# Orbital dynamics of three-dimensional bars: II. Investigation of the parameter space

**ABSTRACT** We investigate the orbital structure in a class of 3D models of barred galaxies. We consider different values of the pattern speed, of the strength of the bar and of the parameters of the central bulge of the galactic model. The morphology of the stable orbits in the bar region is associated with the degree of folding of the x1-characteristic. This folding is larger for lower values of the pattern speed. The elongation of rectangular-like orbits belonging to x1 and to x1-originated families depends mainly on the pattern speed. The detailed investigation of the trees of bifurcating families in the various models shows that major building blocks of 3D bars can be supplied by families initially introduced as unstable in the system, but becoming stable at another energy interval. In some models without radial and vertical 2:1 resonances we find, except for the x1 and x1-originated families, also families related to the z-axis orbits, which support the bar. Bifurcations of the x2 family can build a secondary 3D bar along the minor axis of the main bar. This is favoured in the slow rotating bar case. Comment: 11 pages, 20 figures, 1 table, to appear in MNRAS

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**ABSTRACT:**Stellar bars are the most common non-axisymmetric structures in galaxies and their impact on the evolution of disc galaxies at all cosmological times can be significant. Classical theory predicts that stellar discs are stabilized against bar formation if embedded in massive spheroidal dark matter halos. However, dark matter halos have been shown to facilitate the growth of bars through resonant gravitational interaction. Still, it remains unclear why some galaxies are barred and some are not. In this study, we demonstrate that co-rotating (i.e., in the same sense as the disc rotating) dark matter halos with spin parameters in the range of $0 \le \lambda_{\mathrm{dm}} \le 0.07$ - which are a definite prediction of modern cosmological models - promote the formation of bars and boxy bulges and therefore can play an important role in the formation of pseudobulges in a kinematically hot dark matter dominated disc galaxies. We find continuous trends for models with higher halo spins: bars form more rapidly, the forming slow bars are stronger, and the final bars are longer. After 2 Gyrs of evolution, the amplitude of the bar mode in a model with $\lambda_{\mathrm{dm}} = 0.05$ is a factor of ~6 times higher, A_2/A_0 = 0.23, than in the non-rotating halo model. After 5 Gyrs, the bar is ~ 2.5 times longer. The origin of this trend is that more rapidly spinning (co-rotating) halos provide a larger fraction of trailing dark matter particles that lag behind the disc bar and help growing the bar by taking away its angular momentum by resonant interactions. A counter-rotating halo suppresses the formation of a bar in our models. We discuss potential consequences for forming galaxies at high-redshift and present day low mass galaxies which have converted only a small fraction of their baryons into stars.Monthly Notices of the Royal Astronomical Society 04/2013; 434(2). · 5.52 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We demonstrate that in N-body simulations of isolated disc galaxies there is numerical vertical heating which slowly increases the vertical velocity dispersion and the disc thickness. Even for models with over a million particles in a disc, this heating can be significant. Such an effect is just the same as in numerical experiments by Sellwood (2013). We also show that in a stellar disc, outside a boxy/peanut bulge, if it presents, the saturation level of the bending instability is rather close to the value predicted by the linear theory. We pay attention to the fact that the bending instability develops and decays very fast, so it couldn't play any role in secular vertical heating. However the bending instability defines the minimal value of the ratio between the vertical and radial velocity dispersions $\sigma_z / \sigma_R \approx 0.3$ (so indirectly the minimal thickness) which could have stellar discs in real galaxies. We demonstrate that observations confirm last statement.Monthly Notices of the Royal Astronomical Society 06/2013; 434(3). · 5.52 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**Using the potential from N-body simulations, we construct the Galactic bar models with the Schwarzschild method. By varying the pattern speed and the position angle of the bar, we find that the best-fit bar model has pattern speed $\Omega_{\rm p}=40\ \rm{km\ s^{-1}\ kpc^{-1}}$, and bar angle $\theta_{\rm bar}=45^{\circ}$. $N$-body simulations show that the best-fit model is stable for more than 1.5 Gyrs. Combined with the results in Wang et al. (2012), we find that the bar angle and/or the pattern speed are not well constrained by BRAVA data in our Schwarzschild models. The proper motions predicted from our model are slightly larger than those observed in four fields. In the future, more kinematic data from the ground and space-based observations will enable us to refine our model of the Milky Way bar.Monthly Notices of the Royal Astronomical Society 08/2013; 435(4). · 5.52 Impact Factor

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arXiv:astro-ph/0204078v1 4 Apr 2002

Mon. Not. R. Astron. Soc. 000, 1–11 (2001)Printed 1 February 2008(MN LATEX style file v1.4)

Orbital dynamics of three-dimensional bars:

II. Investigation of the parameter space

Ch. Skokos,1P.A. Patsis,1E. Athanassoula2

1Research Center of Astronomy, Academy of Athens, Anagnostopoulou 14, GR-10673 Athens, Greece

2Observatoire de Marseille, 2 Place Le Verrier, F-13248 Marseille Cedex 4, France

Accepted .... Received ....; in original form ....

ABSTRACT

We investigate the orbital structure in a class of 3D models of barred galaxies. We

consider different values of the pattern speed, of the strength of the bar and of the pa-

rameters of the central bulge of the galactic model. The morphology of the stable orbits

in the bar region is associated with the degree of folding of the x1-characteristic. This

folding is larger for lower values of the pattern speed. The elongation of rectangular-

like orbits belonging to x1 and to x1-originated families depends mainly on the pattern

speed. The detailed investigation of the trees of bifurcating families in the various mod-

els shows that major building blocks of 3D bars can be supplied by families initially

introduced as unstable in the system, but becoming stable at another energy interval.

In some models without radial and vertical 2:1 resonances we find, except for the x1

and x1-originated families, also families related to the z-axis orbits, which support the

bar. Bifurcations of the x2 family can build a secondary 3D bar along the minor axis

of the main bar. This is favoured in the slow rotating bar case.

Key words: Galaxies: evolution – kinematics and dynamics – structure

1INTRODUCTION

Barred galaxies have bars of very different strength, ranging

from the weak bars of SAB galaxies to the strong bars of e.g.

NGC 1365 (Lindblad 1999). They may have large, small, or

no bulge components in their centers. The possibility that

bars in late-type barred spiral galaxies end at their inner

Lindblad resonance (hearafter ILR) has also been considered

(Lynden-Bell 1979, Combes & Elmegreen 1993, Polyachenko

& Polyachenko 1994) and this would imply that in some

cases bars may have corotation far beyond their ends. It is

thus important to understand whether and to what extent

the orbital structure changes with the basic parameters in

the models. We investigate this here using a class of models,

the individual representatives of which differ in their central

mass concentration and in the pattern speed and strength

of the bar.

