . p;’ = py- -cx”x++-p 1 * 7 I1tl ,,(-ky- c,\kX+ + 4d ,,(-'2k + ... , -1 = m,-ky- - ,-Akx+ _ 4d3 .-.,-2k+... pk 3 > I1kl = ,,(-ky- - c,\kx+ - 4d ,,(-2k + ... , Quasiattractors 215 Figure 15. A homoclinic tangency, the last in the sequence of bifurcations in the Figure 15. A homoclinic tangency, the last in the sequence of bifurcations in the development of the Smale’s horseshow (this is the tangency corresponding to the development of the Smale's horseshow (this is the tangency corresponding to the case shown in the Figure 13~). case shown in the Figure 13c). Note, that we have found the intervals where the map fP possesses the attractive single-round Note, that we have found the intervals where the map 11-' possesses the attractive single-round periodic orbit and this is the main element of the proof of Theorem 4 in Section 2. periodic orbit and this is the main element of the proof of Theorem 4 in Section 2. The bifurcation set of the map Tk is contained in the interval [pi’, &“I, where The bifurcation set of the map Tk is contained in the interval [/-ttl, /-t~8], where At /I = &, the fixed point of Tk has the last homoclinic tangency (Figure 15) and an invariant At /-t /-t~8, the fixed point of Tk has the last homo clinic tangency (Figure 15) and an invariant set similar= to those of the Smale’s horseshoe example arises after this bifurcation. Note, that set similar to those of the Smale's horseshoe example arises after this bifurcation. Note, that these bifurcational intervals do not intersect each other for different k. these bifurcational intervals do not intersect each other for different k. Clearly, in addition to the orbits that intersect II+ each time in the same strip, the map fp also hasClearly, orbits thatin addition jump among to the theorbits strips that with intersect various n+ indices. each time The inbifurcation the same strip,intervals the correspond-map 11-' also inghas toorbits these that orbits jump can among now overlap. the strips This with is thevarious case indices.already forThe orbits bifurcation that jump intervals among correspond two strips ui,ing u:to theseand their orbits images, can now the overlap. horseshoes This isTiU,O the andcase Tjcjalready ‘. Figure for orbits 16 showsthat jump the caseamong where two stripsthere exista?, aJ completely and their developedimages, theSmale horseshoes horseshoes Tia? on and ~9 Tjo-J.and uy Figurebut the 16upper shows horseshoe the case intersects where there the lowerexist completelystrip in a “nonregular” developed Smalemanner, horseshoes and new on structurallya? and aJ butunstable the upper orbits horseshoe can arise intersects as a result. the Inlower particular, strip in usinga "nonregular" this construction, manner, oneand can new obtain structurally new heteroclinic unstable orbits(Figure can 16a) arise or ashomoclinic a result. In(Figure particular, 16b) tangencies. using this construction,Moreover, there one existcan obtainhere also new periodic heteroclinic orbits (Figure“jumping” 16a) fromor homoclinic one strip to(Figure another 16b) (they tangencies. correspond Moreover, to the fixedthere points exist hereof the also double-round periodic orbits return "jumping" map TjTi from : up one --) stripup). Theto another regions (theyof stability correspond of these to thedouble-round fixed points periodic of the double-roundorbits can overlap return for mapvarious TjT i and: a? j,--+ evena?). aThe countable regions ofnumber stability of theseof these regions double-round may have periodic common orbits points. can In overlap particular, for variousin the i setand of j, maps even witha countable the homoclinic number oftangency these