Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models

Abstract

The distinction between chaotic and regular behavior of orbits in galactic
models is an important issue and can help our understanding of galactic
dynamical evolution. In this paper, we deal with this issue by applying the
techniques of the Smaller (and Generalized) ALingment Indices, SALI (and GALI),
to extensive samples of orbits obtained by integrating numerically the
equations of motion in a barred galaxy potential. We estimate first the
fraction of chaotic and regular orbits for the two-degree-of-freedom (DOF) case
(where the galaxy extends only in the (x,y)-space) and show that it is a
non-monotonic function of the energy. For the three DOF extension of this model
(in the z-direction), we give similar estimates, both by exploring different
sets of initial conditions and by varying the model parameters, like the mass,
size and pattern speed of the bar. We find that regular motion is more abundant
at small radial distances from the center of the galaxy, where the relative
non-axisymmetric forcing is relatively weak, and at small distances from the
equatorial plane, where trapping around the stable periodic orbits is
important. We also find that the variation of the bar pattern speed, within a
realistic range of values, does not affect much the phase space's fraction of
regular and chaotic motions. Using different sets of initial conditions, we
show that chaotic motion is dominant in galaxy models whose bar component is
more massive, while models with a fatter or thicker bar present generally more
regular behavior. Finally, we find that the fraction of orbits that are chaotic
correlates strongly with the bar strength.

Full text

Page 1

arXiv:1102.1157v2 [astro-ph.GA] 16 Mar 2011
Mon. Not. R. Astron. Soc. 000, 1–16 (2011) Printed 17 March 2011(MN LaTEX style file v2.2)
Regular and chaotic orbits in barred galaxies - I. Applying
the SALI/GALI method to explore their distribution in
several models
T. Manosa,b,c⋆and E. Athanassoulaa
aLAM, UMR6110, CNRS/Universit´ e de Provence, 38 rue Joliot Curie, 13388 Marseille C´ edex 13, France.
bCenter for Research and Applications of Nonlinear Systems, Department of Mathematics, University of Patras, GR–26500, Patras, Greece.
cUniversity of Nova Gorica, School of applied sciences, Vipavska 11c, SI-5270, Ajdovˇ sˇ cina, Slovenia.
Released 2011 March 16
ABSTRACT
The distinction between chaotic and regular behavior of orbits in galactic models is an
important issue and can help our understanding of galactic dynamical evolution. In this
paper, we deal with this issue by applying the techniques of the Smaller (and Gener-
alized) ALingment Indices, SALI (and GALI), to extensive samples of orbits obtained
by integrating numerically the equations of motion in a barred galaxy potential. We
estimate first the fraction of chaotic and regular orbits for the two–degree–of–freedom
(DOF) case (where the galaxy extends only in the (x,y)–space) and show that it is a
non–monotonic function of the energy. For the three DOF extension of this model (in
the z–direction), we give similar estimates, both by exploring different sets of initial
conditions and by varying the model parameters, like the mass, size and pattern speed
of the bar. We find that regular motion is more abundant at small radial distances
from the center of the galaxy, where the relative non-axisymmetric forcing is relatively
weak, and at small distances from the equatorial plane, where trapping around the
stable periodic orbits is important. We also find that the variation of the bar pattern
speed, within a realistic range of values, does not affect much the phase space’s fraction
of regular and chaotic motions. Using different sets of initial conditions, we show that
chaotic motion is dominant in galaxy models whose bar component is more massive,
while models with a fatter or thicker bar present generally more regular behavior.
Finally, we find that the fraction of orbits that are chaotic correlates strongly with
the bar strength.
Key words: galaxies: kinematics and dynamics - galaxies: structure
1 INTRODUCTION
Exploring the nature of orbits in galaxies constitutes a very
important issue, not only because of the evident astronom-
ical interest in classifying the types of orbits that exist in
such systems, but also because orbits are needed for con-
structing self-consistent models of galaxies. In order to study
and understand the structure and dynamics of a galaxy it
is necessary to identify the type of dynamics characterizing
the motion of stars (regular or chaotic) and estimate the
percentages of each type in the phase space of the galaxy.
From previous works that describe galaxies and their
star motion, it is well-known that the analysis of periodic or-
⋆E-mail:thanosm@master.math.upatras.gr (TM); lia@oamp.fr
(EA)
bits, and their stability, can provide very useful information
about galaxy structure. Stable periodic orbits are associated
with regular motion, since they are surrounded by tori of
quasi-periodic motion. Thus regular orbits are trapped in
the vicinity of the parent periodic orbit. On the other hand
unstable periodic orbits breed chaos. If we pick an orbit in
the immediate phase-space vicinity of a chaotic one, we find
that the two orbits will diverge exponentially with time.
Such chaotic orbits fill up all the phase space region that
is available to them. Nevertheless, recent results in galactic
dynamics show that there are chaotic orbits that can sup-
port galaxy features, even thin features like the outer parts
of bars, spirals and rings (Kaufmann & Contopoulos 1996;
Patsis et al. 1997; Patsis 2006; Romero-G´ omez et al. 2006,
2007; Voglis et al. 2006a,b, 2007; Contopoulos & Harsoula
2008; Athanassoula et al. 2009a,b,2010;

Page 2

2
Manos & Athanassoula
Harsoula & Kalapotharakos
These orbits are often called “sticky” and their true chaotic
nature takes very long to be revealed (Contopoulos 2002).
Athanassoula et al. (2010) termed these orbits “confined
chaotic orbits” because the structures they generate are well
confined in configuration space. There are also several recent
related results showing that strong local instability does not
mean diffusion in phase space (e.g. Giordano & Cincotta
2004; Cincota & Giordano 2008; Cachucho et al. 2010).
In our next paper, paper II, we will focus on this special
category of chaotic orbits discussing their significance from
an observational point of view.
Ferrers’ potentials (Ferrers 1877) have proven very ef-
ficient for studying the main properties of real bars. The
detection of periodic orbits and their stability in these
models have been studied in detail by many researchers
(e.g. Athanassoula et al. 1983; Pfenniger 1984; Skokos et al.
2002a,b; Patsis et al. 2002, 2003a,b). It is well-known that
stable periodic orbits of low period possess in their neighbor-
hood sizeable domains of quasiperiodic (or regular) motion.
A number of questions arises, therefore, concerning these
regions of stability: How far from the stable periodic orbit,
can regular motion be sustained? Where are chaotic regions
located in phase space and configuration space? What are
the model’s parameters that favor large islands of stability
around the main stable periodic orbits? In this paper we
plan to address such questions and to provide some tenta-
tive answers.
Chaos arises in many areas of dynamical systems and
its effect is generally related to the type of nonlinear terms
present in the model’s equations of motion (see Contopoulos
2002 for a review). The dynamics of stars in galactic po-
tentials is a particular case, where nonlinear terms play an
important role. The distinction between regular and chaotic
motion is not trivial and becomes more subtle in systems
of many DOF. Thus, it is necessary to use fast and precise
methods to identify the nature of orbits in such models.
Many methods have been developed over the years deal-
ing with this problem. The inspection of successive intersec-
tions of an orbit with a Poincar´ e surface of section (PSS)
(Lieberman & Lichtenberg 1992) has been particularly use-
ful for two DOF Hamiltonian systems. One of the most
popular methods of chaos detection is the computation of
the maximal “Lyapunov Characteristic Exponent” (LCE)
(Oseledec 1968; Benettin et al. 1980a,b; Skokos 2010). Along
the same lines as LCE, several other methods of chaos detec-
tion have been proposed based on the study of the behavior
of deviation vectors: We may mention, e.g. the “Fast Lya-
punov Indicator” (Froeschl´ e et al. 1997; Froeschl´ e & Lega
1998) and the “Mean Exponential Growth of Nearby Or-
bits” (Cincota & Sim´ o 2000; Cincota et al. 2003). On the
other hand, there also exist methods based on the anal-
ysis of time series constructed by the coordinates of each
orbit such as the “Frequency Map Analysis” (Laskar 1990;
Laskar et al. 1992; Laskar 1993). More details about these
and other relative methods can be found in the book of
Contopoulos (2002) and in the review of Skokos (2010) .
Inthe present paper,
the “Smaller ALingment
the properties of two deviation vectors of an orbit
for the quick and efficient distinction of chaotic mo-
tion, originally introduced by Skokos (2001). It has
2009; Tsoutsis et al. 2009).
we use a method called
Index”(SALI), based on
been successfully
tems (Skokos
2002; Skokos et al. 2003a,b; Kalapotharakos et al. 2004;
Skokos et al. 2004; Panagopoulos et al. 2004; Sz´ ell et al.
2004; Antonopoulos & Bountis 2006; Antonopoulos et al.
2006; Capuzzo-Dolcetta et al. 2006; Voglis et al. 2006a,
2007; Kalapotharakos et al.
Manos & Athanassoula2008;
Macek et al. 2010). In every case, it was confirmed to
be a fast and reliable indicator of the chaotic or ordered
nature of orbits. We also use SALI’s recent generalization,
the so-called “Generalized ALignment Index” (GALI)
introduced by Skokos et al. (2007), using a set of p initially
linearly independent deviation vectors of the system.
Thus, following more deviation vectors, we manage to
acquire more extra information about the complexity of
the regular motion, i.e. the dimensionality of the invariant
torus (or simply torus from now on) on which the orbit lies
(Skokos et al. 2008; Manos et al. 2008b,c; Bountis et al.
2009; Manos & Ruffo 2010).
The paper is organized as follows: Sections 2 and 3
are devoted to the chaos detectors we use, i.e. the Lya-
punov spectra and the SALI/GALI methods. In Section 4,
we present in detail the model, which is composed by a bulge,
a disk and a Ferrers bar (Ferrers 1877). In Section 5, we
present our results for the two DOF restriction of the gen-
eral model, calculating the regular and chaotic orbits and
constructing also charts of the different types of motion in
the associated phase space. Section 6 is dedicated to the
study of the full three DOF model in terms of computing
percentages of regular and chaotic orbits, selecting different
sets of initial conditions and varying the parameters of the
bar component. Thus, with the help of the SALI method,
we first distinguish the true nature of the studied orbits in
phase space (Section 7) and correlate the bar force with the
presence of larger amount of chaotic motion in phase space.
Finally, in Section 8, we summarize our conclusions.
applied to different dynamical sys-
2001; Skokos et al.2002; Voglis et al.
2008; Manos et al.
Str´ ansk´ y et al.
2008a;
2009;
2 LYAPUNOV EXPONENTS
The Lyapunov Characteristic Exponents (LCEs) or Lya-
punov Characteristic Numbers (LCN) or more simple
Characteristic Exponents are very important for the
study of dynamical systems, for distinguishing between
regular and chaotic behavior of orbits in phase space
(Benettin et al. 1980a,b; Lieberman & Lichtenberg 1992;
Pettini 2007; Skokos 2010). In practice, the LCEs describe
the rate of separation of infinitesimally close trajectories.
The mathematical definition of LCE relies on Oseledeˇ cs mul-
tiplicative theorem (Oseledec 1968).
A flow x(t) generated by an autonomous first-order sys-
tem is given by:
dx(t)
dt
= F(x(t)), (1)
where F is its velocity field. Let us consider a trajectory in
M–dimensional phase space together with a nearby trajec-
tory, with initial conditions x0 and x0+ ∆x0, respectively.
These evolve with time yielding the tangent vector ∆x(x0,t)
with its Euclidean norm:
d(x0) = ?∆x(x0,t)?. (2)

