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Journal ArticleDOI

Periodic orbits and non-integrability in a cosmological scalar field

10 Jan 2012-Journal of Mathematical Physics (AIP Publishing)-Vol. 53, Iss: 1, pp 012702-012702
TL;DR: In this article, the authors apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing a universe filled with a scalar field which possesses three parameters, and show the non-integrability of these cosmological systems in the sense of Liouville-Arnol'd.
Abstract: We apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing a universe filled with a scalar field which possesses three parameters. The main results are the following. First, we provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system has periodic orbits, the number of such periodic orbits and their stability change with the values of the parameters. These periodic orbits live in the whole phase space in a continuous family of periodic orbits parameterized by the Hamiltonian level. Second, under convenient assumptions we show the non-integrability of these cosmological systems in the sense of Liouville–Arnol'd, proving that there cannot exist any second first integral of class C1. It is important to mention that the tools (i.e., the averaging theory for studying the existence of periodic orbits and their kind of stability, and the multipliers of the...

Summary (2 min read)

1. Introduction and statements of main results

  • To the cosmological model here studied the authors suggest to the reader to look at the paper of Maciejewski et al. [14] and references therein for a detailed deduction and implications about the importance of this model.
  • For more details in this direction see the book [22].
  • The authors main result about the periodic orbits of the Hamiltonian system (5) is summarized as follows.
  • In [6] and [8] also the problem of non-existence of any additional meromorphic first integral in the Hamiltonian system (5) was considered.

2. Proof of Theorem 1

  • For proving Theorem 1 the authors shall apply Theorem 6 to the Hamiltonian system (5).
  • The functions F11 and F12 are analytical.
  • Furthermore they are 2π-periodic in the variable θ, the independent variable of system (12).
  • The authors obtain the existence of one, two, three, fourth or five periodic orbits for positive and negative values of the Hamiltonian level h if the solutions in case (I)-(IV) exists.
  • Now the statements of Theorem 1 follow easily.

3. Proof of Theorem 2

  • The authors assume that they are under the assumptions of Theorem 1, and that one of the founded periodic orbits corresponding to the solutions of cases (I)-(IV) exist.
  • Their associated Jacobians (15)-(18) are different from 1 playing with the energ level h.
  • Since these Jacobians are the product of the four multipliers of these periodic orbits with two of them always equal to 1, the remainder two multipliers cannot be equal to 1.
  • Hence under the assumptions of Theorem 1, by Theorem 5, either the conformal coupled field Hamiltonian systems cannot be Liouville–Arnol’d integrable with any second first integral C, or the system is Liouville-Arnol’d integrable and the differentials of H and C are linearly dependent on some points of these periodic orbits.

4. Proof of Theorem 3

  • In particular, in the regions I1 and I3 the eigenvalues are imaginary pure so the authors can only affirm that the family of periodic orbits is linearly stable.
  • While in the region I2 the eigenvalues are real, then the family of periodic orbits is unstable.
  • In the regions II1 and II3 the family of periodic orbits is linearly stable.
  • While in the region II2 the family of periodic orbits is unstable.

5. Conclusions

  • The authors have used two important tools in the area of dynamical systems.
  • First the averaging method for studying the existence of periodic orbits and their stability of the Hamiltonian systems (5) in their Hamiltonian levels.
  • The second tool allows to study the non–integrability in the sense of Liouville–Arnol’d of the Hamiltonian systems (5), for any second first integral of class C1, see Theorem 2.
  • The scale transformation introduced in the section 2 does not change the topology of the system, thus these results are valid for all ε, and in particular for ε = 1.
  • The two Hamiltonian systems (5) and (8) with ε 6= 0 have qualitatively the same phase portrait.