regions(in the maycase have c > common0, d > 0) points. the maps In particular,with a countable in the setnumber of maps of withattractive the homoclinicperiodic orbits tangency of this (in type the arecase dense c > 0,[44,45]. d> 0) the maps with a countable number of attractiveThe geometric periodic construction orbits of thiswith type two are horseshoes dense [44,45]. was also a basic element of the proof of Theo- remThe 1. geometricFigure 17 constructionshows the bihorseshoe with two horseshoesused for the was proof. also aIn basicthis elementsituation, ofthe the invariant proof of Theoset of rem 1. Figure 17 shows the bihorseshoe used for the proof. In this situation, the invariant set of 216 S. V. GONCHENKO et al. (a)(4 (‘4(b) Figure 16. This figure shows how new heteroclinic or homoclinic tangencies are ob- Figure 16. This figure shows how new heteroc1inic or homoc1inic tangencies are ob tained. Here, on the strips up and CT: there are already developed Smale’s horseshoes tained. Here, on the strips u? and O'J there are already developed Smale's horseshoes for the maps Ti and T’, respectively, but the upper horseshoe intersects the lower for the maps T; and Tj, respectively, but the upper horseshoe intersects the lower strip “nonregularly.” In Figure (a), the manifold W’(Oi) is tangent to II”( In strip "nonregularly." In Figure (a), the manifold W"(Od is tangent to W8(Oj). In Figure (b), a piece W”(Oi) n 0: of the unstable manifold of the point Oi lies just Figure (b), a piece W"(Oi) n O'J of the unstable manifold of the point 0; lies just slightly above the stable manifold of the point Oj and the curve Tj(W”(Oi) 17 u$‘) slightly above the stable manifold of the point OJ and the curve Tj(W"(O,) n uJ) which is a part of the manifold W“(Oi) is tangent to Ws(Oi); i.e., a homoclinic W"(Oi) is W8(Oil; tangencywhich is aof partthe invariantof the manifoldmanifolds of Oi takestangent place. to i.e., a homoc1inic tangency of the invariant manifolds of Oi takes place. the map Ti on 0: is a completely developed Smale horseshoe. The map Tj on 0; is cl.ose to the the map Ti on is a completely developed Smale horseshoe. The map T on is close to the moment of the a?last tangency; i.e., the value of the parameter ,U is close to j$j”“. aJAt this moment, moment of the last tangency; i.e., the value of the parameter J.L is close to J.Ljs. At this moment, unstable whiskers of the hyperbolic set that lies in 0: are tangent, at points of some smooth unstable whiskers of the hyperbolic set that lies in are tangent, at points of some smooth curve, to stable whiskers of the hyperbolic set that liesa? in cj.’ The latter, in intersection with curve, to stable whiskers of the hyperbolic set that lies in The latter, in intersection with the curve of tangency, form a specific (thick) Cantor set, what,aJ. as Newhouse has shown, is the the curve of tangency, form a specific (thick) Cantor set, what, as Newhouse has shown, is the reason for the nonremovable nature of the tangency. reason for the nonremovable nature of the tangency. If we use not two, but a larger number of strips, then we can obtain degenerate homoclinic If tangencieswe use andnot periodictwo, but orbits. a larger In numberparticular, of strips,when three then horseshoeswe can obtain are used,degenerate then cubic homoclinic orbits cantangencies be formed. and periodicFigure 18 orbits. shows Inthree particular, horseshoes when where three WU(Oi)horseshoes and areWS(Oj) used, arethen quadratically cubic orbits WU(Oi) WS(Oj) tangent,can be formed.as are WU(Oj)Figure 18and shows Ws three(Ok). horseshoesThe next figurewhere (Figure 19)and illustrates arehow qua.draticallyfrom