We follow the evolution of all the families of periodic

orbits we think may play a role in the dynamics and mor-

phology of bars and peanuts. We believe we indeed have all

the main families for two reasons. First, the edge-on profiles

of the galaxies are mainly affected by the vertical bifurca-

tions up to the 4:1 vertical resonance. Beyond this resonance

the orbits of the bifurcating 3D families remain close to the

equatorial plane and thus do not characterize the edge-on

morphology. Second, families bifurcated at the n:1 radial

resonances for n > 4 do not in general support the bar (e.g.

Contopoulos & Grosbøl 1989, Athanassoula 1992).

The models presented here are static, but they may be

viewed as corresponding to individual phases of an evolu-

tionary process of the dynamical evolution of a galaxy within

a Hubble time. Therefore, a complete investigation of the dy-

namical system is necessary in order to find all orbits pos-

sibly associated with the presence of specific morphological

features.

In the first paper of this series (Skokos, Patsis &

Athanassoula 2002, hereafter paper I) we presented the basic

families in a model composed of a Miyamoto disc of length

scales A=3 and B=1, a Plummer sphere of scale length 0.4

for a bulge and a Ferrers bar of index 2 and axial ratio

a : b : c = 6 : 1.5 : 0.6. The masses of the three components

satisfy G(MD + MS + MB) = 1 and are given in Table 1.

The length unit is 1 kpc, the time unit 1 Myr and the mass

unit 2 × 1011M⊙.

In the present paper we compare the orbital structure

of our basic model with those encountered in five more mod-

els. Our models, including the fiducial model A1 of paper I,

are described in Table 1. G is the gravitational constant,

MD, MB, MS are the masses of the disk, the bar and the

bulge respectively, ǫs is the scale length of the bulge, Ωb is

the pattern speed of the bar, Ej(r-IILR) and Ej(v-ILR) are

the values of the Jacobian for the inner radial ILR and the

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Ch. Skokos et al.

Table 1. Parameters of our models. G is the gravitational constant, MD, MB, MSare the masses of the disk, the bar and the bulge

respectively, ǫs is the scale length of the bulge, Ωbis the pattern speed of the bar, Ej(r-IILR) is the Jacobian for the inner radial ILR,

Ej(v-ILR) is the Jacobian for the vertical 2:1 resonance, Rc is the corotation radius. The comment in the last column characterizes the

model in order to facilitate its identification.

model nameGMD

GMB

GMS

ǫs

Ωb

Ej(r-IILR)

Ej(v-ILR)

Rc

comments

A1

A2

A3

B

C

D

0.82

0.82

0.82

0.90

0.82

0.72

0.1

0.1

0.1

0.1

0.1

0.2

0.08

0.08

0.08

0.00

0.08

0.08

0.4

0.4

0.4

–

1.0

0.4

0.0540

0.0200

0.0837

0.0540

0.0540

0.0540

-0.441

-0.470

-0.390

–

–

-0.467

-0.360

-0.357

-0.364

–

-0.364

-0.440

6.13

13.24

4.19

6.00

6.12

6.31

fiducial

slow bar

fast bar

no bulge

extended bulge

strong bar

vertical 2:1 resonance respectively, and Rc is the corotation

radius.

This paper is organized as follows: In section 2 we dis-

cuss models with fast, or with slow bars. Section 3 introduces

a model with no 2:1 resonances, section 4 a model with ver-

tical but no radial ILR, and section 5 a model with a massive

bar. We conclude in section 6.

2 THE EFFECT OF PATTERN SPEED

2.1A slow rotating bar

Model A2 is the same as model A1 in everything, except for

the pattern speed, Ωb= 0.02, which is less than half that of

model A1. The corotation in this model is at 13.24, rather

than at 6.13 as in model A1, and the outer inner Lindblad

resonance (OILR) is now at 6.1, i.e. roughly the end of the

bar or the corotation distance of the models with Ωb= 0.054.

The changes in the dynamical behaviour are much more

important than a stretching of the corotation radius by a fac-

tor larger than two and an enlargement of the x2-x3 loop in

the characteristic and in the stability curves of the model.

New bifurcations and new gaps are introduced, while the

morphology of some of the existing families changes dras-

tically. The differences are so big as to introduce nomen-

clature issues. Let us start our examination of the main

simple-periodic families and of their bifurcations with the

help of the characteristic diagram for planar orbits, shown

in Fig. 1. Following the convention introduced in paper I, we

draw with a black line the parts of the characteristics which

correspond to stable parts of the families, while grey is asso-

ciated with instability. There are two main characteristics, or

rather families of characteristics. The lower one is confined

to the region below x ≈ 5.5. It is divided from the upper

characteristic by a gap, occurring roughly at Ej = −0.128.

There are are also a number of 3D families bifurcating from

these characteristics, of which the most important ones will

be described at the end of this section.

The main feature of the characteristic diagram is a con-

tinuous curve constituted by the simple-periodic 2D families

x1, x2 and x3. We will follow it counter-clockwise. Starting

close to Ej = −0.5 for x = 0 we walk along the characteristic

of the typical x1 family. The orbits there are elliptical-like

and support the bar.

At the first S →U transition of x1 the family x1v1 is

bifurcated. That means we have reached at this energy the

vertical 2:1 resonance. It has a similar evolution as in the

Figure 1. Characteristic diagram for the 2D families of model

A2. Grey parts of the lines show the unstable parts of the families.

In (b) we give an enlargement of the area included in the frame

in (a).

fiducial case (paper I), but it is complex unstable for a con-

siderably smaller energy range. This affects strongly the ver-

tical profile of the model (Patsis, Skokos & Athanassoula,

2002a hereafter paper III). Since it is a 3D family it is not

included in Fig. 1.

The first radial bifurcation occurs at Ej ≈ −0.31 and

c ? 2001 RAS, MNRAS 000, 1–11

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Orbital dynamics of three-dimensional bars:II. Investigation of the parameter space

3

gives the family o1. This is stable for a tiny Ej interval, just

after the S→U transition. It then follows a S→U→S→U

sequence and ends again on x1. Thus this family builds a

bubble, both in the characteristic and the stability diagram,

together with x1 or with its indices, as did family t1 in model

A1. Its morphology, however, shows that it is related to a ra-

dial 1:1 resonance (Papayannopoulos & Petrou 1983), since

both cuts with the y = 0 axis are for x > 0 (alternatively

x < 0), so that it can be viewed as a distorted circle. It nev-

ertheless has three tips or ‘corners’, of which the two close

to the y axis are very sharp and for large Ej values they

develop loops. This morphological evolution is reflected by

the small orbits drawn close to the o1 curve on Fig. 1a.