Page 3

Regular and chaotic orbits in barred galaxies - I.
3
Writing ∆x = (∆x1,...,∆xM) ≡ w, the time evolution for
w if found by linearizing (1), to obtain the variational equa-
tions:
dw
dt
= J(x(t))w, (3)
where J(x(t)) = ∂F/∂x is the Jacobian matrix of the F(x).
The mean exponential rate of divergence of two initially
close trajectories is:
σ(x0,w) = lim
t→∞(1
t)lnd(x0,t)
d(x0,0). (4)
It can be shown that σ exists and is finite. Furthermore,
there is an M–dimensional basis ˆ ei of w such that for any
w, σ takes one of the M (possibly non-distinct) values
σi(x0) = σ(x0, ˆ ei),∀i = 1,2,...,M, which are the Lya-
punov characteristic exponents. These can be ordered by
size σ1 ? σ2... ? σM.
3 THE METHOD OF THE SALI (AND GALI)
SPECTRA
Let us consider a Hamiltonian flow of N DOF, an orbit
in the 2N–dimensional phase space with initial condition
P(0) = (x1(0),x2(0),...,x2N(0)) and two deviation vectors
w1(0), w2(0) from the initial point P(0). In order to com-
pute the SALI for that orbit one has to follow the time evo-
lution of the orbit itself as well as the two deviation vectors
w1(t),w2(t) which initially point in two arbitrary directions.
The evolution of the deviation vectors is given by the varia-
tional equations (3) of the flow. At every time step the two
deviation vectors w1(t) and w2(t) are normalized by setting:
ˆ wi(t) =
wi(t)
?wi(t)?,i = 1,2 (5)
and the SALI is computed as (Skokos 2001):
SALI(t) = min{?ˆ w1(t) + ˆ w2(t)?,?ˆ w1(t) − ˆ w2(t)?}. (6)
The properties of the time evolution of the SALI rapidly
distinguish between ordered and chaotic motion as follows:
The SALI fluctuates around a non-zero value for ordered or-
bits, while it tends exponentially to zero for chaotic orbits
(Skokos et al. 2003a,b). In general, two different initial devi-
ation vectors become tangent to different directions on the
torus, producing different sequences of vectors, so that the
SALI always fluctuates around positive values. On the other
hand, for chaotic orbits, any two initially different deviation
vectors in time tend to align in the direction defined by the
maximal LCE (mLCE). Hence, they either coincide with
each other, or become opposite, which leads to the SALI
falling exponentially to zero. Thus, this completely different
behavior of the SALI helps us distinguish between ordered
and chaotic motion in Hamiltonian systems of any dimen-
sionality. An analytical study of SALI’s behavior for such
orbits was carried out in Skokos et al. (2004), where it was
shown that SALI ∝ e−(σ1−σ2)t, σ1,σ2 being the two largest
Lyapunov exponents.
The usual technique to decide whether an orbit can be
called chaotic or regular is to check, after some time inter-
val, if its SALI has become less than a very small threshold
value. In the following we will take this value to be equal
to 10−8. Depending on the location of the orbit, this limit
can be reached more or less fast, as there are phenomena
that can hold off the final characterization of the orbit, and
certain orbits behave as regular for long times before finally
drifting away from regular regions and starting to wander in
a chaotic domain.
The generalized alignment index of order p (GALIp) is
determined through the evolution of 2 ? p ? 2N initially
linearly independent deviation vectors wi(0),i = 1,2,...,p,
so it is related to the computation of many LCEs rather
than just the maximal one. The evolved deviation vectors
wi(t) are normalized every few time steps in order to avoid
overflow problems, but their directions are left intact. Then,
according to Skokos et al. (2007), GALIpis defined to be the
volume of the p-parallelogram having as edges the p unitary
deviation vectors ˆ wi(t),i = 1,2,...,p:
GALIp(t) =? ˆ w1(t) ∧ ˆ w2(t) ∧ ... ∧ ˆ wp(t) ? .
From the definition of GALIp it becomes evident that if at
least two of the deviation vectors become linearly dependent,
the wedge product in Eq. (7) becomes zero and the GALIp
vanishes.
In the case of a chaotic orbit, all deviation vectors tend
to become linearly dependent, aligning in the direction de-
fined by the maximal Lyapunov exponent and GALIp tends
exponentially to zero following the law:
(7)
GALIp(t) ∼ e−[(σ1−σ2)+(σ1−σ3)+...+(σ1−σp)]t,
where σ1 > ... > σpare approximations of the first p largest
Lyapunov exponents. In the case of regular motion, on the
other hand, all deviation vectors tend to fall on the N–
dimensional tangent space of the torus, where the motion is
quasiperiodic. Thus, if we start with p ? N general deviation
vectors, these will remain linearly independent on the N–
dimensional tangent space of the torus, since there is no par-
ticular reason for them to become aligned. As a consequence,
GALIp in this case remains practically constant for p ? N.
On the other hand, for p > N, GALIp tends to zero, since
some deviation vectors will eventually become linearly de-
pendent, following power laws that depend on the dimension-
ality of the torus. According to Christodoulidi & Bountis
(2006) and Skokos et al. (2008) one obtains the following
formula for the GALIp, associated with quasiperiodic orbits
lying on k–dimensional tori (where k is potentially equal to,
or smaller than the system’s size):
(8)
GALIp(t) ∼



constant,
1
tp−k,
1
t2(p−N),
if 2 ? p ? k
if k < p ? 2N − k
if 2N − k < p ? 2N.
(9)
In the case where k = N, GALIp remains constant for 2 ?
p ? N and decreases to zero as ∼ 1/t2(p−N)for N < p ? 2N.
An efficient way to calculate GALIp is by multiplying the
singular values zi,i = 1,...,p, computed through a Singular
Value Decomposition procedure of the matrix formed by the
deviation vectors ˆ wi,i = 1,...,p (Antonopoulos & Bountis
2006; Skokos et al. 2008):
GALIp =
p
?
i=1
zi. (10)
The method has been applied successfully in several Hamil-
tonian systems like the FPU lattice (Skokos et al. 2008) and
coupled symplectic maps (Bountis et al. 2009) for the de-
tection not only of regular and chaotic motion but also the