one of thesetangent, structurally as are WU(Oj)unstable and contours WS(Ok)' one The can, next by afigure small (Figureperturbation, 19) illustratesobtain ahow cubic from tangency one of theseof the structurallymanifolds IV‘(Oi) unstable and contours IV’(Ok). one can, by a small perturbation, obtain a cubic! tangency ofTaking the manifolds into account WU(Oi) a largerand W8(Ok)'number of strips is a quite complicated problem. We bypass the difficultiesTaking intoif note, account instead a larger of calculating number of thestrips multi-round is a quite returncomplicated map, problem.that due toWe Theorem bypass the2, homoclinicdifficulties iftangencies note, instead of high of orders calculating can appear the multi-roundwhen a piece return of IV‘ map,makes that many due rounds to Theorem along the 2, initialhomoc1inic homoclinic tangencies orbit ofI?. high Therefore, orders canthe appear multi-round when a returnpiece ofmaps WU makescan presumably many rounds be modelledalong the byinitial the homoclinicfirst return orbitmaps r.near Therefore, orbits of the highly multi-round degenerate return tangencies. maps can presumably be modelled byThese the first maps return are easily maps calculated. near orbits Indeed, of highly let degeneratea two-dimensional tangencies. diffeomorphism f have an orbit of Thesehomoclinic maps aretangency easily calculated.of some order Indeed, s. In letthis a two-dimensionalcase, the local map diffeomorphism TO still has fthe have lform an givenorbit ofby homo(4.1),(4.2); clinic tangencythe global ofmap some can order be written s. In thisin the case, form the local map To still has the form given by (4.1),(4.2); the global map can be written in the form T--x:+ =az+b(y-y-)+---) x x+ =ax+b(y y-)+ ... , (4.9) g = cz + d(y - y-y+1 -!-. . * ) (4.9) f} ex + d(y - y-)s+1 + ... , where, in the first equation, the dots stand for the second (and more) order terms and, in the secondwhere, equation,in the first for equation, the terms the of dotsthe orderstand o(lxl for the+ Iy second - y-I’+l). (and more) order terms and, in the second equation, for the terms of the order o(JxJ + Iy - y-Js+1). Quasiattra.ctorsQussiattractors 217217 O.1 r----+--~~--~~------~~------~~~~ -1 s t----+---t-~-+------+-----I---I«-'~TXWloJ Figure 17. The bihorseshoe used for the proof of Theorem 1. In this situation, the invariantFigure 17.set Theof thebihorseshoe map Ti onused up foris thethe proofdeveloped of TheoremSmale horseshoe.1. In this situation,The map theTj oninvariant a: is close set ofto thethe mapmoment Ti onof theu? islast the tangency; developed i.e., Smale the valuehorseshoe. of the parameterThe map Tjp ison close uJ isto close p$“. toAt the this moment moment of unstablethe last tangency;whiskers ofi.e., the thehyperbolic value of theset parameteron up touch p. theis closestable to whiskersJ.Lj". At thisof some moment hyperbolic unstable subset whiskers on cry. of the hyperbolic set on u? touch the stable whiskers of some hyperbolic subset on C1J. Consider an s-parameter family fE, E = (~0, . . . , &+I), of maps close to f (fs = f) where Consider an s-parameter family C = (co, ... ,cs-I), of maps close to (fo f) where parameters E are chosen such that theyIe, provides a general unfolding of the given Itangency == between parameters c are chosen such that they provides a general unfolding of the given tangency between IV“ and IV5 (see formula (2.1)). In this case, the global map takes the form W U and W 8 (see formula (2.1)). In this case, the global map takes the form ~-cc+=az+b(y-y-)+~~*) x - x+ = ax + b(y - y-) + ... , (4.10) g=cr++EgfE1(y-y-)+“’ + es-1(y - y-y-’ + d(y - y-y+1 + * * * . (4.10) y = ex + eo + CI(Y - Y-) + ... + es-l(Y - y-)s-l + d(y _ y-)s+1 + .... Let us now consider the first return map Tk(&). The following lemma shows that it is close to a polynomialLet us now one-dimensionalconsider the first map. return map Tk(c). The following lemma shows that it is close to a. polynomial one-dimensional map. LEMMA 2. The map Tk can be brought to the form LEMMA 2. The map Tk can be brought to the form z = y +o Xkyk+y-k’s , x = y + 0 (()..k"(k + "(-k/S)> , (4.11) g = I& + E1y +. * * + E,-tys-l + dys+’ + 0 (Xkrk + y-klJ) ) (4.11) y = Eo + ElY + ... + Es_1ys-l + dyS+l + 0 ()..k"(k + "(-k/S) , 218 S.S.V. V. GONCHENKO et al. (j~ 1 (j~ J Figure 18. The geometric construction by which it is possible to obtain cubic tangen- Figure 18. The geometric construction by which it is possible to obtain cubic ta.ngen cies. Three horseshoes are shown, where W”(Oi) and Ws(Oj), as well as WU(Oj) cies. Three horseshoes are shown, where WU(Oi) and W"(Oj), as well as WU(Oj) and w’(ok) are tangent. a.nd WS(O",) are tangent. by a linear transformation of the coordinates and the parameters. Here I& = yk(lS.l/S)(eo - by a = 'Yk(l+l/s)(eO - r-ky- linear+ . . .),transformation ,l$ = yky-“14~-‘)ei. of the coordinates and the parameters. Here Eo 'Y-ky- + ... ), Ei = 'Yk'Y-k/s(i-l)ei. PROOF. By (4.2),(4.10), the map Tk is written in the following form (see the proof of Lemma 1): PROOF. By (4.2),(4.10), the map Tk is written in the following form (see the proof of Lemma 1): 3 - xf = aX%(l + * ’ *) + b(y - y-) +. . . ) x - x+ ::;:: aAkx(l + ... ) + b(y - y-) + ... , r-“$! (1 + y-kqk(ft,jj)) = dkx(l + .*.) 'Y-ky (1 + 'Y- k7]k(X,y)) = c-\kx(l + ... ) +EfJ+El(y-y-)+... + Es-1(y - y-)“-’ + d(y - y-)S+’ +. . . . + eo + el(y - y-) + ... + es-l(Y - y-)s-l + d(y _ y-)S+1 + .... By the shift of the origin z -+ x + z+, y -+ y + y-, this map is brought to the form By the shift of the origin x -+ x + x+, Y -+ Y + Y- , this map is brought to the form z = by + 0 (A’) + 0 (y2), x = by + 0 (-\k) + 0 (y2) , -f-kg + y2”0(g) = (El-J- yky- + cx”x+ + *. .) + Ely + * * * + Es-lys-l + clys’-1 'Y-ky + 'Y-2kO(y) ::;:: (eo - 'Y-ky- + CAkx+ + ... ) + elY + ... + es_lys-l + dy 8+1 + 0 (ys+2) + XkO (1x1 + Iyl) . + 0 (yS+2) + -\kO (Ixl + Iyl) . If we rescale the variables and the parameters as follows: If we rescale the variables a.nd the parameters as follows: x -+ by-“lSx, Y + Y-k’SY, x -+ b'Y- k/sx, y -+ 'Y-k/sy, (Eo _ ykY- + CAkx+ + . . . ) --) y-“(l+‘/s)Eo, (eo 'Y-ky- c-\kx+ -+ 'Y-k(l+l/s) Eo, + -kyk/s(i-l)E,+ ... ) Ei --+ Y 0 k k 8 ei -+ 'Y- 'Y / (i-l) E i , then the map takes form (4.11). The lemma is proved. then the map takes form (4.11). The lemma is proved. Returning to the initial quadratic homoclinic tangency, we see that for large numbers of rounds alongReturning the homoclinic to the initial orbit, quadratic the multi-round homo clinic return tangency, maps weare see close that to forarbitrary large numbersone-dimensional of rounds polynomialalong the homoclinicmaps in someorbit, regionsthe multi-round of the parameter return mapsspace are and close the to degreearbitrary of theone-dimensional polynomials becomespolynomial arbitrarily maps inlarge some when regions the numberof the parameterof rounds increases.space and Thus, the degreethese multi-round of the polynomials maps in abecomes neighborhood arbitrarily of a largesingle when homoclinic the number tangency of rounds represent increases. the whole Thus, one-dimensional these multi-round dynamics. maps in a neighborhood of a single homoclinic tangency represent the whole one-dimensional dynamics. Quasiattractors 219 (4(a) (b) (c) (d) ~-.~-.