Orbital dynamics of threedimensional 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 x1characteristic. This folding is larger for lower values of the pattern speed. The elongation of rectangularlike orbits belonging to x1 and to x1originated 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 x1originated families, also families related to the zaxis 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

Article: Studying the global dynamics of conservative dynamical systems using the SALI chaos detection method
[Show abstract] [Hide abstract]
ABSTRACT: We use the Smaller ALignment Index (SALI) method of chaos detection, to study the global dynamics of conservative dynamical systems described by differential or difference equations. In particular, we consider the wellknown 2 and 4dimensional symplectic standard map, as well as an autonomous Hamiltonian system of 2 and 3 degrees of freedom describing the motion of a star in models of barred galaxies. The application of SALI helped us to compute rapidly and accurately the percentage of regular and chaotic motion for particular values of the parameters of these systems. We were also able to perform a computationally efficient determination of the dependence of these percentages on the variation of several parameters of the studied models.Nonlinear Phenomena in Complex Systems. 01/2008; 11:171176.  SourceAvailable from: Inma MartinezValpuesta[Show abstract] [Hide abstract]
ABSTRACT: We discuss the formation and evolution of boxy/peanut bulges (B/Ps) and present new simulations results. Orbital structure studies show that B/Ps are parts of bars seen edgeon, they have their origin in vertical instabilities of the disc material and they are somewhat shorter in extent than bars. When the bar forms it is vertically thin, but after a time of the order of a Gyr it experiences a vertical instability and buckles. At that time the strength of the bar decreases, its inner part becomes thicker, so that, seen edgeon, it acquires a peanut or boxy shape. A second buckling episode is seen in simulations with strong bars, accompanied by a further thickening of the B/P and a weakening of the bar. Quantitatively, this evolution depends considerably on the properties of the halo and particularly on the extent of its core. This influences the amount of angular momentum exchanged within the galaxy, emitted by nearresonant material in the bar region and absorbed by nearresonant material in the halo and in the outer disc. Haloes with small cores generally harbour stronger bars and B/Ps and they often witness double buckling.03/2008;  SourceAvailable from: P. A. Patsis[Show abstract] [Hide abstract]
ABSTRACT: The central, or x1, family of periodic orbits is the most important one in almost all twodimensional numerical models of galactic bars in the literature. However, we present evidence that in twodimensional models with sufficiently large bar axial ratios (a/c 6), stable orbits having propeller shapes play the dominant role. In our models this propeller family is in fact a distant relative of the x1 family. There are also intermediate cases in which both families are important. The dominance of one family over the other may have direct consequences on the morphological properties of the bars that can be constructed from them, properties such as faceon bar thinness and strength as well as the boxiness of the outer isophotes.The Astrophysical Journal 12/2008; 624(2):693. · 6.73 Impact Factor
Page 1
arXiv:astroph/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 threedimensional bars:
II. Investigation of the parameter space
Ch. Skokos,1P.A. Patsis,1E. Athanassoula2
1Research Center of Astronomy, Academy of Athens, Anagnostopoulou 14, GR10673 Athens, Greece
2Observatoire de Marseille, 2 Place Le Verrier, F13248 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 x1characteristic. This
folding is larger for lower values of the pattern speed. The elongation of rectangular
like orbits belonging to x1 and to x1originated 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 x1originated families, also families related to the zaxis 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 latetype barred spiral galaxies end at their inner
Lindblad resonance (hearafter ILR) has also been considered
(LyndenBell 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 edgeon 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 edgeon
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(rIILR) and Ej(vILR) are
the values of the Jacobian for the inner radial ILR and the
c ? 2001 RAS
Page 2
2
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(rIILR) is the Jacobian for the inner radial ILR,
Ej(vILR) 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(rIILR)
Ej(vILR)
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 x2x3 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
simpleperiodic 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 simpleperiodic 2D families
x1, x2 and x3. We will follow it counterclockwise. Starting
close to Ej = −0.5 for x = 0 we walk along the characteristic
of the typical x1 family. The orbits there are ellipticallike
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
Page 3
Orbital dynamics of threedimensional 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 ellipselike 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 peanutlike 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 pseudogap. 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 x3like 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 x13like 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 x3like. 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 steplike feature in the curve (Fig. 1b) and be
c ? 2001 RAS, MNRAS 000, 1–11
Page 4
4
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 quasicircular 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 quasicircular. 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 rectangularlike orbits
found in the 4:1 region (cf. Fig. 2g in paper I) are for this
model squarelike. 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 ellipticallike 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 ellipticallike 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 simpleperiodic families of the x1tree 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 quasicircular (or have quasicircular projec
tions on the equatorial plane). The rectangularlike orbits
in this case are almost squares. The model also includes a
simple periodic x2like 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 semimajor axis of the
largest x2 orbit is 0.63 kpc. This means, that the x2 orbits
c ? 2001 RAS, MNRAS 000, 1–11
Page 5
Orbital dynamics of threedimensional 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 x1characteristic 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 x1characteristic 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).
