text: 3 and 2) that the most relevant modes for the expansion of the full wavefunction Ψ of a single electron are given by p orbitals, where z is the direction transverse to the bilayer, as these are the orbitals that contribute the most to z conduction. In formulas ∼ | (cid:105) | (cid:105) | (cid:105) | (cid:105) ψ λ p ; A1 + λ p ; B1 + λ p ; A2 + λ p ; B2 , (5) 1 z 2 z 3 z 4 z where A, B correspond to inequivalent types of carbon atoms in the same plane, and the index 1, 2 corresponds to diﬀerent layers. Given the form (4) of the Hamiltonian, we can recognise that B1 and A2 are the two atoms that are directly aligned along the transversal direction. Solving the eigenvalue equation Hλ = Eλ we ﬁnd that there are no | | (cid:54) non-trivial eigenspinors with λ = 0 when κ = 0. The component λ can then be set to 1 as the eigenvalue equation 1 1 is homogeneous. The full solution follows from linear algebra E E κ λ2 = (cid:126)v κ¯ , λ3 = σ (cid:126)v κ¯ , λ4 = σ κ¯ , (6) F F ± where σ = 1 is a factor labelling diﬀerent bands that appear in the consistency condition satisﬁed by the energy − |(cid:126) | − E2 v κ 2 σγE = 0 . (7) F (cid:126) | | In [16] we expanded the solution around the Fermi points in the dimensionless parameter (cid:15) = vF κ . For the metallic E(±) bands one obtains (cid:0) (cid:1) γ 1 (cid:126) κ (cid:0) (cid:1) λ = + O (cid:15)2 , (8) 2 2mv F (cid:126) ∓ 2 κ λ = + O (cid:15)2 , (9) 3 α 2mv F ∓ 2 κ λ = . (10) 4 α κ¯ For simplicity we set τ = +1 and focus on one of the Fermi points. Then for all values of the constant A the following spinors χ1(t) = e−i(cid:126)2km2t (Aλ3, λ2)T , (11) χ2(t) = e−i(cid:126)2km2t (λ1, Aλ4)T , (12) satisfy the L´evy-Leblond equations [22] (cid:126) (cid:126) i ∂ χ + i v Dχ = 0 , (13) t 2 F 1 − 2mv Dχ2 i (cid:126) F χ1 = 0 , (14) where D = iσjk is the 2–dimensional Dirac operator in phase space and the σj are the Pauli matrices in the standard j basis. As we will discuss in sec. , these are coupled, ﬁrst order equations that are invariant under the Galilei group of non-relativistic mechanics and describe non-relativistic spin 1 fermions. There we will identify the solutions above 2 with linear superpositions of states with diﬀerent values of the pseudospin: this is related to the freedom of choosing the arbitrary constant A. In analogy with the fact that for monolayer graphene the Fermi points are called Dirac points because of the relativistic emergent symmetry, for the bilayer we will use the wording Galilei points. At the Galilei point, i.e. for κ = 0, the solutions have λ = 0, λ = 0. As shown in Figure 1a, this corresponds physically 2 3 to activating only the p orbitals of the A and B atoms: these are not directly aligned along the z direction, thus z 1 2 giving a small overlap of the electron orbitals. ± Conversely, when we perform a similar analysis for the insulating bands E( ) at the same τ = +1 Galilei point we (cid:0) (cid:1) 2 ﬁnd 2mv (cid:0) (cid:1) λ2 = (cid:126)κ¯F + O (cid:15)2 , (15) ± 2 2mv λ3 = α (cid:126)κ¯F + O (cid:15)2 , (16) ± 2 κ λ = . (17) 4 α κ¯ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: that

text: the

text: most

text: relevant

text: modes

text: for

text: the

text: expansion

text: of

text: the

text: full

text: wavefunction

text: Ψ

text: of

text: a

text: single

text: electron

text: are

text: given

text: by

text: p

text: orbitals,

text: where

text: z

text: is

text: the

text: direction

text: transverse

text: to

text: the

text: bilayer,

text: as

text: these

text: are

text: the

text: orbitals

text: that

text: contribute

text: the

text: most

text: to

text: z

text: conduction.

text: In

text: formulas

text: ψ

text: λ

text: p

text: λ

text: p

text: λ

text: p

text: λ

text: p

text: z

text: z

text: z

text: z

text: where

text: A,

text: B

text: correspond

text: to

text: inequivalent

text: types

text: of

text: carbon

text: atoms

text: in

text: the

text: same

text: plane,

text: and

text: the

text: index

text: corresponds

text: to

text: diﬀerent

text: layers.

text: Given

text: the

text: form

text: of

text: the

text: Hamiltonian,

text: we

text: can

text: recognise

text: that

text: and

text: are

text: the

text: two

text: atoms

text: that

text: are

text: directly

text: aligned

text: along

text: the

text: transversal

text: direction.

text: Solving

text: the

text: eigenvalue

text: equation

text: Hλ

text: Eλ

text: we

text: ﬁnd

text: that

text: there

text: are

text: no

text: non-trivial

text: eigenspinors

text: with

text: λ

text: when

text: κ

text: The

text: component

text: λ

text: can

text: then

text: be

text: set

text: to

text: as

text: the

text: eigenvalue

text: equation

text: is

text: homogeneous.

text: The

text: full

text: solution

text: follows

text: from

text: linear

text: algebra

text: E

text: E

text: κ

text: λ2

text: κ¯

text: λ3

text: σ

text: κ¯

text: λ4

text: σ

text: κ¯

text: F

text: F

text: where

text: σ

text: is

text: a

text: factor

text: labelling

text: diﬀerent

text: bands

text: that

text: appear

text: in

text: the

text: consistency

text: condition

text: satisﬁed

text: by

text: the

text: energy

text: v

text: κ

text: σγE

text: F

text: In

text: we

text: expanded

text: the

text: solution

text: around

text: the

text: Fermi

text: points

text: in

text: the

text: dimensionless

text: parameter

text: vF

text: κ

text: For

text: the

text: metallic

text: E(±)

text: bands

text: one

text: obtains

text: γ

text: κ

text: λ

text: O

text: F

text: κ

text: λ

text: O

text: α

text: F

text: κ

text: λ

text: α

text: κ¯

text: For

text: simplicity

text: we

text: set

text: τ

text: and

text: focus

text: on

text: one

text: of

text: the

text: Fermi

text: points.

text: Then

text: for

text: all

text: values

text: of

text: the

text: constant

text: A

text: the

text: following

text: spinors

text: satisfy

text: the

text: L´evy-Leblond

text: equations

text: i

text: χ

text: i

text: v

text: Dχ

text: t

text: F

text: i

text: F

text: χ1

text: where

text: D

text: iσjk

text: is

text: the

text: Dirac

text: operator

text: in

text: phase

text: space

text: and

text: the

text: σj

text: are

text: the

text: Pauli

text: matrices

text: in

text: the

text: standard

text: j

text: basis.

text: As

text: we

text: will

text: discuss

text: in

text: sec.

text: these

text: are

text: coupled,

text: ﬁrst

text: order

text: equations

text: that

text: are

text: invariant

text: under

text: the

text: Galilei

text: group

text: of

text: non-relativistic

text: mechanics

text: and

text: describe

text: non-relativistic

text: spin

text: fermions.

text: There

text: we

text: will

text: identify

text: the

text: solutions

text: above

text: with

text: linear

text: superpositions

text: of

text: states

text: with

text: diﬀerent

text: values

text: of

text: the

text: pseudospin:

text: this

text: is

text: related

text: to

text: the

text: freedom

text: of

text: choosing

text: the

text: arbitrary

text: constant

text: A.

text: In

text: analogy

text: with

text: the

text: fact

text: that

text: for

text: monolayer

text: graphene

text: the

text: Fermi

text: points

text: are

text: called

text: Dirac

text: points

text: because

text: of

text: the

text: relativistic

text: emergent

text: symmetry,

text: for

text: the

text: bilayer

text: we

text: will

text: use

text: the

text: wording

text: Galilei

text: points.

text: At

text: the

text: Galilei

text: point,

text: i.e.

text: for

text: κ

text: the

text: solutions

text: have

text: λ

text: λ

text: As

text: shown

text: in

text: Figure

text: this

text: corresponds

text: physically

text: to

text: activating

text: only

text: the

text: p

text: orbitals

text: of

text: the

text: A

text: and

text: B

text: atoms:

text: these

text: are

text: not

text: directly

text: aligned

text: along

text: the

text: z

text: direction,

text: thus

text: z

text: giving

text: a

text: small

text: overlap

text: of

text: the

text: electron

text: orbitals.