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PERIODIC ORBITS AND NON–INTEGRABILITY
IN A COSMOLOGICAL SCALAR FIELD
JAUME LLIBRE
1
AND CLAUDIO VIDAL
2
Abstract. We apply the averaging theory of first order to study the periodic orbits
of Hamiltonian systems describing an universe filled with a scalar field which possesses
three parameters. The main results are the following.
First, we provide sufficient conditions on the parameters of these cosmological
model, which guarantee that at any positive or negative Hamiltonian level, the Hamil-
tonian system has periodic orbits, the number of such periodic orbits and their sta-
bility change with the values of the parameters. These periodic orbits live in the
whole phase space in a continuous family of periodic orbits parameterized by the
Hamiltonian level.
Second, under convenient assumptions we show the non–integrability of these cos-
mological systems in the sense of Liouville–Arnol’d, proving that cannot exist any
second rst integral of class C
1
.
It is important to mention that the tools (i.e the averaging theory for studying the
existence of periodic orbits and their kind of stability, and the multipliers of these
periodic orbits for studying the integrability of the Hamiltonian system) used here
for proving our results on the cosmological scalar field, can be applied to Hamiltonian
systems with an arbitrary number of degrees of freedom.
1. Introduction and statements of ma in r esults
For a good introduction to the cosmological model here studied we suggest to the
reader to look at the paper of Maciejewski et al. [14] and references therein for a
detailed ded uction and implications about the importance of this model.
The foundation of homogeneous and isotopic cosmological models is the Friedmann-
Robertson-Walker (FRW) universe, described by the metric
(1) ds
2
= a(η)
2
2
+
dr
2
1 Kr
2
+ r
2
d
2
2
,
where a is the scalar factor, d
2
2
is the line element on a two-sphere, and we chose to
use the conformal time η. As it is known from the previous metric, the scalar factor
represents the relative change in the distance of two points whose spatial coordinates
are fixed. It depends only on the time, so that the whole universe is deformed in
a homogeneous fashion. Depending on the m atter components one obtains various
evolution of the scalar factor a, as given by the general action
(2) I =
c
4
16πG
Z
R
1
2
α
ψ
α
ψ + V (ψ) + ξR|ψ|
2
ρ
gd
4
x,
where R is the Ricci scalar, Λ is the cosmological constant, V the field’s potential, ξ
the coupling constant and ρ is th e density of the perfect uid. The potential usually
2000 Mathematics Subject Classification. Primary 34C10, 34C25.
Key words and phrases. periodic orbits, integrability, cosmological scalar fields, averaging theory.
1
This is a preprint of: “Periodic orbits and non-integrability in a cosmological scalar field”, Jaume
Llibre, Claudio Vidal, J. Math. Phys., vol. 53, 012702, 2012.
DOI: [10.1063/1.3675493]