The next S→U transition brings in the system x1v3.

This is a 3D family, so again it is not included in the dia-

gram. When x1 becomes again stable close to Ej ≈ −0.134,

its orbits have developed loops along the major axes of the

ellipses. Since we are already at the area included in the

frame in Fig. 1a, it is easier to follow the evolution of the

families on the characteristic in Fig. 1b. We observe that

close to Ej = −0.115 the curve has a bend and continues

towards lower Ej and higher x values. On the bend x1 orbits

are still very elongated with loops at the y axis, as noted

by a x1 orbit drawn there. The x1 family has developed

these loops well before the bend. Between Ej = −0.115 and

Ej = −0.13, at the rising part of the characteristic, towards

lower Ej values, the x1 orbits become again ellipse-like and

their loops vanish (for the time being we forget about the

gray branch we observe at the same area). Meanwhile, the

characteristic curve has two more bends at x roughly 3 and

5 respectively and for almost the same value of Ej ≈ −0.13,

and then follows the long branch towards lower Ej values,

which reaches Ej ≈ −0.47. On this branch and close to

Ej = −0.13 the x1 orbits have small ellipticities and become

even rounder as we move to lower Ej values. Finally after

Ej ≈ −0.17 the orbits are elongated along the minor axis

of the bar, and are stable (except for −0.23 < Ej < −0.2),

i.e. they belong to the x2 family. At Ej ≈ −0.47 the curve

folds again and continues its journey towards larger Ej val-

ues. The orbits at this branch are typical x3 orbits and exist

until Ej ≈ −0.29, where they change multiplicity. Thus in

this model the x2 and the x3 families are continuations of

the x1, the transition being made by circular and circular-

like orbits, rather than by a gap as in the standard cases

(Contopoulos & Grosbøl 1989, Athanassoula 1992a, paper I

etc.).

At Ej ≈ −0.23 the stability index associated with the

3D bifurcations intersects the −2 axis. So we have the bifur-

cation of a new family with the same multiplicity. We call

this family x2v1. We emphasize the fact that this is a simple

periodic family, since in model A1 (paper I) we had already

encountered a 3D bifurcation of x2 (family x2mul2), which,

however, is of multiplicity 2. Since this new family is a direct

bifurcation of x2 at the S→U transition close to Ej = −0.23,

as we move towards larger values of Ej, it inherits the sta-

bility of the parent family, i.e. it is stable. It stays stable for

a large energy interval, −0.23 < Ej < −0.18, which means

that it is a family that can affect the morphology of the

galaxy. Its morphology can be seen in Fig. 2. As we can see

this family can support a peanut-like feature, which, how-

ever, is elongated not along the major but along the minor

axis of the main bar. If such orbits are populated in a real

Figure 2. Stable orbit of the x2v1 family.

Figure 3. Successive x13 orbits. They are all unstable.

galaxy, then they will support a 3D stellar inner bar with a

‘x2 orientation’.

Close to the part of the x1 characteristic for −0.13 <

Ej < −0.115, where the curve folds and extends towards

lower energies (Fig. 1b), we have, besides the ‘x1 part’, a

gray branch (unstable orbits) that bridges the main loop

with another branch of the characteristic diagram existing

at the same energies and for larger x values. If this bridge

was missing then we would have a classical type 2 gap as

at the radial 4:1 resonance regions (Contopoulos & Grosbøl

1989). What we have now could be called a pseudo-gap. The

orbits of this branch are unstable and belong to a family we

call x13, since it starts for low x values as x1 at point ‘A’

(Fig. 1b) and reaches at ‘B’ a horizontal branch, which is

the characteristic curve of a x3-like family. x13 is a radial

bifurcation in ˙ x, so the curve we give in Fig. 1 for this family

is just the projection of its characteristic in the (Ej,x) plane.

The morphology of these orbits is expected to be related

with inclined ellipses, whose major axis shifts from being

parallel to the bar major axis (for members on or near the

major loop characteristic) to parallel to the bar minor axis

(for members on or near the x3′characteristic). The shift

happens in a small energy interval, in which the x1 orbits

have the longest projections on the y axis. Successive orbits

of x13, as we move from ‘A’ to ‘B’ (Fig. 1b), are given in

Fig. 3. The evolution of the stability indices of x1 in this

area follow every possible complication one could imagine

in order to avoid bifurcating a stable family with similar

morphology. Due to this ‘conspiracy’ it was not possible for

us to find a stable x13-like family.

The second part of the characteristic diagram, at the

same energies as the ‘x1 part’ and for higher x values, has

orbits which are x3-like. These orbits are ellipses elongated

along the minor axis of the bar and are almost everywhere

unstable, except for a tiny part of the characteristic for

Ej ≈ −0.175. We thus called them x3′. Moving along the

x3′branch of the characteristic towards larger Ej values, we

encounter a step-like feature in the curve (Fig. 1b) and be-

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Ch. Skokos et al.

Figure 4. Stable orbits of the families x1′v4 (a), and x1′v5 (b).

yond it we have planar orbits, which can be easily described

as prograde quasi-circular orbits. Their general dynamical

properties and their relation with other families at the area

resemble those of the x1 family. So this family is a kind of

continuation of x1, which we call x1′(as we called, for lower

energies, the continuation of the x3 family x3′). The stability

indices of x1′oscillate and at the tangencies with the b = −2

axis the 3D families x1′v4, x1′v5 etc. are born. We call them

like this because their morphology on the (x,z) and (y,z)

projections resembles the morphology of the x1v4 and x1v5

families of the fiducial case. The bifurcated 3D families re-

main as stable close to the equatorial plane, i.e. they do not

characterize the vertical profile of the model, although they

have large stable parts. It is important to note that in this

case the shape of the x1′orbits – and of the (x,y) projec-

tions of x1′v4 and x1′v5 as well – are not elongated along the

major axis of the bar, but are quasi-circular. Thus, they do

not enhance the bar towards the corotation radius (13 kpc).

This can be seen in Fig. 4.