text: Conversely,

text: when

text: we

text: perform

text: a

text: similar

text: analysis

text: for

text: the

text: insulating

text: bands

text: E(

text: at

text: the

text: same

text: τ

text: Galilei

text: point

text: we

text: ﬁnd

text: λ2

text: O

text: λ3

text: α

text: O

text: κ

text: λ

text: α

text: κ¯

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 a) spherical b) hyperbolic FIG. 4: Conﬁgurations with curvature for the three-layer graphene system exploitable for curvatronics, to tune the electronic properties of few-layer graphene by geometrical eﬀects. (a) Spherical (positive) curvature. (b) Hyperbolic (negative) curvature. (cid:18) (cid:19) In terms of these the chirality matrix is calculated as − ∗ σ 0 Γ = i Γ0 Γ1 Γ2 Γ3 = 3 , (65) 0 σ 3 showing that the spinor Ψˆ built from the combination a = 1, b = 0 has a deﬁnite chirality, which by convention we can call right, and the one built using a = 0, b = 1 has opposite chirality, say left. The role of curvature In [16] we discussed the role of curvature in modifying the electronic band dispersions of bilayer graphene with AB stacking. We expect that the same conclusions apply in the case of few layers AB stacked graphene. When the radius of curvature is small enough, the L´evy-Leblond equations (13)-(14) are replaced by their covariant counterpart, where ∇ ∇ the symbol D is now given by D = σj and is now the 2D spinorial covariant derivative that includes the spin j connection. Correspondingly the 4D massless Dirac equation will include the spin connection of the curved version of the metric (63) g dxµdxν = g dqidqj + 2dudv , (66) µν ij with qi = (x, y). A scalar and vector potential can be added to the metric above, and the reduction procedure from 4 to 2 dimensions is detailed in [16]. An important eﬀect that we expect to be physically observable is that, in the presence of a curved 2D sheet of few–layer graphene, the uncoupled L´evy-Leblond equation (56) becomes (cid:126) (cid:126) (cid:126) − 2 ∇ 2 i ∂ χ = 2χ + R χ , (67) t 2 2m 2 8m 2 where R is the scalar curvature, and the eﬀective mass m has a positive or negative sign depending on the energy band − being of type E(+) or E( ). The last term in equation (67) is a contribution to the local energy of states that depends on the scalar curvature and on the type of band. Possible positive or negative curvatures of the three-layer graphene are depicted in Figure 4. For positive (spherical) curvature the contribution tends to create or increase a gap between − the E(+) and E( ), while for negative (hyperbolic) curvature the tendency is to make the bands closer. Calculating the net eﬀect requires solving (67) with given boundary conditions, because the presence of local curvature is related to the global structure of the surface. We intend to do this quantitative analysis in a separate work. Leaving aside for the moment the subject of global properties of solutions, we can learn from (67) about the order of magnitude of the energy contributions due to curvature. On the one hand, for a radius of curvature smaller than 23nm the local correction is larger than 1meV and therefore should be detectable using existing experimental techniques such as ARPES. On the other hand, if the radius of curvature becomes smaller than 1.16nm the local correction becomes larger than the interlayer hopping energy γ, and therefore it is no longer guaranteed that we can still use the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

label: paragraph

text: a)

text: spherical

text: b)

text: hyperbolic

text: FIG.

text: Conﬁgurations

label: caption

text: with

text: curvature

text: for

text: the

text: three-layer

text: graphene

text: system

text: exploitable

text: for

text: curvatronics,

text: to

text: tune

text: the

text: electronic

text: properties

text: of

text: few-layer

text: graphene

text: by

text: geometrical

text: eﬀects.

text: (a)

text: Spherical

text: (positive)

text: curvature.

text: (b)

text: Hyperbolic

text: (negative)

text: curvature.

text: In

text: terms

text: of

text: these

text: the

text: chirality

text: matrix

text: is

text: calculated

text: as

text: σ

text: Γ

text: i

text: Γ0

text: Γ1

text: Γ2

text: Γ3

text: σ

text: showing

text: that

text: the

text: spinor

text: Ψˆ

text: built

text: from

text: the

text: combination

text: a

text: b

text: has

text: a

text: deﬁnite

text: chirality,

text: which

text: by

text: convention

text: we

text: can

text: call

text: right,

text: and

text: the

text: one

text: built

text: using

text: a

text: b

text: has

text: opposite

text: chirality,

text: say

text: left.

text: The

text: role

text: of

text: curvature

text: In

text: we

text: discussed

text: the

text: role

text: of

text: curvature

text: in

text: modifying

text: the

text: electronic

text: band

text: dispersions

text: of

text: bilayer

text: graphene

text: with

text: AB

text: stacking.

text: We

text: expect

text: that

text: the

text: same

text: conclusions

text: apply

text: in

text: the

text: case

text: of

text: few

text: layers

text: AB

text: stacked

text: graphene.

text: When

text: the

text: radius

text: of

text: curvature

text: is

text: small

text: enough,

text: the

text: L´evy-Leblond

text: equations

text: are

text: replaced

text: by

text: their

text: covariant

text: counterpart,

text: where

text: the

text: symbol

text: D

text: is

text: now

text: given

text: by

text: D

text: σj

text: and

text: is

text: now

text: the

text: spinorial

text: covariant

text: derivative

text: that

text: includes

text: the

text: spin

text: j

text: connection.

text: Correspondingly

text: the

text: massless

text: Dirac

text: equation

text: will

text: include

text: the

text: spin

text: connection

text: of

text: the

text: curved

text: version

text: of

text: the

text: metric

text: g

text: dxµdxν

text: g

text: dqidqj

text: µν

text: ij

text: with

text: qi

text: (x,

text: y).

text: A

text: scalar

text: and

text: vector

text: potential

text: can

text: be

text: added

text: to

text: the

text: metric

text: above,

text: and

text: the

text: reduction

text: procedure

text: from

text: to

text: dimensions

text: is

text: detailed

text: in

text: An

text: important

text: eﬀect

text: that

text: we

text: expect

text: to

text: be

text: physically

text: observable

text: is

text: that,

text: in

text: the

text: presence

text: of

text: a

text: curved

text: sheet

text: of

text: few–layer

text: graphene,

text: the

text: uncoupled

text: L´evy-Leblond

text: equation

text: becomes

text: i

text: χ

text: 2χ

text: R

text: χ

text: t

text: where

text: R

text: is

text: the

text: scalar

text: curvature,

text: and

text: the

text: eﬀective

text: mass

text: m

text: has

text: a

text: positive

text: or

text: negative

text: sign

text: depending

text: on

text: the

text: energy

text: band

text: being

text: of

text: type

text: E(+)

text: or

text: E(

text: The

text: last

text: term

text: in

text: equation

text: is

text: a

text: contribution

text: to

text: the

text: local

text: energy

text: of

text: states

text: that

text: depends

text: on

text: the

text: scalar

text: curvature

text: and

text: on

text: the

text: type

text: of

text: band.

text: Possible

text: positive

text: or

text: negative

text: curvatures

text: of

text: the

text: three-layer

text: graphene

text: are

text: depicted

text: in

text: Figure

text: For

text: positive

text: (spherical)

text: curvature

text: the

text: contribution

text: tends

text: to

text: create

text: or

text: increase

text: a

text: gap

text: between

text: the

text: E(+)

text: and

text: E(

text: while

text: for

text: negative

text: (hyperbolic)

text: curvature

text: the

text: tendency

text: is

text: to

text: make

text: the

text: bands

text: closer.

text: Calculating

text: the

text: net

text: eﬀect

text: requires

text: solving

text: with

text: given

text: boundary

text: conditions,

text: because

text: the

text: presence

text: of

text: local

text: curvature

text: is

text: related

text: to

text: the

text: global

text: structure

text: of

text: the

text: surface.

text: We

text: intend

text: to

text: do

text: this

text: quantitative

text: analysis

text: in

text: a

text: separate

text: work.

text: Leaving

text: aside

text: for

text: the

text: moment

text: the

text: subject

text: of

text: global

text: properties

text: of

text: solutions,

text: we

text: can

text: learn

text: from

text: about

text: the

text: order

text: of

text: magnitude

text: of

text: the

text: energy

text: contributions

text: due

text: to

text: curvature.

text: On

text: the

text: one

text: hand,

text: for

text: a

text: radius

text: of

text: curvature

text: smaller

text: than

text: the

text: local

text: correction

text: is

text: larger

text: than

text: and

text: therefore

text: should

text: be

text: detectable

text: using

text: existing

text: experimental

text: techniques

text: such

text: as

text: ARPES.

text: On

text: the

text: other

text: hand,

text: if

text: the

text: radius

text: of

text: curvature

text: becomes

text: smaller

text: than

text: the

text: local

text: correction

text: becomes

text: larger

text: than

text: the

text: interlayer

text: hopping

text: energy

text: γ,

text: and

text: therefore

text: it

text: is

text: no

text: longer

text: guaranteed

text: that

text: we

text: can

text: still

text: use

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Periodic points of a Landen transformation Armengol Gasull(1), Mireia Llorens(1) and V´ıctor Man˜osa (2) (1) Departament de Matem`atiques, Facultat de Ci`encies, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain mllorens@mat.uab.cat, gasull@mat.uab.cat (2) Departament de Matem`atiques Universitat Polit`ecnica de Catalunya Colom 1, 08222 Terrassa, Spain victor.manosa@upc.edu May 30, 2020 Abstract We prove the existence of 3-periodic orbits in a dynamical system associated to a Landen transformation previously studied by Boros, Chamberland and Moll, disproving a conjecture on the dynamics of this planar map introduced by the latter author. To this end we present a systematic methodology to determine and locate analytically isolated periodic points of al- gebraic maps. This approach can be useful to study other discrete dynamical systems with algebraic nature. Complementary results on the dynamics of the map associated with the Landen transformation are also presented. Mathematics Subject Classiﬁcation 2010: 37C25, 33E05. Keywords: Landen transformation, Periodic points, Poincar´e-Miranda theorem. 1 Introduction Given a deﬁnite integral depending on several parameters, a Landen transformation is a map on these parameters that leaves invariant the integral. In [1, 2], G. Boros and V. Moll introduced  the dynamical system given by     5a + 5b + a b + 9 a + b + 6 n n n n n n a = , b = ,  n+1 n+1  (an + bn + 2)4/3 (an + bn + 2)2/3  d + e + c (b + 3)c + (a + 3)e + 2d c + e n n n n n n n n n n c = , d = , e = , n+1 n+1 n+1 (a + b + 2)2/3 a + b + 2 (a + b + 2)1/3 n n n n n n as a Landen transformation associated to the integral 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Periodic

label: title

text: points

text: of

text: a

text: Landen

text: transformation

text: Armengol

label: author

text: Mireia

text: and

text: V´ıctor

text: Man˜osa

text: Departament

text: de

text: Matem`atiques,

text: Facultat

text: de

text: Ci`encies,

text: Universitat

text: Aut`onoma

text: de

text: Barcelona,

text: Bellaterra,

text: Barcelona,

text: Spain

text: mllorens@mat.uab.cat,

text: gasull@mat.uab.cat

text: Departament

text: de

text: Matem`atiques

text: Universitat

text: Polit`ecnica

text: de

text: Catalunya

text: Colom

text: Terrassa,

text: Spain

text: victor.manosa@upc.edu

text: May

text: Abstract

text: We

label: abstract

text: prove

text: the

text: existence

text: of

text: orbits

text: in

text: a

text: dynamical

text: system

text: associated

text: to

text: a

text: Landen

text: transformation

text: previously

text: studied

text: by

text: Boros,

text: Chamberland

text: and

text: Moll,

text: disproving

text: a

text: conjecture

text: on

text: the

text: dynamics

text: of

text: this

text: planar

text: map

text: introduced

text: by

text: the

text: latter

text: author.