-& (4 (e) Figure 19. This figure shows how, from a contour with two quadratic heteroclinic tangenciesFigure 19. (FigureThis figure(a)), showsone can how, obtain from a acubic contour tangency with two(Figure quadratic (d)). heteroc1inicFirst, by a smalltangencies perturbation (Figure we(a», make one LV(Oi)can obtainintersect a cubic Ws(Oj) tangency transversely (Figure (d».and makeFirst, someby a piecesmall ofperturbation the manifold weW”(Oi) make W"(Oi)lie just intersectslightly above W8(Oj} w’(ok) transversely (Figure and(b)). makeThen somewe makepiece ofWy(Oi) the manifoldintersect W"(Oi)WS(Ok) lie injust four slightly points above(Figure W"(Ok)(c)). There(Figure is (b».a special Then path we make(Figure WU(Oi)(e)) from intersect Figure W"(Ok)(b) to Figure in four(c) pointson which (Figure a cubic (c».tangency There isof a.the special manifolds path W“(Oi)(Figure (e»and fromw’(ok) Figure(Figure (b) to (d))Figure takes (c) place.on which a. cubic tangency of the manifolds W"(Od and W8(Ok) (Figure (d» takes place. In conclusion, we look at the structure of the set of strips for the multi-dimensional case. We alsoIn showconclusion, how the we procedure look at theof structureresealing theof thefirst set return of strips map for works the multi-dimensionalhere. case. We' alsoLet show f be how a multidimensionalthe procedure of rescalingCT-diffeomorphism the first return(r > map3) with works a saddlehere. fixed point 0 whose r stableLet fmanifold be a multidimensionalIV’ is m-dimensional C -diffeomorphism and the unstable (r ?:manifold 3) with W”a saddleis n-dimensional. fixed point 0Let whos€: WB S U andstable W” manifold have a quadraticW is m-dimensional tangency at andthe pointsthe unstable of a homoclinic manifold orbitW isl?. n-dimensional. Let WS andA WUsmall have neighborhood a quadratic U tangency of 0 U I’ atis thethe pointsunion of a homoclinicsmall (n + m)-dimensionalorbit r. disc Ua, and at finiteA smallnumber neighborhood of small (n +U m)-dimensional of 0 Uris the neighborhoodsunion of a small of (nthe +points m)-dimensional of r which disclie outside Uo, and Ua a, ,, Likefinite in number the two-dimensional of small (n + mcase, )-dimensional we denote neighborhoodsthe restriction offlu, the aspoints Ta. Theof r standardwhich lie formoutside of theUo " mapLike inTO the corresponds two-dimensional to the coordinates case, we denote at which the restrictionthe local stable fluo asand To. unstable The standard manifolds form of of0 thElare straightened:map To corresponds W,,,7J = to{x the = 0,ucoordinates = O}, wa,, at which= {y =the 0, vlocal = 0) stable in some and coordinatesunstable manifolds (2, y, U, v).of 0This are straightened: WI~c = {x = O,u = O}, WI~c = {y = O,v = O} in some coordinates (x,y,u,v). ThiE; 220 S.s. V. GONCHENKOGONCHENKO etet al.al. Figure 20. The case of a three-dimensional map where the multipliers Al, X2, and ~1 Figure 20. The case of a three-dimensional map where the multipliers AI, A2, and ")'1 of the fixed point 0 are such that 0 < A2 < X1 < 1 < 71. Here the strips 0: c rI+ of the fixed point 0 are such that 0 < A2 < Al < 1 < ")'1. Here the strips 0'£ C n+ are three-dimensional “plates,” accumulating on W’ n II+ as k -t 00. The strips uk are three-dimensional "plates," accumulating on W· n n+ as k - 00. The strips O'k lie in a wedge abutting W” rl II-, asymptotically contracted along the nonleading lie in a wedge abutting W" n n-, asymptotically contracted along the nonleading coordinate u and tangent to the leading plane IA = 0 everywhere on W” n II-. coordinate 'U and tangent to the leading plane 'U = 0 everywhere on W" n n-. allows one to write 2’0 in the form allows one to write To in the form ti = A~‘u, + f21h