Page 4

4
Manos & Athanassoula
dimensionality of the torus on which a regular trajectory lies
on.
Practically, the SALI is equivalent to GALI2 and the
distinction between regular and chaotic motion in the 2D
Ferrers barred model (two DOF) can be done with either
one of them. For the full 3D version of the model, we also
use GALI3, depending on the properties of the model (if its
phase space is dominated by large chaotic or regular areas)
and on our goals. GALI3generally demands more CPU time,
since it is necessary to follow the evolution of three deviation
vectors instead of two. For the case of chaotic trajectories,
however, GALI3 decays much faster and the calculation can
stop well before the end of total time interval. Summarizing
therefore, if someone had to estimate the amount of chaotic
vs. regular regions in phase space, GALI3 would be more
efficient for models where chaos is dominant, while SALI
would be preferable for models with large regions of order. In
our runs, we have used both SALI (GALI2) for the general
description of chaotic vs. ordered regions, and GALI3 to
follow specific regular orbits and study the dimensionality
of the torus on which they lie.
4 THE MODEL POTENTIAL
The motion of a test particle in a 3D rotating model of a
barred galaxy is governed by the Hamiltonian:
H =1
2(p2
x+ p2
y+ p2
z) + V (x,y,z) − Ωb(xpy− ypx). (11)
The bar rotates around its z–axis (short axis), while the x–
direction is along the major axis and the y along the interme-
diate axis of the bar. The px,py and pz are the canonically
conjugate momenta, V is the potential, Ωb represents the
pattern speed of the bar and H is the total energy of the
orbit in the rotating frame of reference (Jacobi constant).
The corresponding equations of motion are:
˙ x = px+ Ωby,˙ y = py− Ωbx,˙ z = pz,
(12)
˙ px = −∂V
∂x+ Ωbpy,˙ py = −∂V
∂y− Ωbpx,˙ pz = −∂V
∂z.
The potential V of our model consists of three components:
(i) A disc,
(Miyamoto & Nagai 1975):
represented by a Miyamoto-Nagaidisc
VD = −
GMD
?
x2+ y2+ (A +√z2+ B2)2
, (13)
where MD is the total mass of the disc, A and B are its hor-
izontal and vertical scale-lengths, and G is the gravitational
constant.
(ii) A bulge, which is modeled by a Plummer sphere
(Plummer 1911) whose potential is:
VS = −
GMS
?x2+ y2+ z2+ ǫ2
s
,(14)
where ǫs is the scale-length of the bulge and MS is its total
mass.
(iii) A triaxial Ferrers bar (Ferrers 1877), the density ρ(x)
of which is:
?
0
ρ(x) =
ρc(1 − m2)2
,m < 1
,m ? 1, (15)
where ρc =
mass of the bar and
m2=x2
105
32π
GMB
abc
is the central density, MB is the total
a2+y2
b2+z2
c2,a > b > c > 0, (16)
with a,b and c being the semi-axes. The corresponding po-
tential is:
ρc
n + 1
λ
VB = −πGabc
?∞
du
∆(u)(1 − m2(u))n+1, (17)
where
m2(u) =
x2
a2+ u+
y2
b2+ u+
z2
c2+ u,
(18)
∆2(u) = (a2+ u)(b2+ u)(c2+ u), (19)
n is a positive integer (with n = 2 for our model) and λ is
the unique positive solution of:
m2(λ) = 1, (20)
outside of the bar (m ? 1), and λ = 0 inside the bar.
The corresponding forces are given analytically by Pfenniger
(1984).
This model has been used extensively for orbital stud-
ies by Pfenniger (1984), Skokos et al. (2002a,b) and by
Patsis et al. (2002, 2003a,b). Our so–called standard (S)
model has the followingvalues of parameters G=1,
Ωb=0.054 (54 km·sec−1·kpc−1), a=6, b =1.5, c=0.6, A=3,
B=1, ǫs=0.4, MB=0.1, MS=0.08, MD=0.82, both for its
two DOF and three DOF versions. The units used are: 1
kpc (length), 1000 km · sec−1(velocity), 1 Myr (time),
2 × 1011M?(mass). The total mass G(MS + MD + MB)
is set equal to 1.
Our study is mainly focused on understanding the ef-
fect of the bar on the dynamics of the model’s phase space,
by varying the bar mass MB, its semiaxis lengths b,c and
its pattern speed Ωb. In order to do this we have con-
sidered three more models: models C, B and M whose
short z–semiaxis (c–parameter), intermediate y–semiaxis
(b–parameter) and mass of the bar (MB–parameter) are
twice those of the initial reference model. For each of these
four (S,C,B and M) models we launch three different
sets of initial conditions (distributions I,II,III). Hence,
we consider the three standard models (IS,IIS,IIIS)
and their variations (IC,IIC,IIIC), (IB,IIB,IIIB) and
(IM,IIM,IIIM), depending on the varying parameter.
These sets of initial conditions will be discussed in detail
in Section 7. The maximal time of integration of the orbits
through the equations of motion and the variational equa-
tions is set to be T=10,000Myr (10 billion yrs), that corre-
sponds to a time less than, but of the order of one Hubble
time. We chose to take a maximum integration time that
is the same for all orbits, independent of their energy. As
a result, some of the orbits – and particularly the ones in
the innermost regions of the galaxy – may have been cal-
culated for a larger multiple of their own dynamical times
than others. This choice, however, is necessary because it