text: To

text: this

text: end

text: we

text: present

text: a

text: systematic

text: methodology

text: to

text: determine

text: and

text: locate

text: analytically

text: isolated

text: periodic

text: points

text: of

text: al-

text: gebraic

text: maps.

text: This

text: approach

text: can

text: be

text: useful

text: to

text: study

text: other

text: discrete

text: dynamical

text: systems

text: with

text: algebraic

text: nature.

text: Complementary

text: results

text: on

text: the

text: dynamics

text: of

text: the

text: map

text: associated

text: with

text: the

text: Landen

text: transformation

text: are

text: also

text: presented.

text: Mathematics

text: Subject

text: Classiﬁcation

text: Keywords:

text: Landen

text: transformation,

text: Periodic

text: points,

text: Poincar´e-Miranda

text: theorem.

text: Introduction

text: Given

text: a

text: deﬁnite

text: integral

text: depending

text: on

text: several

text: parameters,

text: a

text: Landen

text: transformation

text: is

text: a

text: map

text: on

text: these

text: parameters

text: that

text: leaves

text: invariant

text: the

text: integral.

text: In

text: G.

text: Boros

text: and

text: V.

text: Moll

text: introduced

text: the

text: dynamical

text: system

text: given

text: by

text: a

text: b

text: a

text: b

text: n

text: n

text: n

text: n

text: n

text: n

text: a

text: b

text: (an

text: bn

text: (an

text: bn

text: d

text: e

text: c

text: (b

text: (a

text: c

text: e

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: c

text: d

text: e

text: (a

text: b

text: a

text: b

text: (a

text: b

text: n

text: n

text: n

text: n

text: n

text: n

text: as

text: a

text: Landen

text: transformation

text: associated

text: to

text: the

text: integral

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: This polynomial has degree 371, and using once more Sturm’s method we get that it has exactly 16 diﬀerent real non-zero roots. Lemma 5. Let I , with i = 1, . . . , 16, be disjoint intervals, each one of them containing a unique i real root of d . Then, any real solution (m, n, r) of system (11) is contained in one of the 163 sets 17 I × × ∈ { } := I I I , i, j, k 1, . . . , 16 . (13) i,j,k i j k Proof. Let (m , n , r ) be a real solution of (11). We want to show that it is also a solution of 0 0 0 { } d (m) = 0, d (n) = 0, d (r) = 0 . Observe that by construction, n must be a root of d . 17 17 17 0 15 From Lemma 4, we have that (m , n ) must be also a zero of d and d . Hence n must be also 0 0 13 14 0 a zero of d , and therefore of gcd (d (n), d (n)) which is a polynomial of degree 1087 with the 16 15 16 factor n716. Since we are interested in its non-zero roots, we remove this factor, obtaining that n 0 must be a root of d . By using an analogous argument and Lemma 4 again, we can see that m 17 0 and r are also roots of d . 0 17 Since d has 16 diﬀerent real roots, any solution (m, n, r) of system (11) must be contained in 17 a box of the form (13), and each box contains at most one solution. Now we can prove statement (c) of Theorem 1. We follow the steps explained in Section : − Step 1: Recall that d has 16 non-zero real roots. Two of them are n = 1, n = 2. Although two 17 more explicit roots are n = m , i = 1, 2 given in (7), we prefer to take 14 intervals with rational i i − ends and length smaller than 10 20, I , i = 1, 2, . . . , 14, each one of them containing a unique root. i We consider: (cid:20) (cid:21) (cid:20) (cid:21) −4308988841618670568853 −34471910732949364550823 −34411805733101949308435 −17205902866550974654217 I1 = (cid:20) , (cid:21) , I2 = (cid:20) , (cid:21) , 147573952589676412928 1180591620717411303424 1180591620717411303424 590295810358705651712 −9138398550509024508051 −4569199275254512254025 −4416518740855918762195 −8833037481711837524389 I3 = (cid:20) , (cid:21) , I4 = (cid:20) , (cid:21) , 1180591620717411303424 590295810358705651712 590295810358705651712 1180591620717411303424 −994661336537171251825 −3978645346148685007299 −3977374161031280580629 −994343540257820145157 I5 = (cid:20) , (cid:21), I6 = (cid:20) , (cid:21), 295147905179352825856 1180591620717411303424 1180591620717411303424 295147905179352825856 −197879469664271669175 −3166071514628346706799 −3144313156826151948503 −1572156578413075974251 I7 = (cid:20) , (cid:21), I8 = (cid:20) , (cid:21) , 73786976294838206464 1180591620717411303424 1180591620717411303424 590295810358705651712 −399397086201257638833 −1597588344805030555331 −1053526769518098399097 −131690846189762299887 I9 = (cid:20) , ,(cid:21) I10 = (cid:20) , (cid:21) , 295147905179352825856 1180591620717411303424 4722366482869645213696 590295810358705651712 −1064910654630154190265 − 133113831828769273783 1065572958580542810237 532786479290271405119 I11 = (cid:20) , (cid:21) , I12 = (cid:20) , (cid:21) , 9444732965739290427392 1180591620717411303424 9444732965739290427392 4722366482869645213696 128535594827653577343 1028284758621228618745 177910645965499912685 1423285167723999301481 I13 = , , I14 = , . 590295810358705651712 4722366482869645213696 147573952589676412928 1180591620717411303424 − We also introduce the degenerate intervals I = [ 1, 1] and I = [2, 2] containing the exact roots 15 16 − n = 1 and n = 2. By Lemma 5, all the real solutions of system (11) are contained in one of the 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: polynomial

text: has

text: degree

text: and

text: using

text: once

text: more

text: Sturm’s

text: method

text: we

text: get

text: that

text: it

text: has

text: exactly

text: diﬀerent

text: real

text: non-zero

text: roots.

text: Lemma

text: Let

text: I

text: with

text: i

text: be

text: disjoint

text: intervals,

text: each

text: one

text: of

text: them

text: containing

text: a

text: unique

text: i

text: real

text: root

text: of

text: d

text: Then,

text: any

text: real

text: solution

text: (m,

text: n,

text: r)

text: of

text: system

text: is

text: contained

text: in

text: one

text: of

text: the

text: sets

text: I

label: equation

text: I

text: I

text: I

text: i,

text: j,

text: k

text: i,j,k

text: i

text: j

text: k

text: Proof.

text: Let

text: (m

text: n

text: r

text: be

text: a

text: real

text: solution

text: of

text: We

text: want

text: to

text: show

text: that

text: it

text: is

text: also

text: a

text: solution

text: of

text: d

text: (m)

text: d

text: (n)

text: d

text: (r)

text: Observe

text: that

text: by

text: construction,

text: n

text: must

text: be

text: a

text: root

text: of

text: d

text: From

text: Lemma

text: we

text: have

text: that

text: (m

text: n

text: must

text: be

text: also

text: a

text: zero

text: of

text: d

text: and

text: d

text: Hence

text: n

text: must

text: be

text: also

text: a

text: zero

text: of

text: d

text: and

text: therefore

text: of

text: gcd

text: (d

text: (n),

text: d

text: (n))

text: which

text: is

text: a

text: polynomial

text: of

text: degree

text: with

text: the

text: factor

text: Since

text: we

text: are

text: interested

text: in

text: its

text: non-zero

text: roots,

text: we

text: remove

text: this

text: factor,

text: obtaining

text: that

text: n

text: must

text: be

text: a

text: root

text: of

text: d

text: By

text: using

text: an

text: analogous

text: argument

text: and

text: Lemma

text: again,

text: we

text: can

text: see

text: that

text: m

text: and

text: r

text: are

text: also

text: roots

text: of

text: d

text: Since

text: d

text: has

text: diﬀerent

text: real

text: roots,

text: any

text: solution

text: (m,

text: n,

text: r)

text: of

text: system

text: must

text: be

text: contained

text: in

text: a

text: box

text: of

text: the

text: form

text: and

text: each

text: box

text: contains

text: at

text: most

text: one

text: solution.

text: Now

text: we

text: can

text: prove

text: statement

text: (c)

text: of

text: Theorem

text: We

text: follow

text: the

text: steps

text: explained

text: in

text: Section

text: Step

text: Recall

text: that

text: d

text: has

text: non-zero

text: real

text: roots.

text: Two

text: of

text: them

text: are

text: n

text: n

text: Although

text: two

text: more

text: explicit

text: roots

text: are

text: n

text: m

text: i

text: given

text: in

text: we

text: prefer

text: to

text: take

text: intervals

text: with

text: rational

text: i

text: i

text: ends

text: and

text: length

text: smaller

text: than

text: I

text: i

text: each

text: one

text: of

text: them

text: containing

text: a

text: unique

text: root.

