Towards a Theory of Chaos Explained as Travel on Riemann Surfaces

T o w ards a Theory of Chaos Explained as T ra v el on Riemann Surfaces F Calogero 1 , 2 , D G´ omez-Ullate 3 , P M San tini 1 , 2 , and M Sommacal 4 , 5 1 Dipartiment o di Fisica, Unive r sit` a d i Roma “La Sapienza”, Roma, Italy . 2 Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy . 3 Departamen to de F ´ ısica T e´ orica I I, Universidad Complutense, M adrid, Spain. 4 Dipartiment o di Matematica e Informatica, Universit` a degli Studi di Perugia, Pe r ugia, Italy . 5 Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Italy . E-mail: frances co.calogero@roma 1 .infn.it , david.go mez-ullate@fis.ucm.es , paolo.sa ntini@roma1.infn.it , matt eo.somma cal@pg.infn.it Abstract. This paper pr esen ts a more complete version th an hi therto published of our explanation of a transition from r e gular to irre gular motions and more generally of the nature of a certain kind of det erministic c haos . T o this end we int r oduced a simple mo del analogous to a three-b ody problem i n the plane, whose general solution i s obtained via quadr atur es all perf ormed in terms of elementary functions . F or some v alues of the coupling constants the system is iso chr onous and explicit form ulas for t he p erio d of t he sol utions can b e given. F or other v alues, the motions are conﬁned but feature ap eriodi c (in some sense cha otic) motions. This rich phenomeno l ogy can b e understoo d in remark able, quantitative detail in terms of tra vel on a certain (circular) path on the Riemann surfaces deﬁned by the solutions of a r el ated model considered as functions of a co mplex time. This model is meant to pro vide a paradigmatic ﬁrst step to wards a somewhat nov el understanding of a certain kind of chaotic phe nomena. P ACS nu mber s : 05.45-a, 02.30.Hq, 02.30.Ik. T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 2 1. In tro duction The fact that the distinction among inte gr able or noninte gr able b ehaviors of a dynamical sy stem is somehow connected with the analytic st ructur e o f the so lutions of the mo del under consider ation as functions of the independent v ariable “time” (considered a s a c omplex v ariable) is by no means a novel notion. It go es ba c k to classical work by Carl Jaco bi, Henri Poincar´ e, Sophia Kow a levsk ay a, Paul Painlev´ e and o thers. In recent times some of us had the go o d fortune to hear in several o c c a sions such ide a s clearly describ ed by Mar tin Krus k al [22, 2 3]. A s imple-minded rendition of his teachings can be descr ibed as fo llows: for an evolution to be inte gr able , it should be expressible, at least in principle, via for m ula s that a re not exc essively multivalue d in terms of the dep endent v a riable, entailing that, to the extent this ev olution is expressible b y analytic functions of the dep enden t v ariable (considered as a c omplex v ariable), it might p o ssess branch p oints, but it should not feature a n inﬁnity of them that is dense in the c omplex pla ne of the indep endent v aria ble. A num b er o f techniques collec tiv ely known as Painlev´ e analysis have b een resurre c ted and further develop ed o ver the last few decades (for a review see, for instance, [28]). In essence , they co nsider an ansatz of the lo cal b ehaviour of a solution near a singular it y in terms of a Laure nt s eries, in tr o ducing it in the equations and determining the leading or der s and re s onances (terms in the expans ion at which ar bitrary consta nts app ear ). P a inlev´ e analys is has b een ex tended to test for the presence of algebr a ic br anching (weak Painlev´ e prop erty [28]) by cons idering a Puiseaux serie s instea d of a Laur e n t serie s. These a nalytic techniques (which hav e b een algorithmized and a r e now av ailable in computer pack ag es) co ns titute a useful tool in the in vestigation of in tegra bilit y: in man y nonlinear systems where no solution in closed for m is k nown, Painlev´ e analysis provides informa tion o n the type of br anching featured by the general solution or by special classes of solutions. It ha s a lso prov ed useful to identify sp ecial v alues of the pa rameters for which g e nerally chaotic systems such as H´ enon-Heiles or L orenz are inte gr able [4, 1 6]. On the opp osite side of the spectrum lie c haotic dynamical systems and it is natural to in vestigate the singularity structure of their solutions. T ab or and his collab orato rs initiated this study in the early eighties for the Lorenz system [29] and the Henon-Heiles Hamiltonian[1 6]. They realized that the sing ularities of the solutions in complex-time are imp ortant for the rea l-time evolution of the sys tem. The complex time analytic s tructure was studied by extens ions of the Painlev ´ e analysis inv olving the introduction o f log arithmic terms in the expans ion — the so called Ψ- series — which provides a lo cal representation of the solutions in the neighbourho o d of a singularity in the chaotic r egime. Their lo cal analytic appr oach w a s co mplemen ted by nu mer ical techniques develop ed for ﬁnding the lo ca tio n of the singularities in complex time a nd determining the o rder of br anching [14]. In all the chaotic systems under study , they obser ved numerically that the singular ities in complex time clus ter o n a natural b oundary with self-s imilar structure [15]. An analytic ar gument to explain the mechanism that lea ds to recursive sing ularity cluster ing was given in [2 4]. Similar studies r e lating s ing ularity struc tur e, chaos and integrability have b een perfor med by Bountis and his collab ora tors. Go ing be yond the lo c a l tec hnique s describ ed ab ov e, the emphasis is put on a g lobal prop erty o f the solutions: whether their Riemann surface has a ﬁnite or an inﬁnite num b er of sheets. Bountis prop oses to use the term inte gr able for the ﬁrst c a se a nd non-int e gr able for the seco nd [2, 3]. Using mostly numerical evidence he conjectures that in the non-in tegr able cases the Riemann surfaces ar e T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 3 inﬁnitely-sheeted and the pr o jection on the complex pla ne of the singula rities is dense. Combining analytical and numerical res ults for a simple ODE, Bo un tis and F o k as [18] hav e identiﬁed chaotic systems with the pr op erty that the singularities of their solutions are dense. Painlev ´ e analys is and its ex tensions ar e useful and widely applica ble. Howev er, as lo cal techniques, they provide no infor mation o n the glob al prop erties of the Riemann surfaces o f the solutions, s uch as : the num b er and lo cation of the mov a ble branch po in ts a solutio n has, and moreov er how the diﬀerent sheets of the Riemann s ur face are connected together at thos e bra nc h p oint s . Understa nding these g lobal pro per ties is impo r tant for the dynamics; a detailed analy sis of the Riemann surface asso ciated to the solutions of a dynamical s y stem, whenever it can b e done, pro v ides a muc h deep e r understanding than can b e obtained by lo cal tec hniques alone. This is precisely the motiv ation of the inv estiga tion r epo rted he r ein: to intro duce and study a mo del which is simple enough that a full description of its Riemann surface can b e p erformed, yet co mplicated enough to feature a rich b ehaviour, p os sibly including irregular or c ha otic characteristics. Such a mo del was initially presented in [1] a nd in this pap er w e contin ue inv estigating its prop erties. This line of research originates from a “trick” that is conv enient to identify iso chronous systems [7, 9] – a change of dependent and independent v aria bles, with the new indep endent v ariable traveling on a path in the complex plane. La ter it w as sho wn that man y iso chronous s ystems can b e written by a suitable mo diﬁca tio n o f a larg e clas s of complex ODEs [10, 9]. Using loca l analysis and numerical integration in tw o man y - bo dy systems in the plane [1 3, 11], it was discov ere d that outside the is o c hr ony reg ion there exist per io dic solutions with muc h higher perio ds as well as po ssibly a per io dic solutions , and the connection among this phenomenology and the ana ly tic structure of the cor resp onding solutions as functions of complex time w as illuminated. How ever, those s y stems were to o complicated for a complete description of the Riemann surface to b e ac hieved. Recent work alo ng these lines includes pr oblems whos e s o lution is o btained b y inv ersion o f a hype r elliptic integral: the corr e spo nding Riemann surfa ces hav e b een studied in [1 7, 19], together with the implications on the dynamical proper ties of the mo dels. In the pres en t pap e r we pr ovide many details that were rep or ted without pro of in [1], such as the desc r iption of the gener al so lution by quadrature s , and we also exhibit o ther prop erties of the mo del that w e re not pr esent in [1], such as similarity solutions, equilibrium c o nﬁgurations and small oscillations. Our in vestigation of this mo del will contin ue in a s ubsequent publication, [12], where the full description o f the geometrical proper ties of the Riemann surface will b e giv e n. This pap er is organized as follo ws : in Section 2 w e pre sent our model, including in pa rticular the r elationship among its physic al version (indep enden t v aria ble: the r e al time t ) a nd its auxiliary version (indep endent v ar iable: a c omplex v ariable τ ), and we outline the main ﬁndings repor ted in this pap er. In Section 3 w e discuss the equilibrium conﬁgurations of our physic al model, and the b ehavior of this system in the neig h b or ho o d of these solutions, and we also o btain cer tain exact s imilarity solutions of our mo del and dis cuss their stability . In Section 4 we discuss the analytic structure of the so lutions of the auxiliary mo del via lo cal a nalyses ` a la Painlev ´ e , since the analytic str ucture o f these solutions plays a crucia l role in determining the time evolution of our physic al mo del. In Section 5 we show how the gener al solution of our mo del ca n b e achiev ed by quadr atur es a nd in Section 6 we outline the b ehavior of T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 4 our model based o n these results. Finally , in Sectio n 7 we summar ize o ur results and comment on future dev elo pmen ts. This paper also con tains a few Appendices, where certain calculations ar e conﬁned (to a void interrupting inco n venien tly the ﬂow of the presentation) as w ell as certain additional ﬁndings . 