c ? 2001 RAS, MNRAS 000, 1–11
Page 6
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 rectangularlike
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 faceon 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 t2like 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 x1tree. 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 x1tree. This family exists
for Ej > −0.285 and is one of the families of periodic orbits
related to the zaxis family, i.e. to the 1D orbits on the rota
tional axis of the system. The well known bifurcations of the
zaxis family are the sao and uao families (Heisler, Merritt
c ? 2001 RAS, MNRAS 000, 1–11
Page 7
Orbital dynamics of threedimensional 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 zaxis 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 ntimes it can be considered as
nperiodic (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 nperiodic 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 zaxis 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 3periodic families and the family we discuss
here is one of them. The zaxis 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 zaxis family, we can find out from Eq. A8 the Ej values
at which 3periodic bifurcations will appear. Thus we know
that for Ej = −0.285, a bifurcation of the zaxis 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 zaxis 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 x1tree, 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 selfconsistent 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,
c ? 2001 RAS, MNRAS 000, 1–11
Page 8
8
Ch. Skokos et al.
Figure 14. Stability diagram for x1, x1v1 and x1v3 in model C.
these orbits, since they are bifurcations of the zaxis, 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 Ωκ/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 dasheddotted 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 rectangularlike, 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
c ? 2001 RAS, MNRAS 000, 1–11
Page 9
Orbital dynamics of threedimensional 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 Nbody 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 x1characteristic
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 x1tree bifurcate from x1 at an S→U transi
tion before the local maximum of the x1characteristic at the
radial 4:1 resonance and have initial conditions (x,z,0,0).
(iv) The bars can be supported not only by x1originated
families but, depending on the model, by 3D orbits bifur
cated from families related with the zaxis 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 rectangularlike 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 squarelike and farther out circles and orbits with
circularlike (x,y) projections. Thus in the slow bar case
the bar is supported only by ellipticallike orbits of the x1
tree. The different elongations of the rectangularlike orbits
c ? 2001 RAS, MNRAS 000, 1–11
Page 10
10
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
nonaxisymmetric part of the forcing is relatively larger near
corotation for the fast bar than for the slow one.
(vii) The decreasing part of the x1characteristic is in
most cases unstable, except for the strong bar case (model
D). This favours the presence of rectangularlike 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 rectangularlike orbits can also
be found in the case of the fast rotating model A3, where
rectangularlike 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 rectangularlike 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 x1tree.
(ix) In the x1tree 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 edgeon 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
LyndenBell 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, GauthierVillars, 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 19941999; 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 socalled 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)
c ? 2001 RAS, MNRAS 000, 1–11
Page 11
Orbital dynamics of threedimensional 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 1periodic. 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 1periodic 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
View other sources
Hide other sources
 Available from arxiv.org
 Available from Charalampos Skokos · May 31, 2014