text: i

text: We

text: consider:

text: We

text: also

text: introduce

text: the

text: degenerate

text: intervals

text: I

text: and

text: I

text: containing

text: the

text: exact

text: roots

text: n

text: and

text: n

text: By

text: Lemma

text: all

text: the

text: real

text: solutions

text: of

text: system

text: are

text: contained

text: in

text: one

text: of

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: and 3368785687756582636246263551756811406295236320753178521304454421527 b = , 10710654937528498667637446691242283113536911386660380934878003200 6579659546399575461418490144259606329620802274396204966850496409 b = . 20919247924860348960190099800217926294342327605140376818548736 By using the decimal approximation we get, ∈ (cid:39) − − a [a,a] [ 25210.658115921519312682, 25210.658115921519312679], ∈ (cid:39) b [b,b] [314.5265819322469464743,314.5265819322469464749], − − (cid:39) × − where we observe that max(a a, b b) 2.8 10 18. Applying Lemma 7 to each of the 12 boxes (14), we obtain rational bounds for the components of each periodic point of minimal period 3, that are summarized in the following tables, where only the decimal expression of some signiﬁcative digits is given. In all the cases the maximum length of − the interval localizing the 3-periodic points is smaller than 10 17, so the given expression of both ends of the intervals coincide. O O 1 a b 2 a b -25210.658115921519313 314.52658193224694647 -25080.503857555317449 314.36115078061939834 -11.080089229288244821 -29.194152462502174029 -11.094342178650567807 -29.143225143670723223 1.0164106270635353803 -3.0178440371837045505 1.0179782228602330827 -3.0165421366176918413 O O 3 a b 4 a b -550.35997876621370288 84.580855473468510676 -500.96942815695686889 80.145842594816842809 -13.613164340185764400 -7.6737642167841728949 -13.481597649423988848 -7.4104176831057891201 0.13590789992610542444 -2.1255835876361107899 0.088325991394389446424 -2.0994294342645985249 5 Dynamics associated to the saddle point P 2 In this section we study the invariant sets of the saddle point P . First, Proposition 8 character- 2 izes the stable set of P . We will also give numerical evidences of the existence of homoclinic orbits, 2 that is, initial conditions on the local unstable manifold, whose orbit converges to P . Finally, we 2 will provide numerical evidences of the existence of points on the local unstable manifold, whose orbits end in the non-deﬁnition set. 5.1 The stable set of P 2 { ∈ R We denote the stable set of the ﬁxed point P as W s(P ) = (a, b) 2 : lim →∞ Gn(a, b) = 2 2 n } P . This set is not necessarily a manifold. For hyperbolic points, like P there is also the so called 2 2 ⊂ local stable manifold W s (P ) W s(P ), that is an actual manifold and is only deﬁned in a small loc 2 2 neighborhood of the ﬁxed point. Our ﬁrst result characterizes totally the stable set of P and its 2 local stable manifold. (cid:0) (cid:1) Proposition 8. It holds that ∪ { ∈ R ∃ ≥ ∈ ∪ } \ { − − } W s(P ) = L (a, b) 2 : n 0 : Gn(a, b) R C ( 1, 1) , 2 2 1 1 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: b

text: b

text: By

text: using

text: the

text: decimal

text: approximation

text: we

text: get,

text: a

text: [a,a]

text: b

text: [b,b]

text: where

text: we

text: observe

text: that

text: max(a

text: a,

text: b

text: b)

text: Applying

text: Lemma

text: to

text: each

text: of

text: the

text: boxes

text: we

text: obtain

text: rational

text: bounds

text: for

text: the

text: components

text: of

text: each

text: periodic

text: point

text: of

text: minimal

text: period

text: that

text: are

text: summarized

text: in

text: the

text: following

text: tables,

text: where

text: only

text: the

text: decimal

text: expression

text: of

text: some

text: signiﬁcative

text: digits

text: is

text: given.

text: In

text: all

text: the

text: cases

text: the

text: maximum

text: length

text: of

text: the

text: interval

text: localizing

text: the

text: points

text: is

text: smaller

text: than

text: so

text: the

text: given

text: expression

text: of

text: both

text: ends

text: of

text: the

text: intervals

text: coincide.

text: O

text: O

text: a

text: b

text: a

text: b

text: O

text: O

text: a

text: b

text: a

text: b

text: Dynamics

text: associated

text: to

text: the

text: saddle

text: point

text: P

text: In

text: this

text: section

text: we

text: study

text: the

text: invariant

text: sets

text: of

text: the

text: saddle

text: point

text: P

text: First,

text: Proposition

text: character-

text: izes

text: the

text: stable

text: set

text: of

text: P

text: We

text: will

text: also

text: give

text: numerical

text: evidences

text: of

text: the

text: existence

text: of

text: homoclinic

text: orbits,

text: that

text: is,

text: initial

text: conditions

text: on

text: the

text: local

text: unstable

text: manifold,

text: whose

text: orbit

text: converges

text: to

text: P

text: Finally,

text: we

text: will

text: provide

text: numerical

text: evidences

text: of

text: the

text: existence

text: of

text: points

text: on

text: the

text: local

text: unstable

text: manifold,

text: whose

text: orbits

text: end

text: in

text: the

text: non-deﬁnition

text: set.

text: The

text: stable

text: set

text: of

text: P

text: R

text: We

text: denote

text: the

text: stable

text: set

text: of

text: the

text: ﬁxed

text: point

text: P

text: as

text: W

text: s(P

text: (a,

text: b)

text: lim

text: Gn(a,

text: b)

text: n

text: P

text: This

text: set

text: is

text: not

text: necessarily

text: a

text: manifold.

text: For

text: hyperbolic

text: points,

text: like

text: P

text: there

text: is

text: also

text: the

text: so

text: called

text: local

text: stable

text: manifold

text: W

text: s

text: (P

text: W

text: s(P

text: that

text: is

text: an

text: actual

text: manifold

text: and

text: is

text: only

text: deﬁned

text: in

text: a

text: small

text: loc

text: neighborhood

text: of

text: the

text: ﬁxed

text: point.

text: Our

text: ﬁrst

text: result

text: characterizes

text: totally

text: the

text: stable

text: set

text: of

text: P

text: and

text: its

text: local

text: stable

text: manifold.

text: Proposition

text: It

text: holds

text: that

text: R

text: W

text: s(P

text: L

text: (a,

text: b)

text: n

text: Gn(a,

text: b)

text: R

text: C

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text: The proof has three steps, namely (A)–(C): (A) From the properties (i) and (ii), we conclude − −∞ − g (( 2, 0)) = ( , 4]. −∞ − −∞ − (B) Using (i) and (ii) again, we conclude that g (( , 2)) = ( , 4], hence the interval −∞ − ( , 2) is invariant by g. We will study the dynamics in this interval. ∈ − (cid:39) − Let (cid:96) (p, 2) be the unique value in this interval such that g((cid:96)) = m ((cid:96) 2.6675). By using − ⊂ the monotony of g, the interval [m, (cid:96)] is invariant. Indeed, g([m, (cid:96)]) = [g((cid:96)), g(m)] = [m, 4] ∈ −∞ ∪ − [m, (cid:96)], see again Figure 3. Now we claim that for all t ( , m) ((cid:96), 2) there exists n > 0 such ∈ that t = gn(t) [m, (cid:96)]. n − − −∞ Indeed, by the monotonicity of g in ((cid:96), 2) we have that g(((cid:96), 2)) = ( , m). Since for all ∈ −∞ − ∈ − ∈ −∞ t ( , m), we have g(t) < 4 < (cid:96), then g(t) / ((cid:96), 2), hence g(t) ( , (cid:96)). We only need, ∈ −∞ ∈ −∞ therefore to prove the claim in t ( , m). We proceed by contradiction. Consider t ( , m) 0 ∈ ∈ −∞ and suppose that none iterate t [m, (cid:96)], so that for all n > 0 we have t ( , m). From (iv), n n { } the sequence t is increasing, and as we are assuming that it is bounded from above by m, the n sequence must have a limit that, by continuity, must be a ﬁxed point, which is a contradiction −∞ because there is no ﬁxed point in ( , m]. Hence the claim is proved, and we only have to study the dynamics of g in [m, (cid:96)]. (C) We study now the dynamics on the interval [m, (cid:96)]. We denote m := m and (cid:96) := (cid:96), and 0 0 consider the sequences (cid:96) = g(m − ) = g2((cid:96) − ) and m = g((cid:96) ) = g2(m − ). (15) k k 1 k 1 k k k 1 ≥ Observe that since g is strictly decreasing in [m, (cid:96)], for k 1 we obtain − [m − , (cid:96) ] := g2k 1 ([m, (cid:96)]) and [m , (cid:96) ] := g2k ([m, (cid:96)]) . k 1 k k k { } { } We will prove that m and (cid:96) are increasing and decreasing sequences, respectively, that k k converge to the ﬁxed point p, thus proving the result. − Some computations show that (cid:96) = g(m) = 4 < (cid:96) = (cid:96) , and that m = m < m = 1 0 0 1 − g((cid:96) ) = g( 4). We proceed by induction, assuming that m − < m and (cid:96) < (cid:96) − . Using 1 k 1 k k k 1 (15), as g is decreasing and m − < m , we have (cid:96) = g(m − ) > g(m ) = (cid:96) . Likewise, since k 1 k k k 1 k k+1 { } { } (cid:96) < (cid:96) we have m = g((cid:96) ) > g((cid:96) ) = m . Therefore, the sequences m and (cid:96) are k+1 k k+1 k+1 k k k k monotonous increasing and decreasing, respectively. Since both sequences are bounded, and using the expressions in (15), we have that both converge to a ﬁxed point of g2. But since there are not 2-periodic points, except the ﬁxed point p, we have lim m = lim (cid:96) = p. →∞ k →∞ k k k 17

text: The

text: proof

text: has

text: three

text: steps,

text: namely

text: (A)–(C):

text: (A)

text: From

text: the

text: properties

text: (i)

text: and

text: (ii),

text: we

text: conclude

text: g

text: (B)

text: Using

text: (i)

text: and

text: (ii)

text: again,

text: we

text: conclude

text: that

text: g

text: hence

text: the

text: interval

text: is

text: invariant

text: by

text: g.