2 J. LLIBRE AND C.VIDAL
includes at least a quadratic term m
2
|ψ|
2
, where m is the so called mass of the field.
When ξ = 0 we say that the field is minimally coupled-it does not uncouple since the
determinant of the metric g multiplies the whole Lagrangian density. The case when
ξ =
1
6
is the so called conform al coupling. Under geometric assumptions, the above
action can be simplified so that it allows the Hamiltonian approach with th e phase
variables depen ding only on conformal time η. After including an add itional matter
component ρ is equivalent to considering different energy levels. Namely for ρ a
4
(which is the case of radiation) a constant is added to the Hamiltonian, thus imitating
its nonzero value. This is the justification for studying the integrability of these systems
on a generic en er gy hypersurface.
As was saw in [14] the general action (2) for conformally coupled scalar fields must
includes th e following part
(3) I =
c
4
16πG
Z
R
1
2
α
ψ
α
ψ +
m
2
~
|ψ|
2
+
1
6
R|ψ|
2
λ
4!
|ψ|
4
gd
4
x,
where the additional coupling to gravity through the Ricci scalar R, and a quartic
potential term with constant λ are considered. After some algebraic manipulations,
assuming that the constant angular momentum is null, and un der the use of convenient
variables the Hamiltonian associated to the action (3) assumes the form
(4) H = H(q
1
, q
2
, p
1
, p
2
) =
1
2
(p
2
1
+ p
2
2
) +
1
2
k(q
2
1
+ q
2
2
) + m
2
q
2
1
q
2
2
+
1
4
Λq
4
1
+ λq
4
2
,
with k {−1, 0, 1} K = k|K| is associated to the index of curvature of the space),
λ, Λ, m
2
R. Notice that th e kinetic part is of natural form, albeit indefinite, and the
potential associated is a polynomial of degree four.
In th is paper we study the case k = 1, so the Hamiltonian system is given by
(5)
˙q
1
= p
1
,
˙q
2
= p
2
,
˙p
1
= q
1
m
2
q
1
q
2
2
Λq
3
1
,
˙p
2
= q
2
m
2
q
2
1
q
2
λq
3
2
.
As usual the dot denotes derivative w ith respect to th e independent variable t R, the
time. According to [14] we name (5) the conformal coupled scalar field Hamiltonian
systems with three parameters.
The periodic orbits are the most simple non–trivial solutions of a differential system.
Their study is of particular interest because the motion in their neighborhood can be
determined by their kind of stability. Furthermore, if the system is n on –integrable in
the sense of Liouville –Arnol’d, the existence of isolated periodic orbits in the energy
levels of a Hamiltonian system with multipliers different from 1 is related with the non-
existence of any second rst integral of class C
1
, so the study of these periodic orbits
for a differential system becomes relevant.
In general is very difficu lt to study analytically the existence of periodic orbits and the
kind of their stability for a given Hamiltonian system. In this work we use th e averaging
