The Lyapunov exponents and the neighbourhood of periodic orbits
MMNRAS , 1–5 (2015) Preprint 30 April 2020 Compiled using MNRAS L A TEX style file v3.0
The Lyapunov exponents and the neighbourhood ofperiodic orbits
D. D. Carpintero, , (cid:63) and J. C. Muzzio , Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina Instituto de Astrof´ısica de La Plata – UNLP-Conicet, Paseo del Bosque s/n, 1900 La Plata, Argentina
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We show that the Lyapunov exponents of a periodic orbit can be easily obtained fromthe eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponentsof simply stable periodic orbits are all zero, simply unstable periodic orbits have onlyone positive Lyapunov exponent, doubly unstable periodic orbits have two differentpositive Lyapunov exponents and the two positive Lyapunov exponents of complexunstable periodic orbits are equal. We present a numerical example for periodic or-bits in a realistic galactic potential. Moreover, the center manifold theorem allowedus to show that stable, simply unstable and doubly unstable periodic orbits are themothers of families of, respectively, regular, partially and fully chaotic orbits in theirneighbourhood.
Key words: chaos – instabilities – galaxies: kinematics and dynamics
In several previous articles (Muzzio, Carpintero & Wachlin2005; Muzzio 2006; Muzzio, Navone & Zorzi 2009; Zorzi &Muzzio 2012; Carpintero, Muzzio & Navone 2014; Carpin-tero & Muzzio 2016) we have investigated the role that chaosplays in the dynamics of stellar systems. Since these systemscan be described by autonomous Hamiltonians, their orbitshave always two Lyapunov exponents equal to zero, and theremaining four are always two pairs of opposite real num-bers (e.g., Benettin et al. 1976). This means that there maybe zero, one or two positive Lyapunov exponents. No pos-itive Lyapunov exponents means that there is no directionin phase space along which two initially infinitesimally sepa-rated orbits diverge exponentially, that is, the original orbitis regular. Otherwise, the orbit is chaotic. But chaotic or-bits are evidently not all the same: there are those with onlyone positive Lyapunov exponent, called partially chaotic or-bits, and those with two positive Lyapunov exponents, calledfully chaotic orbits. One of the main results from our above-mentioned works was that fully chaotic orbits are a disjointfamily from the partially chaotic orbits, even in their spatialdistribution inside the system.Contopoulos, Galgani & Giorgilli (1978) and Pettini &Vulpiani (1984) had also reported the finding of partiallychaotic orbits in other autonomous Hamiltonian systems,but their existence was denied by Froeschl´e (1970, 1971) (cid:63)
E-mail: [email protected] (DDC) and by Lichtenberg & Lieberman (1992). Nevertheless, morerecently, Muzzio (2017, 2018) proved their existence over, atleast, time intervals of 50 million Hubble times.On the other hand, studies of the stability of periodicorbits in the three-body problem (Hadjidemetriou 1975) andin triaxial potentials (Magnenat 1982; Contopoulos & Mag-nenat 1985; Contopoulos 2002) have yielded a wealth ofphenomena of interest. The different classes of instability ofthese orbits are determined according to the eigenvalues ofthe monodromy matrix of the periodic orbit. In particular,Contopoulos & Magnenat (1985) have classified the orbitsinto four categories: stable, unstable, doubly unstable andcomplex unstable. The question naturally arises of whetherthese orbits are somehow related to the regular, partiallyand fully chaotic orbits. Therefore, we decided to investigatethe relationship between the eigenvalues of the monodromymatrix of periodic orbits and the Lyapunov exponents ofthose orbits and we found that, in fact, the latter can becomputed from the former. Besides, the stability of the pe-riodic orbits, influences decisively the phase space in theirneighbourhood and it turns out that, just as stable periodicorbits are surrounded by regular orbits, simply unstable pe-riodic orbits are surrounded by partially chaotic orbits anddoubly and complex unstable periodic orbits are surroundedby fully chaotic orbits. The present paper presents our re-sults. Section 2 gives our analytical proof, section 3 presentsa numerical example and our conclusions are described inSection 4. c (cid:13) a r X i v : . [ phy s i c s . pop - ph ] A p r D. D. Carpintero and J. C. Muzzio