Page 5

Regular and chaotic orbits in barred galaxies - I.
5
allows comparison with, or applications to a galaxy or a
simulation, where all information refers to one time only,
the same for all orbits. It was therefore also adopted in
previous orbital structure studies in galactic models (e.g.
El-Zant & Shlosman 2002, 2003; Valluri et al. 2010). This
choice is essential in the case of some sticky orbits which may
appear regular if calculated till e.g. 10,000Myr, but show
clear signs of chaoticity when calculated for much longer
times, i.e. orbits whose diffusion time scale is equal to many
Hubble times. Such times, however, are irrelevant for our
applications, and thus these orbits should not be counted as
chaotic, even if in the long run they may become so. These
issues will be discussed further in Paper II.
The relative strength of the non-axisymmetric forces
can be estimated by the quantity Qt, defined as:
Qt(r) = (∂Φ(r,θ)/∂θ)max/(r∂Φ0/∂r), (21)
The maximum of Qt over all radii shorter than the bar ex-
tent is often referred to as Qb, and used as a measure of the
bar strength (e.g. Buta et al. 2003, 2004; Laurikainen et al.
2004; Buta et al. 2005; Durbala et al. 2009). From now on,
we will for simplicity and conciseness, often refer to the
“strong non-axisymmetric forcings” simply as “strong bars”.
In Fig. 1 we show the above quantity for the 4 models we are
going to study and discuss in this paper. From the quantity
Qt it becomes clear that the increase of the mass of the bar
component (model M: dotted line) gives rise to a “stronger
bar” compared to the initial standard model (model S: solid
line). On the other hand, an increase of the short z–semiaxis
(c–parameter - model C: dashed line) or intermediate y–
semiaxis (b–parameter - model B: dot-dashed line) has as
a result “weaker bars”. Hence, how do “strong bars” affect
the relative regular and chaotic motion in the phase space of
the full three DOF model? Before answering this question,
let us first study and comprehend the dynamics of the two
DOF model.
5THE TWO DOF MODEL POTENTIAL
The two DOF Ferrers model is described by the Hamiltonian
Eq. (11) setting (z,pz) = (0,0), i.e.:
H =1
2(p2
x+ p2
y) + V (x,y) − Ωb(xpy− ypx).(22)
In the two bottom inset figures of Fig. 2a, we present two
typical orbits in the (x,y)–plane of the standard model.
One is regular (bottom left inset), with initial condition:
(x,y,px,py) = (0,−0.625,−0.201,−0.06)
the other chaotic (bottom right inset), with initial condi-
tion: (x,y,px,py) = (0,−0.625,−0.002,−0.24)
both for the Hamiltonian value H = −0.360. Their qual-
itatively different behavior is also shown in the PSS in
Fig. 2a ((y,py)–plane). The chaotic orbit C (gray points)
tends to fill with scattered points the chaotic region, while
the ordered orbit R (black points) creates a set of points
that form a closed invariant curve, on the left part of
the picture. In the top inset figure of Fig. 2a, we show
the different morphologies of three regular orbits around
the periodic orbits in the centers of the three largest is-
lands of regular motion present in the associated PSS. The
model’s basic barred shape, in the (x,y)–plane is mainly
provided by trajectories around the main family of periodic
(orbit R) and
(orbit C),
Figure 1. The quantity Qtas a function of radius. Its maximum
value corresponds to Qb, which gives an estimate of the relative
strength of the non-axisymmetric forces. We see that the increase
of the mass of the bar component (model M: dotted line) corre-
sponds to “stronger bar” compared to the initial standard model
(model S: solid line). On the other hand, when we increase the
short z–semiaxis (c–parameter - model C: dashed line), or the in-
termediate y–semiaxis (b–parameter model B: dash-dotted line)
the bar becomes weaker.
orbits, the so–called x1 family (following the nomenclature
of Contopoulos & Papayannopoulos 1980) which are orbits
elongated along the bar, i.e along the x–axis in our case, and
inside corotation. The stability island on the right gives rise
to elliptical–like orbits, around the x2 family, elongated per-
pendicular to the bar, while the island on the left contains
orbits which are elliptical-like but retrograde, very slightly
elongated perpendicular to the bar and around the family
x4.
We then apply the SALI method to the orbits R and C
and present their different typical evolution behaviors: For
the chaotic orbit C (gray curve in Fig. 2b), the SALI tends
to zero (≃ 10−16) exponentially after some transient time,
while for the regular orbit R, it fluctuates around a positive
number (black curve in Fig. 2b). The corresponding mLCE
and σ1, for these orbits is shown in Fig. 2c, where the σ1 for
the regular orbit tends to zero, while for the chaotic orbit it
tends to a positive number after a long integration time.
This comparison shows clearly an advantage of SALI
(GALI2), namely its ability to detect the chaotic charac-
ter of an orbit faster than by simply following the mLCE,
which typically takes a long time to converge. Even in our
two DOF model, SALI starts to decrease exponentially af-
ter a relatively short time (in Fig. 2b, t ? 103) while the
corresponding mLCE σ1 starts to converge to some positive
non-zero value for t ? 103(see gray curve in Fig. 2c).
Exploiting the efficiency of SALI, we take initial con-
ditions on the (y,py)–plane (with x = 0) and calculate the
values of the index to detect very small regions of stability
(or instability) more globally. We are thus able to construct
a map of chaotic and regular regions, very similar to what
is depicted in a PSS, but with more accuracy and higher
resolution (albeit at a somewhat longer CPU time). Fur-
thermore, as a byproduct of our application, we obtain an

Page 6

6
Manos & Athanassoula
Figure 2. Analysis of orbits in the two DOF case: a) Poincar´ e surface of section in the (y,py)–plane for a regular (R–black color) and a
chaotic (C–gray color) orbit (H = −0.36). The points of the successive intersections of the regular orbit R with this plane create a closed
curve, while the points of the chaotic orbit C fill with scattered points all its available region of motion. The orbits can be seen in the
two inset figures in the bottom of the panel. In the top inset figure, we show examples of the three different morphologies of the regular
orbits around the three main periodic orbits in the centers of the three main islands of stability. Note that the basic barred shape in the
(x,y)–plane is mainly provided by trajectories from the central stable region, around the family of periodic orbits x1 (orbits elongated
along the bar’s major axis). The other two stability islands give either elliptical-like orbits elongated perpendicular to the bar orbits (right
island - family x2), or orbits which are elliptical-like but retrograde with respect the bar’s pattern speed (left island - family x4). b) The
corresponding evolution of the SALI for the R and C orbits of panel a. For the chaotic one C (gray line), the SALI decays exponentially
fast to zero, while for the ordered one R (black line) it fluctuates around a non-zero number. c) The corresponding evolution of the
maximal Lyapunov exponents σ1 for the orbits R and C. For the chaotic orbit C (gray line) this tends after some transient time to a
non-zero value, while for the regular orbit R (black color) it tends linearly to zero. Note that the Log(SALI) in y–axis of panel b ranges
from -16 to 2, while the Log(σ1) in panel c ranges from -5 to 0.
accurate estimate of the percentage of regular to chaotic or-
bits on a surface of section of the given energy.
For example, choosing different values of the energy and
using a sample of 50,000 initial conditions equally spaced
on the same (y,py)–plane, we plot on the left column of
Fig. 3 the PSS for H = −0.360,−0.335,−0.300,−0.260 and
on the right the corresponding final SALI values obtained
from the selected grid of initial conditions, where each point
is colored according to its SALI values at the end of the
integration. In the SALI plots, the light gray color cor-
responds to regular orbits, the black color represents the
chaotic orbits/regions, while the intermediate gray shades
between the two represent orbits with small rate of local
exponential divergence, and/or orbits whose chaotic nature
is revealed after long times, like for example the so-called
sticky orbits, i.e. orbits that ”stick” onto quasiperiodic tori
for long times. Note that the fraction of these orbits (which
lie mainly around the borders of the islands of stability) is
very small (few % of the total amount of initial conditions)
and can be discerned by eye in Fig. 3 only if one focuses on
these particular regions. Thus, calculating from all these ini-
tial conditions the percentage of regular orbits, we are able
to follow how this fraction varies as a function of the total
energy. This problem was already addressed for four mod-
els by Athanassoula et al. (1983) but with fewer orbits, i.e.
larger error bars. In the present model, the chosen energy
values start from a regime of total order, at H = −0.46, up
to values near the escape energy, Hesc = −0.20. The distinc-
tion between chaotic and regular orbit is that SALI< 10−8
for chaotic orbits and ? 10−8for regular ones. In the latter
range one does, of course, include “sticky” chaotic orbits as
well. As is clear from Fig. 4, although the percentages of the
regular orbits decreases sharply as the energy grows above
Figure 4. Percentages of regular orbits for several values of
the energy (H = -0.460,-0.435,-0.410,-0.360,-0.335,-0.300,-0.260,-
0.240) in the two DOF Ferrers model. Note that while the fraction
of the regular orbits decreases sharply as the energy grows above
H = -0.4, this tendency is reversed at higher energy values.
H = −0.4, this tendency is reversed at higher energy values.
Clearly, for H > −0.3, despite the fact that many stability
islands have vanished, the size of the ordered domain in-
creases because the main big island on the left of the PSS
(see Fig. 3f,h) which corresponds to retrograde orbits, has
become larger.

Page 7

Regular and chaotic orbits in barred galaxies - I.
7
Figure 3. The PSS (left column) and SALI (right column) methods are in good agreement when surveying the same projection of the
phase space. In the left column we show the PSS for the two DOF model potential with H = −0.360 (panel a), H = −0.335 (panel c),
H = −0.300 (panel e) and H = −0.260 (panel g). In the right column we present regions of different values of the SALI for 50,000 initial
conditions on the (y,py)–plane for the same values of the Hamiltonian (panels b,d,f and h, respectively). The light gray colored areas
correspond to regular orbits, while the dark black ones to chaotic. Note the excellent agreement between the two methods as far as the
gross features are concerned, as well as the fact that the SALI can easily pick out small regions of stability which the PSS has difficulties
detecting, like those around the central islands in panels b,d,f and h.