method of first order to compute periodic orbits and their kind of stability as it was
established in [7], see Appendix for a summary of this method. This method allows to
find periodic orbits of our cosmological model (5), up to rst order in ε, at any non-zero
Hamiltonian level as a function of the parameters λ, Λ and m. Roughly speaking, this
method reduces the problem of finding periodic solutions of some d ifferential system
to the one of finding zeros of some convenient finite d im ensional function. In [10] and

PERIODIC SOLUT IONS AND NON–INTEGRABILITY IN A COSMOLOGICAL MOD E L 3
[11] some of us applied these techniques to the Henon–Heiles and to the Yang–Mills
Hamiltonians. As we have mention in the abstract the averaging theory for studying
the existence of periodic orbits and their kind of stability, and the multipliers of these
periodic orbits for studying the integrability of the Hamiltonian s y stem are the tools
used here for proving our results on the cosmological scalar eld. We remark that these
tools can be applied to Hamiltonian systems with an arbitrary number of degrees of
freedom.
As we shall see one of the main problems for applying the averaging theory for
studying the periodic orbits of a given differential system is to nd the changes of
variables which allow to write the original differential systems in the normal form for
applying the averaging theory. For more details in this direction see the book [22].
Next we will define some subsets of the (λ, Λ)-plane for each nonzero fixed value of
m (see Figure 1-2). The straight lines:
l
1
: λ =
m
2
3
;
l
2
: λ = m
2
;
l
3
: Λ =
m
2
3
.
l
4
: Λ = m
2
.
The regions:
R
1
: λ < m
2
, Λ > m
2
, λ + Λ > 2m
2
;
R
2
: λ > m
2
, Λ < m
2
, λ + Λ < 2m
2
;
R
3
: λ <
m
2
3
, Λ >
m
2
3
, λ + Λ >
2m
2
3
;
R
4
: λ >
m
2
3
, Λ <
m
2
3
, λ + Λ <
2m
2
3
;
1
: λ < m
2
, Λ > m
2
, λ + Λ < 2m
2
;
2
: λ > m
2
, Λ < m
2
, λ + Λ > 2m
2
;
3
: λ <
m
2
3
, Λ >
m
2
3
, λ + Λ <
2m
2
3
;
4
: λ >
m
2
3
, Λ <
m
2
3
, λ + Λ >
2m
2
3
.
Also we define the half-lines:
s
13
= (R
1
R
3
) l
3
;
s
31
= (R
1
R
3
) {λ + Λ =
2m
2
3
};
s
24
= (R
2
R
4
) {λ + Λ = 2m
2
};
s
42
= (R
2
R
4
) l
1
;
˜s
13
= (Ω
1
3
) {λ + Λ = 2m
2
};
˜s
31
= (Ω
1
3
) l
3
;
˜s
24
= (Ω
2
4
) l
4
;
˜s
42
= (Ω
2
4
) {λ + Λ =
2m
2
3
},
where means boundary.
Our main result about the periodic orbits of the Hamiltonian system (5) is summa-
rized as follows .
Theorem 1. For every m 6= 0 at every positive Hamiltonian level the conformal coupled
scalar field Hamiltonian system (5) has at least
(a1) one periodic orbit if (λ, Λ) s
31
s
42
;
(a2) two periodic orbits if (λ, Λ) s
13
s
24
;
(a3) three periodic orbits if (λ, Λ) [R
1
\ (R
1
R
3
)] [R
2
\(R
2
R
4
)];
(a4) four periodic orbits if (λ, Λ) [R
3
\ (R
1
R
3
)] [R
4
\(R
2
R
4
)];