Y, VIZ + f22(~ Y, u, ~b, fx = Ala:A1x + 111fil@, (x, Y,y, ~)a:v)x + f12b/!2(X, Y,y, U,u, 21)~v)U, u = A2u + h1(X,y,V)X + h2(X,y,U,v)u, (4.12) g = B1z + g11(z, ‘1L, Y>Y + 912(T Y, 219 VI% v = B2V + gm(& U, Y)Y + 922(? Y, U, v)TJ, (4.12) fi = B1X + 911(X, U, y)y + 912(X, y, u, v)v, ii = B2V + 921 (x, U, y)y + 922(X, y, u, v)v, where fij and gij vanish at the origin. Here, the eigenvalues of the matrices Al and Bi are where lij and 9ij vanish at the origin. Here, the eigenvalues of the matrices Al and Bl are the leading multipliers of 0, and the eigenvalues of A2 and Bs are the nonleading multipliers. the leading multipliers of 0, and the eigenvalues of A2 and B2 are the nonleading multipliers. Correspondingly, z and y are leading coordinates and u and v are nonleading coordinates. If Correspondingly, x and y are leading coordinates and u and v are nonleading coordinates. If Xi is real, the matrix A1 has the form A1 = (Xi), and it has the form Al = X ?‘+smp - cosvsinp >'1 is real, the matrix Al has the form Al (>'1), and it has the form Al ,\ ((c~s", -sin",)> for complex Ai. For real yi, the matrix Br has the form Bi = yi, and it has the sm",form cos'"Bi = for cos1/)complex-sin+ >'1. For real'")'l, the matrix Bl has the form Bl '")'1, and it has the form B1 Y if yi is complex. ( COS'"sin+ - cosq!Jsin",)) . f' 1 '")' ( sin", cos '" 1 '")'1 IS comp ex. Like it was done in [2,44], it can be shown that the multidimensional map Tc reduces to a Like it was done in [2,44], it can be shown that the multidimensional map To reduces to a form that is analogous in a sense to expression (3.4) which we have for the two-dimensional case. formNamely, that the is analogousfollowing identitiesin a sense holdto expression in some U-l-coordinates: (3.4) which we have for the two-dimensional case. Namely, the following identities hold in some Or-I-coordinates: fillt=O = 0, fljI(~=o,v=o) = 0, lillx=o == 0, /!jl(y=o,v=O) = 0, (4.13) gilly=o = 0, Slj I(z=O,tL=O) = 0. (4.13) 9illy=0 == 0, 91j 1(x=O,,,=O) = 0. Analogously to the two-dimensional case, in such coordinates the map T,k is linear in the lowest order.Analogously Specifically, to the two-dimensionalthe map Tt : (zo, case, yo,uo,vo) in such H coordinates(zk, yk, ‘&, the‘&) mapfor sufficiently Tt is linear large in the k canlowest be order. Specifically, the map Tt : (xo, Yo, Uo, vo) ...... (Xk' Yk, Uk, Vk) for sufficiently large k can be QuasiattractorsQuasiattrsctors 221.221 x v Figure 21. The three-dimensional case where the multipliers X1, 71, and 72 of the fixedFigure point 21. The0 are three-dimensionalsuch that 0 < X1 case < 1where < y1 the< 72.mUltipliers Here the AI,strips 1'1, andai C1'2 II- of theare three-dimensionalfixed point 0 are such“plates,” that accumulating0 < Al < 1 11" 11+ (70 K 0 O'K+1 0 M+ Figure 22. The projections of the multidimensional strips 0: and u: on the leading Figure 22. The projections of the multidimensional strips a~ and a~ on the leading plane (u,v) = 0 in Case (1,l). These projections look the same as in the two- plane (u,v) = 0 in Case (1,1). These projections look the same as in the two dimensional case. dimensiona.l case. manifolds Mk on which the map Tk is close to the one-dimensional parabola map (see (3.1)), manifolds Mk on which the map Tk is close to the one-dimensional parabola map {see (3.1)), while along the directions complementary to such a manifold there is contraction or expansion while along the directions complementary to such a manifold there is contraction or Elxpansion that is stronger than on Mk. The manifold Mk is not a global invariant manifold seizing all that is stronger than on Mk. The manifold Mk is not a global