In a 3D potential, a regular orbit is defined as an orbit whichobeys al least three isolating integrals of motion (e.g., Binney& Tremaine 2008); otherwise, it is irregular. On the otherhand, a chaotic orbit is defined as an orbit which has sen-sitivity to the initial conditions, that is, if its initial phasespace position w ≡ ( x , v ) is infinitesimally perturbed,then the new orbit (hereafter called perturbed orbit) di-verges exponentially from the original one. Though thereis no proof that every irregular orbit is chaotic, we will stickto the widespread custom of considering both sets as thesame.The standard gauge to measure the rate of divergencebetween an orbit and its perturbed sister is the set of Lya-punov exponents, sometimes called Lyapunov characteristicnumbers or Lyapunov characteristic exponents. If we choosesix independent directions of the phase space e i , i = 1 , . . . , j by theamount δw j (0), then the Lyapunov exponents are definedby (e.g., Lichtenberg & Lieberman 1992) λ i = lim t →∞ t ln | δw i ( t ) || δw i (0) | , i = 1 , . . . , , (1)where δw i ( t ) is the component of the deviation at time t of the initial i th component of the deviation δw i (0), andthe norm | · | is any norm of the phase space, normally theEuclidean one.Numerically, the deviation δ w ( t ) can be computedstarting from the equations of motion ˙w = F ( w ) , (2)where w ( t ) is the phase point of the orbit, ˙w ( t ) its velocity,and F are the functions that define the dynamical system.Developing Eqs. (2) in a Taylor series around the unper-turbed orbit and retaining only the first order, we obtainthe so called variational equations:dd t ( δ w ) = ∂ F ∂ w (cid:12)(cid:12)(cid:12)(cid:12) w · δ w . (3)The set of variational equations (3) for the six com-ponents of the deviation is, then, a system of linear, ho-mogeneous, ordinary differential equations. For this kind ofsystems, a fundamental matrix S ( t ) is defined as a matrixwhose columns are linearly independent solutions of the sys-tem (e.g., Roxin & Spinadel 1976). In our case, S ( t ) = ( δ w , . . . , δ w ) , (4)where each δ w i is an independent solution of Eq. (3) andrepresents a column, is the fundamental matrix of the vari-ational equations. It is then clear that ˙S ( t ) = ∂ F ∂ w (cid:12)(cid:12)(cid:12)(cid:12) w · S ( t ) . (5)Since the δ w i are linearly independent solutions, then S · S T = diag( | δ w | , . . . , | δ w | ) = S , (6)where ( · ) T indicates transposition. If we choose S ( t ) suchthat S (0) = , i.e., the identity matrix (in which case S ( t ) is called main fundamental matrix), Eq. (1) shows that the setof Lyapunov exponents can be expressed as the eigenvaluesof a (diagonal) matrix L where L = lim t →∞ t ln( S ) = lim t →∞ t ln | S | , (7)where | S | = diag( | δ w | , . . . , | δ w | ) (e.g., Benettin et al.1980).Now, let w ( t ) = G ( t, w ) (8)be the solution of Eq. (2) with initial condition w (0) = w ,and let the matrix M be defined by M ( t ) = ∂ G ( t, w ) ∂ w = ∂ w ( t ) ∂ w , (9)i.e. M ( t ) is the matrix that evolves the initial perturbationuntil time t : δ w ( t ) = M ( t ) · δ w . (10)Now, by applying the chain rule, we have ˙M = ∂ ˙w ∂ w = ∂ ˙w ∂ w · ∂ w ∂ w = ∂ F ∂ w · M , (11)i.e., M turns out to be the fundamental matrix S of thevariational equations (cf. Eq. (5)). We now specialise in periodic motion. Let the solution ofEq. (2) represent a periodic orbit of period T . The stabilityof such an orbit can be established studying the behavior ofa second orbit obtained by perturbing the initial conditions,i.e. by integrating the variational