On the detuned 2:4 resonance
OOn the detuned 2:4 resonance
Heinz Hanßmann
Mathematisch Instituut, Universiteit UtrechtPostbus 80010, 3508 TA Utrecht, The Netherlands
Antonella Marchesiello
Faculty of Information Technology, Czech Technical University in PragueTh´akurova 9, 16000 Prague, Czech Republic
Giuseppe Pucacco
Dipartimento di Fisica, Universit`a Tor Vergata RomaVia della Ricerca Scientifica 1, 00133 Roma, Italy
May 21, 2020
Abstract
We consider families of Hamiltonian systems in two degrees of freedom with an equi-librium in 1:2 resonance. Under detuning, this “Fermi resonance” typically leads tonormal modes losing their stability through period-doubling bifurcations. For cubicpotentials this concerns the short axial orbits and in galactic dynamics the resultingstable periodic orbits are called “banana” orbits. Galactic potentials are symmetricwith respect to the co-ordinate planes whence the potential — and the normal form— both have no cubic terms. This Z × Z –symmetry turns the 1:2 resonance intoa higher order resonance and one therefore also speaks of the 2:4 resonance. In thispaper we study the 2:4 resonance in its own right, not restricted to natural Hamil-tonian systems where H = T + V would consist of kinetic and (positional) potentialenergy. The short axial orbit then turns out to be dynamically stable everywhereexcept at a simultaneous bifurcation of banana and “anti-banana” orbits, while itis now the long axial orbit that loses and regains stability through two successiveperiod-doubling bifurcations. Keywords : normal modes, period doubling bifurcation, symmetry reduction, in-variants, normal forms, perturbation analysis
MSC Codes : 37J35, 70H06, 70H33, 70K45, 70K75
Symmetries play a fundamental role in the mathematical modeling of physical systems.Either exact or approximate, they produce extra conservation laws or constrain the struc-ture of relevant equations indicating the way to solve the problem at hand [21]. A partic-ularly striking example is provided by Hamiltonian systems close to resonance around anelliptic equilibrium. The structure of the normal form is largely determined by discrete1 a r X i v : . [ n li n . C D ] M a y ymmetries affecting the degree of the lowest order resonant terms [35]. Consider in twodegrees of freedom the lowest-order genuine 1:2 resonance [11]: its prototype is the Fermiresonance and a simple mechanical example is the spring-pendulum [7]. When enforcingapproximate reflectional symmetry with respect to both degrees of freedom a higher-ordernormal form becomes necessary. Indeed, the cubic resonant terms are removed from thenormal form and the first non-vanishing resonant terms are of 6th order — squaring cubicterms yields invariance under reflections. We follow [9] and denote the resulting problemas 2:4 resonance. However, it shares several features of the lowest-order case and can beinvestigated with analogous techniques.A classical example is that of the motion of a star in an elliptical galaxy whose gravita-tional potential possesses mirror reflection with respect to each symmetry plane [36, 23].When the flattening is small, motion in the core is well approximated by a perturbedsymmetric 1:1 oscillator [38, 29]. But when the flattening is high, the dynamics can becloser to the symmetric 1:2 resonance [27, 22]. Axial orbits of arbitrary amplitude existand may suffer instability at some threshold. At such a threshold a periodic orbit in gen-eral position [31] bifurcates off from the axial orbit together with a symmetric counterpartforming a mirror-symmetric pair. This has interesting consequences for the structure ofthe system. Remark 1.1
For the banana orbits it is straightforward to “see” the two mirror-symmetricmembers of the pair. The two trajectories of the anti-banana (figure-eight) pair are insteadsimply going in opposite direction on the same orbit in configuration space.Let us consider a family of Z × Z –symmetric Hamiltonian systems in two degrees offreedom close to an elliptic equilibrium, which is equivariant with respect to the reflectionalsymmetries (cid:37) : ( x , x , y , y ) (cid:55)→ ( − x , x , − y , y ) (1a) (cid:37) : ( x , x , y , y ) (cid:55)→ ( x , − x , y , − y ) (1b)where ( x, y ) denote the canonical co-ordinates. Assuming the Hamiltonian to be ananalytic function in a neighbourhood of the equilibrium, its series expansion about theequilibrium point can be written as H ( x, y ; δ ) = ∞ (cid:88) j =0 H j ( x, y ; δ ) (2)where H j are homogeneous polynomials of degree 2( j + 1) in the co-ordinates ( x, y ); wediscuss the dependence on the parameter δ ∈ R below. Note that in force of the reflectionalsymmetries (1), odd degree terms are not present in the expansion. The quadratic part H ( x, y ; δ ) = ω x + y ) + ω x + y ) (3)of (2) describes two oscillators with frequencies ω j = ω j ( δ ) ∈ R , j = 1 , x , y )–plane and the( x , y )–plane both consist of periodic orbits and the rest of the phase space is foliatedby invariant 2–tori. Our aim is to understand what happens under addition of higherorder terms. Persistence of invariant 2–tori is addressed by kam Theory. The linearapproximation H of H has a constant frequency mapping and thus fails to satisfy theKolmogorov condition (5). To obtain an integrable approximation of H that does satisfythe Kolmogorov condition we compute a truncated normal form K with respect to H ,see [6] and references therein. If there are no resonances k ω + k ω = 0 , (cid:54) = k ∈ Z of order | k | := | k | + | k | ≤ n , then the normal form of order n depends on ( x, y ) only asa function of the invariants τ = x + y τ = x + y . (4)Such a Birkhoff normal form K = ω τ + ω τ + ω τ + ω τ τ + ω τ + . . . generically satisfies the Kolmogorov conditiondet( ω ij ) ij = ω ω − ω (cid:54) = 0 (5)and/or the iso-energetic non-degeneracy condition2 ω ω ω − ω ω − ω ω (cid:54) = 0 . (6)Therefore, our analysis of the unperturbed system essentially remains valid for (2). Thenon-degeneracy conditions (5) and (6) require the computation of the Birkhoff normalform of order n ≥
4, hence these considerations do not apply to resonances up to or-der 4. Next to the 1:1 and 1: − ± ± | k | ≤ | k | . The resulting dynamics depend thus on the lower-order resonance athand. Also for higher-order resonances a reliable approximation of the dynamics of (2)might require the normalization to be performed at least up to the order at which thefirst resonant term appears [9]. Here we consider the problem of determining the phase space structure of the perturbationof a 1:2 resonant oscillator invariant under the symmetries (1). To catch the main featuresof the orbital structure, we make our parameter δ a detuning parameter [36] by assuming ω = (cid:18)
12 + δ (cid:19) ω (7)3nd proceed as if the unperturbed harmonic part were in exact 1:2 resonance, thus in-cluding the detuning inside the perturbation. In this way we turn (3) into H = ω (cid:16) τ τ (cid:17) (8a)while the perturbation becomes+ δω τ + ∞ (cid:88) j =1 H j ( x, y ; δ ) (8b)where the dependence of H j , j ≥ δ may be arbitrary, e.g. polynomial; for definitenesswe assume that the parameter δ ∈ R only appears in the detuning with all H j = H j ( x, y )independent of δ (and discuss below in how far this captures the behaviour of general 1–parameter families).We aim at a general understanding of the bifurcation sequences of periodic orbits ingeneral position from the normal modes, parametrised by the “energy” E , the detuningparameter δ and the independent coefficients characterising the nonlinear perturbation.This problem was already studied in the case of “natural Hamiltonians” [22], i.e. in casethe potential depends only on the “spatial” variables x , and therefore H j = H j ( x ) for j ≥