Page 8

8
Manos & Athanassoula
6THE THREE DOF MODEL POTENTIAL
We now turn to the three DOF model and begin a study
of ordered and chaotic domains. Before discussing the more
general results about the study of the phase space of the
model, we first present in detail the behavior of our chaos
indicators for some typical orbits. Let us choose a regular
and a chaotic orbit and compare the efficiency of the GALI
method with that of the mLCE’s.
In Fig. 5a,b we show the behavior of a chaotic
orbit (C1), with initial condition: (x,y,z,px,py,pz)
(0.5875,0.0,0.33333,0.0,0.2,0.0)
bit (R1), with initial condition: (x,y,z,px,py,pz)
(0.97917, 0,0.04167, 0,−0.17778,0). For the chaotic one, the
GALI2 (or SALI) has become almost zero for t ≃ 104. How-
ever, as it is clear in Fig. 5a, the higher order GALIk, k =
5,6, indicate the chaoticity of the orbit already by t ≃ 103,
since they have reached very small values at that time. Note
the good agreement between the predicted slopes related
to its two largest Lyapunov exponents (σ1 ≈ 0.00910 and
σ2 ≈ 0.00345) given by Eq. (8). In Fig. 5b we show the GALI
indices for the regular orbit R1, where both GALI2,3 ∝ con-
stant and the GALI4,5,6 decay following the power laws:
=
and ofa regularor-
=
GALI4(t) ∝1
obtained from Eq. (9) for m = 3, indicating regular motion
which lies on a 3D torus.
One example of a low dimensional motion (lower than 3)
is provided by the regular orbit (R2), with initial condition:
(x,y,z,px,py,pz) = (0.5875,0.0,0.29770, 0.0,0.33750, 0.0).
In Fig. 5c, we show the behavior of the Log(GALI) indices
and their slopes for the regular orbit R2. Note that only
GALI2 remains constant while the GALI3,4,5,6 tend to zero
following power laws predicted by Eq. (9):
t2, GALI5(t) ∝1
t4, GALI6(t) ∝1
t6, (23)
GALI2(t) ∝ const.,
GALI5(t) ∝1
for m = 2. This implies that the orbit’s motion lies on a 2D
torus even though in three DOF Hamiltonian systems one
generally expects the dimension of the torus to be three.
In Fig. 6 we show the corresponding (x,y) and (x,z)
projections for the above C1 (1st column), R1 and R2 orbits
(2nd and 3rd column respectively). The chaotic one fills up
its available regime. Regarding the two regular ones, there
is a clear difference reflected in their projections. The “com-
plexity” of R1 (quasiperiodic motion on a 3D torus) is more
pronounced than the one of the orbit R2 (quasiperiodic mo-
tion on a 2D torus). Thus, in such cases GALI3 offers an
extra advantage in helping us detect different “degrees” of
regularity and does not serve only to distinguish between
chaotic and regular motion.
GALI3(t) ∝1
GALI6(t) ∝1
t,
GALI4(t) ∝1
t2,
t4,
t6, (24)
7 THE DISTRIBUTION OF ORBITS IN
PHASE SPACE
In this section we focus on the detection and quantification
of the different kinds of orbital motion (regular and chaotic)
in phase (and configuration) space, as some parameters of
the bar component vary. A similar study, in different poten-
tials, was done in El-Zant & Shlosman (2002, 2003), using
Figure 5. Distinguishing chaotic from regular motion with GALI
method. a) Exponential decay of all GALIs for the chaotic orbit
C1 (lin-log scale) with the predicted slopes related to its two
largest Lyapunov exponents (σ1≈ 0.00910 and σ2≈ 0.00345). b)
Slopes of GALIs for the regular orbit R1 showing that its motion
lies on a 3D torus, where GALI2,3∝ constant while the GALI4,5,6
decay following a power law. c) Slopes of GALIs for the regular
orbit R2 lying on a 2D torus with only GALI2∝ constant.
the Lyapunov spectra as a tool. Here we concentrate mainly
in the bar component and in understanding how its differ-
ent properties can affect the general stability of the system.
We use a relatively large number of trajectories (50,000),
more than one set of initial conditions for the survey of the
phase space, and the SALI method as a chaos detector for

Page 9

Regular and chaotic orbits in barred galaxies - I.
9
Figure 6. Two projections ((x,y) and (x,z)–planes) of the chaotic orbit C1 (1st column) and the quasiperiodic orbits R1 (2nd column)
and R2 (3rd column). Note the different “complexity” between R1, whose regular motion lies on a 3D torus, and R2, which lies on a 2D
as GALI indices detected.
the reasons discussed in the previous sections. We discuss
the relative fraction of regular and chaotic motion not only
as a function of its spatial location but also as a function of
the total energy and to correlate it with the bar strength,
i.e. the relative non-axisymmetric forcing.
7.1 Initial conditions
We now need to define the sample of orbits whose properties
(chaos or regularity) we will examine. The best of course
would have been to draw these from a distribution function
of the model. This, however, is not available, so we choose
initial conditions on different grids in phase space or from
the density distribution and the available energy range in
the model and we measure the corresponding fraction of
different kinds of motion. These percentages can depend on
the choice of the sets of initial conditions and a priori are
not expected to be equal. For this reason, before varying any
other parameter we should consider as carefully as possible
what initial conditions to choose and how well this choice
“spans” the allowed phase system of the system.
We first considered sets of orbits in such a way as to
favor the dynamics near the main family of periodic orbits
x1 (building blocks of the bar). To this end, we launched
initial conditions along the bar’s major x–axis with posi-
tive momenta py and position (z) or momentum values (pz)
in the z–direction (distributions I and II below). We also
considered initial conditions which follow the model’s total
Table 1. Varying parameters for all sets of initial conditions.
distribution/model
MB
bc
IS,IIS,IIIS
0.11.50.6
IC,IIC,IIIC
0.1 1.51.2
IB,IIB,IIIB
0.1 3.0 0.6
IM,IIM,IIIM
0.21.5 0.6
density ρ in positions (x,y,z) while their positions are arbi-
trary (distribution III below). More specifically, the three
different classes (distributions) of initial conditions we use
here are:
• distribution I: 50,000 orbits equally spaced in the space
(x,z,py) with x ∈ [0.0,7.0], z ∈ [0.0,1.5], py ∈ [0.0,0.45] and
(y,px,pz) = (0,0,0).
• distribution II: 50,000 orbits equally spaced in the
space (x,py,pz) with x ∈ [0.0,7.0], py ∈ [0.0,0.35], pz ∈
[0.0,0.35] and (y,z,px) = (0,0,0).
• distribution III: 50,000 orbits whose spatial coordi-
nates are chosen randomly from the density distribution of
our model, using the rejection method (Press et al. 1986)
within the rectangular box −a ? x ? a, −b ? y ? b,
−c ? z ? c. We then draw a random value of the Hamilto-

Page 10

10
Manos & Athanassoula
Figure 7. Percentages of regular orbits as a function of (a) the energy, (b) the averaged < |z| > value, (c) the initial spherical radius
Rspherical, and (d) the averaged < Rspherical> radius of the orbit’s evolution, for models IS, IB, IC and IM.
nian H (total energy) in the range [0, Hmax], where Hmax
is 1.1 times the escape energy Hesc = −0.201defined by the
zero velocity curve going through the L1 or L2 Lagrangian
point. Then we set (py,pz) = (0,0) and we calculate the px–
momenta by the relation: px = H(x,y,z,py,pz) (keeping the
positive solution). Note that in this case instead of giving
py–momenta we tried px. This choice allows us to check how
a different way of populating the phase space may affect the
relative fraction of regular and chaotic motion. This third
distributions has a better representation of the density, but
still is arbitrary with respect to the velocities.
Details of our distributions/models can be found in Ta-
ble 1. In the next sections we start by studying the dis-
tribution of regular and chaotic motion in phase space, for
different choices of initial conditions, different bar parame-
ters (sizes, mass and pattern speed). We also briefly study
the distribution as a function of energy and of location in
configuration space.
1The reason we choose an extended range of energies is because
we wish to study some orbits that are in this range but don’t
necessarily escape during our adopted integration time.
7.2Percentages of regular orbits vs. energy
We next examine the variation of the percentages of regular
and chaotic orbits as a function of the energy and we show
the results in Fig. 7a. We first divided the energy interval
into 30 subintervals, each containing an equal number of or-
bits. In every subinterval we calculated the percentages of
regular and chaotic orbits. In Fig. 7a we plot the percentage
of regular orbits for energies up to H ≃ −0.15, i.e. up to
a value somewhat beyond the escape energy for all models
under study. Generally, we observe that the fraction of reg-
ular orbits decreases as the energy increases. An important
peak, however, occurs before the escape energy, so that for
(H > −0.20) the fraction of regular orbits increases again!
This non–monotonic behavior is related to the growth of
the islands around some basic stable periodic orbits in the
phase space and is similar to what was observed in the two
DOF model, comparing the fraction of regular orbits for the
common energy interval, i.e. up to H ? −0.24 (see Fig. 4).
Picking another set of initial conditions we find similar
qualitative results with some small quantitative differences.
Hence, we find the same behavior when studying the models
of distribution II, i.e. regular motion is dominant for small
generally energies, then a decrease and then a peak. Dis-
tribution III manifests the same trend for small energies
while the peak is less intense compared to the other two
distributions.