4 J. LLIBRE AND C.VIDAL
Figure 1. Regions R
1
, R
2
, R
3
, R
4
for h > 0 and m 6= 0.
Figure 2. Regions
1
,
2
,
3
,
4
for h < 0 and m 6= 0.
(a5) five periodic orbits if (λ, Λ) [R
1
R
3
] [R
2
R
4
].
For m 6= 0 and for every negative Hamiltonian level the conformal coupled scalar field
Hamiltonian system (5) has at least
(b1) one periodic orbit if (λ, Λ) ˜s
31
˜s
42
;
(b2) two periodic orbits if (λ, Λ) ˜s
13
˜s
24
;
(b3) three periodic orbits if (λ, Λ) [Ω
1
\(Ω
1
3
)] [Ω
2
\ (Ω
2
4
)];

PERIODIC SOLUT IONS AND NON–INTEGRABILITY IN A COSMOLOGICAL MOD E L 5
(b4) four periodic orbits if (λ, Λ) [Ω
3
\(Ω
1
3
)] [Ω
4
\ (Ω
2
4
)];
(b5) five periodic orbits if (λ, Λ) [Ω
1
3
] [Ω
2
4
].
Theorem 1 is proved in section 2 using the averaging theory.
From Theorem 1 it follows that when m 6= 0 (fixed) and h > 0, we have periodic
solutions in our cosmological model for any choice of the parameters (λ, Λ) / l
1
l
2
.
The same conclusion holds f or the case h < 0 if (λ, Λ) / l
3
l
4
. Thus combining
these results we obtain that there are four values of the parameters (λ, Λ), namely,
P
1
=
m
2
3
,
m
2
3
l
1
l
3
, P
2
=
m
2
,
m
2
3
l
2
l
3
, P
3
=
m
2
, m
2
l
1
l
4
and P
4
=
m
2
3
, m
2
l
2
l
4
, where our arguments do not guarantee the existence
of periodic solutions for the Hamiltonian system (5).
The periodic solutions of Theorem 1 are of the form
(6)
q
1
(t) = r
cos(t) + O(ε), p
1
(t) = r
sin(t) + O(ε)
q
2
(t) = ρ
cos(t + α
) + O(ε), p
2
(t) = ρ
sin(t + α
) + O(ε),
where r
, ρ
, α
are functions of the parameters h, m, λ, Λ, and ε is a small parameter
which will be defined later on . In the definition of the following four cases we assume
that m and h are fixed with m h 6= 0.
Families (I): h > 0 and (λ, Λ) / l
1
l
2
. We have two families of periodic solu-
tions (6) generated by a circle on the plane (q
2
, p
2
) with r
= 0, ρ
=
h, α
=
±
1
2
arccos
2m
2
+3λ
m
2
.
Families (II): h < 0 and (λ, Λ) / l
3
l
4
. We have two families of periodic solutions (6)
but they are generated by a circle on the (q
1
, p
1
)-plane with r
=
2h, ρ
= 0, α
=
±
1
2
arccos
2m
2
+3Λ
m
2
.
Family (III): Either h > 0 and (λ, Λ) R
1
R
3
, or h < 0 and (λ, Λ)
1
2
. We
have one parametric family of periodic solutions generated by the periodic solution (6)
with r
=
q
2h(m
2
+λ)
2m
2
+λ
, ρ
=
q
2h(m
2
+Λ)
2m
2
+λ
, α
= 0.
Families (IV): Either h > 0 and (λ, Λ) R
2
R
4
, or h < 0 and (λ, Λ)
2
4
. Here
we have two families of periodic solutions generated by the periodic solutions (6) with
r
=
q
2h(m
2
+3λ)
2m
2
+3(λ+Λ)
, ρ
=
q
2h(m
2
+3Λ)
2m
2
+3(λ+Λ)
, α
= ±
π
2
.
It is well known that integrable and non–integrable Hamiltonian systems can have
infinitely many periodic orbits. However it is difficult to nd a whole family of periodic
orbits in an analytical way, specially if the Hamiltonian system is non–integrable. Here
we find them up to rst order in ε. Once we have shown that at any non-zero Hamilton-
ian level there exist periodic orbits, we can use these p articular periodic orbits to prove
our second main result about the non–integrability in the sense of Liouville–Arnol’d of
our conformal coupled scalar field Hamiltonian system (5).
Theorem 2. Assume that the conformal coupled scalar field Hamiltonian system (5)
satisfies the assumptions of one of the statements of Theorem 1, and denote b y () this
statement. Then, under the assumption of statement (),
(a) either the conformal coupled scalar fields Hamiltonian system (5) is Liouville–
Arnol’d integrable and the gradients of the two constants of motion are linearly
dependent on some points of the periodic orbits found in statement () of The-
orem 1,