1. Here we consider the more general system (8). We follow a different, geometricapproach that allows not only to reproduce the results of [22], but also to extend these and,under certain assumptions, to deduce the generic behavior of (8). The results obtainedare summarized in theorems 5.2 and 5.3. Actually the value E of the Hamiltonian H doesnot always correspond to the energy of the system now, but colloquially we shall still call E the (generalized) energy.As remarked above, in presence of symmetries the minimal truncation order necessaryto include at least one resonant term in the normal form depends not only on the order | k | of the resonance, but also on the symmetries at hand. For the reflectional symmetries (1)the minimal truncation order increases to 2 | k | , see [9, 16, 31]. Thus, in this point of view,the symmetric 1:2 resonance behaves as a higher order resonance, and as said we shallspeak of 2:4 resonance. The approach we take to study the 2:4 resonance has become rather standard, comparewith [25, 12, 20, 24, 14] and references therein. Normalizing about the periodic flow of theresonant oscillator introduces an extra continuous symmetry, cf. [6, 31], while preservingalready existing symmetries of the system. Studying the normal form dynamics in theirown right allows to reduce to one degree of freedom, cf. [10, 13]. We follow the treatmentof resonant normal modes in the 3 D H´enon–Heiles family in [18] and first consider theinsufficient 4th order normal form before turning to the 6th order normal form necessaryfor the fine structure, see also [35]. Aspects of the dynamics that are persistent underaddition of higher order normalized terms have a chance to persist also when “perturbing4ack” to the original system (of which normal forms of increasingly high order form anincreasingly close approximation).The normal form turns out to have an S × Z –symmetry, where the second factor isinherited from the second factor of the original Z × Z –symmetry and the first factor S is an improvement upon the original Z due to normalization. Reducing this symmetry bymeans of invariants allows to get a global picture of the (reduced) dynamics, see Figs. 1,3 and 4 below. The cuspidal form of the singularity corresponding to the family of shortaxial orbits explains why here bifurcations of banana and anti-banana orbits now happensimultaneously. This perspective also allows to decide at once that going to any higherorder than 6 in the normalization process does not again lead to qualitative changes, butonly to quantitative ones.We introduce the (truncated) normal form for the system (8) in section 2 and re-duce the dynamics to one degree of freedom. We do this in two steps, first reducing the S –symmetry and then the remaining Z –symmetry. Then by a geometric approach westudy the equilibria of the reduced system and describe the possible bifurcation sequencesin sections 3 and 4. In section 3 we restrict to the normal form of order 4 while theimprovements due to the normal form of order 6 are presented in section 4. The resultsso obtained are used in section 5 to deduce the dynamics of the original system. Sec-tion 6 demonstrates our results for a specific class of examples. Some final comments andconclusions follow in section 7. Let us zoom in on the neighbourhood of the equilibrium at the origin by introducing aperturbing parameter ε >
0, scaling co-ordinates as( x, y ) (cid:55)→ ( εx, εy ) (9a)and also the detuning (7) as δ (cid:55)→ ε δ , (9b)so that it can be treated as a second order term in the perturbation. Scaling furthermoretime as t (cid:55)→ ε ω t (9c)no ε remains in the unperturbed resonant oscillator (8a) while we get ω = 2 for thefrequencies in the Hamiltonian (8), thereby turning (8) into H = τ + 2 τ + 2 ε δτ + ∞ (cid:88) j =1 ε j H j . (10)5he system defined by (10) is in general not integrable, even after truncation of theconvergent series. The flow ϕ H t of the unperturbed system (8a) yields the S –action ϕ H on R ∼ = C given by ϕ H : S × C −→ C ( (cid:96), ( z , z )) (cid:55)→ (e − i (cid:96) z , e − (cid:96) z ) (11)where z j = x j + i y j , j = 1 , . The perturbed Hamiltonian (10) is in general not invariant under this action, however wecan normalize H so that the resulting normal form does have the oscillator symmetry (11).A set of generators of the Poisson algebra of ϕ H –invariant functions is given by τ = z ¯ z , τ = z ¯ z σ = Re z ¯ z , σ = Im z ¯ z τ ≥ τ ≥ R ( τ, σ ) := 2 τ τ − ( σ + σ ) = 0 . (13)See [10, 11, 16] for more details. The normalization allows us to reduce the dynamicsto one degree of freedom as the Poisson bracket on R induced by (4) and (12) has twoCasimir elements, namely R and H = τ + 2 τ . For a fixed value η ≥ H we caneliminate τ = ( η − τ ). The dynamics are constrained to the reduced phase space V η = (cid:8) ( τ , σ , σ ) ∈ R : R η ( τ , σ , σ ) = 0 , ≤ τ ≤ η (cid:9) (14)with Poisson structure { f, g } = (cid:104)∇ f × ∇ g | ∇ R η (cid:105) , where R η ( τ , σ , σ ) = ( η − τ ) τ − ( σ + σ ) . The normal form for the 2:4 resonance (10), truncated at order 6 in the original vari-ables ( x, y ), has the general structure K ( τ, σ ; δ ) = K ( τ ) + ε K ( τ ; δ ) + ε (cid:20) µ σ − σ νσ σ + K ( τ ; δ ) (cid:21) (15)with K = H = τ + 2 τ = ηK = 2 δτ + α τ + α τ + α τ τ K = ρ δτ + ρ δτ + ρ δτ τ + α τ τ + α τ τ + α τ + α τ . µ, ν , ρ j , j = 1 , , α i , i = 1 , . . . , H j in the original Hamiltonian (10). To keep our analysis as general aspossible, here and in the following we prefer to work with the normal form (15) with themost generic coefficients. Afterwards, in section 6 we give an application to an explicitclass of systems. We assume at least one of the coefficients µ and ν to be non-vanishing,otherwise the first order at which the normal form yields stabilized dynamics would behigher. Remark 2.1
The δ –dependent terms with coefficients ρ j in K are an artefact of thenormalization procedure and if we decide to normalize to higher order also the resulting K , K , . . . are going to depend on the detuning δ . In case the H j in (8b) do dependon δ , we develop these dependencies into series and adjust the passage from the H j in (8)to the H j in (10) according to the scaling (9b). While this does affect the quantitativevalues of the ρ j , once these changed values are computed there are no further adjustmentsto be made and in particular the qualitative statements in the sequel remain unchanged.The Z × Z –symmetry of (10) generated by (1) is inherited by the normal form (15). Infact, for (cid:96) = π the S –action (11) yields the reflectional symmetry (1a); correspondingly,none of the invariants in (4) and (12) changes under (1a). The remaining symmetry (1b)becomes ( τ, σ ) (cid:55)→ ( τ, − σ ) (16)whence the normal form (15) depends on σ , σ only via ( σ − σ ) and σ σ . We performa further reduction to explicitly divide out this symmetry, by introducing variables [19] u := τ v := ( σ − σ ) w := σ σ . (17)Note that, since the reduced phase space is a surface of revolution, by rotation we canalways eliminate one of the two variables v, w from the Hamiltonian (recall that we donot consider the case µ = ν = 0 here). For definiteness we assume from now on µ > ν = 0. Remark 2.2