2. Presen tation of the mo del In this sectio n we intro duce the mo del treated in this pa per , and we outline our main ﬁndings that ar e then pro ven and further discussed in subs e quen t sections. 2.1. The auxiliary mo del The auxiliary mo del on which we fo cus in this pap er is characterized by the following system of thr e e coupled no nlinear ODEs: ζ ′ n = g n +2 ζ n − ζ n +1 + g n +1 ζ n − ζ n +2 . (1) Notation : here and hereafter indices such as n, m range from 1 to 3 and ar e deﬁned mod (3); τ is the ( c omplex ) indep endent v ariable of this aux iliary mo del; the t hr e e functions ζ n ≡ ζ n ( τ ) ar e the dependent v ar iables of this auxiliary mo del, and w e assume them to b e as well c omplex ; a n app ended prime a lways denotes diﬀerentiation with r esp ect to the argument o f the functions it is app ended to (her e, of course, with resp ect to the c omplex v ariable τ ); and the thr e e quantities g n are a rbitrary “coupling constants” (p ossibly also c omplex; but in this pap er w e restrict considera tio n mainly to the c a se with r e al coupling constants; this is in particula r herea fter assumed in this section). In the following we will often focus on the “semisymmetrical case” characterized by the equalit y of two of the thr e e co upling consta n ts, say g 1 = g 2 = g , g 3 = f , (2) since in this case the treatment is simpler yet still adequate to exhibit most asp ects of the phenomenolo gy w e are in ter ested in. More sp ecial cases are the “fully symmetrical”, or “ in teg r able”, one characterized by the equality of al l thr e e co upling constants, g = f , g 1 = g 2 = g 3 = g , (3) and the “tw o- bo dy” o ne , with only one non v anishing coupling co nstant, say g 1 = g 2 = g = 0 , g 3 = f 6 = 0 . (4 a ) In this latter case clea rly ζ ′ 3 = 0 , ζ 3 ( τ ) = ζ 3 (0) (4 b ) (see (1)) and the rema ining two-b o dy problem is trivially solv able, ζ s ( τ ) = 1 2 [ ζ 1 (0) + ζ 2 (0)] − ( − ) s  1 4 [ ζ 1 (0) − ζ 2 (0)] 2 + f τ  1 / 2 , s = 1 , 2 , (4 c ) while the justiﬁcation for lab eling the fully symmetrical case (3) as “integrable” will be clear fro m the following (or see Section 2.3.4 .1 of [8]). B efore introducing o ur physic al mo del, let us no te that the aux iliary s ystem (1) is invariant under translatio ns of bo th the indep enden t v ariable τ (indeed, it is autonomous ) and the dep endent v ar iables ζ n ( τ ) ( ζ n ( τ ) ⇒ ζ n ( τ ) + ζ 0 , ζ ′ 0 = 0), and it is mo r eov er invariant under an a ppropriate simult a neous r escaling o f the independent and the dep endent v aria bles. T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 5 2.2. The trick and the physic al mo del The tr ick mentioned ab ov e, relating the auxiliary mo del to the phy sic al mo del, amounts in o ur pr esent case to the in tro duction o f the ( r e al ) indep endent v aria ble t (“physical time”), as well a s the t hr e e ( c omplex ) dep endent v aria bles z n ≡ z n ( t ), via the follo wing pos itio ns: τ = exp(2 i ω t ) − 1 2 i ω , (5 a ) z n ( t ) = exp( − i ω t ) ζ n ( τ ) . (5 b ) W e hereafter as sume the constant ω to b e r e al (for deﬁniteness, p ositive, ω > 0; note that for ω = 0 the change of v ariables disapp ear s), and w e asso ciate to it the p e rio d T = π ω . (5 c ) Note that this change of v ar iables entails that the initia l v alues z n (0) of the “pa r ticle co ordinates” z n ( t ) co incide with the initial v alues ζ n (0) of the dependent v ariables of the auxilia ry mo del (1): z n (0) = ζ n (0) . (6) It is easily s een that, via this change of v ariables, (5), the eq uations o f motion (1) satisﬁed by the quantities ζ n ( τ ) ent ail the following ( autonomous ) equations of motion (in the r e al time t ) for the pa r ticle co or dinates z n ( t ): ˙ z n + i ω z n = g n +2 z n − z n +1 + g n +1 z n − z n +2 . (7) Here a nd herea fter sup erimp o sed dots indicate diﬀerentiations with resp ect to the time t. So, this mo del (7) describ es the “physical evolution” which we study . Note that its equatio ns o f motion, (7), are of Aristotelian , r ather than Newtonian , type: the “velocities” ˙ z n , ra ther than the “a ccelerations” ¨ z n , of the moving particles are determined by the “for ces”. In App endix D we discuss the connection of this mo de l with more cla ssical many-b o dy problems, character ized by Newtonian equa tions of motion. Let us immedia tely emphasiz e tw o imp ortant qualitative asp ects o f the dynamics of our physic al model (7). The “one-b o dy force” repr esent e d by the second term, i ω z n , in the left-hand side of the equations o f motion (7) b ecomes dominant w ith resp ect to the “tw o-b o dy forces ” app earing in the right-hand side in deter mining the dynamics whenever the ( c omplex ) co ordina te z n of the n -th particle b ecomes lar ge (in mo dulus). Hence when | z n ( t ) | is very lar ge, the solution z n ( t ) of (7) is approximated by the solution of ˙ z n + i ω z n ≈ 0 implying that z n ( t ) is characterize d by the b ehavior z n ( t ) ≈ c exp ( − i ω t ), therefor e the tra jectory of the n - th particle tends to rotate (clo ckwise, with per io d 2 T ) on a (large) circle. This eﬀect causes al l motions of our physic al mo del, (7), to b e c onﬁne d . Secondly , it is clear that the t wo-b o dy forces (see the rig ht-hand s ide of (7)) cause a singularity whenever there is a c ol lision of t wo (o r all thr e e ) of the particles a s they mov e in the c omplex z -plane, and b ecome do minan t whenever two or thr e e par ticles get very close to each other, namely in the case of ne ar c ol lisions . But if the thr e e pa rticles mov e ap erio dic al ly in a c onﬁne d r e gion (near the o rigin) of the complex z -plane, a lot o f ne ar c ol lisions shall indeed o ccur . And since the outcome of a ne ar c ol lision is likely to b e quite diﬀerent dep ending on w hich T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 6 side tw o pa rticles sca tter past ea ch other – and this, espe c ially in the ca se of very close ne ar c ol lisions , dep ends sensitively o n the initial data of the tra jectory under consideratio n – we see here a mechanism complicating the motion, indeed ca using some kind of chaos as so ciated with a sensitive dep endenc e of the motion on its initial data. This sug gests that our mo del (7 ), in s pite of its simplicity , is likely to b e rich enough to cause an interesting dynamical evolution. W e will see that this is indeed the case. But b efore pro ceeding with this inv e s tigation let us interject tw o remarks (somewhat r elated to each other). Remark 1 This system (7) is stil l invariant under tr anslations of the indep endent variable t (inde e d, it is again auto nomous ) but, in c ontr ast to (1), it is no longer invariant under tr anslations of t he dep endent variables z n ( t ) nor under a simple r esc aling of the indep endent variabl e t and of the dep endent variables z n ( t ) . Remark 2 The gener al s olut ion of the e quations of motion (7) has the stru ctur e z n ( t ) = z C M ( t ) + ˇ z n ( t ) , (8 a ) wher e the thr e e functions ˇ z n ( t ) satisfy themselves the same e quations of motion (7) as wel l as the additional r estriction ˇ z 1 ( t ) + ˇ z 2 ( t ) + ˇ z 3 ( t ) = 0 (8 b ) which is cle arly c omp atible with these e quations of motion, and c orr esp ondingly z C M ( t ) is the c enter of mass of the system (7), z C M ( t ) = z 1 ( t ) + z 2 ( t ) + z 3 ( t ) 3 , (9 a ) and it evolves ac c or ding to the simple formula z C M ( t ) = z C M (0) exp ( − i ω t ) = Z exp ( − i ω t ) . (9 b ) In Sec tion 3 (and App e ndix A) we determine the eq uilibrium co nﬁgurations o f our physic al mo del, namely the v alues z (eq) n of the thr e e particle co ordinates z n such that z n = z (eq) n , ˙ z n = 0 (10) satisfy the equations of motion (7 ), and we ascer tain the b ehavior o f our system in the neighborho o d o f these conﬁguratio ns. In the second part of Section 3, and then almost always in the rest of this pa per (and throughout the res t o f this section) we restrict for simplicity considera tion to the semisymmetrica l case , see (2). A main ﬁnding in Sectio n 3 (and Appendix A) is that in the semisymmetr ical case our mo del (7) po ssesses generally two equilibrium conﬁgura tions z (eq) n . W e moreover determine the thr e e exp onents γ ( m ) characterizing the small os c illations of o ur sy stem in the neighborho o d o f each of thes e two conﬁgurations, deﬁned acco rding to the standar d formulas (see Section 3) z n ( t ) = z (eq) n + ε w n ( t ) , ( 1 1 a ) w ( m ) n ( t ) = exp( − i γ ( m ) ω t ) v ( m ) n , (11 b ) where of course ε is an inﬁnitesimal ly sm al l parameter and the quantities v ( m ) n are time-indepe nden t. W e ﬁnd that the ﬁrst two of these thr e e exp onents take in b oth cases the simple v alue s γ (1) = 1 , γ (2) = 2 ; (12) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 7 the ﬁrst of these corresp onds of cour se to the cen ter-o f-mass motion, see (9 b ). As for the thir d exp onent γ (3) , w e ﬁnd for one eq uilibrium conﬁgur ation γ (3) = f + 8 g f + 2 g = 1 µ , (13 a ) and for the other γ (3) = f + 8 g 3 g = 2 1 − µ . (13 b ) Here w e ha ve introduced the constant µ, µ = f + 2 g f + 8 g (14) whose v alue, as we shall se e, plays an impor tant r o le in de ter mining the dy namical evolution of our mo del: in particular, this evolution does lar gely dep e nd on whether or not µ is a r e al r ational num b er, and if it is r ational , µ = p q (15) with p and q c oprime inte gers (and q p ositive , q > 0), on whether the tw o natural nu mber s | p | and q are large or small. A hin t of this is alr eady appa r ent from the results we just rep orted: w hile the s olutions w (1) n ( t ) and w (2) n ( t ) , see (11), ar e b oth p erio dic with p erio d 2 T (see (5 c ); in fa c t w (2) n ( t ) is p erio dic with p er io d T ), the solution w (3) n ( t ) , s e e (1 1), is perio dic with the p erio d ˜ T , ˜ T = 2 T γ (3) , (16) which is clear ly c ongruent to T only if µ is r ational , s ee (15) and (13 ) – implying then that the small os cillations around the equilibrium conﬁg urations are always c ompletely p erio dic with a p erio d whic h is a ﬁnite inte ger multiple of T . In Section 3 w e a ls o intro duce the sp ecial class of ( exact and c ompletely explicit ) “similarity” solutions of our equa tions of motion, (7), and ana lyze their stability , namely the solutions of our system in the immediate neig h b or ho o d of these similar ity solutions. 