invariant manifold s.eizing all dynamics of the system in the neighborhood of the tangency, but it is an invariant manifold for dynamics of the system in the neighborhood of the tangency, but it is an invariant manifold for the map Tk defined on the single strip ok.o Nevertheless, the presence of these invariant Imanifolds the map Tk defined on the single strip ol Nevertheless, the presence of these invariant manifolds allows one to reduce some questions to the study of two-dimensional maps Tklmb. In this way, allows one to reduce some questions to the study of two-dimensional maps TklMk' In this way, the multidimensional version of Theorem 1 was proved in [30]. the multidimensional version of Theorem 1 was proved in [30j. At the same time, there exists here a countable number of nonstandard strips, on which the At the same time, there exists here a countable number of nonstandard strips, on which the map Tk is essentially multidimensional. Thus, if the product D of all the leading multipliers is map Tk is essentially multidimensional. Thus, if the product D of all the leading multipliers is less than unity, then for a countable number of strips D: the first return map is close to one of less than unity, then for a countable number of strips the first return map is close to one of maps given by formulae (3.2)-(3.4), f or some resealed coordinates0'2 (we write only that part of the maps given by formulae (3.2)-{3.4), for some rescaled coordinates (we write only that part of the map which corresponds to nontrivial behavior: for the other variables the map Tk acts as strong map which corresponds to nontrivial behavior: for the other variables the map Tk acts as strong contraction or strong expansion). contraction or strong expansion). We explain this statement in more detail for Case (2,l) at D = X2y < 1 and Xy > 1. For (2,1) D = A2,,/ 1 A,,/ theWe sake explain of simplicity, this statement we suppose in more that detailthere forare Caseno nonleading at multipliers; < i.e.,and we consider> 1. Forthe three-dimensionalthe sake of simplicity, case wewhere suppose the multipliers that there of are 0 areno nonleadingX1,2 = Xe*@ multipliers; and y (here i.e., 0 1). LEMMA 3. In the case under consideration, there exist infinitely many strips ui for which the mapLEMMA Tk takes3. In thethe form case under consideration, there exist infinitely many strips 0"2 for which the map Tk takes the form 31 = 21 + Elk(Zlr 22, Y), Xl = Xl + elk(Xl, X2, y), 32 = Y +&Zk(Zl,ZZ,Y), (4.15) X2 = Y + e2k(Xl, X2, y), (4.15) &? = M - Y2 - Bzl + &3k(xl, 22, ?4), iJ = M - y2 - BXl + e3k(Xl! X2, y), in some resealed coordinates. Here M and B are resealed parameters which can take arbitrary finitein some values rescaled for k coordinates.large enough; Herethe functionsM and B E&are tend rescaled to zero parameters as k + co. which can take arbitrary finite values for k large enough; the functions eik tend to zero as k -> 00. PROOF. By (4.12),(4.13), the map TO has the form PROOF. By (4.12),(4.13)' the map To has the form 21 = X(x1 coscp - 22sincp) + 0 (llz1121yl) , Xl A(Xl COS * iJ = 'YY + 0 (1I xlllyI2) . Take a pair of homoclinic points M-(0,0, y-) E IV;, and M’(a:,z$,O) E PVC,. Since W“ Takeand Ws a pairhave ofa homoquadratic clinic tangency points M-{O,O,y-)at M +, the globalE WI~c map and Tl M+(xt,xt,O)acting from a smallE WI~c' neighborhood Since WU and W8 have a quadratic tangency at M+, the global map T1 acting from a small neighborhood Quasiattractors 223s223; y . 0• .