If the system is reversible, then ν = 0 from the start, but µ might benegative and in applications it is not always helpful to actually perform a π –rotation toachieve µ >
0. Therefore we sometimes also comment on the case µ <
0. For the samereason we do not simply scale to µ = 1.The normal form (15) then becomes, after neglecting constant terms and scaling one moretime by ε , K η ( u, v, w ; δ ) = (2 δ + αη ) u + λu + ε [ µv + K η ( u ; δ )] (18)where K η ( u ; δ ) = β δu + β δηu + γ u + γ ηu + γ η u (19)7 P η isotropy subgroup dynamics period | z | = η (cid:54) = 0, z = 0 = τ Q Z first normal mode long τ = 0 = z , | z | = η (cid:54) = 0 Q { , π } × Z second normal mode shortTable 1: Isotropy subgroups of the S –action (11), H = η (cid:54) = 0, combined with the Z –action (16). Note that in case η → P η shrinks to the equilibrium at the origin,with corresponding isotropy subgroup S × Z . and λ = α + α − α , α = α − α , β = ρ + ρ − ρ , β = ρ − ρ , (20a) γ = − α α α − α , γ = α − α α , γ = α − α . (20b)To prevent that for any η ≥ δ = − αη for which K η = O ( ε ) wemake the genericity assumption λ (cid:54) = 0 on the coefficients of the 4th order terms H ( x, y )in (2)/(10). Remark 2.3
We follow a perturbative approach, introducing ε as a small perturbingparameter and looking for the bifurcation curves in the ( δ, η )–plane as power series in ε .The results obtained are reliable only for small values of ε , i.e. if the original system isnot too far from the equilibrium at the origin and if the resonance ratio (7) is not too farfrom the 2:4 resonance — the detuning being scaled as in (9b).The (twice) reduced phase space P η = (cid:8) ( u, v, w ) ∈ R : S η ( u, v, w ) = 0 , ≤ u ≤ η (cid:9) (21)has the Poisson structure { f, g } = (cid:104)∇ f × ∇ g | ∇ S η (cid:105) , where S η ( u, v, w ) = ( η − u ) u − v + w ) . Correspondingly, the equations of motion take the formdd t uvw = ∇ K η × ∇ S η on P η ⊆ R whence the singular points Q := (0 , ,
0) and Q := ( η, ,
0) are alwaysequilibria for the reduced system. The corresponding isotropy subgroups of the S × Z –action combining (11) with (16) are given in table 1.8 emark 2.4 Note that H is an integral of motion for the reduced system (15) and notfor the original system (10), which is in general not integrable, the Hamiltonian beingits only integral of motion. In [7] the value η of H is also referred to as distinguishedparameter. In section 5 we shall describe the bifurcations of the original system in termsof the detuning δ and of the (generalized) energy E .We aim at understanding the dynamics on the reduced phase space P η . In particular, welook for the critical curves in the ( δ, η )–plane corresponding to the bifurcations, togetherwith the possible bifurcation sequences (which would then depend also on the coefficientsof the normal form, actually not on all of them, as we shall see). Then, we use theinformation obtained from the normal form to deduce the dynamics of the original system.We start by investigating the equilibria of the system defined by (18). In this section we treat the 2:4 resonance as a higher order resonance. As there are no cubicresonance terms, this means that we work with an approximating Birkhoff normal formof order 4 in the original variables ( x, y ), i.e. we first look at its first order approximation,obtained by neglecting the second order term in ε of (18) and study the dynamics definedby K ηδ ( u, v, w ) = (2 δ + αη ) u + λu (22)on (21) where δ, η are parameters with η distinguished with respect to δ and λ, α arenon-vanishing constant coefficients. Note that this puts the conditions α (cid:54) = α and α (cid:54) = 2 α + α on the α i in (15), but there are no conditions on β and γ in (19); recallthat µ > α + 2 λ (cid:54) = 0, a genericity assumptionthat puts the additional constraint α (cid:54) = 4 α on the α i in (15). The reduced phase space (21) has a cuspidal singularity at Q = (0 , ,
0) and a conicalsingularity at Q = ( η, ,
0) and these are always equilibria (see Fig. 1). The intersectionsof P η with the level sets K ηδ ( h ) := (cid:8) ( u, v, w ) ∈ R : K ηδ ( u, v, w ) = h (cid:9) (23)for h = K ηδ ( Q i ) , i = 1 , Q i , whence both equilibria are stable. Remark 3.1
The origin Q reconstructs to the family of short axial orbits as predictedby Lyapunov’s Centre Theorem [31] and is singular already on V η . For the 1:2 resonancethe point ( τ , σ , σ ) = ( η, ,
0) — which corresponds to the family of long axial orbitsand gets reduced to Q — is not a singular point of V η and correspondingly Lyapunov’s9entre Theorem does not apply here. The extra Z –symmetry turns the 1:2 resonanceinto a 2:4 resonance whence also the family of long axial orbits becomes a normal mode,see again [31]. The relation between normal modes and singular points of the reducedphase space extends to n degrees of freedom and we refer to [26] for more information.The remaining (non-empty) intersections P η ∩ K ηδ ( h ) ⊆ R , h = (2 δ + αη ) u + λu , < u < η (24)yield “great circles” on the surface of revolution P η as the level sets K ηδ ( h ) consist of twovertical planes perpendicular to the u –axis (recall that we assumed λ (cid:54) = 0). From theequations of motion ˙ u = 0˙ v = 4 w ∂K ηδ ∂u ˙ w = − v ∂K ηδ ∂u we infer that these great circles are periodic orbits, except when K ηδ ( h ) is a double planewhere the circle consists of equilibria. Since ∂K ηδ ∂u = 2 δ + αη + 2 λu the corresponding double root is given by u = u := − δ + αη λ (25)and it gives a circle on the reduced phase space only if0 < u < η . (26)This restricts the parameter values to D αλ := (cid:110) ( δ, η ) : − λη − αη < δ < − αη (cid:111) if λ > D αλ := (cid:110) ( δ, η ) : − αη < δ < | λ | η − αη (cid:111) if λ < D αλ of D αλ the dynamics is indeed what is expected [30] from ahigher order resonance: the phase flow consists of a family of periodic orbits extendingbetween the two singular equilibria, which therefore must be stable (see Fig. 1). Higherorder terms in ε clearly change the shape of the intersections (24). However, for ε smallenough the dynamics qualitatively stays the same. In two degrees of freedom the singularequilibria reconstruct to the two normal modes and the periodic orbits reconstruct toa single family of invariant 2–tori satisfying the Kolmogorov and the iso-energetic non-degeneracy condition. We have recovered the description of the dynamics given in theintroduction, which indeed is valid for all higher order resonances.10 v Figure 1:
Possible intersections between the level sets (23) and the reduced phase space. Suchintersections correspond to stable singular equilibria or to periodic orbits. Here we depict onlylevel sets of (23) that do not degenerate into a double plane (for those the circle consists ofequilibra).