equations. Let M be thefundamental matrix of this system; it satisfies (cf. Eq. (11)) ˙M ( t ) = ∂ ˙w ∂ w (cid:12)(cid:12)(cid:12)(cid:12) w · M ( t ) . (12)According to the Floquet theorem (Floquet 1883), thefundamental matrix M ( t ) in this case is also periodic withperiod T . This property, along with Eq. (10) and the as-sumption that M (0) = allow us to write δ w ( T ) = M ( T ) · δ w (0) . (13)The main fundamental matrix evaluated at t = T , M ( T ),is called the monodromy matrix of the periodic orbit (e.g.Contopoulos 2002).Since the motion is periodic with period T , then δ w (2 T ) = M (2 T ) · δ w (0)= M ( T ) · [ M ( T ) · δ w (0)]= [ M ( T )] · δ w (0) , (14)so we have, for n ∈ N , M ( nT ) = [ M ( T )] n . (15)Then, the Lyapunov exponents can be easily computed asthe natural logarithms of the eigenvalues of the monodromymatrix divided by T (cf. Eq.(7)): L = lim t →∞ t ln | M ( t ) | = lim n →∞ nT ln | M ( nT ) | = 1 T ln | M ( T ) | , (16) MNRAS , 1–5 (2015) yapunov exponents of periodic orbits where in the last line we have used Eq. (15). If we let (cid:96) i , i = 1 , . . . , M ( T ), then we have λ i = 1 T ln | (cid:96) i | . (17)As usual, the (cid:96) i ’s are obtained as the six roots of thecharacteristic polynomial of M ( T ), a (cid:96) + a (cid:96) + a (cid:96) + a (cid:96) + a (cid:96) + a (cid:96) + a = 0 , (18)where the a i are real numbers. The two null Lyapunov ex-ponents imply that two of the eigenvalues are always unity.After dividing by ( (cid:96) − , the remaining polynomial we writeas c (cid:96) + c (cid:96) + c (cid:96) + c (cid:96) + c = 0 . (19)We also know that the four remaining Lyapunov exponentscome in pairs of opposite numbers, so the correspondingeigenvalues are pairs of reciprocal numbers. Let { (cid:96) , (cid:96) = (cid:96) − , (cid:96) , (cid:96) = (cid:96) − } be those eigenvalues. Then, the polynomial(19) can be written (cid:96) + α(cid:96) + β(cid:96) + α(cid:96) + 1 = 0 , (20)where α = − ( (cid:96) + (cid:96) + (cid:96) + (cid:96) ) ,β = (cid:96) (cid:96) + (cid:96) (cid:96) + (cid:96) (cid:96) + (cid:96) (cid:96) + 2 . (21)Following the standard notation initiated by Hadjidemetriou(1975), we let b = − ( (cid:96) + (cid:96) ) ,b = − ( (cid:96) + (cid:96) ) , (22)with which α = b + b ,β = b b + 2 , (23)and, therefore, b = 12 (cid:16) α + √ ∆ (cid:17) ,b = 12 (cid:16) α − √ ∆ (cid:17) , (24)where∆ = α − β − . (25)With this notation, the four roots (eigenvalues) can be writ-ten as (cid:96) , = 12 (cid:18) − b ± (cid:113) b − (cid:19) ,(cid:96) , = 12 (cid:18) − b ± (cid:113) b − (cid:19) . (26) Although the Lyapunov exponents of a periodic motion canbe effortlessly computed from the eigenvalues of the mon-odromy matrix, the stability of its orbits is usually studiedby considering ∆, b and b (e.g. Hadjidemetriou 1975; Mag-nenat 1982; Contopoulos & Magnenat 1985; Patsis & Zachi-las 1990, 1994; Contopoulos 2002), which are combinationsof those eigenvalues:∆ = ( − (cid:96) − (cid:96) + (cid:96) + (cid:96) ) (27)and b , b given by Eq. (22). Using this three indicators, aperiodic orbit is found to be (e.g., Contopoulos 2002): (i) Stable , if ∆ > | b | <
2, and | b | <
2. In this case,all the eigenvalues are complex numbers lying on the unitcircle, and, besides their reciprocal property (cid:96) (cid:96) = (cid:96) (cid:96) = 1,the pairs also obey (cid:96) = (cid:96) ∗ and (cid:96) = (cid:96) ∗ , where the asteriskmeans complex conjugation.(ii) Unstable , if ∆ > | b | <
2, and | b | > > | b | >
2, and | b | <
2. In this case, a pair of reciprocal rootsare complex conjugate lying on the unit circle, and the othertwo roots are real.(iii)
Doubly Unstable , if ∆ > | b | >
2, and | b | >
2. Allfour roots are real.(iv)
Complex Unstable , if ∆ <