The characteristic of x1′, as in the case of model A1 for

x1, has a local maximum at Ej ≈ −0.11. At the decreas-

ing branch (lower x’s for larger Ej values) the orbits of the

family develop ‘corners’. The usually rectangular-like orbits

found in the 4:1 region (cf. Fig. 2g in paper I) are for this

model square-like. x1′has a stable part just after the turning

point, while in model A1 the decreasing branch is almost ev-

erywhere unstable. For yet larger energies, when the orbits

at their four apocentra have loops, x1′is unstable.

As can be seen in Fig. 1b the gap at Ej ≈ −0.11 is a

real type 2 gap, the upper branch of which has stable circu-

lar orbits at the ‘increasing x’ part and unstable hexagonal

orbits at the decreasing part following it at larger Ej val-

ues. The latter are not much elongated along the y axis.

Due to this morphological evolution of the x1 family there

are no elliptical-like orbits elongated along the y axis to ex-

tend the bar towards corotation. The elongated orbits which

reach the farthest out in the y direction are elliptical-like or-

bits with loops, reaching y ≈ 9.4, surrounded by a roundish

structure reaching the corotation region (Fig. 5).

Before closing our description of model A2, we should

mention that the family x1v4, initially bifurcated as dou-

ble unstable, becomes stable for larger energies and provides

the system with 3D orbits with low |z|. The orbit we give in

Fig. 6 has Ej = −0.12, while the family x1v4 bifurcates for

Ej ≈ −0.245 at a D→U transition of x1. The evolution of

Figure 5. Stable orbits for model A2.

Figure 6. Stable orbit of the x1v4 family of model A2.

the stability indices of this family in model A2 is less com-

plicated than in model A1. It nevertheless shows all kinds of

instabilities we encounter in 3D Hamiltonian systems and fi-

nally ends again on x1. This means that it can be considered

both as a direct and as an inverse bifurcation of x1.

Summarizing the main differences of the orbital be-

haviour of the slow rotating bar model from that in the

fiducial case, we underline the existence of a complicated

common characteristic of the x1, x2 and x3 families. As a

consequence the simple-periodic families of the x1-tree ap-

pear in two parts. The second part consists of x1′and its 3D

bifurcations. The families of the x1′-tree have large stable

parts, but they do not help the bar reach closer to corotation

since they are quasi-circular (or have quasi-circular projec-

tions on the equatorial plane). The rectangular-like orbits

in this case are almost squares. The model also includes a

simple periodic x2-like 3D family. Other differences in the

orbital behaviour from model A1 that should be mentioned

is the small complex unstable part of x1v1 and the bifurca-

tion of the family x1v4 at a D→U stability transition.

2.2A fast rotating bar

Model A3 has a fast rotating bar. Its pattern speed is 0.0837,

which brings corotation to 4.2 kpc, i.e. closer to the center

than the end of the imposed bar. All other parameters re-

main as in models A1 and A2.

The major effect, as expected, is that the OILR ap-

proaches the IILR, and the size of what we would call ‘x2-

region’ is drastically reduced. In model A3 both x2 and x3

families still exist. The size of the semi-major axis of the

largest x2 orbit is 0.63 kpc. This means, that the x2 orbits

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Orbital dynamics of three-dimensional bars:II. Investigation of the parameter space

5

Figure 7. Stability diagram for the family x1 in model A3. The

black bold curves at the right part of the figure (−0.265 < Ej<

−0.235) are the stability indices of the family x1v8. Light grey

curves indicate instability.

could support features of sizes about ≈ 1.2 kpc. In other

words in such models, the x2 family could play a role in the

dynamics of the innermost 1 kpc of the system if the corre-

sponding orbits are populated, despite the fact that the x2-

loops we find are tiny (∆Ej ≈ 0.01) in comparison to those

of models A1 and of course A2. In Fig. 7 we see the evolution

of the stability indices of this model. Note the small ellip-

tical features around Ej ≈ −0.385, which are made from

the combination of the stability indices of x2 and x3. The

stability indices of these two families do not have any other

cuts or tangencies with the b = 2 or b = −2 axes and thus

this model has no 3D families oriented perpendicular to bar

major axis and cannot form a peanut with this orientation.

The oscillations of the b1 and b2 curves of x1 bring

in the system the families x1v1, x1v3 and x1v5 as stable.

Their dynamical behaviour, and thus their importance for

the global dynamics of the system, do not differ from what

we encountered in the fiducial case, and so we do not discuss

it further here. In this model x1v4 is not significant. It re-

mains unstable until its orbits reach high z values above the

equatorial plane. The curves indicated by

the orbits at the branch of the characteristic of x1, after the

bend of the curve towards lower energies for Ej ≈ −0.235

(see Fig. 8 below). Light gray indicates also in Fig. 7 unsta-

ble orbits. The lower index almost goes through the point

of intersection of x1 with the −2 axis. However, because of

the location of the second index, we do not have a loop that

closes on x1 there.

The new elements that the study of this model brings to

the investigation of the orbital dynamics of barred potentials

are focused at the region of the (type 2) gap at the 4:1

resonance. In Fig. 8 we show what is new in this model on a

characteristic (Ej,x) diagram of x1. We have also included

the (Ej,x) projections of a planar family (q0), which has

˙ x0 ?= 0 in the initial conditions, and a 3D family (x1v8),

which is unstable in model A1.

Let us start from the latter. As can be seen in Fig. 7,

the stability index associated with the vertical bifurcations

has its seventh cut with the b = −2 axis at Ej ≈ −0.265

←

x1 correspond to

Figure 8. Part of the characteristic diagram of model A3. It

shows the curve of the family x1 at the 4:1 resonance region, and

the (Ej,x) projections of families q0 and x1v8. Light gray color

indicates unstable orbits.