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"Periodic orbits and non-integrabili..." refers background in this paper

  • ...For more details in this direction see the book [22]....

    [...]

Book
01 Jan 1990
TL;DR: In this article, the Poincare-Bendixson theorem is applied to the analysis of two-dimensional linear systems with first integrals and integral manifolds, and the Lagrange standard form is used.
Abstract: 1 Introduction.- 1.1 Definitions and notation.- 1.2 Existence and uniqueness.- 1.3 Gronwall's inequality.- 2 Autonomous equations.- 2.1 Phase-space, orbits.- 2.2 Critical points and linearisation.- 2.3 Periodic solutions.- 2.4 First integrals and integral manifolds.- 2.5 Evolution of a volume element, Liouville's theorem.- 2.6 Exercises.- 3 Critical points.- 3.1 Two-dimensional linear systems.- 3.2 Remarks on three-dimensional linear systems.- 3.3 Critical points of nonlinear equations.- 3.4 Exercises.- 4 Periodic solutions.- 4.1 Bendixson's criterion.- 4.2 Geometric auxiliaries, preparation for the Poincare-Bendixson theorem.- 4.3 The Poincare-Bendixson theorem.- 4.4 Applications of the Poincare-Bendixson theorem.- 4.5 Periodic solutions in ?n.- 4.6 Exercises.- 5 Introduction to the theory of stability.- 5.1 Simple examples.- 5.2 Stability of equilibrium solutions.- 5.3 Stability of periodic solutions.- 5.4 Linearisation.- 5.5 Exercises.- 6 Linear Equations.- 6.1 Equations with constant coefficients.- 6.2 Equations with coefficients which have a limit.- 6.3 Equations with periodic coefficients.- 6.4 Exercises.- 7 Stability by linearisation.- 7.1 Asymptotic stability of the trivial solution.- 7.2 Instability of the trivial solution.- 7.3 Stability of periodic solutions of autonomous equations.- 7.4 Exercises.- 8 Stability analysis by the direct method.- 8.1 Introduction.- 8.2 Lyapunov functions.- 8.3 Hamiltonian systems and systems with first integrals.- 8.4 Applications and examples.- 8.5 Exercises.- 9 Introduction to perturbation theory.- 9.1 Background and elementary examples.- 9.2 Basic material.- 9.3 Naive expansion.- 9.4 The Poincare expansion theorem.- 9.5 Exercises.- 10 The Poincare-Lindstedt method.- 10.1 Periodic solutions of autonomous second-order equations.- 10.2 Approximation of periodic solutions on arbitrary long time-scales.- 10.3 Periodic solutions of equations with forcing terms.- 10.4 The existence of periodic solutions.- 10.5 Exercises.- 11 The method of averaging.- 11.1 Introduction.- 11.2 The Lagrange standard form.- 11.3 Averaging in the periodic case.- 11.4 Averaging in the general case.- 11.5 Adiabatic invariants.- 11.6 Averaging over one angle, resonance manifolds.- 11.7 Averaging over more than one angle, an introduction.- 11.8 Periodic solutions.- 11.9 Exercises.- 12 Relaxation Oscillations.- 12.1 Introduction.- 12.2 Mechanical systems with large friction.- 12.3 The van der Pol-equation.- 12.4 The Volterra-Lotka equations.- 12.5 Exercises.- 13 Bifurcation Theory.- 13.1 Introduction.- 13.2 Normalisation.- 13.3 Averaging and normalisation.- 13.4 Centre manifolds.- 13.5 Bifurcation of equilibrium solutions and Hopf bifurcation.- 13.6 Exercises.- 14 Chaos.- 14.1 Introduction and historical context.- 14.2 The Lorenz-equations.- 14.3 Maps associated with the Lorenz-equations.- 14.4 One-dimensional dynamics.- 14.5 One-dimensional chaos: the quadratic map.- 14.6 One-dimensional chaos: the tent map.- 14.7 Fractal sets.- 14.8 Dynamical characterisations of fractal sets.- 14.9 Lyapunov exponents.- 14.10 Ideas and references to the literature.- 15 Hamiltonian systems.- 15.1 Introduction.- 15.2 A nonlinear example with two degrees of freedom.- 15.3 Birkhoff-normalisation.- 15.4 The phenomenon of recurrence.- 15.5 Periodic solutions.- 15.6 Invariant tori and chaos.- 15.7 The KAM theorem.- 15.8 Exercises.- Appendix 1: The Morse lemma.- Appendix 2: Linear periodic equations with a small parameter.- Appendix 3: Trigonometric formulas and averages.- Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3.- Appendix 5: Bifurcations of self-excited oscillations.- Appendix 6: Normal forms of Hamiltonian systems near equilibria.- Answers and hints to the exercises.- References.

1,290 citations

Journal ArticleDOI
TL;DR: The connection between nonlinear partial differential equations solvable by inverse scattering transforms and nonlinear ODEs of P-type (i.e., no movable critical points) is discussed in this article.
Abstract: We develop here two aspects of the connection between nonlinear partial differential equations solvable by inverse scattering transforms and nonlinear ordinary differential equations (ODE) of P‐type (i.e., no movable critical points). The first is a proof that no solution of an ODE, obtained by solving a linear integral equation of a certain kind, can have any movable critical points. The second is an algorithm to test whether a given ODE satisfies necessary conditions to be of P‐type. Often, the algorithm can be used to test whether or not a given nonlinear evolution equation may be completely integrable.

995 citations


"Periodic orbits and non-integrabili..." refers methods in this paper

  • ...They were found by applying the so called ARS algorithm based on the Painlevé analysis [1]....

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Frequently Asked Questions (2)
Q1. What are the contributions in "Periodic orbits and non–integrability in a cosmological scalar field" ?

The authors apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing an universe filled with a scalar field which possesses three parameters. The main results are the following. First, the authors provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system has periodic orbits, the number of such periodic orbits and their stability change with the values of the parameters. Second, under convenient assumptions the authors show the non–integrability of these cosmological systems in the sense of Liouville–Arnol ’ d, proving that can not exist any second first integral of class C. 

The second tool allows to study the non–integrability in the sense of Liouville–Arnol ’ d of the Hamiltonian systems ( 5 ), for any second first integral of class C1, see Theorem 2.