Physics Letters A 375 (2011) 3184–3187
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Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems
Konstantin E. Starkov
1
CITEDI-IPN, Av. del Parque 1310, Mesa de Otay, Tijuana, BC, Mexico
article info abstract
Article history:
Received 27 March 2011
Received in revised form 24 June 2011
Accepted 28 June 2011
Available online 2 July 2011
Communicated by C.R. Doering
In this Letter we prove that all compact invariant sets of the Bianchi VIII Hamiltonian system are con-
tained in the set described by several simple linear equalities and inequalities. Moreover, we describe
invariant domains in which the phase flow of this system has no recurrence property and show that
there are no periodic orbits and neither homoclinic, nor heteroclinic orbits contained in the zero level
set of its Hamiltonian. Similar results are obtained for the Bianchi IX Hamiltonian system.
©
2011 Elsevier B.V. All rights reserved.
1. Introduction
Some features of global dynamics of the Bianchi VIII and
Bianchi IX Hamiltonian systems are treated in this Letter. These
systems which are also known as Mixmaster universe models be-
long to the Bianchi class A cosmological models and may be de-
rived with help of solving vacuum Einstein’s field equations after
postulating a certain symmetry of space–time, see [8].Theimpor-
tance of studies of Bianchi models is discussed in many books and
papers, e.g. in [18,2,6] and is related to the fact that they serve as
a spatially homogeneous and anisotropic generalization of spatially
homogeneous and isotropic Friedmann–Robertson–Walker models.
Besides, it is worthwhile to notice that considering of Bianchi A
cosmological models as Hamiltonian systems allows us to apply
powerful analytical methods of the qualitative theory of ordi-
nary differential equations and dynamical systems theory. Particu-
larly through this approach one can attack chaoticity/nonchaoticity
problems, study integrability/nonintegrability properties, find solu-
tions with a specific behaviour, etc.
In this Letter the localization problem of compact invariant sets
of the Bianchi VIII and Bianchi IX Hamiltonian systems is exam-
ined. Studies related to the localization problem of compact invari-
ant sets [14] and, more specifically, to the nonexistence of periodic
orbits have appeared recently in cosmology and may be helpful
for a deeper understanding of the global dynamics in long peri-
ods of time. For example, the nonexistence of periodic orbits of
the Bianchi IX Hamiltonian system located in three invariant co-
ordinate hyperplanes has been established in [1]. The localization
problem for the Einstein–Yang–Mills equations has been examined
in [17] in which the nonexistence of periodic (homoclinic; hetero-
clinic) orbits has been stated in the most of cases. The characteri-
E-mail addresses: konst@citedi.mx, konstarkov@hotmail.com.
1
The mailing address: CITEDI-IPN, 482 W. San Ysidro Blvd #1861, San Ysidro,
CA 92173, USA.
zation of the long-time behavior of Bianchi VIII and IX Hamiltonian
systems is related to the question about chaoticity/nonchaoticity of
these models, see a discussion in [15].Inparticular,itwasproved
in [5] that the phase flow of the Bianchi IX Hamiltonian systems
has no recurrence property. Since this property is the standard
ingredient in the definition of the deterministic chaos one may
conclude that we do not meet the deterministic chaos in global
in this case.
Further, it should be mentioned that these Bianchi systems at-
tracted attention of many specialists in integrability theory who
applied different techniques from the Darboux theory of integra-
bility, the differential Galois theory, etc. One may consult in [1,
15,16,3,9–12] and others. However, dynamics of Bianchi Hamilto-
nian systems has a physical sense only inside the zero level set
H
−
1
(
0
)
of the corresponding Hamiltonian H, see e.g. in [4].There-
fore information concerning the nonintegrability of Bianchi models
in the whole space R
6
is not sufficient to state the nonintegrability
in H
−
1
(
0
)
and questions concerning to the existence/nonexistence
of the deterministic chaos in H
−
1
(
0
)
or in invariant domains in
H
−
1
(
0
)
remain.
Below we examine the Bianchi VIII Hamiltonian system
˙
y
1
=
1
2
y
1
(
z
2
−
z
1
),
˙
y
2
=
1
2
y
2
(
z
1
+
z
2
),
˙
y
3
=
y
3
z
3
,
˙
z
1
=
2
(
y
1
+
y
2
)(
y
1
−
y
2
−
y
3
),
˙
z
2
=
2y
3
(−
y
1
+
y
2
+
y
3
),
˙
z
3
= (
y
1
+
y
2
)
2
−
y
2
3
,
(1)
with the Hamiltonian
H
= (
y
1
+
y
2
)
2
+
y
3
(
2y
2
−
2y
1
+
y
3
) −
z
2
z
3
+
1
4
z
2
1
−
z
2
2
.
0375-9601/$ – see front matter
©
2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2011.06.064