text: We

text: will

text: study

text: the

text: dynamics

text: in

text: this

text: interval.

text: Let

text: (p,

text: be

text: the

text: unique

text: value

text: in

text: this

text: interval

text: such

text: that

text: m

text: By

text: using

text: the

text: monotony

text: of

text: g,

text: the

text: interval

text: [m,

text: is

text: invariant.

text: Indeed,

text: g([m,

text: g(m)]

text: [m,

text: [m,

text: see

text: again

text: Figure

text: Now

text: we

text: claim

text: that

text: for

text: all

text: t

text: m)

text: there

text: exists

text: n

text: such

text: that

text: t

text: gn(t)

text: [m,

text: n

text: Indeed,

text: by

text: the

text: monotonicity

text: of

text: g

text: in

text: we

text: have

text: that

text: m).

text: Since

text: for

text: all

text: t

text: m),

text: we

text: have

text: g(t)

text: then

text: g(t)

text: hence

text: g(t)

text: We

text: only

text: need,

text: therefore

text: to

text: prove

text: the

text: claim

text: in

text: t

text: m).

text: We

text: proceed

text: by

text: contradiction.

text: Consider

text: t

text: m)

text: and

text: suppose

text: that

text: iterate

text: t

text: [m,

text: so

text: that

text: for

text: all

text: n

text: we

text: have

text: t

text: m).

text: From

text: (iv),

text: n

text: n

text: the

text: sequence

text: t

text: is

text: increasing,

text: and

text: as

text: we

text: are

text: assuming

text: that

text: it

text: is

text: bounded

text: from

text: above

text: by

text: m,

text: the

text: n

text: sequence

text: must

text: have

text: a

text: limit

text: that,

text: by

text: continuity,

text: must

text: be

text: a

text: ﬁxed

text: point,

text: which

text: is

text: a

text: contradiction

text: because

text: there

text: is

text: no

text: ﬁxed

text: point

text: in

text: m].

text: Hence

text: the

text: claim

text: is

text: proved,

text: and

text: we

text: only

text: have

text: to

text: study

text: the

text: dynamics

text: of

text: g

text: in

text: [m,

text: (C)

text: We

text: study

text: now

text: the

text: dynamics

text: on

text: the

text: interval

text: [m,

text: We

text: denote

text: m

text: m

text: and

text: and

text: consider

text: the

text: sequences

text: g(m

text: and

text: m

text: k

text: k

text: k

text: k

text: k

text: k

text: Observe

text: that

text: since

text: g

text: is

text: strictly

text: decreasing

text: in

text: [m,

text: for

text: k

text: we

text: obtain

text: [m

text: ([m,

text: and

text: [m

text: ([m,

text: k

text: k

text: k

text: k

text: We

text: will

text: prove

text: that

text: m

text: and

text: are

text: increasing

text: and

text: decreasing

text: sequences,

text: respectively,

text: that

text: k

text: k

text: converge

text: to

text: the

text: ﬁxed

text: point

text: p,

text: thus

text: proving

text: the

text: result.

text: Some

text: computations

text: show

text: that

text: g(m)

text: and

text: that

text: m

text: m

text: m

text: g(

text: We

text: proceed

text: by

text: induction,

text: assuming

text: that

text: m

text: m

text: and

text: Using

text: k

text: k

text: k

text: k

text: as

text: g

text: is

text: decreasing

text: and

text: m

text: m

text: we

text: have

text: g(m

text: g(m

text: Likewise,

text: since

text: k

text: k

text: k

text: k

text: k

text: we

text: have

text: m

text: m

text: Therefore,

text: the

text: sequences

text: m

text: and

text: are

text: k

text: k

text: k

text: k

text: k

text: monotonous

text: increasing

text: and

text: decreasing,

text: respectively.

text: Since

text: both

text: sequences

text: are

text: bounded,

text: and

text: using

text: the

text: expressions

text: in

text: we

text: have

text: that

text: both

text: converge

text: to

text: a

text: ﬁxed

text: point

text: of

text: But

text: since

text: there

text: are

text: not

text: points,

text: except

text: the

text: ﬁxed

text: point

text: p,

text: we

text: have

text: lim

text: m

text: lim

text: p.

text: k

text: k

text: k

text: k


text: Proof. Due to the particular form of the linear part of F , the local unstable manifold W u (0, 0) loc is given by a smooth function of the form y = w(x) = w x2 + w x3 + w x4 + w x5 + O(x6), 2 3 4 5 that is, a point is on the local stable manifo(cid:0)ld if it is of(cid:1) the form (x, w(x)). Imposing that F (x, w(x)) = (F (x, w(x)), F (x, w(x))) is also on this curve we get that the points on the local 1 2 unstable manifold must satisfy F (x, w(x)) = w F (x, w(x)) . The result follows by comparing the 2 1 terms in the Taylor development of both members of the last equation. Acknowledgements The authors are supported by Ministry of Economy, Industry and Competitiveness of the Span- ish Government through grants MINECO/FEDER MTM2016-77278-P (ﬁrst author) and DPI2016- 77407-P (AEI/FEDER, UE, second and third author). The ﬁrst author is also supported by the grant 2014-SGR-568 from AGAUR, Generalitat de Catalunya. The third author is supported by the grant 2014-SGR-859 from AGAUR, Generalitat de Catalunya. References [1] G. Boros, V. H. Moll. A rational Landen transformation. The case of degree six, in M. Knopp et al. eds, Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis, Con- temporary Mathematics 251. AMS, Providence RI 2000, 83–91. [2] G. Boros, V. H. Moll. Landen transformations and the integration of rational functions. Math- ematics of Computation 71 (2001), 649–668. [3] M. Chamberland, V. H. Moll. Dynamics of the degree six Landen transformation. Discrete and Continuous Dynamical Systems A 15 (2006), 905–919. [4] J. Garc´ıa-Saldan˜a, A. Gasull, H. Giacomini. Bifurcation diagram and stability for a one– parameter family of planar vector ﬁelds. J. Math. Anal. Appl. 413 (2014), 321–342. [5] L. Gardini. Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding pe- riodic points Nonlinear Analysis: Theory, Methods & Applications 23 (1994), 1039–1089. [6] L. Gardini, I. Sushko, V. Avrutin, M. Schanz. Critical homoclinic orbits lead to snap-back repellers. Chaos, Solitons & Fractals 44 (2011), 433–449. [7] J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin 1997. [8] W. Kulpa. The Poincar´e-Miranda Theorem. Amer. Math. Month. 104 (1997), 545–550. 22

text: Proof.

text: Due

text: to

text: the

text: particular

text: form

text: of

text: the

text: linear

text: part

text: of

text: F

text: the

text: local

text: unstable

text: manifold

text: W

text: u

text: loc

text: is

text: given

text: by

text: a

text: smooth

text: function

text: of

text: the

text: form

text: y

text: w(x)

text: w

text: w

text: w

text: w

text: that

text: is,

text: a

text: point

text: is

text: on

text: the

text: local

text: stable

text: if

text: it

text: is

text: the

text: form

text: (x,

text: w(x)).

text: Imposing

text: that

text: F

text: (x,

text: w(x))

text: (F

text: (x,

text: w(x)),

text: F

text: (x,

text: w(x)))

text: is

text: also

text: on

text: this

text: curve

text: we

text: get

text: that

text: the

text: points

text: on

text: the

text: local

text: unstable

text: manifold

text: must

text: satisfy

text: F

text: (x,

text: w(x))

text: w

text: F

text: (x,

text: w(x))

text: The

text: result

text: follows

text: by

text: comparing

text: the

text: terms

text: in

text: the

text: Taylor

text: development

text: of

text: both

text: members

text: of

text: the

text: last

text: equation.

text: Acknowledgements

text: The

text: authors

text: are

text: supported

text: by

text: Ministry

text: of

text: Economy,

text: Industry

text: and

text: Competitiveness

text: of

text: the

text: Span-

text: ish

text: Government

text: through

text: grants

text: MINECO/FEDER

text: (ﬁrst

text: author)

text: and

text: (AEI/FEDER,

text: UE,

text: second

text: and

text: third

text: author).

text: The

text: ﬁrst

text: author

text: is

text: also

text: supported

text: by

text: the

text: grant

text: from

text: AGAUR,

text: Generalitat

text: de

text: Catalunya.

text: The

text: third

text: author

text: is

text: supported

text: by

text: the

text: grant

text: from

text: AGAUR,

text: Generalitat

text: de

text: Catalunya.

text: References

text: G.

text: Boros,

text: V.

text: H.

text: Moll.

text: A

text: rational

text: Landen

text: transformation.

text: The

text: case

text: of

text: degree

text: six,

text: in

text: M.

text: Knopp

text: et

text: al.

text: eds,

text: Analysis,

text: Geometry,

text: Number

text: Theory:

text: The

text: Mathematics

text: of

text: Leon

text: Ehrenpreis,

text: Con-

text: temporary

text: Mathematics

text: AMS,

text: Providence

text: RI

text: G.

text: Boros,

text: V.

text: H.

text: Moll.

text: Landen

text: transformations

text: and

text: the

text: integration

text: of

text: rational

text: functions.