0. In this case, besides (cid:96) (cid:96) = (cid:96) (cid:96) = 1, we have (cid:96) = (cid:96) ∗ and (cid:96) = (cid:96) ∗ , that is, thereciprocal and conjugate pairs are different.Now, we want to write these four types of stability interms of the Lyapunov exponents. According to Eq. (17),any pair of reciprocal eigenvalues (cid:96) i = (cid:96) − j will yield λ i = 1 T ln | (cid:96) i | = 1 T ln | (cid:96) − j | = − T ln | (cid:96) j | = − λ j , (28)and any pair of conjugate eigenvalues (cid:96) i = (cid:96) ∗ j will yield λ i = 1 T ln | (cid:96) i | = 1 T ln | (cid:96) ∗ j | = 1 T ln | (cid:96) j | = λ j . (29)The different stability cases then yield:(i) Stable orbits. Both pairs of roots are simultaneouslyreciprocal and conjugate. For the first pair { (cid:96) , (cid:96) } we have λ = − λ and λ = λ , and the same for { (cid:96) , (cid:96) } . Therefore, λ = λ = λ = λ = 0 , (30)that is, stable periodic orbits have all their Lyapunov expo-nents equal to zero.(ii) Unstable orbits. Two of the roots are reciprocal andconjugate, so the previous analysis apply. The other two areonly reciprocal. Therefore, we have λ = λ = 0 ,λ = − λ (31)if | b | <
2, or with the pairs interchanged if | b | <
2. Thus,unstable periodic orbits have only two non-zero and oppositeLyapunov exponents.(iii) Doubly Unstable orbits. Now both pairs are only re-ciprocal, so we have λ = − λ ,λ = − λ . (32)Therefore, there are four non-zero Lyapunov exponents thatare opposite in pairs for doubly unstable orbits.(iv) Complex Unstable orbits. In this case we have λ = − λ , λ = − λ , λ = λ , λ = λ , and therefore λ = − λ = λ = − λ , (33)that is, a complex unstable orbit has two equal pairs of op-posite non-zero Lyapunov exponents. Thus far, we have only dealt with the periodic orbits them-selves, but the theorem of existence of center manifolds (e.g.Guckenheimer & Holmes 2013; Berglund 2001) allows us to
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D. D. Carpintero and J. C. Muzzio extend our results to the neighbourhood of those orbits, andwill lead us to an important conclusion. Let us consider the4D Poincar´e map around the fixed point of a simply unsta-ble periodic orbit which has one positive, one negative andtwo zero eigenvalues. Thus, according to the theorem, thereexist in the neighbourhood of the fixed point a 1D unstablelocal invariant manifold, a 1D stable local invariant man-ifold and a 2D centre invariant manifold . Therefore, theLyapunov exponents of the orbits in that neighbourhoodwill be one negative, one positive and four zero (two due tothe centre invariant manifold and two due to the conserva-tion of energy), i.e., they are partially chaotic orbits. Thesame reasoning can be applied to the stable and to the dou-bly unstable orbits. In brief, stable and simply and unstableperiodic orbits are the mothers of families of, respectively,regular and partially chaotic orbits, while both doubly andcomplex unstable periodic orbits are the mothers of familiesof fully chaotic orbits. Patsis & Zachilas (1990) investigated the stability of the pe-riodic orbits along the axis of rotation of a model galaxyusing the monodromy matrix. They modeled the potentialwith a disc of the Miyamoto & Nagai (1975) type and atriaxial logarithmic halo. We used their potential togetherwith the
Liamag routine, kindly provided by D. Pfenniger(see Udry & Pfenniger 1988) to obtain the Lyapunov expo-nents. We selected initial conditions for orbits along the axisof rotation with different energies and values of the angularvelocity; the integration time was 10 time units. Two pos-itive Lyapunov exponents were considered equal when theydiffered by less than 1 . × − .Fig. 1 presents our results and it can be compared withFigure 2 of Patsis & Zachilas (1990). The stable and simplyunstable regions of their diagram agree very well with theregions of our Fig. 1 that correspond, respectively, to our re-gions with all null and with just one positive Lyapunov expo-nents. The comparison of the doubly unstable and complexunstable regions of Patsis and Zachilas with our regions oc-cupied by orbits with two positive Lyapunov exponents and,respectively, λ (cid:54) = λ and λ = λ shows, however, somesmall disagreements. For a rotationless galaxy our resultsgive orbits with λ = λ > − . − .