The dynamics become more intricate (and interesting) if the equation defining (24)has two coinciding roots, i.e. where ( δ, η ) ∈ D αλ . In this case, second order terms in thereduced Hamiltonian (18) are not negligible and they are needed to describe the phaseportrait of the system. We defer this full treatment of (18), with ε >
0, to section 4 below.Note that for λ = 0, which we excluded, the reduced phase space P η consists of equilibriaof (22) when 2 δ + αη = 0 and then all aspects of the dynamics of (18) are determined bythe higher order terms. The reduced dynamics on (21) is governed by the parameters δ and η , the latter beingdistinguished with respect to the former, while the coefficients ( λ, α ) ∈ R obtained fromthe Birkhoff normal form via (20a) determine the shape of D αλ and thus where bifurcationstake place. Indeed, the double planes pass through the singular point Q = (0 , , ∈ P η when 2 δ + αη = 0 and through Q = ( η, , ∈ P η when 2 δ + ( α + 2 λ ) η = 0. This yieldsthe bifurcation diagram in Fig. 2, with structurally stable dynamics on P η for ( δ, η ) / ∈ D αλ and a great circle of equilibria for ( δ, η ) ∈ D αλ .Note that the red boundary cannot be vertical since α (cid:54) = 0 and the requirement α + 2 λ (cid:54) = 0, i.e. α (cid:54) = 4 α , ensures that the blue boundary cannot be vertical as well. Thisis an additional genericity assumption, ensuring that next to the value η = − δα at Q (28)also the value η = − δα + 2 λ at Q (29)is finite. This requires δα ≤ δ ( α +2 λ ) ≤
0, respectively, since η cannot be negative.The boundary ∂ D αλ marks the transition from the regime where the 2:4 resonancebehaves as a higher order resonance to the regime where the inclusion of a 6th order11 η Figure 2:
Bifurcation diagram for ε = 0, depicted with parameter values α = − . λ = 0 . Q is degenerate. The green sector represents the set D αλ defined in (27) — for ( δ, η ) ∈ D αλ the dynamics has one great circle of regular equilibria nextto the singular equilibria Q and Q . The equilibrium Q is degenerate along the blue line. InFig. 5 this corresponds to the case V-III (below-middle). resonance term to dissolve the continuum of equilibria becomes crucial. A bifurcationsequence along a straight line passing through D αλ consists of the structurally stable flowdeveloping a degenerate singular equilibrium at ∂ D αλ , the resulting great circle of equilibriamoving through the reduced phase space P η to the other singular equilibrium which thenbecomes degenerate and leading back to the structurally stable flow, but now with thedirection of the periodic orbits reversed. By kam Theory, most of the invariant 2–tori reconstructed from the family of greatcircles extending between the two singular equilibria Q and Q persist the perturbationfrom (22) to the original system (10), while it is generic for resonant tori to break up andnot persist the perturbation. The great circles that consist of equilibria already break upunder the integrable perturbation from (22) to (18), subject to the genericity conditions µ > λ (cid:54) = 0 (recall that we furthermore assume α (cid:54) = 0 and α + 2 λ (cid:54) = 0). We thereforeaim at understanding the dynamics around the degenerate case h = h := − (2 δ + αη ) λ (30)when the equation in (24) has two coinciding roots (25) satisfying (26). What happenswhen we look at the normal form up to second order terms in the perturbation, i.e. at (18)with ε >
0, is that single vertical planes K ηδ ( h ), h away from h , get replaced by almostvertical surfaces that still lead to intersections with the reduced phase space P η thatare periodic orbits, while near the double vertical planes K ηδ ( h ) these level sets becomealmost parabolic cylinders, touching P η at elliptic equilibria where the rest of the levelset lies outside of P η and at hyperbolic equilibria where part of the level set lies inside of12 η ⊆ R . For energy levels between these equilibria, the parabolic cylinders intersect P η in periodic orbits circling around such an elliptic equilibrium.From the elliptic and hyperbolic equilibria the so-called banana and anti-banana orbitsare reconstructed. In astronomical systems the stable orbits are usually called bananasand the unstable ones anti-bananas. Here we consider more general systems, and preferto follow a different nomenclatura, by calling anti-bananas the figure eight orbits thatcorrespond to tangencies on the upper part of P η . We call banana orbits the orbitscorresponding to tangencies on the lower part of P η , independent of whether they arestable or not; compare with [27]. Let us start by investigating the stability of the singular equilibria. In particular, wewant to find the critical values of η (if any) that correspond to a stability/instabilitytransition of the singular equilibria. Indeed, while the mechanism how this happens ismore transparent when varying δ , the parameter η is distinguished with respect to thedetuning δ and this point of view allows to look at bifurcations when solely changing theinitial conditions. Such instability transitions produce new (regular) equilibria for thereduced system, bifurcating off from the singular equilibria. If the corresponding criticalvalues of η are not too high, this reflects in the bifurcation of periodic orbits from thenormal modes in the original system. We shall discuss this point in sections 4.4 and 5.Since we are now looking at the system near h = h , we consider the level sets K η,h δ,ε ( k ) := (cid:8) ( u, v, w ) ∈ R : K η ( u, v, w ; δ ) = h + ε k (cid:9) which give a family of third order curves when intersecting with the ( u, v )–plane, withequation v ( u ) = 1 µ (cid:20) k − λε ( u − u ) − K η ( u ; δ ) (cid:21) , (31)where u was obtained in (25) in the first order approximation. The ε in the denominatorlets the parabolic part of the curve (31) dominate over the cubic part K η .At Q = (0 , ,
0) the reduced phase space section P η ∩ { w = 0 } has a cuspidalsingularity. Suppose that (31) passes through the origin ( u, v ) = (0 ,
0) with non-vanishingfirst derivative (see Fig. 3). Let us denote the corresponding value of k by k . Recall thatwe assumed for definiteness that µ is positive and first take λ >
0, so λµ > u >
0, compare with Fig. 3 (right). Hence, values of k higher than k shift (31) upward and correspond near Q to empty intersections of theenergy levels K η,h δ,ε ( k ) with the reduced phase space P η and thus to no dynamics. Valuesof k lower than k shift (31) downward and lead to periodic orbits around Q ; in both casesthere may furthermore be periodic orbits where the second leaf of the parabolic-cylinder-like level set K η,h δ,ε ( k ) intersects P η , again compare with Fig. 3 (right). For u < k higher than k yield periodic orbits (as v (cid:48) (0) is negative) and there are no additionalintersections for k < k , compare with Fig. 3 (left). The equilibrium Q is thereforestable for v (cid:48) (0) (cid:54) = 0 and it can be unstable only if the curve (31) passes through the origin13 v uv Figure 3:
Possible configurations between the (thick black) phase space section P η ∩ { w = 0 } and a second order approximation of (31) for δ = 0 . α = − λ = 0 . µ = 0 . ε = 0 . η = 0 . η = 0 . k corresponding to the (thick) red curve wehave a stable equilibrium at the origin (left) or a stable equilibrium at the origin and a periodicorbit around it (right). For values of k slightly different (thin grey curves) we can have periodicorbits around the origin or no dynamics; in the right figure we furthermore have periodic orbitsaround a regular equilibrium. ( u, v ) = (0 ,
0) with vanishing first derivative. This happens for v (cid:48) (0) = − µ (cid:20) δ + αηε + β δη + γ η (cid:21) = 0 . (32)Since we are following a perturbative approach, we look for a solution of this equation inthe form of a power series η = η + ε η in ε . For α (cid:54) = 0 and δα ≤ η = η := − δα + 2 ε δ α ( β α − γ ) , (33)with η from (28) in the first order approximation. Remark 4.1