(the depth and size of the unstable region is very small; we

observe in Fig. 7 that the depth and size of the successive

unstable regions decreases with increasing energy). At this

point a new stable family is born. Fig. 8 shows that this

family is bifurcated just before the local maximum of the

x1 characteristic curve. The (x,z) and (y,z) morphology of

this new family is similar to that of family x1v8 of our fidu-

cial model (cf. Fig. 17c in paper I) and thus, according to

the rules set in paper I, we call it x1v8, although it emerges

at the seventh vertical bifurcation. In model A3 the suc-

cession of appearance of the bifurcating families associated

with the vertical 5:1 resonance is reversed compared with the

families of the corresponding instability strip in the fiducial

case. Now this instability strip is located before the local

maximum of the x1-characteristic at Ej ≈ −0.26 (Fig. 8),

while in model A1 it is located beyond the corresponding

local maximum. As discussed in paper I when the evolution

of the stability index of x1 associated with the vertical bi-

furcations has successive cuts with the b = −2 axis giving

rise to a S→U→S sequence in its stability, a stable and an

unstable family are introduced in the system. In model A1

for all instability strips at the vertical resonances before the

local maximum of the characteristic curve, the families in-

troduced as stable at the S→U transition are bifurcations in

z, and the unstable ones, bifurcated at the U→S part, are

bifurcations in ˙ z. The opposite is true for the 5:1 resonance

instability strip located beyond the local maximum. There

we had a stable family bifurcated in ˙ z, which we called x1v7

and an unstable one bifurcated in z we called x1v8 (paper

I). In the present model the corresponding instability strip

of the vertical 5:1 resonance is located before the local max-

imum of the x1-characteristic for Ej ≈ −0.26 (Fig. 8) and

the family introduced as stable is a bifurcation in z. Since

we keep the nomenclature introduced in the fiducial model

throughout this series of papers, this is family x1v8 and

the bifurcation in ˙ z, unstable in the present model, is x1v7.

We note that while the x1v7 family of model A1 very soon

gets orbits with large |z|’s, family x1v8 is stable everywhere

and its orbits remain confined close to the equatorial plane

(Fig. 9).

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6

Ch. Skokos et al.

Figure 9. Orbits of the family x1v8 of model A3. From top to

bottom: Ej= −0.26, Ej= −0.25 and Ej= −0.24 respectively.

Figure 10. A sequence of three stable orbits of the q0 family of

model A3, showing its morphological evolution. They have (from

left to right) Ej= −0.2615,−0.260 and −0.255 respectively.

Almost at the local maximum of the x1 characteris-

tic, at Ej ≈ −0.26, we have another S→U transition of x1

(Fig. 8). There we have a radial bifurcation with ˙ x ?= 0 .

We call the resulting family q0, since it bifurcates at the 4:1

resonance close to the local maximum, and its morphology

is different than that of the q1, q2 families of model A1. Its

morphological evolution, as we move along the (Ej,x) pro-

jection of its characteristic, is given in Fig. 10. In Fig. 8 we

see that q0 is stable almost everywhere with only two small

unstable zones. The one closer to the bifurcating point is

bridged by a family (not shown in Fig. 8) existing just in

this interval. The orbits of this family are slightly asymmet-

ric with respect to the corresponding unstable orbits of q0

at the same Ej values. Practically one could say that q0

is stable even there. The second zone of instability is in an

area where the loops of q0 are big, so that the orbits are less

interesting because of their morphology. Thus we conclude

that practically all the morphologically interesting parts of

q0 are stable.

Concluding about model A3, we can say that its dif-

ferences with respect to the fiducial case are focused at the

dynamics close to the local maximum of the characteristic at

the radial 4:1 resonance. In general, in most other models we

examined, the decreasing branches of the x1 characteristics

beyond the local maximum harbor mainly unstable families.

In that respect, model A3 is an exception, because at this

region we can find decreasing branches with large stability

regions offered not by x1 itself but by q0 and x1v8. We note

that q0 and x1v8 are the most elongated rectangular-like

orbits we found in any of the models we studied. In model

A3 the bar is supported by the x1 and q0 families up to 3.8

kpc, with corotation at 4.2 kpc (For more details about the

supported face-on morphologies of our models, see Patsis,

Skokos & Athanassoula, 2002b hereafter paper IV). Further

differences are the size of the x2 region, which in model A3 is

very small and the insignificance of the x1v4 family. A final

difference of A3 from the rest of the models we examined is

the lack of a ‘bow’ structure in the stability diagrams. The

rest of the orbital structure is similar to that of model A1.

3 A MODEL WITHOUT 2:1 RESONANCES

All models presented until now include an explicit bulge

component in the form of a Plummer sphere. In order to

investigate the influence of central concentrations on the

dynamics of the bar we consider a model without this com-

ponent, and we increase the mass of the disc accordingly

so that the total mass stays the same as that of the other

models. This is model B, in which all other parameters are

as in model A1. We note that this particular case has been

studied by Pfenniger (1984).

The model is characterized by the lack of radial as well

as vertical 2:1 resonances. The stability indices of x1 have

their first tangency with the b = −2 axis at Ej ≈ −0.240.

We call the family bifurcated at this point x1v5 because

the (x,z) and (y,z) projections of its orbits have the same

morphology as that of the x1v5 family of model A1. x1v5

exists up to Ej ≈ −0.219 where it rejoins x1. This family

corresponds to the Bz1 family of Pfenniger (1984). Another

3D orbit is bifurcated from x1 at Ej ≈ −0.217, and is mor-

phologically similar to x1v5 so that we name it x1v5′. This

family has stable orbits with low |z| over a reasonable energy

range, i.e it is an important family of the system, as was al-

ready pointed out by Pfenniger (1984) who named it Bz2.

At Ej ≈ −0.215 the x1v7 family is bifurcated, which cor-

responds to the B˙ z3 family of Pfenniger (1984). The overall

evolution of the stability indices in this model is character-

ized by a complicated ‘bow’, around Ej ≈ −0.22, reminis-

cent of the bow in model A1. The bow is at the center of the

3:1 region, which in this model is rather extended. The val-

ues of the indices of the 3:1 families remain smaller than 0,

and all bifurcations are simple periodic families. The model

has t1, t2 and 3D 3:1 orbits with t1- and t2-like projections.

Its 4:1 gap is of type 2, and beyond this gap, towards coro-

tation, the orbital behaviour resembles that of model A1.

In this model we found one more family, which has large

stable parts over a very extended energy range. This fam-

ily has morphological similarities with x1v4, but it is not

related with the x1-tree. This means that at least as far as

we have followed, it does not bifurcate from or be linked

with a family belonging to the x1-tree. This family exists

for Ej > −0.285 and is one of the families of periodic orbits

related to the z-axis family, i.e. to the 1D orbits on the rota-

tional axis of the system. The well known bifurcations of the

z-axis family are the sao and uao families (Heisler, Merritt

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Orbital dynamics of three-dimensional bars:II. Investigation of the parameter space

7

Figure 11. Part of the stability diagram of model B. It includes

z3.1s, z3.1u and parts of z3.

