# Periodic orbits and non-integrability in a cosmological scalar field

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)

Jump to: [1. Introduction and statements of main results] – [2. Proof of Theorem 1] – [3. Proof of Theorem 2] – [4. Proof of Theorem 3] and [5. Conclusions]

### 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 ﬁrst 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

−dη

2

+

dr

2

1 − Kr

2

+ r

2

d

2

Ω

2

,

where a is the scalar factor, d

2

Ω

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 − 2Λ −

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 ﬂuid. 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 − 2Λ −

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 ﬁrst 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 ﬁrst 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 ﬁeld. 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 ﬁnd 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

;

Ω

1

: λ < −m

2

, Λ > − m

2

, λ + Λ < −2m

2

;

Ω

2

: λ > −m

2

, Λ < − m

2

, λ + Λ > −2m

2

;

Ω

3

: λ < −

m

2

3

, Λ > −

m

2

3

, λ + Λ < −

2m

2

3

;

Ω

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

∩ Ω

3

) ∩ {λ + Λ = −2m

2

};

˜s

31

= ∂(Ω

1

∩ Ω

3

) ∩ l

3

;

˜s

24

= ∂(Ω

2

∩ Ω

4

) ∩ l

4

;

˜s

42

= ∂(Ω

2

∩ Ω

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 Ω

1

, Ω

2

, Ω

3

, Ω

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

∩ Ω

3

)] ∪ [Ω

2

\ (Ω

2

∩ Ω

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

∩ Ω

3

)] ∪ [Ω

4

\ (Ω

2

∩ Ω

4

)];

(b5) ﬁve periodic orbits if (λ, Λ) ∈ [Ω

1

∩ Ω

3

] ∪ [Ω

2

∩ Ω

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) ∈ Ω

1

∪ Ω

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+Λ

, ρ

∗

=

q

2h(m

2

+Λ)

2m

2

+λ+Λ

, α

∗

= 0.

Families (IV): Either h > 0 and (λ, Λ) ∈ R

2

∪R

4

, or h < 0 and (λ, Λ) ∈ Ω

2

∪Ω

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 ﬁnd a whole family of periodic

orbits in an analytical way, specially if the Hamiltonian system is non–integrable. Here

we ﬁnd them up to ﬁrst 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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01 Jan 1967

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Abstract: Introduction Foreward by Tudor Ratiu and Richard Cushman Preliminaries Differential Theory Calculus on Manifolds Analytical Dynamics Hamiltonian and Lagrangian Systems Hamiltonian Systems with Symmetry Hamiltonian-Jacobi Theory and Mathematical Physics An Outline of Qualitative Dynamics Topological Dynamics Differentiable Dynamics Hamiltonian Dynamics Celestial Mechanics The Two-Body Problem The Three-Body Problem.

3,561 citations

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25 Oct 1985

TL;DR: The history of the theory of averaging can be found in this paper, where a 4-dimensional example of Hopf Bifurcation is presented. But the history of averaging is not complete.

Abstract: Basic Material and Asymptotics.- Averaging: the Periodic Case.- Methodology of Averaging.- Averaging: the General Case.- Attraction.- Periodic Averaging and Hyperbolicity.- Averaging over Angles.- Passage Through Resonance.- From Averaging to Normal Forms.- Hamiltonian Normal Form Theory.- Classical (First-Level) Normal Form Theory.- Nilpotent (Classical) Normal Form.- Higher-Level Normal Form Theory.- The History of the Theory of Averaging.- A 4-Dimensional Example of Hopf Bifurcation.- Invariant Manifolds by Averaging.- Some Elementary Exercises in Celestial Mechanics.- On Averaging Methods for Partial Differential Equations.

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

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

[...]

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01 Jan 1990TL;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

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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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###### Q2. What future works have the authors mentioned in the paper "Periodic orbits and non–integrability in a cosmological scalar field" ?

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.