• • ...... ", " I " I .... " I . ,. / . " I / I / " I I I I I I Figure 23. The three-dimensional strips ug and LT~ in Case (2,1), where the fixed pointFigure 0 23.has Themultipliers three-dimensional 0 Xl.2 = Xe*iv strips and0'2 andy1 > u~1. inHere Case the (2,1), strips whereug c theII+ fixedare three-dimensionalpoint 0 has multipliers‘Lplates” 0 '>'1,2accumulating := .>.e±i'P andon Wa 1'1 rl> II+1. Hereas k the-+ co. strips The u2strips c n+cr: arelie three-dimensionalin the involuted roll, "plates" wound accumulatingonto the segment on W·W” n fln+ II-. as k -> 00. The strips O'~ lie in the involuted roll, wound onto the segment WU n n-. of M- into a small neighborhood of M+, has the form of M- into a small neighborhood of M+, has the form 21 -x1 + = bl(Y - y-) + a1121 + a1222 + * * * ) Xl - xt = bl(y - y-) + aUXl + al2X2 + ... , 22 - x2+ = bz(y - y-) + a2121 + a2222 + . .* , (4.17) X2 - xt = b2(y - y-) + a21Xl + a22 x2 + ... , (4.17) a = p + Cl21 + c252 + d(y - y-y + . . . , fi = /.L + CIXI + C2X2 + d(y - y-)2 + ... , where bl + bz # 0, cf + cl # 0 since TI is a diffeomorphism, and d # 0 since the tangency is quadratic;where b~ +p b~is =1=the 0, splittingc~ + c~ ::f:.parameter. 0 since Tl is a diffeomorphism, and d ::f:. 0 since the tangency is quadratic;We may /.Lassume is the blsplitting # 0. By parameter. the orthogonal coordinate transformation We may assume b1 =1= O. By the orthogonal coordinate transformation x1 4 x1cosa+x2sincr, 22 -+x2coscr--xlsina, 224 S. V. GONCHENKO et al. Figure 24. The three-dimensional strips c$ and ui in Case (1,2), where the fixed pointFigure 0 24.has Themultipliers three-dimensional 0 < X1 < 1 strips and 71,~ 0'2 =and ye*“+. O't in HereCase the (1,2), strips whereu: c theII- fixedare three-dimensionalpoint 0 has multipliers“plates” 0 Figure 25. The four-dimensional strips up and uk in Case (2,2), where the fixed pointFigure 0 25.has Themultipliers four-dimensional 0 Xl.2 = Xe*i’f’ strips and crZ 71and , 2 =crk 7efi$. in CaseHere (2,2), the wherestrips crithe c fixedII- pointlie in the0 hasinvoluted multipliers roll wound0 >'1,2 ==onto >.e±i'P the andtwo-dimensional ')'1,2 == ,),e±i1/J.area Here W” the13 stripsII-. The 0'1 cstrips n lieu: inlie thein the involuted involuted rollroll, wound wound onto onto the thetwo-dimensional W’ n II+. area WU n n-. The strips 0"2 lie in the involuted roll, wound onto the W' n n+. 21 = by + A”0 (1141) + 0 (Y2) I Xl = by + )..kO (11xll) + 0 (y2) , 22 = @?J”Xl + azzXkz2 + 0 (y2) + Xko (~~z~~), X2 a21)..k a22)..kx2 (y2) )..ko(lIxID, -k = x1 + + 0 + (4.20) 1 g+ o(l%l + ll~ll> = Ml + dyky2 + (b)khk(dZ1 + (h)lc@2k(+2 (4.20) 17+ (~)0 Y -k 0(1171 + Ilxll) = Ml +d'ly2 + ()..')')k,Blk(~)Xl + ()..')')k,B2k(~)X2 + Xk-ikO(11412 + IYI * 1141)+ Yk4Y2)> + )..k')'kO (11 x11 2 + Iyl . Ilxll) + ')'kO(y2) , where where Ml = -yk (P + A’Plk(‘p) Resealing the variables Rescaling the variables b b Zl + -- qyk, Q + -- u2&2y-k, Y + -;YTk, d d 226 S. V. GONCHENKO et al.ai, we get the following expression for the map Tk: 31Xl ==y+... Y + ... ), 22X2 = 321Xl + .... *. ,) (4.21) fjg = M - y2y2 - &cl+BXl + (,\2'Y)k(x2$ k PZk(cp)~Z{32dip)X2 + .... . * , where the dots stand for the terms which tend to zero as k -+ 00; M = -dykMl, B = where the dots stand for the terms which tend to zero as k ---t 00; M = -d-yk M l , B = -b@lk(V+~Y)“.-b{31k( ip)('\'Y)k. Recall that we consider the case Xy'\1' > 1, X2y,\21' < 1. Therefore, (,\2'Y)k(X2y)k «:< 1 and (AT)~(A'Y)k »B 1 at large k. Thus, the term with z2X2 in the third equation of (4.21) is small, so thethe map is now brought to form (4.15). The coefficient B is the product of the large quantity (A'Y)k(XT)~ and thethe value & = cl cos kcp + c2 sin kp. When the ratio z is abnormally (exponentially) well approximated (3u Cl cos kip + C2 sin kip. 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