text: Math-

text: ematics

text: of

text: Computation

text: M.

text: Chamberland,

text: V.

text: H.

text: Moll.

text: Dynamics

text: of

text: the

text: degree

text: six

text: Landen

text: transformation.

text: Discrete

text: and

text: Continuous

text: Dynamical

text: Systems

text: A

text: J.

text: Garc´ıa-Saldan˜a,

text: A.

text: Gasull,

text: H.

text: Giacomini.

text: Bifurcation

text: diagram

text: and

text: stability

text: for

text: a

text: one–

text: parameter

text: family

text: of

text: planar

text: vector

text: ﬁelds.

text: J.

text: Math.

text: Anal.

text: Appl.

text: L.

text: Gardini.

text: Homoclinic

text: bifurcations

text: in

text: n-dimensional

text: endomorphisms,

text: due

text: to

text: expanding

text: pe-

text: riodic

text: points

text: Nonlinear

text: Analysis:

text: Theory,

text: Methods

text: Applications

text: L.

text: Gardini,

text: I.

text: Sushko,

text: V.

text: Avrutin,

text: M.

text: Schanz.

text: Critical

text: homoclinic

text: orbits

text: lead

text: to

text: snap-back

text: repellers.

text: Chaos,

text: Solitons

text: Fractals

text: J.

text: Guckenheimer,

text: P.

text: Holmes.

text: Nonlinear

text: Oscillations,

text: Dynamical

text: Systems,

text: and

text: Bifurcations

text: of

text: Vector

text: Fields,

text: Springer,

text: Berlin

text: W.

text: Kulpa.

text: The

text: Poincar´e-Miranda

text: Theorem.

text: Amer.

text: Math.

text: Month.


text: (cid:80) (cid:80) ∈ I M for all x . If it holds that P (x) = M (x) < M =: U < 0, then we can discard (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) I the box . To compute the bounds M and M , we use the following straightforward result, which can (cid:96) (cid:96) (cid:80) be easily implemented in any computer algebra software. · · · I × Lemma 2. Consider P (x) = (cid:96) M(cid:96)(x) where M(cid:96)(x) = a(cid:96)x(cid:96)11x(cid:96)22 x(cid:96)nn, and a box = [u1, v1] × · · · × { } (cid:80) (cid:80) [u , v ] [u , v ]. Set O+ = (x , . . . , x ), such that x > 0 for all i = 1, . . . , n . Then, 2 2 n n 1 n i I ⊂ ⊂ R ∈ I ≤ ≤ (i) If O+ n, then for all x , M P (x) M , where (cid:96) (cid:96) (cid:96) (cid:96) · · · · · · (a) M = a u(cid:96)1u(cid:96)2 u(cid:96)n and M = a v(cid:96)1v(cid:96)2 v(cid:96)n if a > 0. (cid:96) (cid:96) 1 2 n (cid:96) (cid:96) 1 2 n (cid:96) · · · · · · (b) M = a v(cid:96)1v(cid:96)2 v(cid:96)n and M = a u(cid:96)1u(cid:96)2 u(cid:96)n if a < 0. (cid:101) (cid:96) (cid:96) 1 2 n (cid:96) (cid:96) 1 2 n (cid:96) I (cid:54)⊂ ∈ Q I (ii) If O+ we can always take a number ξ > 0, ξ such that the new box = [u + ξ, v + (cid:101) 1(cid:101) 1 × × · · · × ⊂ I ξ] [u + ξ, v + ξ] [u + ξ, v + ξ] O+, and then ﬁnd bounds for P on , using the 2 2 n n − − − I bounds given in item (i) for P (x , x , . . . , x ) := P (x ξ, x ξ, . . . , x ξ) on . ξ 1 2 n 1 2 n We try to apply the discard procedure until the number of remaining boxes coincides with our hopes. These hopes usually came from a previous numerical study of the problem. We start trying to prove that the ﬁrst function f does not vanish in the given box. It may happen that it is easier 1 to try to prove the same with another f . Notice also that sometimes to discard a box we must go j to the Step 1 and start with smaller boxes. Step 3: Once it is achieved an optimized list of non-discarded boxes, we identify those boxes that correspond to either ﬁxed points or periodic points with a period being a divisor of p, which we assume that we already know, and we also discard them. Step 4: From the non-discarded boxes list obtained in the previous step, we try to show that each box actually contains a solution by applying the Poincar´e-Miranda theorem. For completeness, we ⊂ R recall it. As usual, B and ∂B denote, respectively, the closure and the boundary of a set B n. I { ∈ R ≤ ≤ } Theorem 3 (Poincar´e-Miranda). Set = x = (x , . . . , x ) n : L < x < U , 1 i n . 1 n i i i I → (cid:54) ∈ I Suppose that f = (f , f , . . . , f ) : Rn is continuous, f(x) = 0 for all x ∂ , and for 1 2 n ≤ ≤ 1 i n, · ≤ f (x , . . . , x − , L , x , . . . , x ) f (x , . . . , x − , U , x , . . . , x ) 0, i 1 i 1 i i+1 n i 1 i 1 i i+1 n ∈ I Then, there exists s such that f(s) = 0. It is clear that when we deﬁne f to try to apply Poincar´e-Miranda theorem, the order of the components matters. So, sometimes to be under the hypotheses of the theorem it is better to 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: I

text: M

text: for

text: all

text: x

text: If

text: it

text: holds

text: that

text: P

text: (x)

text: M

text: (x)

text: M

text: U

text: then

text: we

text: can

text: discard

text: I

text: the

text: box

text: To

text: compute

text: the

text: bounds

text: M

text: and

text: M

text: we

text: use

text: the

text: following

text: straightforward

text: result,

text: which

text: can

text: be

text: easily

text: implemented

text: in

text: any

text: computer

text: algebra

text: software.

text: I

text: Lemma

text: Consider

text: P

text: (x)

text: where

text: and

text: a

text: box

text: [u

text: v

text: [u

text: v

text: Set

text: O+

text: (x

text: x

text: such

text: that

text: x

text: for

text: all

text: i

text: n

text: Then,

text: n

text: n

text: n

text: i

text: I

text: R

text: I

text: (i)

text: If

text: O+

text: n,

text: then

text: for

text: all

text: x

text: M

text: P

text: (x)

text: M

text: where

text: (a)

text: M

text: a

text: and

text: M

text: a

text: if

text: a

text: n

text: n

text: (b)

text: M

text: a

text: and

text: M

text: a

text: if

text: a

text: n

text: n

text: I

text: Q

text: I

text: (ii)

text: If

text: O+

text: we

text: can

text: always

text: take

text: a

text: number

text: ξ

text: ξ

text: such

text: that

text: the

text: new

text: box

text: [u

text: ξ,

text: v

text: I

text: ξ]

text: [u

text: ξ,

text: v

text: ξ]

text: [u

text: ξ,

text: v

text: ξ]

text: O+,

text: and

text: then

text: ﬁnd

text: bounds

text: for

text: P

text: on

text: using

text: the

text: n

text: n

text: I

text: bounds

text: given

text: in

text: item

text: (i)

text: for

text: P

text: (x

text: x

text: x

text: P

text: (x

text: ξ,

text: x

text: ξ,

text: x

text: ξ)

text: on

text: ξ

text: n

text: n

text: We

text: try

text: to

text: apply

text: the

text: discard

text: procedure

text: until

text: the

text: number

text: of

text: remaining

text: boxes

text: coincides

text: with

text: our

text: hopes.

text: These

text: hopes

text: usually

text: came

text: from

text: a

text: previous

text: numerical

text: study

text: of

text: the

text: problem.

text: We

text: start

text: trying

text: to

text: prove

text: that

text: the

text: ﬁrst

text: function

text: f

text: does

text: not

text: vanish

text: in

text: the

text: given

text: box.

text: It

text: may

text: happen

text: that

text: it

text: is

text: easier

text: to

text: try

text: to

text: prove

text: the

text: same

text: with

text: another

text: f

text: Notice

text: also

text: that

text: sometimes

text: to

text: discard

text: a

text: box

text: we

text: must

text: go

text: j

text: to

text: the

text: Step

text: and

text: start

text: with

text: smaller

text: boxes.

text: Step

text: Once

text: it

text: is

text: achieved

text: an

text: optimized

text: list

text: of

text: non-discarded

text: boxes,

text: we

text: identify

text: those

text: boxes

text: that

text: correspond

text: to

text: either

text: ﬁxed

text: points

text: or

text: periodic

text: points

text: with

text: a

text: period

text: being

text: a

text: divisor

text: of

text: p,

text: which

text: we

text: assume

text: that

text: we

text: already

text: know,

text: and

text: we

text: also

text: discard

text: them.

text: Step

text: From

text: the

text: non-discarded

text: boxes

text: list

text: obtained

text: in

text: the

text: previous

text: step,

text: we

text: try

text: to

text: show

text: that

text: each

text: box

text: actually

text: contains

text: a

text: solution

text: by

text: applying

text: the

text: Poincar´e-Miranda

text: theorem.

text: For

text: completeness,

text: we

text: R

text: recall

text: it.

text: As

text: usual,

text: B

text: and

text: ∂B

text: denote,

text: respectively,

text: the

text: closure

text: and

text: the

text: boundary

text: of

text: a

text: set

text: B

text: n.

text: I

text: R

text: Theorem

text: (Poincar´e-Miranda).