2.3. Conserve d quantities It is imp ortant to note at this point that the auxiliar y mo del (1) p ossess e s conserved quantities, which w ill be used in Sec tio n 5 to obtain its general so lutio n by quadratur e s . Firstly , due to the tr anslational inv ariance it is ob vious that the quant ity Z = 1 3 3 X n =1 ζ n , (17) do es not dep end on τ . In addition, the analys is of Section 5 shows that in the semisymmetrical case there exists a n extra c onserved q ua n tity given by ˜ K = (2 ζ 3 − ζ 1 − ζ 2 ) − 2 " 1 − ( ζ 1 − ζ 2 ) 2 + ( ζ 2 − ζ 3 ) 2 + ( ζ 3 − ζ 1 ) 2 2 µ (2 ζ 3 − ζ 1 − ζ 2 ) 2 # µ − 1 . (18) Here the constant µ is deﬁned in terms of the c oupling constants g and f , see (2), by (14). W e alrea dy mentioned that the v alue of this parameter (in particular, whether T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 8 or not µ is a r ational num b er) pla ys an important role in deter mining the dynamical evolution of our mo del. A hint of this is now provided by the app ear ance of this n umber µ as an exponent in the rig h t-ha nd side of (18), s ince this exp onent characterizes the m ultiv aluedness of the dep endence of the constant ˜ K on the co o rdinates ζ n . 3. Equilibrium conﬁgurations, small oscil lations and si m ilarit y solutions of the physic al mo del In this sectio n we discuss, ﬁrstly , the equilibrium conﬁgur ations of our physic al mo del, (7), a nd its be havior near equilibrium, and secondly , a sp ecial, explicit “similarity” solution of our model and its stability . The e qu ilibrium c onﬁgur ations of o ur physic al mo del (7), z n ( t ) = z (eq) n , ˙ z n ( t ) = 0 , (19) (see (10)) are clearly characterized by the algebraic equations i ω z (eq) n = g n +1 z (eq) n − z (eq) n +2 + g n +2 z (eq) n − z (eq) n +1 . (20) These algebraic equations en tail z (eq) 1 + z (eq) 2 + z (eq) 3 = 0 . (21) It is now conv enient to set z (eq) n = (2 i ω ) − 1 / 2 α n , (22) so tha t the equilibrium equations (2 0) read as fo llows: α n 2 = g n +1 α n − α n +2 + g n +2 α n − α n +1 . (23) These algebraic equations can b e conv eniently (see b elow) r e w r itten as follows: α n = β n +1 ( α n − α n +2 ) + β n +2 ( α n − α n +1 ) , (24 a ) via the p osition β n = 2 g n ( α n − 1 − α n +1 ) 2 . (24 b ) W e now note that, in orde r that the three equations (24 a ) (which are linear in the three unknowns α n , although o nly apparently so, se e (24 b )) hav e a nonv anishing so lutio n, the quan tities β n m us t ca use the following determina n t to v anish:       β 2 + β 3 − 1 − β 3 − β 2 − β 3 β 3 + β 1 − 1 − β 1 − β 2 − β 1 β 1 + β 2 − 1       = 0 . (25) T o analy ze the smal l oscil lations of our sys tem (7 ) around its equilibrium conﬁguratio ns we now set z n ( t ) = z (eq) n + ε w n ( t ) , ( 2 6 a ) (see (11 a )) and we then get (linearizing by tr eating ε as an inﬁnitesimally small parameter) ˙ w n + i ω w n + i ω β n +1 ( w n − w n +2 ) + β n +2 ( w n − w n +1 ) = 0 . (26 b ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 9 Therefore the three exp onents γ ( m ) characterizing the small oscillations around equilibrium via the formula w ( m ) n ( t ) = exp( − i γ ( m ) ω t ) v ( m ) n , (27) providing thr e e indep enden t solutions of the s ystem of line ar ODEs (26 b ), are the thr e e e igenv alues o f the symmetrical matrix B =   β 2 + β 3 + 1 − β 3 − β 2 − β 3 β 3 + β 1 + 1 − β 1 − β 2 − β 1 β 1 + β 2 + 1   , (28) and the thr e e 3-vectors ~ v ( m ) ≡  v ( m ) 1 , v ( m ) 2 , v ( m ) 3  are the corresp onding eigen vectors, 3 X ℓ =1 B nℓ v ( m ) ℓ = γ ( m ) v ( m ) n . (29) Hence the thr e e exp onents γ ( m ) are the thr e e ro ots o f the “secular equation”       β 2 + β 3 + 1 − γ − β 3 − β 2 − β 3 β 3 + β 1 + 1 − γ − β 1 − β 2 − β 1 β 1 + β 2 + 1 − γ       = 0 . (30) Clearly these thr e e ro ots are given by the following formulas: γ (1) = 1 , γ (2) = 2 , γ (3) = 2( β 1 + β 2 + β 3 ) . (31) Indeed the determinan t (30) v anishes for γ = γ (1) = 1 (when each line sums to zero ) and for γ = γ (2) = 2 (see (2 5)), a nd the third s olution, γ (3) = 2( β 1 + β 2 + β 3 ) , (32) is then implied b y the tra ce condition trace [ B ] = 3 + 2( β 1 + β 2 + β 3 ) = γ (1) + γ (2) + γ (3) . (33) The ﬁr st of these 3 solutions, γ (1) = 1 , corre s po nds to the center of ma ss motion (it clearly en tails v (1) n = v (1) , see (27) and (28)). In the semisymmetrical c ase (2) the equations (2 3 ) (or equiv alent ly (24)) characterizing, via (22), the equilibrium conﬁgura tions can b e solved explicitly (see Appendix A). One ﬁnds that ther e are two distinct equilibrium co nﬁgurations (in fact four, if one takes account of the trivia l p oss ibilit y to exchange the roles of the tw o “equal” pa rticles w ith lab els 1 and 2) , the ﬁrst o f which reads simply z (eq) 3 = 0 , z (eq) 1 = − z (eq) 2 = z (eq) , ( z (eq) ) 2 = f + 2 g 2 i ω , (34) while the second has a slightly more co mplicated expression (see Appendix A). Note how ever that, in b oth c ases , there holds the relation ( z (eq) 1 − z (eq) 2 ) 2 + ( z (eq) 2 − z (eq) 3 ) 2 + ( z (eq) 3 − z (eq) 1 ) 2 = 3 ( f + 2 g ) i ω . (35) Moreov e r, in b oth cases the cor resp onding v alues for the eigenv alue γ (3) , see (32), a re easily ev a lua ted. The ﬁrs t solution yields (see (13 a )) γ (3) = f + 8 g f + 2 g = 1 µ = q p , (36 a ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 10 where, for future reference, we express ed γ (3) not o nly in terms of the parameter µ , see (14), but as well in terms of its ra tional expr ession (15) (whenever a pplica ble), while the second solution likewise yields γ (3) = f + 8 g 3 g = 2 1 − µ = 2 q q − p . (36 b ) Note that this implies that in the “integrable” ca se (3 ) bo th these formulas, (3 6 a ) and (36 b ), yield γ (3) = 3; but it is ea sily seen that in this case only the ﬁr st equilibr ium conﬁguratio n (34) actually exists. So in the “integrable” ca se the o scillations around the (only) equilibrium conﬁguratio n (34) are the linea r s uper pos ition o f three per io dic motions (see (27)) with resp ective perio ds 2 T , T and 2 T 3 (see (5 c )). Also in the “tw o-b o dy” case (4) the second equilibr ium conﬁguratio n do es not exist, while the ﬁrst formula, (36 a ), yields γ (3) = 1 , so in this case the small oscilla tions around the (only) equilibrium conﬁguration (34) a re the linear s uper po s ition of tw o perio dic motions, with p erio ds 2 T and T (see (5 c ); co nsistently with the explicit solution, easily obtainable from (4 c ) via (5)). As can b e easily veriﬁed, the equilibrium conﬁg ur ations (19) with (20) are mer ely the sp ecial case co rresp onding to z C M (0) = 0 , c = 0 of the following t wo-parameter family of ( exact ) “similarity” s o lutions o f our equa tions of motion (7): z n ( t ) = z C M ( t ) + ˜ z n ( t ; c ) , (37 a ) ˜ z n ( t ; c ) ≡ z (eq) n [1 + c exp ( − 2 i ω t ) ] 1 / 2 , (37 b ) with the center o f mass co or dina te z C M ( t ) evolving acco rding to (9 b ). The tw o arbitr ary ( c omplex) cons tan ts featured b y this solution a r e of cours e z C M (0) = Z (see (9 b )) a nd c , while the constants z (eq) n ’s a re deﬁned as in the pr eceding s ection, se e (20). Clearly these ( exact ) so lutions corr e s po nd, via the trick (5), the relation (22) (whic h is clear ly consistent with (23) and (20)) and the simple rela tio n τ b = c − 1 2 i ω , (38) to the tw o-pa rameter family ζ n ( τ ) = Z + α n ( τ − τ b ) 1 / 2 , (39) of ( exact ) solutions of (1). Let us now discuss the stability of this solutio n, (3 7 b ). T o this e nd we set z n ( t ) = ˜ z n ( t ; c ) + ε ˜ w n ( t ) , (40 a ) and we inser t this ansatz in our equa tions o f mo tion (7), linear izing them by trea ting ε as an inﬁnitesimally s ma ll para meter. W e th us get · ˜ w n + i ω ˜ w n + i ω [ β n +1 ( ˜ w n − ˜ w n +2 ) + β n +2 ( ˜ w n − ˜ w n +1 )] 1 + c exp ( − 2 i ω t ) = 0 , (40 b ) having used the deﬁnition (24 b ). Cle arly the solution of this system of ODEs r eads ˜ w n ( t ) = exp ( − i ω t ) χ n ( ϑ ) , (41 a ) with ϑ ≡ ϑ ( t ) = t − (2 i ω ) − 1 log  1 + c exp ( − 2 i ω t ) 1 + c  (41 b ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 11 and the functions χ n ( ϑ ) s olutions of the auto nomous linear s y stem of ﬁrst-or der ODEs χ ′ n + i ω [ β n +1 ( χ n − χ n +2 ) + β n +2 ( χ n − χ n +1 )] = 0 , (41 c ) where the primes denote of co urse diﬀerentiation with res pect to ϑ. Hence (see (26 b )) the thr e e indep endent so lutions of this linea r sys tem are χ ( m ) n ( ϑ ) = exp ( i ω ϑ ) w ( m ) n ( ϑ ) , (41 d ) with the functions w ( m ) n deﬁned by (27) (of cour se with t r eplaced by ϑ ), yielding via (27) and (41 a ) with (41 b ) the fo llowing tw o eq uiv alent ex pressions for the t hr e e independent solutions o f the linear sys tem (40 b ): ˜ w ( m ) n ( t ) =  1 + c exp ( − 2 i ω t ) 1 + c  ( γ ( m ) − 1) / 2 exp( i γ ( m ) ω t ) ˜ v ( m ) n , (42 a ) ˜ w ( m ) n ( t ) =  exp (2 i ω t ) + c 1 + c  ( γ ( m ) − 1) / 2 exp( i ω t ) ˜ v ( m ) n . (42 b ) Here the thr e e exp onents γ ( m ) are deﬁned as above, see (31 ), and lik ewise the “eigenv ecto r s” ˜ v ( m ) n coincide with those deﬁned ab ov e up to ( arbitr ary ) nor ma lization constants c ( m ) , ˜ v ( m ) n = c ( m ) v ( m ) n . (42 c ) Note the equiv alence o f the tw o expressions (42 a ) and (42 b ) (the motiv ation for writing these t wo versions of the same form ula will be immediately clear ). F or m = 1 , 2 , 3 these s olutions, see (42 b ), are p erio dic functions o f the ( r e al ) time t with p erio d 2 T if | c | > 1 . If instead | c | < 1, the solutio ns (see (42 a )) with m = 1 resp ectively m = 2 are p erio dic with p erio ds 2 T respe ctiv e ly T (see (31)). The solutio n with m = 3 is p erio dic if γ (3) is r e al, but with the per io d 2 T | γ (3) | which is not congruent to T if γ (3) is irr ational ; it grows exp onentially with increasing time if Im  γ (3)  < 0, implying instability of the solutio n (37) in this case, a nd it instead decays exp onentially if Im  γ (3)  > 0, implying a limit cyc le be havior in conﬁg ur ation space, namely asymptotic a pproach to a s o lution c ompletely p erio dic with p erio d T o r 2 T depe nding whether the cen ter of mass of the system is ﬁxed at the origin or itself moving with p erio d 2 T ; but note that in this pa per we restrict our attent io n to the case with r e al coupling constants. 