220 as well as for energies larger than − . λ (cid:54) = λ in theupper right region of our figure do not extend to energieslarger than − . λ and λ are very small in those regions and close to theprecision of our computations. The differences between thetwo positive Lyapunov exponents of the orbits in the upper-most right lane, for example, are about 1 . × − , i.e. verynear our limiting value of 1 . × − . The method of the Figures 1.1.5 and 1.1.6 of Wiggins (1990) provide nice 3D ex-amples of orbits for the case of stable and unstable manifolds.
Figure 1.
Types of orbits in the angular velocity vs. energy di-agram. Orbits with null Lyapunov exponents are shown as dots(black in the electronic version) and orbits with only one positiveLyapunov exponent as filled squares (red in the electronic ver-sion). The blank space corresponds to orbits with two equal non-zero Lyapunov exponents and plus signs (green in the electronicversion) represent orbits with two non-zero Lyapunov exponentsthat are not equal. monodromy matrix seems, therefore, to be better than Lya-punov exponents to distinguish doubly unstable from com-plex unstable periodic orbits, but that is not a problem forus because we did not intend to replace the method of themonodromy matrix by the use of Lyapunov exponents.
We have proven that stable periodic orbits have null Lya-punov exponents, simply unstable periodic orbits have onlyone positive Lyapunov exponent, doubly unstable peri-odic orbits have two different positive Lyapunov exponentsand complex unstable periodic orbits have two equal posi-tive Lyapunov exponents. A corollary of our result is thatcomplex instability does not exist in systems with two-dimensional (2D) configuration spaces, an assertion thatContopoulos (2002, p. 287) gives without proof, becausecomplex instability demands an exponential expansion intwo dimensions but only one is available in 2D autonomoussystems (the other one has a zero Lyapunov exponent).The most important result of our study is that the sta-bility of the periodic orbits (revealed by their Lyapunov ex-
MNRAS , 1–5 (2015) yapunov exponents of periodic orbits ponents) drastically affects the phase space in their neigh-borhood and, as a result, stable, simply unstable and bothdoubly and complex unstable periodic orbits should be sur-rounded by families of, respectively, regular, partially andfully chaotic orbits. Thus, our result gives further supportto the existence of partially chaotic orbits.We have investigated the presence of chaos in manygalactic models in the past and our experience is that λ isusually much larger than λ when both are not zero. Thus,it was surprising to find that almost 98 per cent of the or-bits with two non-zero Lyapunov exponents in Figure 1 have λ = λ . The most likely explanation for this oddity is thatall the orbits in that sample are periodic, while in our mod-els it would have been almost impossible to find a periodicorbit by chance. ACKNOWLEDGEMENTS
The comments of an anonymous referee were very useful toimprove the original version of this paper and are gratefullyacknowledged. We are very grateful to D. Pfenniger for theuse of his code. We acknowledge support from grants fromthe Universidad Nacional de La Plata, Proyecto 11/G153,and from the CONICET, PIP 0426.
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