Page 11

Regular and chaotic orbits in barred galaxies - I.
11
Figure 8. Slices of the (x,py)–plane for different z values. Dark regions correspond to chaotic initial conditions, light gray to regular
ones and intermediate colors to orbits with a small rate of local exponential divergence, and/or orbits whose chaotic nature is revealed
after long times, like for example the so-called sticky orbits. The gray-scale bar represents the values of the SALI in logarithmic scale.
7.3 Spread of regular orbits in configuration space
We also explored the way in which regular and chaotic orbits
are distributed along the z–direction of the configuration
space. Following the evolution of each orbit, we calculated
the mean absolute value of their z–coordinate (< |z| >) and
plot in Fig. 7b the fraction of regular orbits as a function of
the < |z| > for the models of distribution I. It clearly fol-
lows from these results that the intervals relatively near the
(x,y)–plane (< |z| > < 0.35) contain mainly regular orbits,
while those with larger values of mean absolute distance z
are mostly chaotic. Similar qualitative results are obtained
for distribution II (not shown here). Checking distribution
III we find good match for the relative fraction of regular
motion for relatively small < |z| > (up to 0.4) with the other
models. For intermediate values (0.4 < < |z| > < 1.5) the
motion is mostly chaotic while we don’t have orbits reaching
larger values, which is due to the way the initial conditions
are given.
We also looked at these percentages as a function of
the initial spherical radius (Rspherical =
and the mean spherical radius over the evolution of the
orbits (< Rspherical >). Again, dividing the total range
of the Rspherical values in 30 subintervals, we calculated
the percentages of regular orbits as a function of the mean
Rspherical value in each subinterval. We find that the frac-
tion of regular orbits for all models decreases sharply with
increasing Rsphericalup to Rspherical< 1.5, where it reaches
a minimum (see Fig. 7c). For 1.5 < Rspherical< 7 it starts to
increase gradually almost monotonically for model IS and
with some rather week fluctuations for IC. Note that the
model IB possesses a relatively large fraction of regular mo-
tion within the interval 2.5 < Rspherical< 4.5 as well. Model
IM follows a similar trend, but with a different scale than
IB, containing its larger fraction of regular motion in the
same roughly intervals, while for larger Rspherical the mo-
tion is mainly chaotic. This result is in good agreement with
the results in Fig. 7d, where the horizontal axis corresponds
to the value of the mean spherical radius over the evolution
time of the orbits. As previously, distribution II confirms
?x2
0+ y2
0+ z2
0)

Page 12

12
Manos & Athanassoula
well this behavior while distribution III is in good agree-
ment for relatively small radial distances from the center of
the galaxy and diversifies for larger ones.
7.4 Fraction of regular motion as a function of
distance from the equatorial plane
We then examine whether the orbital motion is more “reg-
ular” near the (x,y)–plane or far from it and how this re-
lates to their behavior as a function of < |z| > discussed
in the subsection 7.3 and Fig. 7b. For this, we took the set
of initial conditions from distribution I (version IC in this
example) with initial conditions given in the space (x,z,py)
and create a mesh in the (x,py)–plane with different slices
in the z–direction that start from z = 0 up to z = 1.25 with
step = 0.25. In Fig. 8, we see that as the distance of the
initial conditions from the equatorial plane increases (from
top to bottom row and from left to right panels), large is-
lands of stability in the (x,py)–plane start to shrink, when
x is between 0 and 2. This trend is almost monotonic up to
the z–slice z ≈ 1 (Fig. 8, third row, left panel). For z = 1.25
(Fig. 8, third row, right panel) the stability region, lying be-
tween x = 1.5 and x = 2 starts to grow. This result turns
out to be in good agreement with the one in Fig. 7b, where
we plot the percentages of regular orbits as a function of
< |z| >. This trend, as expected, is similar for the other
two different sets of initial conditions (distributions IIC and
IIIC), since the model parameters remain the same.
7.5 Percentages of regular orbits vs pattern speed
Furthermore, choosing one class of initial conditions (distri-
bution IS), we compare different models for several values
of the bar’s pattern speed Ωb, keeping the remaining param-
eters constant. In this study, we try to explore the effect of
the value of the pattern speed on the percentages of globally
ordered or chaotic behavior of the system.
Based on the structure and crowding of the periodic or-
bits, Contopoulos (1980) showed that bars have to end be-
fore corotation, i.e. that rL > a, where rL the Lagrangian,
or corotation, radius. A similar result was reached from the
calculation of the self-consistent response to a bar forcing
(Athanassoula 1980). These two results are very useful, since
they set a lower limit to the corotation radius, which can
thus not be smaller than the bar length, but give no in-
formation on an upper limit. Comparing the shape of the
observed dust lanes along the leading edges of bars to that
of the shock loci in hydrodynamic simulations of gas flow in
barred galaxy potentials, Athanassoula (1992a,b) was able
to set both a lower and an upper limit to the corotation ra-
dius, namely rL = (1.2 ± 0.2)a. This restricts the range of
possible values of the pattern speed for our model, from
Ωb = 0.0367, that corresponds to the Lagrangian radius
rL = 1.4a, to Ωb = 0.0554, that corresponds to rL = 1.0a.
Using these extreme values, and the three intermediate fre-
quencies: Ωb = 0.0403, Ωb = 0.0444 and Ωb = 0.0494, that
correspond to the Lagrangian radii rL = 1.3a, rL = 1.2a
and rL = 1.1a, respectively, we investigated how the value
of the pattern speed affects the system and found that as
Ωb increases the percentage of the regular orbits decreases
relatively weakly. The corresponding results are given in Ta-
ble 2. It turns out that the variation of the pattern speed
Table 2. Models with different pattern speeds Ωb.
modelΩb
rL
% of Regular orbits
IΩ1
0.0367032 1.4a 28.39
IΩ2
0.04030141.3a 27.63
IΩ2
0.04443651.2a 27.26
IΩ4
0.04936541.1a26.55
IΩ5
0.0554349 1.0a25.55
does not affect drastically the relative fraction of regular
and chaotic motion. One would probably need to try more
extreme values of Ωb, beyond the upper and lower limits of
the corotation radius, in order to see a significant change.
Such values, however, would be unrealistic for real galaxies.
7.6Fraction of regular and chaotic trajectories
In this section and in Fig. 9, we investigate the amount of
regular motion in the phase space for all the three sets of
initial conditions (distributions I, II and III), varying the
axial ratios (b/a or c/a parameters) and the mass (MB pa-
rameter) of the bar component, as described in Table 1. In
all distributions we used 50,000 initial conditions and we
employed the bootstrap–method (Press et al. 1986) on sev-
eral subsamples to make sure that this number is sufficiently
high to provide reliable information. Computing the distri-
butions of regular and chaotic orbits for the above classes of
initial conditions we find the following:
7.6.1 Percentages for distribution I
Before starting with the full 3D model, we first measured the
percentages of regular orbits in the 2D model. We use initial
conditions equally spaced in (x,py), with (y,px) = (0,0),
x ∈ [0.0,7.0] and py ∈ [0.0,0.45]. We find relative fractions
of regular motion equal to 45.40% for model IM, 53.43%,
for model IS, 92.43% for model IB and 64.43% for model
IC. This shows that at least in the 2D case, the massive
bar model (IM) has the highest fraction of chaotic orbits,
followed by the standard distribution IS. The two models
with the extended short (IC), or intermediate axis (IB)
have the least chaos.
Let us now turn to the full 3D orbital coverage. In
Fig. 9a we present the percentages of the regular orbits for
the various sets of initial conditions of the distribution I.
Comparing the percentages, we see that increasing the mass
of the bar in model IM increases chaotic behavior. This con-
firms the results obtained above and in Athanassoula et al.
(1983) for the two DOF case. On the other hand, when the
bar is thicker (model IC), i.e. the length of the z–axis larger,
the system becomes more regular. Finally, the correspond-
ing results for a fatter bar, i.e. with larger y–axis (model
IB) demonstrate that the increase of the intermediate axis
of the bar also provides the system with more ordered be-
havior. Comparing the 2D and 3D cases presented above,
we note that the trends we find are the same, but that the