4. Analytic structure of the soluti ons of the auxil iary mo del In this s e c tion we discuss the pr ope r ties of analyticity as functions o f the complex v ariable τ of the s olutions ζ n ( τ ) of the aux iliary model (1) (with arbitr ary v alues of the 3 coupling constants g n , i. e. not r estricted by the semisymmetrica l co ndition (2): except whe n this is explicitly speciﬁed, see b elow). In particular we show ﬁrs t of all that, for appropr ia te initial data character ized by suﬃciently lar ge v alues of the mo duli of al l thr e e in ter particle distances, na mely by the condition (s e e (6)) that the quantit y ζ min = min n,m =1 , 2 , 3; n 6 = m | ζ n (0) − ζ m (0) | (43) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 12 be ade quately lar ge , the so lutions ζ n ( τ ) are holomorphic in a disk D 0 of ( arbitr arily lar ge ) radius d 0 centered at the origin, τ = 0 , of the complex τ - plane (of cours e the “adequately large” v alue o f the quantit y ζ min depe nds on d 0 , and o n the mag nitude of the thr e e c o upling constants g n ; see (51) b elow). W e moreover discuss via a lo ca l analysis a la Painlev´ e the na ture of the sing ularities of the so lutio ns ζ n ( τ ) of the auxiliary mo del (1) as functions of the c omplex v ar iable τ a nd w e thereb y justify the assertions made in this res pect in Section 2. T o prov e the ﬁrst p oint, set σ n ( τ ) = ζ n ( τ ) − ζ n (0) , (44 a ) so tha t these quantities σ n ( τ ) v anish initially , σ n (0) = 0 , (44 b ) and, as a consequence of (1), s atisfy the equations of motion σ ′ n ( τ ) = g n +1 ζ n (0) − ζ n +2 (0) + σ n ( τ ) − σ n +2 ( τ ) + g n +2 ζ n (0) − ζ n +1 (0) + σ n ( τ ) − σ n +1 ( τ ) . (44 c ) A s tandard theorem (see, for instance , [20]) guarantees then that these quantities σ n ( τ ) – hence a s well the functions ζ n ( τ ) , see (44 a ) – ar e holomorp hic in τ (at least) in a disk D 0 centered at the orig in τ = 0 in the c omplex τ -plane, the radius d 0 of which is b ounded b elow by the ine q uality d 0 > b 4 M (45) (this formula coincides with the last equation of Section 1 3.21 of [20], with the assignments m = 3 a nd a = ∞ , the ﬁrst of which is justiﬁed by the fact that the system (44 c ) features 3 co upled e quations, the second o f which is justiﬁed by the autonomous character of the equations of motion (44 c )). The tw o p ositive quantities b and M in the right-hand side of this inequalit y are deﬁned as follows. The quan tity b is r equired to guara n tee that the right-hand sides of the equations of motion (44 c ) be holomorphic (as functions of the dep endent v aria ble s σ n ) provided these quantities satisfy the three inequalities | σ n | ≤ b ; (46) clearly in o ur case a suﬃcient condition to gua r antee this is provided by the single restriction b < ζ min 2 , (47) with ζ min deﬁned by (43). The second quantit y in the right-hand side of (45), M ≡ M ( b ) , is the u pp er bo und o f the right-hand sides of (44 c ) when the qua n tities σ n satisfy the inequality (46); but of cour se the inequa lit y (45) holds a fortiori if w e ov erestima te M , a s we shall prese n tly do. Indeed clear ly the equatio ns of motio n (44 c ) with (4 6) and (47) e n tail M < 4 G ζ min − 2 b , (48) with G = max n =1 , 2 , 3 | g n | . (49) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 13 Insertion of (48) in (45) yields d 0 > b ( ζ min − 2 b ) 16 G , (50) hence, setting b = ζ min 4 (to maximize the right - hand side; note the consistency of this assignment with (47)) , d 0 > ζ 2 min 128 G , (51) conﬁrming the as sertion made above (that d 0 can be made arbitr arily lar ge b y choo sing ζ min ade quately lar ge ). Next, let us sho w, via a loca l analysis ` a la Painlev ´ e , that the singularities as functions of the complex v ariable τ of the gener al solutions ζ n ( τ ) of our auxiliary mo del (1) asso cia ted with a coinc idenc e of two o f the thr e e comp onents ζ n are squ ar e- r o ot br anch p oints (recall that a singular it y at ﬁnite τ of a solution ζ n ( τ ) o f the evolution equations (1) may only o cc ur when the rig h t- ha nd side o f thes e equations diverges). Such a sing ula rity o ccurs for those v alues τ b of the indep e nden t v ariable τ such that t wo of the thr e e functions ζ n coincide, fo r instance ζ 1 ( τ b ) = ζ 2 ( τ b ) 6 = ζ 3 ( τ b ) . (52) The squ ar e-ro ot character of these branch p oints is evident from the following ansatz characterizing the b ehavior of the solutions o f (1) in the neighborho o d of these singularities: ζ s ( τ ) = ζ b − ( − 1) s α ( τ − τ b ) 1 / 2 + v s ( τ − τ b ) + ∞ X k =3 α ( k ) s ( τ − τ b ) k/ 2 , s = 1 , 2 ζ 3 ( τ ) = ζ 3 b + v 3 ( τ − τ b ) + ∞ X k =3 α ( k ) 3 ( τ − τ b ) k/ 2 , (53 a ) with α 2 = g 3 , v 3 = − g 1 + g 2 ζ b − ζ 3 b , v s = g s + 5 g s +1 6 ( ζ b − ζ 3 b ) , s = 1 , 2 mo d(2) , (53 b ) and the constants α ( k ) n determinable (in principle) re c ur sively (for k = 3 , 4 , ... ) by inserting this ansatz in (1), s o that, to b egin with α (3) 3 = 2 α ( g 2 − g 1 ) 3 ( ζ b − ζ 3 ) 2 , (53 c ) α (3) s = − ( − 1 ) s α 36 ( ζ b − ζ 3 ) 2 " 3 ( g s − 7 g s +1 ) + ( g 1 − g 2 ) 2 g 3 # , s = 1 , 2 mo d(2) , (53 d ) and so on. The diligent reader will verify the consis tency of this pro ce dure, for any assignment o f the t hr e e constants τ b , ζ b , and ζ 3 b , which r emain undetermined except for the obvious r estrictions τ b 6 = 0 , ζ b 6 = 0 , ζ 3 b 6 = ζ b . The fact that (53) contains thr e e a rbitrary ( c omplex ) consta n ts – the maxima l num b er of integration c onstants compatible with the system o f thr e e ﬁrst-o rder ODEs (1) – shows that this ansatz is indeed adequate to r epresent lo cally , in the neig h b or ho o d of its sing ularities o ccurring at τ = τ b , the gener al so lution of (1). T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 14 An analo gous ana lysis of the b ehavior of the so lutions of the s y stem (1) near the v alues of the indep enden t v aria ble τ wher e a triple coincidence of a ll thr e e functions ζ n o ccurs (cor r esp onding to the excluded as signment ζ 3 b = ζ b in the above ansatz (53)), indicates, somewhat sur prisingly , tha t such a t r iple coinc idenc e , ζ 1 ( τ b ) = ζ 2 ( τ b ) = ζ 3 ( τ b ) = Z (54) might also o ccur for the gener al solution of the system (1). This conclusion is r eached via a lo cal analysis ana lo gous to that p er fo rmed a b ove, and is then co nﬁrmed (for the semisymmetrica l case, s ee (2)) by the exact trea tmen t of Section 5. Indee d the natural extension of the ab ov e ansatz (53) character iz ing the b ehavior of the solutions of (1) in the neighborho o d o f such singular ities, corr esp onding to a triple coincidence, see (5 4 ), o f the thr e e functions ζ n ( τ ) , reads a s follows: ζ n ( τ ) = Z + η n ( τ − τ b ) (1 − γ ) / 2 + α n ( τ − τ b ) 1 / 2 + o  | τ − τ b | 1 / 2  , (55 a ) provided Re ( γ ) < 0 . (55 b ) Here the thr e e constants α n are determined, as ca n b e easily veriﬁed, just by the thr e e nonlinear alg ebraic equations (23) that were found in the pre c e ding section while inv estigating the equilibrium co nﬁgurations of o ur physic al system (7), while the thr e e constants η n , as well a s the exp onent γ , are r equired to satisfy the algebra ic eq ua tions ( γ − 1 ) η n 2 = g n +1 ( η n − η n +2 ) ( α n − α n +2 ) 2 + g n +2 ( η n − η n +1 ) ( α n − α n +1 ) 2 . (56) These algebraic equations, (23) and (56), can b e c onv eniently rewritten a s follows: α n = β n +1 ( α n − α n +2 ) + β n +2 ( α n − α n +1 ) , (57 a ) ( γ − 1 ) η n = β n +1 ( η n − η n +2 ) + β n +2 ( η n − η n +1 ) , (57 b ) via the int r o duction of the quantities β n , se e (24 b ). Note that in this manner these tw o sets of equatio ns, (57 a ) a nd (57 b ), hav e a quite similar lo o k , which s hould howev er not mislea d the r eader to underestimate their basic diﬀerence: the thr e e equations (57 a ) are merely a co n venient w ay to rewrite, via the deﬁnition (24 b ), the thr e e nonline ar equations (23), which determine (albeit not uniquely , see Appendix A) the thr e e constants α n ; while the equations (57 b ) are thr e e line ar eq uations for the thr e e quantities η n , hence they can determine these thr e e unknowns only up to a common m ultiplica tiv e constant (provided they admit a nontrivial s o lution: se e below). Of cour se these linear equations (57 b ) admit the trivial solution η n = 0 , and it is easily seen that ther e indeed is a sp ecial ( exact ) s o lution of the equa tions of motion (1) having this pr o per t y , see (39) with the constants α n determined by (23) and co mputed, for the s emisymmetrical mo del, in App endix A. This “similarity solution” (39) of the system (1) has been discussed in the preceding s e c tion; but let us emphasize here that it only provides a two -par ameter ( Z and τ b ) clas s of solutions of the equations of motion (1), while the gener al solutio