& Schwarzschild 1982). They are introduced at S→U and

U→S stability transitions of the z-axis family respectively,

at which we have cuts of one of the stability indices with

the b = −2 axis. We find them considering the z = 0 plane

as a surface of section. However, if the orbits of a single pe-

riodic family are repeated n-times it can be considered as

n-periodic (paper I, §2.2). As explained in Appendix I, spe-

cific values of the stability index (Eq. A8) determine the Ej

value at which an n-periodic family will bifurcate. These n-

periodic families are the so called ‘deuxi` eme genre’ families

of Poincar´ e (1899). The family we found to be important in

this model is a bifurcation of the z-axis family when we con-

sider its orbits repeated three times, i.e. of z3, according to

the nomenclature we introduced in paper I. In this case tan-

gencies of the stability indices of z3 with the b = −2 axis will

bifurcate two 3-periodic families and the family we discuss

here is one of them. The z-axis family does not change its

stability at the energies at which the new families are born.

Already by studying the evolution of the stability indices of

the z-axis family, we can find out from Eq. A8 the Ej values

at which 3-periodic bifurcations will appear. Thus we know

that for Ej = −0.285, a bifurcation of the z-axis orbits with

multiplicity 3 will be born. We call this family z3.1s and its

position of birth is seen in Fig. 11.

In Fig. 11 we give the evolution of the stability indices

of z3. As expected at Ej ≈ −0.285 it has a tangency with

the b = −2 axis, and z3.1s is bifurcated. Actually at this

point two families are bifurcated. At Ej ≈ −0.256 the one

initially bifurcated as stable becomes unstable and remains

so thereafter, while the opposite is true for the one bifurcated

as unstable. We call z3.1s the one which is stable for the

largest energy range and z3.1u the one initially bifurcated

as stable. In any case their morphologies are very similar.

We note that neither of the families have one of the stability

indices on the b = −2 axis for some energy interval. Both

stay close to it after the bifurcating point, but not on it, as

one can realize by looking at the appropriate enlargement of

Fig. 11 (not plotted here). The range of energies over which

z3.1s is stable emphasizes its importance (Fig. 11).

The multiplicity of an orbit is associated with the sur-

face of section we use. The z-axis orbits are calculated using

as surface of section the (x,y) plane, so the multiplicity in

Figure 12. The morphological evolution of three stable z3.1s

orbits. In (a) for Ej = −0.25, in (b) for Ej = −0.22, and in (c)

for Ej= −0.2. In the (x,y) projection we include, in grey, the x1

orbit with the same Ej.

Figure 13. A stable z5.1s orbit at Ej≈ −0.225.

this case does not refer to the morphology of the projection

of the orbit on this plane, but to the number of intersec-

tions with this plane. The detailed morphological evolution

is given in Fig. 12. We observe that the multiplicity of the

z3.1s family if we consider as surface of section the y = 0

plane, as we do for all families of the x1-tree, is 1. In the

(x,y) projection we have overplotted with light grey the cor-

responding x1 orbits. The (x,y) projection of z3.1s is always

included inside the curve of the x1 orbit. It is evident that

the morphology of the x1 orbits is similar but not identical

to that of the z3.1s (x,y) projection. We observe also that

the (x,z) and (y,z) projections remain close to the plane

of symmetry of the galactic model, at least for the lowest

energies.

Apart from z3.1s, we have found other families associ-

ated with zns. A case that could be mentioned is a bifur-

cation of z5, the shape of which is given in Fig. 13. Orbits

like this can populate a galactic bulge or the central part of

disks. Indeed, although we have not an explicit bulge compo-

nent in this model, our disc is not flat. Due to the geometry

of the Miyamoto disc one would need in the central part

orbits with projections on the z axis of the order of 1 kpc in

order to build a self-consistent model. Thus orbits like z5.1s

should be considered. In general, however, the tangencies

with the b = −2 axis are for larger energies and as a result,

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Ch. Skokos et al.

Figure 14. Stability diagram for x1, x1v1 and x1v3 in model C.

these orbits, since they are bifurcations of the z-axis, have

big |z| values, and so, even if they have stable parts, are not

interesting building blocks for the disc of our system⋆.

For this model we underline the presence and impor-

tance of the z3.1s orbits and the lack of the 3D families

associated with low order vertical resonances, since the first

vertical bifurcation of x1 is x1v5, a family bifurcated at the

vertical 4:1 resonance.

4 A MODEL WITHOUT RADIAL ILRS

Model C is intermediate between models A1 and B. It has

a Plummer sphere bulge, the scale length of which is 2.5

times larger than the scale length of the bulge of A1. It is

thus considerably less centrally concentrated, and as a result

its Ω-κ/2 curve is less peaked. This model does not have any

radial ILRs, since we have, like in model A1, Ωb=0.054.

On the other hand the model does have a vertical 2:1

resonance, where is bifurcated a x1v1 family (Fig. 14), which

in this model is characterized by a large stable part. After

the usual S→∆ transition at Ej = −0.08 the family remains

always complex unstable. Furthermore, at Ej = −0.26 the

maximum z of the orbits is 1 kpc, and at Ej = -0.225 the

maximum z is 1.5 kpc. This means that it is a very impor-

tant family for the dynamics of the system. x1v3 exists as

well. It has a S→∆→S→U sequence of stability types, but

the ∆→S→U part happens in a very narrow energy range

(Fig. 14). At the final S→U transition the 3D family x1v3.1

depicted in Fig. 15 is bifurcated. In this particular model this

family is just bridging x1v3 with x1v4 at Ej ≈ −0.2232. At

this energy x1v4 becomes stable and x1v3.1 can be consid-

ered as an inverse bifurcation of it. All this is worth mention-

ing because x1v3.1 is a 3D family with a (x,y) projection

resembling that of the family q0 of model A3.

The most important feature of model C is that x1v1

becomes complex unstable for the first time at large energies

and not just after it is born as e.g. in the fiducial model A1.

⋆This is also the case for the zn orbits for n < 6 in all other mod-

els studied in this paper. Either they do not exist (models A1, A2,

A3 and D) or they are not so important because of their stability

in combination with their morphological evolution (model C).

Figure 15. A stable orbit of the family x1v3.1 in model C at

Ej≈ −0.2235.

Figure 16. Characteristic diagram for some important families

in model D. The unstable region of x1v1 bridged by x1v1.1 is

indicated by a dashed-dotted line.

The consequences of this stability evolution for the global

dynamics of the model are described in detail in paper III.