Page 13

Regular and chaotic orbits in barred galaxies - I.
13
Figure 9. Percentages of regular orbits (SALI? 10−8) for distribution I (left panel), II (middle panel) and III (right panel) of initial
conditions for all the different model versions, where the bar mass and the length of its intermediate and minor (vertical) semi-axes are
varied. Generally, when the bar mass increases the model tends to be more chaotic, while when the size of the bar semi-axes (b or c
parameters) increases the fraction of regular motion grows. The small difference in the trends between models IS → IC and IIIS → IIIC
is due to the different way in which the sample of initial conditions in phase space is chosen.
relative fractions of regular orbits are somewhat higher in
the 2D case. We will discuss this further towards the end of
the section.
7.6.2Percentages for distribution II
Similarly, in Fig. 9b, we present percentages of the regular
orbits for initial conditions selected for distribution II and
models S, M, B and C. As before, the increase of the bar
mass (model IIM) causes more extensive chaos, while for a
thicker bar or fatter bar (models IIB and IIC), the system
is again more regular. It is thus clear that the results of
distributions I and II are similar, presumably dues to the
fact that they contain initial conditions that cover the phase
space in a similar manner, i.e. they favor the region around
the x1 family of periodic orbits.
7.6.3 Percentages for distribution III
Finally, we plot in Fig. 9c the percentages of regular orbits
for several sets of initial conditions from distribution III.
A first general observation is that the fraction of ordered
motion is significantly smaller than is distributions I and II,
for all models. The basic reason for this difference is related
to the way that the momenta of the positions are given in
this distribution of initial conditions. In particular, all the
orbits in this case are launched with a pxmomentum instead
of a py as in distributions I and II. As a consequence, the
big island of stability around the main family of periodic
orbits x1 is populate with fewer orbits and thus in general
more chaotic orbits are launched and measured. Neverthe-
less, again the increase of the bar mass in the IIIM model
results in more chaotic behavior. As for the IB model, a
thicker bar in the y–direction turns out to make the sys-
tem more regular. Regarding the case of larger z–semiaxis
(model IIIC), we notice a small decrease in the percent-
ages of regular orbits (9.74% now, from 10.46%). It turns
out that this particular different distribution of initial con-
ditions doesn’t reveal quite the same trend as in the previous
cases (IS → IC and IIS → IIC).
Figure 10. The relative fraction of chaotic motion as a func-
tion of bar strength Qbincreases (see Fig.1). Filled squares (?)
correspond to the models M, S, C and B (from right to left,
respectively) for distribution I. while filled triangles (?) are for
distribution II and filled circles (•) for distribution III.
Concluding this section dedicated to the dynamical
study of regular and chaotic motion in the phase space
is that their corresponding fraction is basically related to
the way that one populates the main stable (or unsta-
ble) periodic orbits of the system. We find that motion
near the plane is generally stable (non chaotic) and that
trajectories spending the largest fraction of time at large
z are usually chaotic (with large Lyapunov exponents).
These results are in agreement with the results obtained by
El-Zant & Shlosman (2002, 2003) for different models. The
increase of the bar mass leads to more chaotic motion in the
phase space as found in Athanassoula et al. (1983) for a two
DOF model, while in general the increase of the bar y or
z–semiaxis leads to more regularity.

Page 14

14
Manos & Athanassoula
7.7 The effect of bar strength
Reviewing the above percentages of regular (and chaotic)
motion in phase space, we see that they change in the same
way when the basic parameters and properties of the model
are varied. In order to understand this better we examine
next the effect of the bar strength, measuring it as dis-
cussed in Section 4, i.e. from the relative strength of the
non-axisymmetric forcing. In Fig. 10 we plot the relative
fraction of chaotic motion as a function of bar strength Qb.
Black filled squares correspond to distribution I, filled trian-
gles to distribution II and filled circles to distribution III.
This figure reveals that, provided the choice of initial condi-
tions for the orbits is the same or similar, there is a tight cor-
relation between the two quantities. the amount of chaos in-
creases with increasing bar strength, as intuitively expected.
What is unexpected, however, is how tight all these correla-
tions are, with rank correlation coefficients ∼ −0.936 for all
distributions.
Fig. 10 also shows that, except for the bar strength,
the distribution of initial conditions chosen is also impor-
tant. Distributions I and II differ little and their best fit-
ting straight lines are roughly parallel and little displaced
from each other. This is in good agreement with our pre-
vious statement (subsect. 7.1) that they both cover well
the regular orbits trapped around the stable orbits of the
x1 family, which are in fact the backbones of the bar
(Athanassoula et al. 1983). On the other hand, distribution
III has a much higher fraction of chaotic orbits, as could
also be seen from Fig. 9c, attesting that the correspond-
ing distribution of initial conditions is more chaotic, for the
reasons we already described.
8 CONCLUSIONS
In this paper, paper I, we studied the detection and distri-
bution of regular and chaotic motion in the phase space of
barred galaxy models. Our results referring to the dynam-
ical properties of the 2D/3D model with a Ferrers bar can
be summarized as follows:
(i) We applied successfully the GALI method to distin-
guish both qualitatively and quantitatively between regular
and chaotic orbits. We showed its efficiency and its advan-
tage in detecting fast and accurately the chaotic nature of a
trajectory.
(ii) In 2D systems, using the SALI (GALI2) method, we
were able to identify efficiently and fast tiny regions of regu-
lar or chaotic motion, which are not clearly visible on PSSs.
(iii) Using GALI2,3 we detected regular motion in low di-
mensional tori, i.e. examples of regular orbits of the three
DOF Ferrers’ model that lie on a 2D torus, while the torus’
expected dimension is generally three. Concerning chaotic
orbits, the GALI3,4,5,6 indices decay faster than SALI and
are able to detect chaos at early times well before this is
evident from the calculation of the mLCE.
(iv) We also tried different models and different distribu-
tions of initial conditions for the orbits to test the fraction of
regular and chaotic motion in various cases. We found that
regular orbits are generally dominant at relatively small ra-
dial distances from the center of the galaxy and at small
distances from the equatorial plane.
(v) We tested the effect of varying the bar pattern speed
for our standard model and found that, within a realistic
range of values, this parameter does not affect much the
phase space dynamics. Within these limits, we found that
the fraction of regular motion varies only by about 3 per
cent, in the sense that the bars at the slow limit of the
realistic range have more regular motion than the bars in
the fast limit.
(vi) We varied the values of the main parameters of our
models, such or the mass and the axial ratio of the bar com-
ponent (in the 3D model). Chaos turns out to be dominant in
galaxy models whose bar component is more massive, while
models with a thicker or fatter bar present generally more
regular behavior although the initial conditions are given
can in general affect somewhat the relative percentages.
(vii) We found a very strong correlation (rank correlation
coefficient at ∼0.94) between the fraction of chaotic orbits
and the relative strength of the non-axisymmetric forcing.
This holds for all our three initial conditions distributions
taken individually, even though they were specifically cho-
sen so as to represent different types of orbits. We can thus
conclude that strong non-axisymmetric forcings (i.e. strong
bars) are the main cause of the presence of large amount of
chaotic motion in phase space.
In our next paper of this series (paper II) we focus on
the significance of the “different degrees” of chaotic mo-
tion, like the strong and weak chaos, and their significance
from an astronomical point of view. The fact that many
“sticky”/chaotic orbits often persist and behave in a reg-
ular manner for very large time intervals, before showing
their chaoticity, makes them astronomically important for
supporting the galaxy structure. We present there a first
approach in filtering these weak chaotic orbits from the
strongly chaotic ones and classify them observationally as
effectively ordered motion.
9 ACKNOWLEDGEMENTS
We would like to thank Ch. Skokos, T. Bountis, A. Bosma,
M. Romero-G´ omez and P.A. Patsis for their fruitful com-
ments and discussions on this work. We also thank the ref-
eree, P.M. Cincotta, for his report, which helped us im-
prove the clarity of the paper. T. Manos acknowledges the
“Karatheodory” graduate student fellowship No B395 of the
University of Patras and funds from the Marie Curie fel-
lowship No HPMT-CT-2001-00338 to Marseille observatory.
Visits between the Marseille and Patras team members were
partially funded by the region PACA in France. This work
was partially supported by the grant ANR-06-BLAN-0172.
REFERENCES
Antonopoulos, Ch., Bountis, T., 2006, Phys. Rev. E, 73,
056206
Antonopoulos, Ch., Bountis, T., 2006, ROMAI Journal,
2(2), 1
Antonopoulos, Ch., Bountis, T., Skokos, Ch., 2006, Int. J.
Bif. Chaos, 16(6), 1777
Athanassoula, E. 1980, A&A, 88, 184