n of this system o f thr e e ﬁrst-or der ODEs must of course feature thr e e arbitr ary par a meters. A gener al solution of the evolution equations (1 ) cor r esp onds instead to the ansatz (55 a ) if the linea r equatio ns (57 b ) for the thr e e co eﬃcients η n admit a nonvanishing solution, b e cause in such a case, as mentioned above, a common sca ling para meter for these thr e e co eﬃcients re ma ins a s an additional ( thir d ) fr e e parameter (b esides Z and τ b ). The condition fo r this to happ en is the v anishing o f the deter mina n t of T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 15 the coeﬃcients of these thr e e linea r equa tions, (57 b ), namely again v alidity of the determinantal co nditio n (30), a cu bic equa tion for the unkno wn γ , which determines, as discussed in the preceding section, the t hr e e v alues (31) of this quantit y . But the ﬁrst tw o of these v alues , γ = γ (1) = 1 and γ = γ (2) = 2 (see (31)), are not consistent with the require men t (5 5 b ). The thir d solutio n, γ = γ (3) = 2( β 1 + β 2 + β 3 ) (see (31)) might instead be consistent with the req uir ement (55 b ), and whenever this happ ens the ansatz (55 ) indicates that the gener al so lution of the system o f ODEs (1) do es feature a “ triple coincidence”, see (54), and identiﬁ es the character of the corres p onding br anch p oint . In the s emisymmetrical case (2) the equa tions characterizing the eq uilibrium conﬁguratio n, (23) or eq uiv alently (57), can b e s olved (see App endix A). One ﬁnds that ther e are two distinct solutions of these nonlinear e quations (23) (in fact four , since ea ch solution ha s a triv ial counterpart obta ined by exchanging the role of the t wo “equa l” particles with la bels 1 and 2) . The ﬁrst solution yields for γ = γ (3) the v alue (36 a ) , which is consistent w ith the condition (55 b ) iﬀ Re ( µ ) < 0 , (58 ) and it yields for the branch po in t exp onent, se e (55 a ), the v alue 1 − γ 2 = µ − 1 2 µ = p − q 2 p ; (59) while the se c ond solution yields for γ (3) the v alue (36 b ), whic h is consistent with the condition (55 b ) iﬀ Re ( µ ) > 1 , (60 ) and it yields for the branch po in t exp onent, se e (55 a ), the v alue 1 − γ 2 = µ + 1 2 ( µ − 1 ) = p + q 2 ( p − q ) . ( 6 1) The last equality in (59) and (61) a re o f course only v alid if µ is ratio nal, µ = p/q . Note tha t these ﬁndings imply that the bra nc h p oint ass o ciated with “ triple coincidences” is no t (only) of squar e-r o ot type, being also characterized, s ee (55 a ), by the exp onent 1 − γ 2 , the v alue of which dep e nds on the v alue of the par ameter µ , s e e (59) and (61); ho wever this kind of branch po in t is not present if 0 < Re ( µ ) < 1 , (62) since in this case neither (58) nor (60) are satis ﬁe d. The results pres e n ted in this section are not entirely rigo rous, since the lo ca l analysis of the singular ities we p erfor med above on the basis of appropria te ans¨ atze should be complemented b y pro ofs that the r elev ant e x pansions co n verge. Moreover these analyse s provide informa tion on the nature of the bra nch p oints, but neither on their num b er no r their lo catio n. But these results are conﬁrmed and complemented below (see Section 5) by the ana lysis of the exact g eneral solution of the e quations of motion (1). Our motiv ation for ha ving nevertheless pres en ted here a dis c ussion of the character o f the singularities o f the so lutio ns of (1) via a lo cal a nalysis ` a la Painlev ´ e is b ecause a n analo gous trea tmen t may b e applicable to mo dels which are not as explicitly solv able a s that treated in this pape r (see for insta nce [13] and [1 1]). T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 16 5. General solutio n b y quadratures In this sectio n we o btain and discuss the gener al solutions of our models, (1 ) and (7). But since the gener al solution of the physic al mo del (7) is easily obtained via the trick (5) from the gener al solution of the auxiliary pro blem (1), we fo cus to b egin with on this model. A ﬁr st constan t o f the motion is provided by the cen ter- of-mass co ordinate Z = ζ 1 + ζ 2 + ζ 3 3 , ( 6 3 a ) since the equatio ns of motion (1) clearly entail Z ′ = 0 (63 b ) hence Z ( τ ) = Z (0) . (63 c ) And cle a rly the gener al solution of (1) reads ζ n ( τ ) = Z + ˇ ζ n ( τ ) , (64 a ) with the set of 3 functions ˇ ζ n ( τ ) providing themselv es a s o lution of (1), indep endent of the v alue of Z and satisfying the (compatible) cons tr aint ˇ ζ 1 ( τ ) + ˇ ζ 2 ( τ ) + ˇ ζ 3 ( τ ) = 0 . (64 b ) It is moreover clear that the equations of motion (1) entail ζ ′ 1 ζ 1 + ζ ′ 2 ζ 2 + ζ ′ 3 ζ 3 = g 1 + g 2 + g 3 , (65 a ) hence there also holds the relation ζ 2 1 + ζ 2 2 + ζ 2 3 = 2 ( g 1 + g 2 + g 3 ) ( τ − τ 0 ) . (65 b ) It is now conv enient to set, as in the Appendix B of [13], ζ s = Z −  2 3  1 / 2 ρ cos  θ − ( − 1) s 2 π 3  , s = 1 , 2 , (66 a ) ζ 3 = Z −  2 3  1 / 2 ρ cos θ . (66 b ) Then, summing the squares o f these three for mulas a nd using the identities cos( θ ) + co s( θ + 2 π 3 ) + cos( θ − 2 π 3 ) = 0 , (67) cos 2 ( θ ) + cos 2 ( θ + 2 π 3 ) + co s 2 ( θ − 2 π 3 ) = 3 2 , (68) one ea sily g e ts ζ 2 1 + ζ 2 2 + ζ 2 3 = 3 Z 2 + ρ 2 (69 a ) or equiv a le n tly ρ 2 = 1 3 h ( ζ 1 − ζ 2 ) 2 + ( ζ 2 − ζ 3 ) 2 + ( ζ 3 − ζ 1 ) 2 i , (69 b ) hence, from (65 b ), ρ 2 = 2 ( g 1 + g 2 + g 3 ) ( τ − τ 0 ) − 3 Z 2 = 2 ( g 1 + g 2 + g 3 ) ( τ − τ 1 ) , (69 c ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 17 τ 1 = τ 0 + 3 Z 2 2 ( g 1 + g 2 + g 3 ) , (69 d ) which also entails ρ ′ ρ = g 1 + g 2 + g 3 . (69 e ) Here we assume that the sum o f the thr e e coupling c o nstants g n do es not v anis h, g 1 + g 2 + g 3 6 = 0 . The sp ecia l case in whic h this sum does instead v anish is tr e ated in Appendix C . The e x pression o f the cons ta n t τ 1 in terms of the initial data is of c ourse (see (69 c )) τ 1 = − ρ 2 (0) 2 ( g 1 + g 2 + g 3 ) , ( 7 0 a ) namely (see (69 c )) τ 1 = − ζ 2 1 (0) + ζ 2 2 (0) + ζ 2 3 (0) − 3 Z 2 2 ( g 1 + g 2 + g 3 ) , (70 b ) or equiv a le n tly (see (6 9 d )) τ 1 = − ( ζ 1 − ζ 2 ) 2 + ( ζ 2 − ζ 3 ) 2 + ( ζ 3 − ζ 1 ) 2 6 ( g 1 + g 2 + g 3 ) . (70 c ) There r emains to compute θ ( τ ) , or rather u ( τ ) = cos θ ( τ ) . ( 7 1) Inserting the ansatz (66) in the equation o f mo tion (1) with n = 3 , one ea s ily gets ρ 2 (cos θ ) ′  4 co s 2 θ − 1  = (4 g 1 + 4 g 2 + g 3 ) c os θ − 4 ( g 1 + g 2 + g 3 ) c os 3 θ + √ 3 ( g 1 − g 2 ) s in θ . (72) F rom now on in this sectio n – for simplicity , and bec a use it is suﬃcient for our purp oses – we restrict atten tion to the semisymmetrica l cas e (2), s o that the last equation becomes simply , via (71), ρ 2 u ′  4 u 2 − 1  = ( f + 8 g ) u − 4 ( f + 2 g ) u 3 . (73) The general case without the restriction (2) is trea ted in Appendix C. This ODE ca n b e easily integrated via a quadra ture (using (69 c )), a nd this leads to the following formula: [ u ( τ )] − 2 µ  u 2 ( τ ) − 1 4 µ  µ − 1 = K ( τ − τ 1 ) , (74) where the par ameter µ is deﬁned b y (14) and K is an integration c onstant. Here we a re of course a ssuming that f + 8 g 6 = 0 (see (14)); the case when this do es n ot happ en is treated in Appendix C. (Also recall that, a s promised a b ove, we sha ll treat in Appendix C the cas e in which the sum of the three coupling constants g n v anishes, namely when f + 2 g = 0, which entails µ = 0 , see (14)). As fo r the quantit y K in (74), it is an ( a priori arbitr ary ) integration constant. It is a matter of elementary algebra to e x press this co nstant in ter ms o f the orig ina l dep endent v ariables ζ n (via (74), (69 c ), (71) and (66 )), and one thereby obtains the r elation K = 12 ( f + 2 g ) ˜ K (75) with ˜ K deﬁned by (1 8). This ﬁnding justiﬁes the asse r tion that ˜ K is a consta n t of motion, see Section 2.3 ; and o f course it deter mines the v alue to b e assigned to T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 18 the co ns tan t K in the context of the initial-value problem. Likewise the v alue to b e assigned, in the context o f the initial-value problem, to the constant τ 1 app earing in the righ t-ha nd side of (74) is given by the formula K τ 1 = − [ u (0)] − 2 µ  u 2 (0) − 1 4 µ  µ − 1 , (76) where (s e e (7 1) and (66 b )) u (0) = −  3 2  1 / 2 ζ 3 (0) − Z ρ (0) (77 a ) namely u (0) = − 2 ζ 3 (0) − ζ 1 (0) − ζ 2 (0) h 2 n [ ζ 1 (0) − ζ 2 (0)] 2 + [ ζ 2 (0) − ζ 3 (0)] 2 + [ ζ 3 (0) − ζ 1 (0)] 2 oi 1 / 2 . (77 b ) Of co urse in these formulas the initial v alues ζ n (0) o f the co ordinates ζ n ( τ ) o f the auxiliary pro blem (1) c an b e r eplaced by the initial v alues z n (0) o f the physic al pr oblem (7), see (6). Let us emphasize that we hav e now reduced, via (66) with (69 c ) and (71), the solution o f our problem (1) to the inv estiga tion of the function u ( τ ) of the c omplex v ariable τ , deﬁned for τ 6 = 0 as the solution of the ( nondiﬀer ential ) equa tion (7 4 ) that evolv es by contin uity from u (0) at τ = 0. T o pro ce ed with our a nalysis an a dditio na l change of v ariables is now convenien t. W e in tro duce the new ( c omplex ) indep endent v ariable ξ by setting ξ = K ( τ − τ 1 ) 4 µ , (78) and the new ( c omplex ) dep enden t v ariable w ≡ w ( ξ ) by setting w ( ξ ) = 4 µ [ u ( τ )] 2 . (79) Thereby the expression of the s o lution (66 ) of o ur origina l problem (1) rea ds ζ s ( τ ) = Z −  f + 2 g 3 K  1 / 2 ξ 1 / 2 n − [ w ( ξ ) ] 1 / 2 + ( − ) s [12 µ − 3 w ( ξ )] 1 / 2 o , s = 1 , 2 , (80 a ) ζ 3 ( τ ) = Z − 2  f + 2 g 3 K  1 / 2 [ ξ w ( ξ )] 1 / 2 , (80 b ) while the ( nondiﬀer ential ) equation that determines the dependenc e of w ( ξ ) on the ( c omplex ) v aria ble ξ r eads [ w ( ξ ) − 1 ] µ − 1 [ w ( ξ )] − µ = ξ . (81) Note tha t this eq uation is indep endent o f the initial data ; it only features the co nstant µ , whic h only dep ends on the coupling constants, see (14). W e conclude that the so lution o f our physic al pro blem (7) as the r e al time v ar iable t evolv es onw ar ds from t = 0 is essentially given, via (80) and (5), by the evolution of the so lution w ( ξ ) of