text: Set

text: x

text: (x

text: x

text: n

text: L

text: x

text: U

text: i

text: n

text: n

text: i

text: i

text: i

text: I

text: I

text: Suppose

text: that

text: f

text: (f

text: f

text: f

text: Rn

text: is

text: continuous,

text: f(x)

text: for

text: all

text: x

text: and

text: for

text: n

text: i

text: n,

text: f

text: (x

text: x

text: L

text: x

text: x

text: f

text: (x

text: x

text: U

text: x

text: x

text: i

text: i

text: i

text: n

text: i

text: i

text: i

text: n

text: I

text: Then,

text: there

text: exists

text: s

text: such

text: that

text: f(s)

text: It

text: is

text: clear

text: that

text: when

text: we

text: deﬁne

text: f

text: to

text: try

text: to

text: apply

text: Poincar´e-Miranda

text: theorem,

text: the

text: order

text: of

text: the

text: components

text: matters.

text: So,

text: sometimes

text: to

text: be

text: under

text: the

text: hypotheses

text: of

text: the

text: theorem

text: it

text: is

text: better

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: Ab-initio study of the Coulomb interaction in Nb Co clusters: x ∗ Strong on-site versus weak non-local screening † ‡ L. Peters,1, E. S¸a¸sıo˘glu,1, I. Mertig,1 and M. I. Katsnelson1 (Dated: May 30, 2020) By means of ab-initio calculations in conjunction with the random-phase approximation (RPA) within the full-potential linearized augmented plane wave method we study the screening of the ≤ ≤ Coulomb interaction in Nb Co (1 x 9) clusters. In addition, these results are compared x with pure bcc Nb bulk. We ﬁnd that for all clusters the onsite Coulomb interaction in RPA is strongly screened whereas the inter-site non-local Coulomb interaction is weakly screened and for some clusters it is unscreened or even anti-screened. This is in strong contrast with pure Nb bulk, where the inter-site Coulomb interaction is almost completely screened. Further, constrained RPA → calculations reveal that the contribution of the Co 3d 3d channel to the total screening of the Co 3d electrons is small. Moreover, we ﬁnd that both the onsite and inter-site Coulomb interaction parameters decrease in a reasonable approximation linearly with the cluster size and for clusters having more than 20 Nb atoms a transition from 0D to 3D screening is expected to take place. PACS numbers: 31.15.A-, 31.15.es, 36.40.-c, 36.40.Cg, 73.22.-f, I. INTRODUCTION This information is crucial in order to obtain a proper fundamental understanding of the system. Namely cor- relation eﬀects among the electrons inhibit in general an The interest in the ﬁeld of clusters is growing due to exact solution. Therefore, approximate methods are re- the increasing demand for nano-technology. Besides the quired in practice. The choice of a suitable approximate relevance for technology, clusters are also fundamentally method requires knowledge of the eﬀective Coulomb in- very interesting. They behave in general very diﬀerent teraction in the system. More precisely, the gradient of from their bulk counterpart. Also their electronic and the eﬀective Coulomb interaction is of importance.13,14 magnetic properties can drastically change by just adding A very small gradient means that the eﬀective Coulomb or removing one atom.1–4 For example, in a recent work interaction is merely constant, while a very large gradi- on Nb Co clusters it is demonstrated that Nb Co and x 5 ent indicates a purely local eﬀective Coulomb interaction. Nb Co are non-magnetic, while Nb Co and Nb Co are 7 4 6 In the former case a mean-ﬁeld treatment, i.e. single- strongly magnetic.5 The physical origin of this behavior particle approach, is probably a good choice, while for can be traced back to the drastic change of the electronic the latter it might be the (generalized) Hubbard model. structure as a function of cluster size. The study of bimetallic clusters oﬀers a broader play- The aim of the present work is the ab-initio determina- ground than for pure clusters. This has resulted in a tion of the Coulomb interaction for NbCo to Nb Co and number of intriguing observations.5–10 For example, from 7 Nb Co clusters. Besides being fundamentally interesting, 9 an anion photo-electron spectroscopy study on bimetal- such information is crucial to select an adequate theoreti- lic Nb Co clusters, it was observed that for x = 6 due to x cal method for a further investigation of the system. The the addition of one Co atom to the Nb host, the elec- x geometries of Nb Co to Nb Co and Nb Co are well estab- 3 7 9 tronic structure resembles that of a typical bulk semi- lished from a comparison of theoretically and experimen- conductor.8 Therefore, this cluster was then proposed as tally obtained vibrational spectra.5 In addition NbCo and a candidate for semiconductor materials. Nb Co are considered, since the number of isomers is very 2 From the theoretical side the bond properties and elec- small. All these clusters are known to be magnetic except tronic structure of (NbCo)x clusters has been investi- Nb5Co and Nb7Co, which are non-magnetic. By employ- gated by means of relativistic density functional theory ing the full-potential linearized augmented plane wave (DFT).11 Further, the geometry, stability and electronic (FLAPW) method using Wannier functions in conjunc- properties of neutral and anionic NbxCo clusters is com- tion with the random-phase approximation (RPA)15–17, pared with pure Nbx clusters within a DFT study.12 Re- it is found that in these clusters the onsite Coulomb in- cently a combined theoretical and experimental investiga- teraction in RPA is well screened, while the inter-site tion has been performed on NbxCo clusters.5 In this work Coulomb interactions are barely screened. Interestingly the geometry is obtained from a comparison of experi- for NbCo the inter-site interaction is unscreened, while mentally and theoretically obtained vibrational spectra. for Nb Co even anti-screening occurs. The important 4 With the geometry established the electronic structure is consequence being that the screened Coulomb interaction investigated in order to explain the magnetic properties is almost constant throughout the clusters. For complete- obtained from magnetic deﬂection experiments. ness these results are compared with pure Nb bulk for To our knowledge a systematic assessment of screening which only the onsite Coulomb interaction is apprecia- and correlation eﬀects in Nb Co clusters does not exist. ble, while the inter-site Coulomb interactions are almost x

text: Ab-initio

text: study

text: of

text: the

text: Coulomb

text: interaction

text: in

text: Nb

text: Co

text: clusters:

text: x

text: Strong

text: on-site

text: versus

text: weak

text: non-local

text: screening

text: L.

text: E.

text: I.

text: and

text: M.

text: I.

text: (Dated:

text: May

label: date

text: By

text: means

text: of

text: ab-initio

text: calculations

text: in

text: conjunction

text: with

text: the

text: random-phase

text: approximation

text: (RPA)

text: within

text: the

text: full-potential

text: linearized

text: augmented

text: plane

text: wave

text: method

text: we

text: study

text: the

text: screening

text: of

text: the

text: Coulomb

text: interaction

text: in

text: Nb

text: Co

text: x

text: clusters.

text: In

text: addition,

text: these

text: results

text: are

text: compared

text: x

text: with

text: pure

text: bcc

text: Nb

text: bulk.

text: We

text: ﬁnd

text: that

text: for

text: all

text: clusters

text: the

text: onsite

text: Coulomb

text: interaction

text: in

text: RPA

text: is

text: strongly

text: screened

text: whereas

text: the

text: inter-site

text: non-local

text: Coulomb

text: interaction

text: is

text: weakly

text: screened

text: and

text: for

text: some

text: clusters

text: it

text: is

text: unscreened

text: or

text: even

text: anti-screened.

text: This

text: is

text: in

text: strong

text: contrast

text: with

text: pure

text: Nb

text: bulk,

text: where

text: the

text: inter-site

text: Coulomb

text: interaction

text: is

text: almost

text: completely

text: screened.

text: Further,

text: constrained

text: RPA

text: calculations

text: reveal

text: that

text: the

text: contribution

text: of

text: the

text: Co

text: channel

text: to

text: the

text: total

text: screening

text: of

text: the

text: Co

text: electrons

text: is

text: small.

text: Moreover,

text: we

text: ﬁnd

text: that

text: both

text: the

text: onsite

text: and

text: inter-site

text: Coulomb

text: interaction

text: parameters

text: decrease

text: in

text: a

text: reasonable

text: approximation

text: linearly

text: with

text: the

text: cluster

text: size

text: and

text: for

text: clusters

text: having

text: more

text: than

text: Nb

text: atoms

text: a

text: transition

text: from

text: to

text: screening

text: is

text: expected

text: to

text: take

text: place.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: This

text: information

text: is

text: crucial

text: in

text: order

text: to

text: obtain

text: a

text: proper

text: fundamental

text: understanding

text: of

text: the

text: system.

text: Namely

text: cor-

text: relation

text: eﬀects

text: among

text: the

text: electrons

text: inhibit

text: in

text: general

text: an

text: The

text: interest

text: in

text: the

text: ﬁeld

text: of

text: clusters

text: is

text: growing

text: due

text: to

text: exact

text: solution.

text: Therefore,

text: approximate

text: methods

text: are

text: re-

text: the

text: increasing

text: demand

text: for

text: nano-technology.

text: Besides

text: the

text: quired

text: in

text: practice.

text: The

text: choice

text: of

text: a

text: suitable

text: approximate

text: relevance

text: for

text: technology,

text: clusters

text: are

text: also

text: fundamentally

text: method

text: requires

text: knowledge

text: of

text: the

text: eﬀective

text: Coulomb

text: in-

text: very

text: interesting.

text: They

text: behave

text: in

text: general

text: very

text: diﬀerent

text: teraction

text: in

text: the

text: system.

text: More

text: precisely,

text: the

text: gradient

text: of

text: from

text: their

text: bulk

text: counterpart.

text: Also

text: their

text: electronic

text: and

text: the

text: eﬀective

text: Coulomb

text: interaction

text: is

text: of

text: magnetic

text: properties

text: can

text: drastically

text: change

text: by

text: just

text: adding

text: A

text: very

text: small

text: gradient

text: means

text: that

text: the

text: eﬀective

text: Coulomb

text: or

text: removing

text: one

text: For

text: example,

text: in

text: a

text: recent

text: work

text: interaction

text: is

text: merely

text: constant,

text: while

text: a

text: very

text: large

text: gradi-

text: on

text: Nb

text: Co

text: clusters

text: it

text: is

text: demonstrated

text: that

text: Nb

text: Co

text: and

text: x

text: ent

text: indicates

text: a

text: purely

text: local

text: eﬀective

text: Coulomb

text: interaction.