Page 15

Regular and chaotic orbits in barred galaxies - I.
15
Athanassoula, E., Bienayme, O., Martinet, L., Pfenniger,
D., 1983, A&A, 127, 349
Athanassoula, E., 1992a, MNRAS, 259, 328
Athanassoula, E., 1992b, MNRAS, 259, 354
Athanassoula, E., Romero-G´ omez, M., Masdemont, J. J,
2009a, MNRAS, 394, 67
Athanassoula, E., Romero-G´ omez, M., Bosma, A., Masde-
mont, J. J., 2009b, MNRAS, 400, 1706
Athanassoula, E., Romero-G´ omez, M., Bosma, A., Masde-
mont, J. J., 2010, MNRAS, 407, 1433
Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.,
1980, Meccanica, 9
Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.-M.,
1980, Meccanica, 21
Bountis, T., Skokos, Ch., 2006, Nucl. Instr. Meth. - Sect.
A, 561, 173
Bountis, T., Manos, T., Christodoulidi, H., 2009, J. Comp.
Appl. Math., 227, 17
Buta, R., Block, D. L., Knapen, J. H., 2003, AJ, 126, 1148
Buta, R., Laurikainen, E., Salo, H., 2004, AJ, 127, 279
Buta, R., Vasylyev, S., Salo, H., Laurikainen, E., 2005, AJ,
130, 506
Capuzzo-Dolcetta, R., Leccese, L., Merritt, D., Vicari, A.,
2007, ApJ, 666, 165
Christodoulidi, H., Bountis, T., 2006, ROMAI Journal,
2(2), 37
Cachucho, F., Cincotta, P. M., Ferraz-Mello, S., 2010,
CMDA, 108, 35
Cincota, P. M., Sim´ o, C., 2000, Astron. Astroph. Suppl.
Ser., 147, 205
Cincota, P. M., Giordano, C. M., Sim´ o, C., 2003, Physica
D, 182, 151
Cincotta, P. M., Giordano, C. M., 2008, In the book Non-
linear Phenomena Research, Nova Science Publishers, 319
Contopoulos, G., Papayannopoulos, Th., 1980, A&A, 92,
33
Contopoulos G., 1980, A&A, 81, 198
Contopoulos, G., Grousousakou, E., Voglis, N., 1995, A&A,
304, 374
Contopoulos, G., Voglis, N., 1996, Celest. Mech. Dyn. As-
tron., 64, 1
Contopoulos, G., Voglis, N., 1997, A&A, 317, 73
Contopoulos, G., 2002, Order and chaos in dynamical as-
tronomy, Springer-Verlag, Berlin
Contopoulos, G., Harsoula, M., 2008, Int. J. Bif. Chaos, 18,
2929
Durbala, A., Buta, R., Sulentic, J. W., Verdes-Montenegro,
L., 2009, MNRAS, 397, 1756
El-Zant, A., Shlosman, I., 2002, ApJ, 577, 626
El-Zant, A., Shlosman, I., 2003, ApJ, 595, L41
Ferrers, N. M., 1877, Quart. J. Pure Appl. Math., 14:1
Froeschl´ e, C., Lega, E., Gonzi, R., 1997, Celest. Mech. Dyn.
Astron., 67, 41
Froeschl´ e, C., Lega, E., 1998, A&A, 334, 355
Giordano, C. M., Cincotta, P. M., 2004, A&A, 423, 745
Harsoula, M., Kalapotharakos, C., 2009, MNRAS, 394,
1605
Kalapotharakos, C., Voglis, N., Contopoulos, G., 2004,
MNRAS, 428, 905
Kalapotharakos, C., Efthymiopoylos, C., Voglis, N., 2008,
MNRAS, 383, 971
Kaufmann, D. E., Contopoulos, G., 1996, A&A, 309, 381
Laskar, J., 1990, Icarus, 88, 266
Laskar, J., Froeschl´ e, C., Celletti, A., 1992, Physica D, 56,
253
Laskar, J., 1993, Physica D, 67, 257
Laurikainen, E., Salo, H., Buta, R., 2004, ApJ, 607, 103
Lieberman, M., Lichtenberg, A., 1992, Regular and
Stochastic Motion, Springer Verlag, Berlin
Macek, M., Dobeˇ s, J., Cejnar, P., 2010, Phys. Rev. C, 82,
014308
Manos, T., Skokos, Ch., Athanassoula, E., Bountis, T.,
2008a, Nonlinear Phenomena in Complex Systems, 11(2),
171
Manos, T., Skokos, Ch., Bountis, T., 2008b, Proceedings of
the conference Chaos, Complexity and Transport: Theory
and Applications, edited by Chandre C., Leoncini X. and
Zaslavsky G., World Scientific Publishing, 356
Manos, T., Athanassoula, E., 2008, Proceedings of the
international conference Chaos in Galaxies, edited by
Contopoulos G. and Patsis P., Springer-Verlag, Berlin-
Heidelberg (ASSP), 115
Manos, T., Skokos, Ch., Bountis, T., 2008c, Proceedings
of the international conference Chaos in Galaxies, edited
by Contopoulos G. and Patsis P., Springer-Verlag, Berlin-
Heidelberg (ASSP), 376
Manos, T., Ruffo, S., 2010, Trans. Th. and Stat. Phys.,
(accepted - eprint arXiv:1006.5341)
Miyamoto, M., Nagai, R., 1975, PASJ, 27, 533
Oseledec, V. I., 1968, Trans. Moscow Math. Soc., 19, 197
Panagopoulos, P., Bountis, T., Skokos, Ch., 2004, J. Vib.
& Acoust., 126, 520
Patsis, P. A., 2006, MNRAS, 369, L56
Patsis, P. A., Athanassoula, E., Quillen, A. C., 1997, ApJ,
483, 731
Patsis, P. A., Skokos, Ch., Athanassoula, E., 2002, MN-
RAS, 337, 578
Patsis, P. A., Skokos, Ch., Athanassoula, E., 2003a, MN-
RAS, 342, 69
Patsis, P. A., Skokos, Ch., Athanassoula, E., 2003b, MN-
RAS, 346, 1031
Pettini, M., 2007, Geometry and topology in Hamilto-
nian dynamics and statistical mechanics, Springer Verlag,
Berlin
Pfenniger, D., 1984, A&A, 134, 373
Plummer, H. C, 1911, MNRAS, 71, 460
Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery,
B. P., Numerical recipes in Fortran 77: The art of scientific
computing, Cambridge: University Press, Second Edition,
Volume 1 of Fortran Numerical Recipes
Romero-G´ omez, M., Masdemont, J. J., Athanassoula, E.,
Garc´ ıa-G´ omez, C., 2006, A&A, 453, 39.
Romero-G´ omez, M., Athanassoula, E., Masdemont, J. J.,
Garc´ ıa-G´ omez, C., 2007, A&A, 472, 63
Romero-G´ omez, M., Masdemont, J. J., Garc´ ıa-G´ omez, C.,
Athanassoula, E., 2009, Communications in Nonlinear
Science and Numerical Simulations, 14, 412
Skokos, Ch. 2001, J. Phys. A: Math. Gen., 34, 10029
Skokos, Ch., Patsis, P. A., Athanassoula, E., 2002a, MN-
RAS, 333, 847
Skokos, Ch., Patsis, P. A., Athanassoula, E., 2002b, MN-
RAS, 333, 861
Skokos, Ch., Antonopoulos, Ch., Bountis, T., Vrahatis, M.,
2002, in Proceedings of the 4th GRACM, edited by Tsa-