this ( nondiﬀer ential ) equation, (81), as the c omple x v ar iable ξ trav els round a nd round on the cir cle Ξ in the c omplex ξ - plane deﬁned by the equation (see (78) and (5 a )) ξ = R exp (2 i ω t ) + ¯ ξ = R [exp (2 i ω t ) + η ] , (82 a ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 19 namely on the c ircle with center ¯ ξ and radius | R | . The par a meters R and ¯ ξ (o r η ) depe nd on the initial data a ccording to the formulas (implied b y (78), (5), (7 5), (18), (70 c )) R = 3 ( f + 8 g ) 2 i ω [2 z 3 (0) − z 1 (0) − z 2 (0)] 2 [1 − κ ] µ − 1 , (82 b ) ¯ ξ = R η, (82 c ) η = i ω n [ ζ 1 (0) − ζ 2 (0)] 2 + [ ζ 2 (0) − ζ 3 (0)] 2 + [ ζ 3 (0) − ζ 1 (0)] 2 o 3 ( f + 2 g ) − 1 , (82 d ) κ = 2 µ [2 ζ 3 (0) − ζ 1 (0) − ζ 2 (0)] 2 [ ζ 1 (0) − ζ 2 (0)] 2 + [ ζ 2 (0) − ζ 3 (0)] 2 + [ ζ 3 (0) − ζ 1 (0)] 2 . ( 8 2 e ) Of co urse in these formulas the initial v alues ζ n (0) o f the co ordinates ζ n ( τ ) o f the auxiliary problem (1) can b e r eplaced by the initial v alues z n (0) of the co o rdinates z n ( t ) o f the physic al problem (7), see (6 ). Let us emphasize that, as the complex v ar iable ξ trav els on the cir cle Ξ – taking the time T to make each round, see (8 2 a ) and (5 c ) – the dep endent v ariable w ( ξ ) trav els on the Riemann surface deter mined by its dep endence o n the c omplex v ariable ξ , as entailed by the equation (8 1) that r elates w ( ξ ) to its argument ξ – star ting at t = 0 from ξ = ξ 0 , ξ 0 = ¯ ξ + R = ( η + 1) R, (83 a ) ξ 0 = i ω R n [ ζ 1 (0) − ζ 2 (0)] 2 + [ ζ 2 (0) − ζ 3 (0)] 2 + [ ζ 3 (0) − ζ 1 (0)] 2 o 3 ( f + 2 g ) (83 b ) (see (82)) and corresp ondingly from w ( ξ 0 ) = w 0 , w 0 = 1 κ = [ ζ 1 (0) − ζ 2 (0)] 2 + [ ζ 2 (0) − ζ 3 (0)] 2 + [ ζ 3 (0) − ζ 1 (0)] 2 2 µ [2 ζ 3 (0) − ζ 1 (0) − ζ 2 (0)] 2 (84) (see (82 e )). Let us therefore now disc us s the structure of this Riemann surface, namely the analytic prop erties o f the function w ( ξ ) deﬁned by (81). Ther e are tw o types o f singularities, the “ﬁxed” ones o ccurring at v alues of the indep endent v ariable ξ , and corres p onding ly of the dep endent v aria ble w , that can b e rea d dir ectly fro m the structure of the equa tion (8 1) under investigation, and the “mov able” ones (this name being g iven to underline their diﬀerence from the ﬁx e d ones) o ccurring a t v alues o f the indep endent a nd dependent v ariables , ξ and w , that cannot be dir ectly read from the structur e o f the equation (81) under inv estiga tion (they “ mov e” as the initial data are modiﬁed). 5.1. Movable singularities T o inv es tig ate their nature it is conv enient to diﬀeren tiate (81), obtaining thereby (using again (81)) ξ w ′ = − w ( w − 1 ) w − µ , (85) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 20 where the prime indica tes of course diﬀerentiation with resp ect to ξ . (Note that this ODE is implied by the nondiﬀerential equation (81), while its solution repro duces the nondiﬀerential equation (81) up to multiplication of its rig h t-ha nd side by an arbitrar y constant). The po sition of the sing ularities, ξ b , a nd the co rresp onding v alues of the dependent v ar iable, w b ≡ w ( ξ b ) , are then characterized by the v anis hing of the denominator in the right-hand side of this formula, yielding the rela tion w b = µ, (86 ) which, combined with (8 1) (at ξ = ξ b ) is easily seen to yield ξ b = ξ ( k ) b = r exp (2 π i µ k ) , k = 1 , 2 , 3 , ..., (87 a ) ξ b = ξ ( k ) b = r exp  i 2 π p k q  , k = 1 , 2 , ..., q, (87 b ) r = ( µ − 1) − 1  µ − 1 µ  µ . (87 c ) In the last, (87 c ), of these form ula s it is understoo d that the principal determination is to b e taken of the µ -th power appear ing in the rig h t-hand side. The ﬁrst o f these formulas, (8 7 a ), shows clea rly that the num b er of these branch p oints is inﬁnite if the parameter µ is irr ational , a nd that they then sit densely o n the circle B in the complex ξ -plane centered at the or igin and having ra dius r , see (87 c ). Note that this entails that the generic point o n the circle B is not a branch po in t (just as a generic r e al nu mber is not r ational ); but every generic p oint o n the circle B has some branch po int (in fact, an inﬁnity of br anch p oints!) arbitr arily clo se to it (just as every generic r e al nu mber has an inﬁnity o f r ational num b ers arbitr arily close to it). As fo r the second of this form ula s, (87 b ), it is instead appropriate to the case in which the parameter µ is r ational , see (15), in which case the branch points sit aga in on the cir cle B in the complex ξ -plane, but ther e are only a ﬁnite n umber , q , of them. These singularities are all squar e r o ot branc h points, as implied by the following standard proo f. Set, for ξ ≈ ξ b , w ( ξ ) = µ + a ( ξ − ξ b ) β + o  | ξ − ξ b | Re ( β )  , (88 a ) with the assumption (immediately veriﬁed, se e b elow) that 0 < Re ( β ) < 1 . (88 b ) It is then immediately seen that the inse r tion o f this ansatz in (8 5 ) (is consistent and) yields β = 1 2 , a 2 = 2 (1 − µ ) ξ b = − 2  µ µ − 1  µ . (88 c ) Note that these results conﬁr m the treatment o f Section 4: the squar e r o ot branch po in ts of w ( ξ ) identiﬁed here, see (86), a re easily seen to co rresp ond, via (80), to the pair coincidence ζ 1 ( τ b ) = ζ 3 ( τ b ) or ζ 2 ( τ b ) = ζ 3 ( τ b ); while there is an additional class of squar e-r o ot branch points whic h only aﬀect ζ 1 ( τ ) and ζ 2 ( τ ), but neither ζ 3 ( τ ) nor w ( ξ ) , and occur at w = 4 µ (89 a ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 21 due to the v anishing of the sec ond squar e-r o ot term inside the curly brack et in the right-hand side of (80 a ), and corresp ond therefore to the coincidence ζ 1 ( τ b ) = ζ 2 ( τ b ). The corresp onding v alues of ξ (as implied by (89 a ) with (81)) are ξ = (4 µ − 1) µ − 1 (4 µ ) µ = 1 4 µ  1 − 1 4 µ  µ − 1 (89 b ) (w e use the plural to refer to these v alues b ecause o f the multiv a luedness of the function in the right-hand side of this form ula ). 5.2. Fixe d singu larities Next, let us consider the “ﬁxed” singularities, whic h clearly ca n o nly o ccur at ξ = ∞ and at ξ = 0 , with co rresp onding v alues for w . Let us in vestigate ﬁrstly the nature of the singula rities at ξ = ∞ . Tw o b ehaviors of w ( ξ ) a r e then pos sible for ξ ≈ ∞ , depending on the v alue of (the real par t of ) µ . The ﬁrst is c ha racterized by the ansatz w ( ξ ) = aξ β + o  | ξ | Re ( β )  , Re( β ) < 0 , (90 a ) and its insertion in (81) yields β = − 1 µ , a µ = − exp( i π µ ) , (90 b ) which is co nsistent with (90 a ) iﬀ Re( µ ) > 0 . (90 c ) The second is c hara cterized by the ansatz w ( ξ ) = 1 + aξ β + o  | ξ | Re ( β )  , Re ( β ) < 0 , (91 a ) and its insertion in (81) yields β = 1 µ − 1 , a µ − 1 = 1 , (91 b ) which is co nsistent with (91 a ) iﬀ Re( µ ) < 1 . (91 c ) W e therefor e co nclude that there are three p ossibilities: if Re( µ ) > 1 , only the ﬁrst ansatz , (90 ), is applicable, and it c har acterizes the nature of the branc h p oint of w ( ξ ) at ξ = ∞ ; if Re( µ ) < 0 , o nly the sec o nd ansatz , (91), is applicable, a nd it character izes the nature o f the branch p oint of w ( ξ ) a t ξ = ∞ ; while if 0 < Re( µ ) < 1 , bo th ans¨ atze , (90) and (91) , ar e applicable, so b oth t yp es of bra nc h points occur at ξ = ∞ . Next, let us in vestigate the nature of the singularity at ξ = 0 . It is then easily seen, by an analo gous trea tmen t, that t wo be haviors are p oss ible, as display e d by the following ans¨ atze : either w ( ξ ) = aξ β + o  | ξ | Re ( β )  , Re( β ) > 0 , (92 a ) β = − 1 µ , a µ = − exp ( i π µ ) , (92 b ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 22 which is a pplicable iﬀ Re( µ ) < 0 ; (92 c ) or w ( ξ ) = 1 + aξ β + o  | ξ | Re ( β )  , Re( β ) > 0 , (93 a ) β = 1 µ − 1 , a µ − 1 = 1 , (93 b ) which is a pplicable iﬀ Re( µ ) > 0 . (93 c ) This analys is shows that the function w ( ξ ) features a br anch p oint a t ξ = 0 the nature of whic h is c ha racterized by the relev ant exp onent β , see (9 2 b ) o r (93 b ), whichever is applicable (see (92 c ) and (93 c )). But let us emphasize that there is n o bra nc h p oint at al l at ξ = 0 if neither o ne of the t wo inequalities (92 c ) and (93 c ) holds , na mely if 0 < Re( µ ) < 1. 5.3. Explicitly solvable c ases Let us end this Section 5 b y noting that the equa tion (8 1) for certain rational v alues of µ r educes to such a low degr ee p olynomial equation that it can be solv ed explicitly . In particular, the p olyno mial equation is of second degr e e if µ = − 1 , 1 / 2 or 2; it is of third degree if µ = − 2 , − 1 / 2 , 1 / 3 , 2 / 3 , 3 / 2, o r 3; while it is o f fourth degr ee if µ = − 3 , − 1 / 3 , 1 / 4 , 1 / 2 , 3 / 4 , 4 / 3 or 4 . The diligent r eader mig h t w is h to use the corr e spo nding explicit so lutio ns for m ula s for thes e cases to verify the v alidit y of the previous discussion. 