This answers an open question from [22] where this critical value for η wasfound with an “empirical” approach. The two families of periodic orbits, namely bananaand anti-banana orbits, bifurcate for the two-degree-of-freedom system defined by thenormal form and up to second order terms in the perturbation this happens simultane-ously, at the same critical value of η . Since Q is a cusp point this has a geometric reasonand in particular subsists through all orders of the perturbation.Note that v (cid:48)(cid:48) (0) = − µ (cid:20) λε + β δ + γ η (cid:21) , (34)therefore we can assume that v (cid:48)(cid:48) (0) does not vanish for small values of ε , thus there is nodegeneracy. In case λ < λµ < u , interchanging the effects of shifting (31) upwards and downwards.The above discussion applies mutatis mutandis , leading to the same formula (33) when α (cid:54) = 0. 14 emark 4.2 Equation (32) is of second order in η , therefore in general it admits twosolutions for η . However, only one of these two solutions is convergent for ε → ε , reads η = − αγ ε + (cid:16) δα − β γ (cid:17) . We aim at an approximation of thedynamics of the original system (10) in a neighbourhood of the origin and at low energies.Therefore here and in the computation of (37) below we disregard solutions that aredivergent for ε → η .Note that η can be related to the (generalized) energy in a similar fashion as in (50).At Q = ( η, ,
0) the reduced phase space has a conical singularity. The intersection ofthe reduced phase space P η with the ( u, v )–plane is given by C η ± = P η ∩ { w = 0 } = (cid:26) ( u, v ) ∈ R : v = ±
12 ( η − u ) u , ≤ u ≤ η (cid:27) (35)whence the slope of the two contour lines constituting (35) at ( u, v ) = ( η,
0) is ∓ η . Bythe same argument we used above, the corresponding equilibrium can be unstable only ifthe slope of the curve (31) at ( u, v ) = ( η,
0) takes values in the interval ] − η , η [. Thus,to find the critical values for η which correspond to stability/instability transitions of theequilibrium, we need to solve v (cid:48) ( η ) = ∓ η . (36)Proceeding as before, we look for solutions of the form η = η + ε η with η from (29).We arrive at the two solutions η = η ± given by η ± := − δα + 2 λ + 2 ε δ ( α + 2 λ ) [2 β ( α + 2 λ ) + β ( α + 2 λ ) − ( γ ± µ )] , (37)where γ = 6 γ + 4 γ + 2 γ . Such solutions are acceptable if α + 2 λ (cid:54) = 0 and δ ( α + 2 λ ) ≤ v (cid:48)(cid:48) ( η ) = − µ (cid:20) λε + β δ + (3 γ + γ ) η (cid:21) and λ (cid:54) = 0 we can (as in (34)) conclude that there is no degeneracy. Note that thedifference between the threshold values in (37) is η − − η + = 4 ε δ µ ( α + 2 λ ) . (38)Therefore, the equilibrium is unstable for η − < η < η + if α + 2 λ < η + < η < η − if α + 2 λ >
0. For µ <
Regular equilibria correspond to points where the level sets K η,h δ,ε ( k ) touch (i.e. are tangentto) the reduced phase space P η . The normal form (18) is independent of the variable w ,15hence the level sets K η,h δ,ε ( k ) are cylinders (consisting of lines parallel to the w –axis) onthe basis of the curve (31). However, a tangent plane to the surface of revolution P η can contain the w –axis only at points ( u, v, w ) with w = 0. Thus, K η,h δ,ε ( k ) and P η can touch each other only at points in the ( u, v )–plane; this is what we achieved whenrotating to ν = 0. The intersection of P η with the ( u, v )–plane is given by (35) whenceregular equilibria correspond to the points u ∈ ]0 , η [ in which (31) and (35) intersect withcoinciding slopes. As we can always adjust the second order part k of the energy to make(31) and (35) intersect where desired, this gives the equation2 δ + αη + 2 λu − ε (cid:20) γ ± µ ) u δβ + η (2 γ ∓ µ )] u + δβ η + γ η (cid:21) = 0for the slopes to coincide. Looking for a solution of the form u = u + ε u , we find thetwo solutions u = u ± := u + ε u ± subject to 0 ≤ u ± ≤ η , where u ± = 4 λ (2 δ + αη )[2 δβ + η (2 γ ∓ µ )] − ηλ ( δβ + γ η ) − γ ± µ )(2 δ + αη ) λ and u as in (25). Solving u ± = η for η = η + ε η we recover (37), while solving u ± = 0we recover (33).Therefore, as expected, the bifurcation of regular equilibria is related to the transitionto instability of singular equilibria. Since at Q = (0 , ,
0) there is a cusp singularity,the corresponding equilibrium is unstable only at η = η . However at this critical value, two tangency points appear/disappear simultaneously and therefore two regular equilibriabifurcate off from the origin. On P η they correspond to two points U ± = ( u ± , v ± ,
0) with v + = −
12 ( η − u + ) u and v − = + 12 ( η − u − ) u − , one lying on the lower and one on the upper contour of the reduced phase space. Thesingular equilibrium Q = ( η, , U − on the upper contour of the reduced phase space for η = η − and to the point U + on the lower contour for η = η + . We discuss the implicationsfor the original system in section 5.Let us conclude this section with the analysis of the stability of the regular equilibria.Once we know that the two curves (31) and (35) touch, to study the stability of thecorresponding equilibrium we need to know “how” they touch. Indeed, for small valueof ε , the curvature of (31) is determined by its second order approximation given by theparabola P defined by v = P ( u ) := 1 µ (cid:20) k − λε ( u − u ) (cid:21) . (39)16 v uvuv uv Figure 4:
Possible tangencies between the parabola (39) and the phase space section P η ∩{ w =0 } in (35) for increasing values of η and fixed values δ = − . µ = 0 . λ = 0 . α = 1 and ε =0 . η = 0 . Q = ( η, , η = 0 .
43. Then an unstable equilibrium appears from the singular equilibrium Q ,which becomes stable after the bifurcation. Lower left: η = 0 .
49. Both regular equilibria aregoing to disappear when reaching the stable singular equilibrium Q = (0 , ,
0) at the cuspsingularity. Lower right: η = 0 .
55. Both regular equilibria disappeared and the only equilibriaare the singular ones, both stable.
Moreover, the smaller the value of ε , the greater in absolute value the curvature of sucha parabola. Along the limit ε → C η − “from outside”, i.e. there is no intersection point other thanthe tangency point. Let k + be the corresponding level for k . We can always assume thecurvature of P to be high enough (in absolute value) so that this can happen only ifthe parabola is upside-down, i.e. concave; as µ > λ >
0. Highervalues of k + then shift the parabola upward and correspond to closed orbits around theequilibrium — which therefore is stable (see Fig. 4 upper right) — until the maximumof P reaches the upper contour.If the parabola (39) is convex, then it touches the lower contour of the phase space“from inside”, i.e. there are two further intersections on the upper contour of the phase17pace. In this case the equilibrium is unstable. This happens for λ < P η andthe energy level set K η,h δ,ε ( k + ), i.e. the surface corresponding to (31) for h = h + ε k + .Similarly, for the stability analysis of the equilibrium on the upper contour of thephase space, we find that it is stable for λ < λ >
0; for µ <
Remark 4.3
The simple geometry of the parabola allows to immediately conclude stabil-ity or instability of the equilibria and how these come into existence through centre-saddleand Hamiltonian flip bifurcations. The corresponding formulas may as well be searchedfor as double roots of the difference of the polynomials describing phase space and energylevel set [14]. For more involved expressions than the present cubic, which is well ap-proximated by a parabola, an algebraic point of view can support the present geometricapproach, relying on the resultant of two polynomials and related tools.