5 A STRONG BAR CASE

Strong perturbations in Hamiltonian systems result in sys-

tems with a larger degree of orbital instabilities, and a larger

amount of chaos. Model D has a bar twice as massive as that

of the other models and a disc accordingly less massive, so

that the total mass is the same. We can see the effect of

this change in Fig. 16, which is a characteristic diagram

of families x1, o1 and also of the (Ej,x) projection of the

characteristic of the 3D family x1v1. The rising part of the

branch of the x1 characteristic, for Ej < −0.205, is steeper

than in model A1. In this model x1 is mainly stable at its

decreasing branch (Ej > −0.205). The morphology of the x1

orbits there is rectangular-like, and this clearly shows that

the model with a stronger bar favours this morphology.

The behaviour of x2 in model D is similar to that in

model A1. The variation of the stability indices of x1 intro-

duces as first bifurcating family in the system family x1v1.

This has first a short stable part and then becomes complex

unstable. The branch on Fig. 16 indicated by x1v1 is just the

(Ej,x) projection of its characteristic curve. On this curve

we note with ∆ the complex unstable part. In the S→∆

transition there is no family inheriting the stability of x1v1

when the latter becomes unstable. As a result, the only sta-

ble family for −0.38 < Ej < −0.338 is the o1 family, which

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Orbital dynamics of three-dimensional bars:II. Investigation of the parameter space

9

Figure 17. Two o1 orbits in model D, symmetric with respect

to the major axis of the bar.

Figure 18. Orbits of the families x1v1 (a) and x1v1.1 (b) in

model D at Ej≈ −0.225 and −0.235 respectively.

we also found in the slow bar case. If, at a given energy,

we consider the two representatives of this family which are

symmetric with respect to the major axis of the bar, we get

the combined morphology shown in Fig. 17.

We should also note that the (x,y) projection of the

x1v1 family, away from the family’s bifurcating point, does

not quite follow the morphological evolution of x1 at the

same energy. The (x,y) projections get squeezed on the sides

already before they become rectangular shaped and thus

tend to take a shape like ‘8’. This happens just before |z|

reaches values larger than 2 kpc. This morphology as well

as the morphology of family x1v1.1, which bridges a small

zone of simple instability of x1v1, can be seen in Fig. 18.

To summarise the specific features of the orbital struc-

ture of model D it is worth underlining that, due to its

stronger bar, the x1v1 family bifurcates at lower energies

than in the other models. After it bifurcates from x1 as sta-

ble it has, as usual, a complex unstable part, but beyond

the ∆→S transition the orbits have still low |z|. Another

interesting feature is the stability of the x1 family at the de-

creasing part of the characteristic. Also, as in model B, fam-

ilies x1v5 and x1v5′exist and have stable representatives.

Finally we note that from the families bifurcated initially as

unstable, only x1v6 has away from the bifurcating point a

small stability part.

6 CONCLUSIONS

In this paper we investigated the orbital structure in a class

of models representing 3D galactic bars. The parameters we

varied are the pattern speed, the strength of the bar and

parameters defining the bulge component of the galactic

model. We found all the families that could play an im-

portant role in the dynamics of 3D bars, and we registered

the main changes which happen as we vary the parame-

ters under consideration. Since evolutionary scenarios of the

morphology of the bars within a Hubble time could include

an increase of the bulge mass and a deceleration of the bar,

as well as an increase or decrease of the bar’s strength, our

models could correspond to discrete phases in the dynam-

ical evolution of a barred galaxy. They could thus be used

to explain the changes in the underling dynamics when the

galaxies evolve. Similarly, they can be used to understand

the dynamics of selected snapshots of N-body simulations.

Our main conclusions in the present paper are:

(i) In all models we examined, the extent of the orbits

which are most appropriate to sustain 3D bars is confined

inside the radial 4:1 resonance. Viewing the models face-

on, the orbits with the longest projections along the major

axis of the bar are either boxy or elongated with loops at

the major axis; these are typical shapes of the orbits in the

radial 4:1 resonance region. This behaviour is common to

both slow and fast rotating bars.

(ii) The evolution of the characteristic of the basic fam-

ily x1 depends heavily on the pattern speed. The slower

the bar rotates, the more complicated the x1-characteristic

curve becomes. In the slowest of our models the families x1,

x2 and x3 share the same characteristic curve. The folding

of the characteristic towards lower energies, with most ex-

treme case the one with the slow rotating bar, corresponds

to a ‘bow’ feature in the evolution of the stability indices as

function of Ej.

(iii) The fast rotating bar model A3 did not have the com-

plicated evolution in the x1 characteristic and in the stability

diagram corresponding to the ‘bow’. In this case all main 3D

families of the x1-tree bifurcate from x1 at an S→U transi-

tion before the local maximum of the x1-characteristic at the

radial 4:1 resonance and have initial conditions (x,z,0,0).

(iv) The bars can be supported not only by x1-originated

families but, depending on the model, by 3D orbits bifur-

cated from families related with the z-axis orbits. This has

been encountered in the case of a model without radial or

vertical 2:1 resonances.

(v) Slow pattern rotation favours the presence of 3D x2-

type orbits along the minor axis of the main bar. These

orbits, which can lead to a 3D inner bar, are typical orbits

of the potentials we studied.

(vi) The most elongated 4:1 rectangular-like orbits have

been encountered in the fast rotating bar model A3. On the

contrary, the corresponding orbits in the slow bar of model

A2 are square-like and farther out circles and orbits with

circular-like (x,y) projections. Thus in the slow bar case

the bar is supported only by elliptical-like orbits of the x1-

tree. The different elongations of the rectangular-like orbits

c ? 2001 RAS, MNRAS 000, 1–11

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Ch. Skokos et al.

can be explained by the fact that we have, in all models

considered, bars of the same length in the imposed potential.

Since the corotation radius changes with pattern speed, the

non-axisymmetric part of the forcing is relatively larger near

corotation for the fast bar than for the slow one.

(vii) The decreasing part of the x1-characteristic is in

most cases unstable, except for the strong bar case (model

D). This favours the presence of rectangular-like orbits at

the outer parts of strong bars, in good agreement with ob-

servations (Athanassoula, Morin, Wozniak et al. 1990). This

could be due to the fact that the bar forcing is stronger in

the strong bar case. Stable rectangular-like orbits can also

be found in the case of the fast rotating model A3, where

rectangular-like stable orbits are provided not by the fam-

ily x1 but by the families q0 and x1v8. This could again

be due to the fact that, in the fast bar case, the forcing in

the corotation region is larger than in other cases. The two

above put together seem to argue that a strong bar forcing

in the region around corotation is necessary for the model

to have stable rectangular-like orbits which are sufficiently

elongated along the bar major axis.

(viii) Models with low mass concentrations at the center

(models B and C) favour the presence of zn bifurcations for

low n, which in some cases may be important for the global

dynamics of the system (e.g. family z3.1s in model B). In

this way we can have bar supporting families unrelated with

the x1-tree.