6. The ph ysical mo del The solution (8 0) can als o be wr itten, via (75), (18) and (5 ), directly for the particle co ordinates z n ( t ) , to read as follows: z s ( t ) = Z e iωt − 2 z 3 (0) − z 1 (0) − z 2 (0) 6 √ µ [ η exp ( − 2 i ω t ) + 1 ] 1 / 2 · ·  − [ ˇ w ( t )] 1 / 2 + ( − ) s [12 µ − 3 ˇ w ( t )] 1 / 2  , s = 1 , 2 , (94 a ) z 3 ( t ) = Z e iωt − 2 z 3 (0) − z 1 (0) − z 2 (0) 3 √ µ [ η exp ( − 2 i ω t ) + 1 ] 1 / 2 [ ˇ w ( t )] 1 / 2 , (94 b ) where the constant η is given in terms of the initial data by (82 d ) and w e set ˇ w ( t ) ≡ w [ ξ ( t )] , (95) so that this depe nden t v ariable is now the so lution o f the nondiﬀerential e q uation (see (81)) [ ˇ w ( t ) − 1 ] µ − 1 [ ˇ w ( t )] − µ = R exp (2 i ω t ) + ¯ ξ = R [exp (2 i ω t ) + η ] , (96) where the co ns tan ts R , ¯ ξ and η are deﬁned in terms of the initial data, see (82) (and recall that the initial data ζ n (0) can b e replaced by the initia l data z n (0), s e e (6)). The depe ndent v aria ble ˇ w ( t ) is o f course the solution o f this equation, (96), identiﬁed by contin uity , as the time t unfolds fro m t = 0 , from the initial da tum ˇ w (0) = w 0 assigned T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 23 at t = 0, see (8 4): this sp eciﬁcation is necessa r y , sinc e genera lly the nondiﬀerential equation (81) has mor e than a single solution, in fa ct p ossibly ev en an inﬁnity of solutions. A discussion of the b ehaviour o f this solution of the initial-v alue problem of our mo del (7) with (2) clearly hinge s on as certaining how the so lutio n ˇ w ( t ) of (96) e volves in time. This equation (96) corres p onds of course to the combination of (8 1) with (82 a ). Hence one must ﬁrstly elucidate the s tructure of the Riemann surface deﬁned by the dependence of w ( ξ ) o n the complex v ariables ξ as determined by the nondiﬀerential equations (81), a nd then understand the conseq ue nc e s of a trav el on this Riemann surface when the complex v ariable ξ evolves acco rding to (82 a ), namely it trav els round a nd round on the circle of center ¯ ξ and radius | R | in the c o mplex ξ -plane. The ﬁrst task is s imple, its foundation b eing provided by the analysis provided in Sections 5.1 and 5 .2. The second task is muc h more de ma nding, inasm uch a s it hinges o n the detailed manner the sheets of the Riemann s ur face are connected via the cuts ass o ciated with the br anch po in ts discussed in Sections 5.1 and 5.2. The main results of this analysis have alr eady b een rep orted (without pro ofs) in [1]; their deriv ation requir es a suﬃciently extended tre a tmen t to sugge s t a separa te presentation [1 2]. T o avoid unnecessary rep etitions, a lso the detailed analysis of the Riemann surface asso ciated to (81) is postp oned to [12]. 7. Outlo ok In this pap er we rep ort a deep er ana lysis of the mo de l intro duced in [1] ex plaining many results that were there rep orted without pro of (such as the deriv ation of the general solution b y quadratures ) while a dding some new material (such as a detailed analysis o f the equilibrium co nﬁgurations, s ma ll oscillations and similarity solutions of the mo del). The nov elty of this approa ch in accounting for a new phenomeno logy asso ciated to chaotic motion in dynamical systems lies in the fact that the solution is a multi-v alued function of (complex) time, and a deta iled ana lysis of its Riema nn surface leads to very sp eciﬁc predictions in the simpler ( µ r ational) cases, while it also un veils a sour ce of irregula r b ehaviour in the more complicated ( µ irrationa l) c a ses – unpredictable inasmuc h as the determina tion of its evolution requir e s knowledge with arbitrar ily large prec ision o f the initia l data. The full ana ly sis o f the dy namics of this mo del – including the geometry o f the asso c iated Riemann surfac e – is po stpo ne d to a future publication [12]. The purp ose of this series of pap ers together with o ther r elated pr o jects [1 7, 19], is to go b eyond the lo cal analysis p erfo r med in the litera ture r elating analytic prop erties of s olutions in co mplex time with dynamical pr op erties of the mo del (Painlev ´ e- Kow alews k a ya and its non-mer omorphic ex tens io ns) and to p erform a full description of the glob al prop erties o f the Riemann surface. This full description requir es no t just ﬁnding the type of bra nc h p oints and their p ositio ns, but sp ecifying how the diﬀerent sheets of the Riemann sur face ar e attached together at those branch p oints. Whenever pos s ible, such an approach provides v er y detailed infor mation on the dynamics that cannot be obtained b y the more classical lo ca l analy ses. T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 24 Ac kno wl edgments W e would like to thank the Ce ntro Internacional de Ciencia s in Cuer nav aca, in particular F ra n¸ c o is Ley vraz and Thomas Seligma n, fo r their supp ort in o rganizing the Scientiﬁc Gatherings on Inte gr able Systems and the T r ansition t o Chaos which provided several opp ortunities for us to meet a nd work together . It is a pleasure to ackno wledge illuminating discussions with Boris Dubrovin, Y uri F edorov, Jean-P ierre F ran¸ coise, Peter Gr inevich, F ran¸ c o is Ley vraz, Alexander Mikha ilov, Thomas Se lig man and Carles Sim´ o. The res earch o f DGU is supp or ted in part by the Ram´ on y Ca jal pro g ram of the Spanish ministr y of Science a nd T echnology and by the DGI under g r ants MTM20 06- 00478 a nd MTM2 006-14 603. App endix A In this app endix we solve, in the s e mis ymmetrical case, see (2), the nonlinear algebraic equations (23) that characterize the equilibrium conﬁgurations a nd we thereby compute the “ eigenv alue” γ (3) , namely we o btain its tw o expressio ns (36 a ) and (36 b ). The equations to be solved read (see (23)) α 1 = 2 g α 1 − α 3 + 2 f α 1 − α 2 , (97 a ) α 2 = 2 g α 2 − α 3 − 2 f α 1 − α 2 , (97 b ) α 3 = 2 g α 3 − α 1 + 2 g α 3 − α 2 , (97 c ) and they of course imply the r e lation α 1 + α 2 + α 3 = 0 . (98) It is now conv enient to set S = α 1 + α 2 , D = α 1 − α 2 , (99 a ) ent a iling α 1 = S + D 2 , α 2 = S − D 2 , α 3 = − S . (99 b ) F rom (the sum of ) (97 a ) and (97 b ) we easily get S  9 S 2 − D 2  = 24 g S, (100) and from this we ge t tw o types of solutions. The ﬁrst solution is characteriz e d by S = 0 , implying (see (99 b ) and (97 a )) α 3 = 0 , α 1 = − α 2 = α, α 2 = f + 2 g , (101) ent a iling (via (22)) the so lution (34) for the equilibr ium conﬁguration, a s well a s (via (24 b ) with (2)) the expr essions β 3 = f 2 ( f + 2 g ) , (102 a ) β 1 = β 2 = 2 g f + 2 g , (102 b ) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 25 hence, via (32), the ﬁrst expressio n, (36 a ), for γ (3) . The se c ond solution is characterized by 9 S 2 − D 2 = 24 g . (103) W e no w subtract (97 b ) from (9 7 a ) a nd we thereby easily get D 2 = − 8 g D 2 9 S 2 − D 2 + 4 f , (104) hence, via the preceding r elation, D 2 = 3 f , S 2 = f + 8 g 3 . (105) And v ia (24 b ) with (2) and (99 b ) this is easily seen to yield β 1 + β 2 + β 3 = f + 8 g 6 g , (106) namely , via (3 2), the second ex pression, (36 b ), o f γ (3) . No te mor eov er that, in b oth c ases , o ne gets the relation ( α 1 − α 2 ) 2 + ( α 2 − α 3 ) 2 + ( α 3 − α 1 ) 2 = 6 ( f + 2 g ) , (107) as c an b e easily veriﬁed fr om (101) as well as from (7) with (105). App endix B In this App endix we co nsider certain nongeneric (classes of ) solutions of o ur physical problem (7), characterized by sp ecial subclass es of initial data . If the initial data ar e suc h that η , hence as well ¯ ξ , vanish , η = ¯ ξ = 0 – and this ent a ils that the initial data satisfy the condition [ z 1 (0) − z 2 (0)] 2 + [ z 2 (0) − z 3 (0)] 2 + [ z 3 (0) − z 1 (0)] 2 = 3 ( f + 2 g ) i ω , (108) see (82) and (6); hence these initial data ar e not generic, dep ending o nly on 2 arbitra ry c omplex parameters ra ther than on 3 suc h para meters (or, equiv alen tly , only on 1 rather than 2 such parameter s b esides the trivial cons ta n t Z that only aﬀects the center-of-mass motio n, see (6)) – then the time evolution of the solution z n ( t ) of our ph y sical pro ble m (7), see (6), is clear ly per io dic with the p erio d T q rather tha n T . The cons equence of this fact are suﬃciently obvious not to require an y additional elab oration. An example of this type is that characterized by the parameters ω = f = 2 π , g = π , ⇒ T = 1 2 , µ = p q = 2 5 , (109 a ) and the initial data z 1 (0) = 0 . 2 , z 2 (0) = − 0 . 7965 8 + 0 . 7 1779i , z 3 (0) = 0 . 59658 − 0 . 7177 9i that imply that the center of mass is initially at the orig in and therefore stays there for a ll time, Z = 0 . These initial data are easily seen to sa tisfy the condition (10 8). T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 26 App endix C In this Appendix w e ex plain how to integrate the ODE (72) in the general case when the three coupling co nstants g n are al l diﬀere n t, namely when the restrictio n (2) ident ifying the semisymmetrica l case do es not a pply , and we also pr ovide the solution of the ODE (72) in the tw o sp ecial cases (b elonging to the semisymmetrical c lass characterized by the restriction (2)) the treatment of which had b een omitted in Section 5, and as well in a nother spec ia l ca se no t b elong ing to the s e misymmetrical class. Solution of e quation ( 72) in the gener al c ase In this subse c tio n o f Appendix C we indica te how the ODE (72) can be integrated in the g eneral case when the three coupling consta nts g n are al l diﬀerent. It is then conv enient to se t V ( τ ) = ta n [ θ ( τ ) ] , (110) so tha t this ODE reads V ′ V  V 2 − 3  ( V 2 + 1) ( A V 3 + C V 2 + A V + C − 2) = 1 ( τ − τ 1 ) (111) with A = √ 3 ( g 1 − g 2 ) 2 ( g 1 + g 2 + g 3 ) , C = 4 g 1 + 4 g 2 + g 3 2 ( g 1 + g 2 + g 3 ) . ( 1 12) T o in tegra te this ODE we set A V 3 + C V 2 + A V + C − 2 = A ( V − V 1 ) ( V − V 2 ) ( V − V 3 ) , (113) so that the thr e e quantities V n are the thr e e ro o ts of this p olyno mial of thir d degree in V . W e then decomp ose this ratio nal function of V in simple fractions, V  V 2 − 3  ( V 2 + 1) ( A V 3 + C V 2 + A V + C − 2) = 5 X j =1 µ j V − V j , (114) where o f co urse V 4 = i, V 5 = − i , (115) and the ﬁve quantities µ j are eas ily ev aluated in terms of the 3 roo ts V n : µ j = V j ( − 3 + V 2 j ) 5 Y k =1 ,k 6 = j ( V j − V k ) − 1 . (116) The integration of the ODE (111) is now tr ivial (using (114)), and it yields (using (115)) the ﬁnal formula [ V ( τ ) − i ] µ 4 [ V ( τ ) + i ] µ 5 3 Y n =1 [ V ( τ ) − V n ] µ n = K ( τ − τ 1 ) , (117) where K is the in tegratio n constant. T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 27 Solution of e quation(72) in t wo sp e cial sub c ases of the semisymmetric al c ase In this subse c tio n of App endix C w e pr ovide the solutio n of the ODE (72) in the t wo sp ecial sub cases (of the semisymmetrical cas e) the treatment of whic h had been omitted in Section 5, a nd as w ell in ano ther sp ecial case not belong ing to the semisymmetrical class. If g 1 + g 2 + g 3 = 0 , (118) ρ is c o nstant (namely τ -indepe ndent, ρ ( τ ) = ρ (0) , see (69 c )). Moreov er , via the restriction (2) characterizing the semisymmetrical class, we get (see also (14)) f = − 2 g , µ = 0 . (119 