The dissolution of the great circle of equilibria into a stable and an unstable equilibrium,with a family of periodic orbits inside the separatrix of the latter surrounding the former,allows to complete the bifurcation diagram obtained in section 3.2. Indeed, the region D αλ — the green sector in Fig. 2 — no longer stands for structurally unstable dynamics. Theblue line in Fig. 2 splits into the two lines η − = η − ( δ ) and η + = η + ( δ ). Between theselines the dynamics is as depicted in Fig. 4 (upper right/lower left). The other boundaryline of D αλ — the red line in Fig. 2 — does not split but gets refined from (28) to (33)and now stands for the simultaneous bifurcation at Q = (0 , ,
0) where the two regularequilibria disappear into the singular equilibrium Q .The resulting possibilities are assembled in Fig. 5. The central ( λ, α )–plane allows todistinguish the six cases I-IV to V-III — when passing through one of the lines { α = 0 } , { λ = 0 } and { α + 2 λ = 0 } the bifurcation diagram in the ( δ, η )–plane changes. Thebifurcation diagrams for α + 2 λ > α + 2 λ < η –axis, together with exchanging the blue and green thresholds. Varying δ through 0 yields a passage through resonance, for reasonably small values of η ≥ δ = 0. Note that since we reduced the dynamics on P η through (17), every regularequilibrium on P η corresponds to two regular equilibria on V η . Namely, the equilibrium U − gives two equilibria A ± on V η . Such equilibria lie on the intersection between V η and theplane σ = 0 and are symmetric with respect to the plane σ = 0. These reconstruct in twodegrees of freedom to the anti-banana orbits. Similarly, the equilibrium U + correspondsto two equilibria B ± on V η ∩ { σ = 0 } , symmetric with respect to the plane σ = 0. Fromthese the banana orbits are reconstructed in two degrees of freedom. In the previous section we have treated the detuning δ as a parameter. However, thevalue η ≥ H is a distinguished parameter with respect to δ and one can18 η δη δη (cid:73) V (cid:45) (cid:73) V (cid:45) (cid:73) IV (cid:45) (cid:73) III (cid:45) (cid:73)
V II (cid:45)
V V (cid:45)
III λαδη δη δη
Figure 5:
Middle: diagram showing on the ( λ, α )–plane the regions corresponding to Cases I–V, in blue for δ < δ >
0. The black line gives the boundary α + 2 λ = 0.Clockwise around: bifurcation diagrams on the ( δ, η )–plane corresponding to the various cases(Case IV-I in the right above, Case III-V in the middle above, etc.). The bifurcation thresholds η , η − and η + are in red, blue and green, respectively.
19n fact consider the three coefficients δ , α (cid:54) = 0 and λ (cid:54) = 0 in (18) as fixed with α + 2 λ (cid:54) = 0,take µ > µ = 1, although we refrain from explicitly performing this re-parametrisation) and ignore the values of β , β , γ , γ , γ in (19) which — for sufficientlysmall values of δ and η , cf. remark 4.2 — do not change the dynamics. Varying η thenyields the bifurcation sequence. While the signs of α and α + 2 λ determine the sign of thefirst order approximation of (33) and (37), the sign of α + 2 λ also decides the bifurcationorder of U + and U − in force of (38). To fix the ideas, let us start by assuming δ <
0. Thefive possible cases I–V, which are described in the following, are in accordance with thelabeling in Fig. 5.Case I. α < < α + 2 λ . In this case the critical value η is not acceptable: the redline in Fig. 5 (lower-right) does not pass through the left quadrant. Bifurcationscan occur only when η passes through the critical values η ± , with η + < η − . Since λµ > η = η + a stable equilibrium U + appears from the singular equilibrium Q that becomesunstable. At η = η − the singular equilibrium Q turns stable again and an unstableequilibrium U − appears. The equilibrium at Q always stays stable. Increasing η beyond η − does increase the size of P η , but the configuration of equilibria remainsqualitatively that of Fig. 4 (upper-right and lower-left).Case II. 0 < α < α + 2 λ . All critical values are positive now, with η + < η − < η . Theparabola P is still concave. As in the previous case, we see first the appareanceof one stable equilibrium U + at η = η + , while Q becomes unstable. Then, anunstable equilibrium U − appears for η = η − and Q comes back to stability. Thedifference is that when η increases up to η = η both equilibria U + and U − disappearon Q . For η > η the only remaining equilibria are the singular ones, both stable.Note that the bifurcation sequence resembles the passage through resonance. Thepossible configurations on the reduced phase space section (35) are shown in Fig. 4for increasing values of η .Case III. 0 < α + 2 λ < α , i.e. λ <
0. In this case the threshold values (33) and (37) arestill all positive, however now η < η + < η − and the parabola (39) is convex, since λ <
0. Therefore we see first the appearance of both equilibria U − and U + from thesingular equilibrium at the origin. Since (39) is convex, the equilibrium U − is stableand U + is unstable now. Such equilibria disappear then on Q . The first equilibriumto disappear is the one at U + , for η = η + , while the equilbrium Q becomes unstable.At η = η − also the the equilibrium U − disappears and the equilbrium Q turnsback to stability. Also here the bifurcation sequence resembles the passage throughresonance.Case IV. α + 2 λ < < α . In this case the only acceptable threshold value is η . Thisimplies that bifurcations can occur only from the equilibrium at the origin. At η = η both equilibria U − and U + bifurcate off from the origin, the equilibrium U + on thelower arc is unstable and U − is stable. No bifurcation occurs from the equilibriumat Q , which is always stable. As in Case I, increasing η beyond η merely increasesthe size of P η , but does not change the configuration of equilibria.20ase V. α < α + 2 λ <
0. All the critical values are not acceptable. Therefore theonly equilibria are Q and Q , both stable. In Fig. 5 this case corresponds to the(empty) left quadrants of the remaining two lower bifurcation diagrams.The regions on the ( λ, α )–plane corresponding to the sequences I–V are displayed in Fig. 5,also for α + 2 λ <
0. In this case, the difference η + − η − changes its sign and the parabolareverses its concavity. As a consequence the equilibria U + and U − exchange their stabilityand exchange themselves in the bifurcation sequence. Moreover, all the inequalities on α, δ, λ must be inverted. For example Case III occurs now for α < α + 2 λ < δ > U + and U − exchanging their role in the bifurcation sequence. For δ > δ = 0 the thresholds satisfy η + = η − = η = 0 as all bifurcation lines originatefrom the origin; recall that α (cid:54) = 0 and α + 2 λ (cid:54) = 0 (next to λ (cid:54) = 0). In Cases I & IV ( α and α + 2 λ do not have the same sign) we also have for δ = 0 the configuration of equilibria U + and U − next to Q and Q as in Fig. 5 (upper-left and lower-right); otherwise ( α and α + 2 λ have the same sign) the situation is that of Case V except that the critical valuesare all zero, i.e. at η = 0 the extreme of the parabola (35) passes through Q . We have seen in the previous section that the bifurcation sequences are determined bythe detuning δ and the coefficients α, λ . The coupling constant µ may take any value,but the degenerate case µ = 0 is not included in the general approach; we could easilyscale µ = 1 and conclude µ >
0, but keep µ in the formulas to allow for fast conclusionsconcerning reversible systems with µ <
0. By the results of the previous section, theadditional coefficients β i , γ j , do not modify the qualitative picture. CA | − | − Figure 6:
The bifurcation plot in the (
C, A )–plane, see (40).