(ix) In the x1-tree we encounter complex instability

mainly in the x1v1 family. It can however, happen (e.g. in

model C) that complex instability appears in large energies

and thus all orbits with |z| < 2 kpc are stable. One must

examine in every case the extent in z of the complex unsta-

ble orbits of x1v1 in order to decide about the significance

of this family for a model.

The connection between the families of periodic orbits

and the observed morphologies in edge-on disc galaxies is

discussed in paper III, and the contribution of orbital theory

to the question of the boxiness of the outer isophotes in early

type bars in paper IV.

REFERENCES

Athanassoula E., 1992a, MNRAS 259, 328

Athanassoula E., Morin S., Wozniak H., Puy D., Pierce M. J.,

Lombard, J., Bosma, A., 1990, MNRAS 245, 130

Broucke R., 1969, NASA Techn. Rep. 32, 1360

Combes F., Elmegreen B.G., 1993, A&A 271,391

Contopoulos G., Grosbøl P., 1989, A&AR 1,261

Contopoulos G., Magnenat P., 1985, Celest. Mech. 37, 387

Heisler J., Merritt D., Schwarzschild M., 1982, ApJ 258, 490

Lindblad P.O., 1999, A&ARv 9, 221

Lynden-Bell D., 1979, MNRAS 187, 101

Patsis P.A., Skokos Ch., Athanassoula E., 2002a (paper III - in

preparation)

Patsis P.A., Skokos Ch., Athanassoula E., 2002b (paper IV - in

preparation)

Papayannopoulos T., Petrou M., 1983, A&A 119, 21

Pfenniger D., 1984, A&A 134, 373

Poincar´ e H, 1899, ‘Les Methodes Nouvelles de la Mechanique Ce-

leste’, Vol. III, Gauthier-Villars, Paris

Polyachenko V.L., Polyachenko E.V., 1994, AstL 20, 416

Skokos Ch., Patsis P.A., Athanassoula E., 2002 MNRAS, paper I

this issue

Yakubovich V. A., Starzhinskii V. M., 1975, ‘Linear Differential

Equations with Periodic Coefficients’, Vol. 1, J. Wisley, New

York

ACKNOWLEDGMENTS

We acknowledge fruitful discussions and very useful com-

ments by Prof. G. Contopoulos. We thank the referee for

useful suggestions that allowed to improve the presentation

of our work. This work has been supported by EΠET II

and KΠΣ 1994-1999; and by the Research Committee of the

Academy of Athens. Ch. Skokos and P.A. Patsis thank the

Laboratoire d’Astrophysique de Marseille, for an invitation

during which essential parts of this work have been com-

pleted.

APPENDIX A:

GENRE’ FAMILIES IN HAMILTONIAN

SYSTEMS

POINCAR´E’S ‘DEUXI`EME

The number n of intersections of a periodic orbit with the

Poincar´ e surface of section, when the orbit has a particu-

lar direction, defines its multiplicity. So a periodic orbit of

multiplicity n has n points of intersection with the Poincar´ e

surface of section and it is called a periodic orbit of period

n.

The linear stability or instability of a periodic orbit is

defined by the eigenvalues of the corresponding monodromy

matrix (see for example Yakubovich & Starzhinskii 1975).

The columns of this matrix are suitably chosen linearly in-

dependent solutions of the so-called variational equations.

These equations describe the time evolution of a small devi-

ation from the periodic orbit. The eigenvalues of the mon-

odromy matrix of a periodic orbit can be grouped as pairs of

inverse numbers, i.e. if λ is an eigenvalue then 1/λ is also an

eigenvalue (Broucke 1969, Contopoulos & Magnenat 1985).

The stability index b that corresponds to a particular pair

of eigenvalues is defined as:

b = −

?

λ +1

λ

?

(A1)

An orbit is stable when both stability indices are real num-

bers in the interval (−2, 2), which equivalently means that

the corresponding eigenvalues are complex conjugate num-

bers on the unit circle.

As a parameter of the dynamical system changes the

eigenvalues move on the complex plane. When two eigenval-

ues moving on the unit circle coincide on λ = 1 and split

along the real axis, the stability type of the orbit changes

from stable to unstable. The corresponding stability index

is negative and decreases below −2. At the same time a new

periodic orbit of the same multiplicity is born. If, on the

other hand, the two eigenvalues continue to lie on the unit

circle, after coinciding on λ = 1, which means that the orbit

remains stable, then two new orbits of the same multiplic-

ity are born. A periodic orbit of multiplicity 1 can be also

considered as a periodic orbit of multiplicity n > 1 if it is re-

peated n times. It has as monodromy matrix Mn the matrix

Mn = Mn

1, (A2)

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Orbital dynamics of three-dimensional bars:II. Investigation of the parameter space

11

Figure A1. The eigenvalues of a stable orbit on the unit circle.

Figure A2. The values of the stability index b that correspond

to bifurcations of period n = 1,2,...,10 given by Eq. (A8).

where M1 is the monodromy matrix of the periodic orbit

considered as 1-periodic. If the periodic orbit of period 1 is

stable, then it has a pair of eigenvalues of the form

λ = cosϑ + isinϑ,

1

λ= cosϑ − isinϑ,(A3)

as seen in Fig. A1. Thus the corresponding stability index

is:

?

λ

b = −λ +1

?

= −2cosϑ. (A4)

Considering this orbit as one of period n its eigenvalues will

be of the form λn, (1/λ)n, so that the corresponding stability

index b(n)becomes:

b(n)= −2cos(nϑ). (A5)

A tangency of b(n)with the line b = −2 gives birth to two

new periodic orbits of period n, while the 1-periodic orbit

remains stable. This bifurcation happens when

b(n)= −2 ⇒ cos(nϑ) = 1 (A6)

This condition is satisfied if we have e.g.

ϑ = 2π1

n,

(A7)

or equivalently when the stability index b of the period 1

periodic orbit crosses the line

b = −2cos

?

2π1

n

?

. (A8)

The bifurcating families of periodic orbits are the ‘deuxi` eme

genre’ families of Poincar´ e (1899). In Fig. A2 we plot the

lines given by (A8) corresponding for n = 1,2,...10. We see

that, as we approach b = −2, the density of the lines as well

as the period of the bifurcating orbit increase.

This paper has been produced using the Royal Astronomical

Society/Blackwell Science LATEX style file.

c ? 2001 RAS, MNRAS 000, 1–11

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