a ) Then (74) is replaced by u ( τ ) ex p  − 2 u 2 ( τ )  = exp  3 f ( τ − τ 0 ) ρ 2 (0)  . (11 9 b ) Let us also note that, if (2 ) were replaced b y g 1 = − g 2 = g , g 3 = 0 , (120 a ) which is a lso co nsistent with (118), then (74) with (71) would b e replaced by θ ( τ ) + s in [2 θ ( τ ) ] = 2 √ 3 g ( τ 0 − τ ) ρ 2 (0) . (120 b ) Returning to the semisymmetrical c a se characterized by v alidit y of the re striction (2) we now consider the second case whose trea tmen t had bee n omitted in Section 5 , namely f = − 8 g . ( 1 21) Note that in this ca se µ div er ges, see (14). Then (74) is r eplaced by u ( τ ) ex p  u 2 ( τ )  = [ K ( τ − τ 1 )] − 1 / 2 . (122) App endix D: relation with more standard (Newtonian) three-b o dy problems In this Appendix w e indicate the relation amo ng the three - bo dy problems treated in this pap er, c ha racterized by equations of motion of Aristotelian type (“the particle velo cities are prop or tional to a ssigned external and interparticle forces”), with analogo us many-bo dy pro blems character ized b y equations of motion of Newtonian t y pe (“the particle ac c eler ations are pro p or tional to a ssigned exter nal and in ter particle forces”). The results r eviewed in this section ar e of in ter est inasmuc h a s they relate the mo del trea ted in this pap er to other , somewhat more physical and cer tainly more classical, many-bo dy problems , including a prototypical thr ee-b o dy mo del introduce d, and shown to b e solv able by q uadratures, by Carl Jac o bi one and a half centuries ag o [21], and the one-dimensional Newtonian ma n y- bo dy problem with t wo-b o dy for ces prop ortional to the inv er se cub e o f the interparticle distance introduced and solved ov er four decades a go (ﬁrstly in the quantal context [5, 6] and then in the clas sical context [25, 2 6]), which contributed to the blo om in the inv estigatio n of integrable dynamical systems of the la st few decades (see for insta nce [2 7, 8]). T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 28 By diﬀerentiating the equations of motion (1) and using them aga in to eliminate the ﬁrst deriv atives in the r ig h t- hand sides one ge ts the fo llowing se c ond-or der equations of motion of Newtonian t y pe: ζ ′′ n = − 2 g 2 n +1 ( ζ n − ζ n +2 ) 3 − 2 g 2 n +2 ( ζ n − ζ n +1 ) 3 + g n +1 ( g n − g n +2 ) ( ζ n − ζ n +2 ) 2 ( ζ n +2 − ζ n +1 ) + g n +2 ( g n − g n +1 ) ( ζ n − ζ n +1 ) 2 ( ζ n +1 − ζ n +2 ) . (123) Likewise from the equations of mo tion (7) one ge ts ¨ z n + ω 2 z n = − 2 g 2 n +1 ( z n − z n +2 ) 3 − 2 g 2 n +2 ( z n − z n +1 ) 3 + g n +1 ( g n − g n +2 ) ( z n − z n +2 ) 2 ( z n +2 − z n +1 ) + g n +2 ( g n − g n +1 ) ( z n − z n +1 ) 2 ( z n +1 − z n +2 ) . (124) Of co urse the so lutio ns of the ﬁrst-or der equations of motio n, (1) resp ectively (7), satisfy as well the corr esp onding se c ond-or der equations of mo tion, (123) resp ectively (124), but they provide only a subset o f the so lutions o f the latter. On the other ha nd it is again true that the solutions o f the second-o rder equations of motion (123) and (124) are rela ted via the trick. In the in teg rable “ equal-particle” case , see (3), these e q uations of motion simplify and co rresp ond r espe c tiv ely to the Newtonian equations o f motion yielded by the t wo standard N -b o dy Hamilto nians H  ζ , π  = N X n =1 π 2 n 2 − N X m,n =1; m 6 = n g 2 2 ( ζ n − ζ m ) 2 , (125) resp ectively H  z , p  = N X n =1 p 2 n + ω 2 z 2 n 2 − N X m,n =1; m 6 = n g 2 2 ( z n − z m ) 2 , (126 ) with N = 3 , the complete integrability of which is by no w a classical result (even in the N -b o dy case with N > 3: s ee for instance [8]). In fact the more g e neral thr e e-b o dy Hamiltonian mo dels H  ζ , π  = 3 X n =1 " π 2 n 2 − g 2 n ( ζ n +1 − ζ n +2 ) 2 # , (12 7) resp ectively H  z , p  = 3 X n =1 " p 2 n + ω 2 z 2 n 2 − g 2 n ( z n +1 − z n +2 ) 2 # , (128) featuring three diﬀer ent co upling constan ts g n , that yield the equations of motion ζ ′′ n = − 2 g 2 n +1 ( ζ n − ζ n +2 ) 3 − 2 g 2 n +2 ( ζ n − ζ n +1 ) 3 , (129) resp ectively ¨ z n + ω 2 z n = − 2 g 2 n +1 ( z n − z n +2 ) 3 − 2 g 2 n +2 ( z n − z n +1 ) 3 , (130) T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 29 are also solv able by quadratures. F or the equations of mo tion (129) this discov ery is due to Ca rl J acobi [21]; while the solutions of the e q uations of motion (1 30) can b e easily obtained from those of the equatio ns of motio n (129) via the trick (5). F or a detailed discus sion of thes e solutions , and additional indications on key contributions to the study o f this pro blem, the in terested reader is r eferred to [13] a nd [8 ]. But we will per haps a lso rev isit this problem, b ecause we b elieve that additional study of these mo dels, (12 9) and (1 3 0), shall shed additional light on the mec ha nism resp onsible for the onset of a certain kind of deterministic chaos , as discussed above. Finally let us reca ll that our Aris totelian mo del (7), as well a s the Newtonian mo dels describ ed in this App endix, des cribing “particles” moving in the c omplex z - plane, can b e easily reformulated a s mo dels descr ibing particles moving in the re al plane, with r otation-invariant (or at least c ovariant ) r e al tw o-vector equations of motion (see for instance C ha pter 4 of [8]). [1] F. Calogero, D . Gomez-Ull ate, P . M. Sant i ni and M. Sommacal, On the transition from regular to irregular motions, explained as trav el on Riemann surfaces, J. Phys. A: Math. Gen . 38 (2005) 8873–8896 . [2] T. Bountis, In vestigating non-integrabilit y and c haos in complex time, Physic a D86 (1995) 256–267. [3] T. Boun tis, L. Drossos, and I. C. P erciv al. Nonintegrable systems with algebraic singularities in complex time. J. Phys. A (1991) 24 3217–3236. [4] T. Bount i s, H. Segur and F. Viv aldi, In tegrable Hamil tonian systems and the Painlev ´ e property , Phys. R ev. A, 25 (1982) 1257–1264. [5] F. Calogero, Solution of a three-b ody pr oblem in one dimension, J. Math. Phys 10 (1969) 2191-2196. [6] F. Calogero, Solution of the One Dimensional N -Bo dy Problem with Quadratic and/or In versely Quadratic P air P otentials, J. Math. Phys. 12 (1971) 419–4 36 ; Erratum: J.Math.Phys . 37 (1996) 3646. [7] F. Calogero, A class of in tegrable Hami ltonian systems whose solutions are (perhaps) all completely p erio dic, J. Math. Phys. 38 (1997) 5711-5719. [8] F. Calogero, Classic al many- b o dy pr oblems amenable to exact t r e atments , Lect ur e Notes i n Ph ysi cs Monograph m 66 , Springer, Berlin 2001. [9] F. Calogero, Iso chr onous systems , Oxford Universit y Press, Oxford 2008. [10] F. Calogero and J.-P . F ran¸ coise. P er i odic motions galore: how to modify nonlinear evolution equations so that they feature a lot of peri odic solutions. J. Nonline ar Math. Phys. , 9 (2002) 99–125. [11] F. Calogero, J.-P . F r an¸ coise and M. Sommacal, Pe r iodi c solutions of a many-rotato r pr oblem in the plane. II. A nalysis of v ari ous motions, J. N online ar Math. Phys. 10 (2003) 157–214. [12] F. Calogero, D. Gomez-Ull ate, P . M. Santini and M. Sommacal, T o wards a theory of c haos as tra vel in Riemann surfaces II, in preparation. [13] F. Calogero and M. Sommacal, Periodic solutions of a system of complex ODEs. I I. Hi gher peri ods, J. Nonline ar Math. Phys. 9 (2002) 1–33. [14] Y. F. Chang and G. Cor liss. Ratio-like and r ecurrence relation tests for con vergence of series. J. Inst. Math. Appl. 25 (1980) 349–359. [15] Y. F. Chang, J. M. Greene, M . T ab or, and J. W eiss. The analytic structure of dynamical systems and self-simi lar natural b oundaries. Physic a D 8 (1983) 183–207. [16] Y. F. Chang, M. T abor, and J. W eiss. Analytic structure of the H´ enon-Heiles Hamiltonian in int egrable and nonintegrable regimes. J. Math. Phys. 23 (1982) 531–538. [17] Y u. F edoro v and D. Gomez-Ullate, A class of dynamical s ystems whose solutions trav el on the Riemann surf ace of an hyperelliptic f unction, Physica D 227 (2007) 120–134. [18] A. S. F ok as and T. Bountis. Order and the ubiquitous o ccurrence of chaos. Phys. A 228 (1996) 236–244. [19] P . Grinevich and P . M . San tini, Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve µ 2 = v n − 1 , n ∈ Z : ergo dicity , i sochron y , per iodicity and fractals, Physic a D 232 (2007) 22–32. [20] E. L. Ince, Or dinary diﬀe rential e quations , Dov er, New Y ork, 1956. T owar ds a The ory of Chaos Explaine d as T r avel on Riemann Surfac es 30 [21] C. Jacobi, Problema trium corp orum mu tuis attractionibus cubis distan tiarum inv erse propor tionali bus recta l inea se mo ven tium, in Gesammelte Werke , v ol. 4 , Berl in, 1866, pp. 533-539. [22] M. D. K rusk al and P . A. Clarkson, The Painlev ´ e-Ko walewski and poly-Painlev ´ e tests for int egrabil it y , St udies Appl. Math. 86 (1992) 87–165. [23] M. D. K rusk al, A . Ramani, and B. Grammaticos. Singularit y analysis and i ts relation to complete, partial and nonintegrabilit y . In Partial ly inte gr able evolution e quations in physics (L es Houches, 1989) , v olume 310 of NA TO A dv. Sci. Inst. Ser. C Math. Phys. Sci. , pages 321–372. Kluw er Acad. Publ., Dordrech t, 1990. [24] G. Levine and M. T ab or. Int egrating the nonin tegrable: analytic structure of the Lorenz system revisited. Physic a D 3 3 (1988) 189–210. [25] C. M arc hior o, Solution of a three-b ody scatering problem in one dim ension, J. Math. Phys. 11 (1970) 2193–2196. [26] J. Moser, Three i n tegrable Hamiltonian systems connected with isosp ectral deformations, A dv. Math. 16 (1975) 197–220. [27] A. M. Perelomov , Inte g r able system of classic al me chanics and Lie algebr as , Birkh¨ auser, Basel 1990. [28] A. Ramani, B. Grammaticos, and T. Boun tis. The Painlev´ e prop ert y and singulari t y analysis of i n tegrable and nonin tegrable systems. Phys. Rep. 180 (1989) 159–245. [29] M. T abor and J. W eiss. A nalytic structure of the Lorenz system. Phys. R ev. A 2 4 ( 1981) 2157–216 7.

Towards a Theory of Chaos Explained as Travel on Riemann Surfaces

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