To give a more abstract view we can introduce — in analogy to what is done in [29]21 the parameters A := − δ + αη λη (40a) C := − µδ λ ( α + 2 λ ) . (40b)The first, the asymmetry parameter, measures how far is the system from the resonancemanifold. The second, the coupling parameter, is a measure of the strength of resonantcoupling. The Jacobi-determinantdet D δ,η (cid:18) CA (cid:19) = det (cid:32) − µ λ ( α +2 λ ) − λη δλη (cid:33) = Cλη of the bifurcation mapping warns us to be careful where δ and/or η vanish. To capturethe main qualitative features of the system we assume that the non-vanishing parametersare δ, α, λ, µ and put β i ≡ γ j ≡
0. The bifurcation thresholds then are η = − δα (41) η ± = − δα + 2 λ ∓ µδ ( α + 2 λ ) . (42)We see that the asymmetry parameter vanishes at the critical value (41), A ( η ) = 0 (43)and that, to first order in δ , A ( η ± ) = 1 ∓ µδ λ ( α + 2 λ ) = 1 ± C . (44)Then, we see that we can plot the straight lines (43) and (44), in the interval − ≤ C ≤ | C | > η cannot stay between η + and η − . Forsufficiently small δ this is ruled out by the assumption λ (cid:54) = 0 that also ensures that A and C are well defined.In this plot, a vertical straight line represents a given system at varying the distin-guished parameter η . Therefore, we can recapitulate the bifurcation scenario in the lightof the plot in Fig. 6. Let us recall the five cases enumerated in the previous subsection.Case I. α < < α + 2 λ , C > η is not acceptable, the red horizontal line disappears from the plot. Considering theparameter (40a), a sequence with growing η goes from top to bottom. Bifurcationsoccur when η passes first through η + (green line), then through η − (blue line).22ase II. 0 < α < α + 2 λ , C >
0. All critical values are acceptable now and the sequencewith growing η still goes from top to bottom: Since η + < η − < η , the sequence isgiven by the green-blue-red passings.Case III. 0 < α + 2 λ < α , C < η < η + < η − and the sequence with growing η goes frombottom to top: the sequence is now given by the red-green-blue passings.Case IV. α + 2 λ < < α , C >
0. Only η is acceptable, whereas η ± are both notacceptable: the only line present in the plot is the red one.Case V. α < α + 2 λ <
0. None of the thresholds is acceptable and the plot isempty, no bifurcations occur. - - x y - - x y - - - - x y - - - - x y - - - - x y - - - - x y Figure 7:
Sequence of x , y surfaces of section (above) and corresponding x , y surfaces ofsection (below) in Case III. In Fig. 7, a typical sequence of surfaces of section corresponding to Case III is shown togive an impression on how the abstract Fig. 6 translates to the concrete dynamics. Thesurfaces of section of the other cases are different, but the way how they relate to thecorresponding part of Fig. 6 is similar.
If a normalization is carried far enough to obtain only isolated equilibria (after symmetryreduction), we know the essential characteristics of the system. Including higher orders23ay shift the positions of the equilibria, but does not alter their number or stability.Therefore the isolated fixed points of (15) correspond to periodic orbits for the originalsystem. The results obtained can be trusted up to low energies, in a neighbourhood ofthe central equilibrium and not too far from the resonance at hand.To deduce the periodic orbits of the system from the equilibria of (15), we introduceaction angle variables z j = (cid:112) τ j e i ϕ j , j = 1 , . (45)The singular equilibria correspond to the normal modes of the system. Namely, Q corresponds to τ = 0 and τ = η , i.e. to the orbit along the x –axis, in the followingalso referred to as short axial orbit. Similarly, Q gives the orbit along the x –axis, alsoreferred to as long axial orbit, determined by τ = η and τ = 0. Remark 5.1
To be consistent with our previous papers, so that the conditions (46)and (47) for banana and anti-banana are the same as in for example [23], one has toexchange sine and cosine in the following.The regular equilibria correspond to periodic orbits in general position. The equilibrium U + has co-ordinates ( u, v, w ) such that 0 < u < η , v < w = 0. From (17) we seethat it must then be v = − σ and σ = 0. By expressing (12) in terms of (45) we get thecondition σ = τ √ τ cos(2 ϕ − ϕ ) = 0 , τ , τ ∈ ]0 , η [ . This implies 2 ϕ − ϕ ∈ (cid:26) π , π (cid:27) . (46)Similarly, we recognize that the equilibrium U − corresponds to the condition σ = τ √ τ sin(2 ϕ − ϕ ) = 0 , τ , τ ∈ ]0 , η [that gives 2 ϕ − ϕ ∈ { , π } . (47)Orbits satisfying (46) and (47) are called, because of their shape in the ( x , x )–plane,banana orbits and figure-eight or anti-banana orbits, respectively [27]. We found in sec-tion 4.1 the critical values (33) and (37) that determine the bifurcations of the reducedsystem. However, η is not a constant for the original system; nevertheless we can use(33) and (37) to find threshold values for the bifurcations in terms of the (generalized)energy E . On the long axial orbit ( τ = η , τ = 0), the normal form (15) reads as K = η + ε (2 δ + α η ) η + ε ( ρ δ + α η ) η . (48)Here and in the following, since we refer to the original system, we express the formulasin terms of the coefficients of the original normal form (15). By the scaling of time (9c)we have ω K + O ( ε ) = H = E (49)24nd combining (48) and (49) we can express the (generalized) energy in terms of η as E = ω (cid:2) η + ε (2 δ + α η ) η + ε ( ρ δ + α η ) η (cid:3) + O ( ε ) . (50)Substituting (37) into (50), we find the critical energy threshold values that correspondto the bifurcations off from the long axial orbit, given to second order in δ by E ± := − ω δ α − α + 4 ω δ (4 α − α ) [(4 α − α )(2 β + β − α + α ) − γ ± µ )] (51)for banana (upper signs) and anti-banana (lower signs) orbits, respectively. Recall β = ρ + ρ − ρ and β = ρ − ρ from (20a) and from (37) that γ = 6 γ + 4 γ + 2 γ =6 α − α . We inverted the detuning scaling (9b), so that (51) and (52) are expressed interms of the original detuning parameter. Similarly, for the bifurcation off from the shortaxial orbit ( τ = 0, τ = η ) we use (33) and find to second order in δ that E = − ω δα − α + 2 ω δ ( α − α ) [( α − α )(2 β + α ) − γ ] (52)where β = ρ − ρ and γ = α − α . However, above a certain threshold oneshould not expect that the formal series developed by the normalization procedure staysclose for a very long time to the solutions of the original problem. Since we pushed thenormalization up to including terms of 6th order in the phase space variables ( x, y ), wecan trust such quantitative predictions on the bifurcation and stability of the periodicorbits up to the second order in the detuning parameter (since, we recall, this is assumedto be a second order term). We can summarize these results as follows. Theorem 5.2
Let us consider the dynamical systems defined by H , cf. (2) and its normalform (15) with respect to the oscillator symmetry (11) . Assume the co-ordinate system tobe rotated so that µ > and ν = 0 in (15) . In a neighbourhood of the central equilibriumand for sufficiently small values of the detuning parameter δ ,i) at the stability/instability transition of a normal mode, periodic orbits in generalposition bifurcate. In particular, at each transition of the long axial orbit, a pair ofperiodic orbits bifurcate (a pair of banana or a pair of anti-banana orbits). At theinstability of the short axial orbit, two pairs of periodic orbits (a pair of banana anda pair of anti-banana orbits) bifurcate concurrently;ii) up to second order in the detuning, the instability/stability transition of the normalmodes occur at the critical energies (51) and (52) for the long and short normalmode, respectively. The coefficients α and λ determine the possible bifurcation sequences according to theprevious section. Recalling that 2( α + 2 λ ) = 4 α − α and 2 α = α − α , the analysis thatresulted into Cases I–V can be rewritten in terms of the periodic orbits of the originalsystem. 25 heorem 5.3 Under the conditions of theorem 5.2 the possible bifurcation sequences aredetermined by the coefficients α , α , α in the normal form (15) . For α (cid:54) = α , α (cid:54) =2 α + α , α (cid:54) = 4 α and δ < we have the following cases.Case I. α − α < < α − α . The short axial orbit is always stable. The long axialorbit changes its stability twice. At first it suffers a transition to instability at thecritical energy E = E + and a pair of stable banana orbits appears. At E = E − apair of anti-banana orbits appears, while the long axial orbit comes back to stability.Case II. < α − α < α − α . While the (generalized) energy passes through thecritical values E = E + and subsequently E = E − , the bifurcation sequence followsthe previous case. However a further bifurcation occurs at E = E , when both pairsof periodic orbits in general position disappear on the short axial orbit.Case III. < α − α < α − α . At E = E a pair of banana and a pair of anti-bananaorbits bifurcate off from the short axial orbit. The banana orbits are unstable andthe anti-banana orbits are stable. At E = E + banana orbits disappear on the longaxial orbits, which becomes unstable. At E = E − anti-banana orbits disappear aswell, and the long axial orbit turns back to stability.Case IV. α − α < < α − α . At E = E a bifurcation occurs from the short axialorbit, and the two pairs of periodic orbits in general position appear. Banana orbitsare unstable and anti-banana orbits are stable. The long axial orbit is always stable.Case V: α < α < α . The only periodic orbits are the normal modes, both stable.The undetuned system for δ = 0 behaves as in Cases I or IV (after the bifurcations) if α − α and α − α have opposite signs and otherwise as in Case V. Assuming the co-ordinate system to be such that ν vanishes in the normal form (15) isnot needed for qualitative predictions, but only for quantitative ones. And even here thenecessary rotation can simply be turned back. In fact, the presence of ν would not changethe possible bifurcation scenario of the system, but would affect the value of the energythresholds (51) and (52) — replacing µ by (cid:112) µ + ν . Remark 5.4
Let us note one more time that for definiteness we assumed µ positive and δ negative. Taking δ > µ (4 α − α ) − , a change in the sign of µ would affect only the bifurcation order of bananaand anti-banana orbits, that consequently would also exchange their stability properties,as would a change in the sign of 4 α − α . To demonstrate our results with an example, let us consider the family of potentials V ( x , x ; q, p ) = 1 p (cid:18) x + x q (cid:19) p/ , < p < < q ≤ . (53)26his gravitational potential is generated by a simple but realistic matter distribution [32,28, 34, 33, 3]. Its astrophysical relevance [4, 5] is based on the ability to describe ina simple way the gross features of elliptical galaxies embedded in a dark matter halo.Here the lower limit p → p → /q and we slightly extend its range from the range < q ≤ q = that the 2:4 resonance occurs. Lower positive values of q can in principle be consideredbut correspond to an unphysical density distribution. The truncated series expansion (2)is “prepared” for normalization by setting q = ω ω = 12 + δ , (54)the canonical variables and time are rescaled according to (9) and we expanded in seriesof the detuning according to (54). The coefficients of the normal form (15) read as α = 32 B , α = 6 B , α = 4 B , ρ = 3 B , ρ = − B , ρ = 0 (55a) α = − B + 9 B , α = −
23 (46 B − B ) , (55b) α = 17 B − B , α = − B − B ) , (55c) µ = 3(2 B − B ) , ν = 0 , (55d)where B = p −
28 and B = ( p − p − , (56)compare with [23]. As the potential is scalar, the ensuing system is reversible with respectto ( x , x ) (cid:55)→ ( x , − x ) (57)which through reduction turns into( u, v, w ) (cid:55)→ ( u, v, − w )and explains why ν = 0. By substituting (55) into (20) we find λ = − α = p − < µ = 132 ( p − < . (58)Note that the π –rotation ( u, v, w ) (cid:55)→ ( u, − v, − w ) still allows to achieve µ > α j , j = 1 , , α − α < < α − α and, concentrating on q > , the detuning δ is positive, the bifurcation sequence follows Case I of theorem 5.3(in which remember to reverse all the inequalities, according to remark 5.4). As µ is27egative, bifurcations occur always from the long normal mode, with bananas appearingat lower energies than anti-bananas. The critical values of the energy that determine thebifurcations can be found by substituing (55d) and (56) into (51) and, expressed in termsof the parameters of (53), in agreement with [23] read as E + = 162 − p (cid:18) q − (cid:19) + 8(41 p − p − (cid:18) q − (cid:19) (59a) E − = 162 − p (cid:18) q − (cid:19) + 8(53 p + 14)3( p − (cid:18) q − (cid:19) (59b)for the bifurcation of banana and anti-banana orbits, respectively. Numerical values ofthe thresholds when applied e.g. to the logarithmic potential (taking p = 0), are in goodagreement with the bifurcation values obtained from numerical computations [27]. Remark 6.1
When modeling the dynamics in a rotating galaxy using the Hamiltonianfunction H ( x, y ) = y + y − Ω( x y − x y ) + V ( x , x ; q, p ) , (60)we generalize (53) which is the limit of (60) as Ω →
0. Due to the rotation of the galaxy,the Hamiltonian (60) does not respect the symmetries (1). However, after diagonalizationof the quadratic part, its series expansion still has the form (2). With the assumptionthat the angular velocity Ω is a small parameter and < q ≤ ν .Modulo a rotation to eliminate ν , theorems 5.2 and 5.3 can be applied. The resultingfamilies of periodic orbits would however correspond to more “fancy” orbits for the originalsystem (60), once the diagonalizing transformation is inverted. We leave a deeper analysisof this problem to future work.Several results of the theory developed above can be extended to a 3–dimensional modelof the form H ( x, y ) = y + y + y V ( x , x , x ) , (61)in the cases in which the mirror symmetries (1) are extended to the third axis whencomposing with the transformation law( x , y ) (cid:55)→ ( − x , − y ) . (62)Each symmetry plane of the potential generates an invariant subset where the dynamicsessentially reduce to those investigated above. By introducing a further detuning pa-rameter associated to the second frequency ratio, bifurcation and stability of periodic28rbit families on the symmetry planes can be deduced. The validity of this approachis supported by analogous results obtained with the 1:1:1 resonance [37, 12, 15]. Fora deeper understanding, 3–dimensional normal forms of the symmetric 1:1:2, 1:2:2 and1:2:4 resonances are necessary [1, 2, 31], which usually provide the properties of periodicorbits in general position. Note that a normal form of the 1:2:2 resonance is always in-tegrable, while already the cubic normal forms of the 1:1:2 and 1:2:4 resonances are notintegrable [8]. However, the discrete symmetries not only make the cubic terms vanish butmay furthermore enforce some of the non-trivial normal forms to be integrable, see [17]. We considered families of Hamiltonian systems in two degrees of freedom with an equilib-rium in 2:4 resonance, a 1:2 resonance with additional discrete symmetry. Under detuning,this typically leads to normal modes losing their stability through period-doubling bifur-cations. This now concerns the long axial orbit, losing and regaining stability through twoperiod-doubling bifurcations. In galactic dynamics one speaks of banana and anti-bananaorbits. The short axial orbit turns out to be dynamically stable everywhere except at asimultaneous bifurcation of banana and anti-banana orbits.We excluded the case µ = 0 from our considerations since it would require furthernormalization. Indeed, for µ = 0 the normal form (15) resembles (22) and leads toa similar degeneracy, which to break requires higher order terms that do depend on v (or on w ). One may speculate that for such a k :2 k resonance, k ≥ Q “turns into” a cusp and the two successive period-doubling bifurcations ofthe long periodic orbit occur simultaneously, as it happens to the two successive period-doubling bifurcations of the short periodic orbit when the 1:2 resonance becomes the2:4 resonance. Acknowledgements
We thank the referees for the suggested improvements of the text. A.M. was supported bythe Grant Agency of the Czech Republic, project 17-11805S. G.P. acknowledges GNFM-INdAM and INFN for partial support.
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