text: Nb

text: Co

text: are

text: non-magnetic,

text: while

text: Nb

text: Co

text: and

text: Nb

text: Co

text: are

text: In

text: the

text: former

text: case

text: a

text: mean-ﬁeld

text: treatment,

text: i.e.

text: single-

text: strongly

text: The

text: physical

text: origin

text: of

text: this

text: behavior

text: particle

text: approach,

text: is

text: probably

text: a

text: good

text: choice,

text: while

text: for

text: can

text: be

text: traced

text: back

text: to

text: the

text: drastic

text: change

text: of

text: the

text: electronic

text: the

text: latter

text: it

text: might

text: be

text: the

text: (generalized)

text: Hubbard

text: model.

text: structure

text: as

text: a

text: function

text: of

text: cluster

text: size.

text: The

text: study

text: of

text: bimetallic

text: clusters

text: oﬀers

text: a

text: broader

text: play-

text: The

text: aim

text: of

text: the

text: present

text: work

text: is

text: the

text: ab-initio

text: determina-

text: ground

text: than

text: for

text: pure

text: clusters.

text: This

text: has

text: resulted

text: in

text: a

text: tion

text: of

text: the

text: Coulomb

text: interaction

text: for

text: NbCo

text: to

text: Nb

text: Co

text: and

text: number

text: of

text: intriguing

text: For

text: example,

text: from

text: Nb

text: Co

text: clusters.

text: Besides

text: being

text: fundamentally

text: interesting,

text: an

text: anion

text: photo-electron

text: spectroscopy

text: study

text: on

text: bimetal-

text: such

text: information

text: is

text: crucial

text: to

text: select

text: an

text: adequate

text: theoreti-

text: lic

text: Nb

text: Co

text: clusters,

text: it

text: was

text: observed

text: that

text: for

text: x

text: due

text: to

text: x

text: cal

text: method

text: for

text: a

text: further

text: investigation

text: of

text: the

text: system.

text: The

text: the

text: addition

text: of

text: one

text: Co

text: atom

text: to

text: the

text: Nb

text: host,

text: the

text: elec-

text: x

text: geometries

text: of

text: Nb

text: Co

text: to

text: Nb

text: Co

text: and

text: Nb

text: Co

text: are

text: well

text: estab-

text: tronic

text: structure

text: resembles

text: that

text: of

text: a

text: typical

text: bulk

text: semi-

text: lished

text: from

text: a

text: comparison

text: of

text: theoretically

text: and

text: experimen-

text: Therefore,

text: this

text: cluster

text: was

text: then

text: proposed

text: as

text: tally

text: obtained

text: vibrational

text: In

text: addition

text: NbCo

text: and

text: a

text: candidate

text: for

text: semiconductor

text: materials.

text: Nb

text: Co

text: are

text: considered,

text: since

text: the

text: number

text: of

text: isomers

text: is

text: very

text: From

text: the

text: theoretical

text: side

text: the

text: bond

text: properties

text: and

text: elec-

text: small.

text: All

text: these

text: clusters

text: are

text: known

text: to

text: be

text: magnetic

text: except

text: tronic

text: structure

text: of

text: (NbCo)x

text: clusters

text: has

text: been

text: investi-

text: and

text: which

text: are

text: non-magnetic.

text: By

text: employ-

text: gated

text: by

text: means

text: of

text: relativistic

text: density

text: functional

text: theory

text: ing

text: the

text: full-potential

text: linearized

text: augmented

text: plane

text: wave

text: Further,

text: the

text: geometry,

text: stability

text: and

text: electronic

text: (FLAPW)

text: method

text: using

text: Wannier

text: functions

text: in

text: conjunc-

text: properties

text: of

text: neutral

text: and

text: anionic

text: NbxCo

text: clusters

text: is

text: com-

text: tion

text: with

text: the

text: random-phase

text: approximation

text: pared

text: with

text: pure

text: Nbx

text: clusters

text: within

text: a

text: DFT

text: Re-

text: it

text: is

text: found

text: that

text: in

text: these

text: clusters

text: the

text: onsite

text: Coulomb

text: in-

text: cently

text: a

text: combined

text: theoretical

text: and

text: experimental

text: investiga-

text: teraction

text: in

text: RPA

text: is

text: well

text: screened,

text: while

text: the

text: inter-site

text: tion

text: has

text: been

text: performed

text: on

text: NbxCo

text: In

text: this

text: work

text: Coulomb

text: interactions

text: are

text: barely

text: screened.

text: Interestingly

text: the

text: geometry

text: is

text: obtained

text: from

text: a

text: comparison

text: of

text: experi-

text: for

text: NbCo

text: the

text: inter-site

text: interaction

text: is

text: unscreened,

text: while

text: mentally

text: and

text: theoretically

text: obtained

text: vibrational

text: spectra.

text: for

text: Nb

text: Co

text: even

text: anti-screening

text: occurs.

text: The

text: important

text: With

text: the

text: geometry

text: established

text: the

text: electronic

text: structure

text: is

text: consequence

text: being

text: that

text: the

text: screened

text: Coulomb

text: interaction

text: investigated

text: in

text: order

text: to

text: explain

text: the

text: magnetic

text: properties

text: is

text: almost

text: constant

text: throughout

text: the

text: clusters.

text: For

text: complete-

text: obtained

text: from

text: magnetic

text: deﬂection

text: experiments.

text: ness

text: these

text: results

text: are

text: compared

text: with

text: pure

text: Nb

text: bulk

text: for

text: To

text: our

text: knowledge

text: a

text: systematic

text: assessment

text: of

text: screening

text: which

text: only

text: the

text: onsite

text: Coulomb

text: interaction

text: is

text: apprecia-

text: and

text: correlation

text: eﬀects

text: in

text: Nb

text: Co

text: clusters

text: does

text: not

text: exist.

text: ble,

text: while

text: the

text: inter-site

text: Coulomb

text: interactions

text: are

text: almost

text: x


text: 2 M. ORIEUX, J.-B. CAILLAU, T. COMBOT, AND J. FÉJOZ where (cid:112) − − 1 µ Ω (t, q) = µ (cid:112) (q + µ cos t)2 + (q + µ sin t)2 1 2 − µ − − − − (q (1 µ) cos t)2 + (q (1 µ) sin t)2 1 2 ∈ is the potential parameterized by the ratio of masses, , and where µ [0, 1/2] ∈ R u 2 is the control, whose amplitude is modulated by the second parameter, ε ≥ 0. Alternatively to time m(cid:90) inimization, minimization of the L2 norm of the control can be considered, tf → u2(t) dt min . 0 This is the so-called energy cost. In the uncontrolled case ( ), it is well ε = 0 known that the Kepler case ( ) is integrable (and geodesic) while there are µ = 0 obstructions to integrability for positive . In the controlled case ( ), the µ ε > 0 Kepler problem for the energy cost has been shown to be integrable (and geodesic) when suitably averaged (see [5] for a survey). The aim of this paper is to study the integrability properties of the Kepler problem for time minimization. The pioneering work of Ziglin in the 80’s [17], followed by the modern formu- lation of diﬀerential Galois theory in the late 90’s by Moralès, Ramis and Simó [11, 12], have led to a very diverse literature on the integrability of Hamiltonian systems. According to Pontrjagin’s Maximum principle, one can turn general op- timization problems with dynamical constraints into Hamiltonian systems, which are generally not everywhere diﬀerentiable. Optimal control theory thus provides an abundant class of dynamical systems, for which integrability naturally is a cen- tral question. Yet, diﬀerential Galois theory has not so often been applied in this context, in part because of the diﬃculty brought by the singularities. Notwith- standing theses singularities (vanishing of the adjoint variable , here), we show p v how to apply these ideas to the system (2). 2. Setting We ﬁrst recall some classical facts on optimal control. We refer for example to the book of Agrachev [1] for more details. Let be an -dimensional smooth M n R manifold and U an arbitrary subset of m. A controlled dynamical system is a smooth family of vector ﬁelds × → f : M U T M (weaker smoothness assumptions may also be made). Let U { (cid:55)→ ∈ ∞ R } = u : t u(t) L ( , U) loc ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: M.

text: ORIEUX,

text: J.-B.

text: CAILLAU,

text: T.

text: COMBOT,

text: AND

text: J.

text: FÉJOZ

text: where

text: µ

text: Ω

text: (t,

text: q)

text: µ

text: (q

text: µ

text: cos

text: (q

text: µ

text: sin

text: µ

text: (q

text: µ)

text: cos

text: (q

text: µ)

text: sin

text: is

text: the

text: potential

text: parameterized

text: by

text: the

text: ratio

text: of

text: masses,

text: and

text: where

text: µ

text: R

text: u

text: is

text: the

text: control,

text: whose

text: amplitude

text: is

text: modulated

text: by

text: the

text: second

text: parameter,

text: ε

text: Alternatively

text: to

text: time

text: inimization,

text: minimization

text: of

text: the

text: norm

text: of

text: the

text: control

text: can

text: be

text: considered,

text: tf

text: dt

text: min

text: This

text: is

text: the

text: so-called

text: energy

text: cost.

text: In

text: the

text: uncontrolled

text: case

text: it

text: is

text: well

text: ε

text: known

text: that

text: the

text: Kepler

text: case

text: is

text: integrable

text: (and

text: geodesic)

text: while

text: there

text: are

text: µ

text: obstructions

text: to

text: integrability

text: for

text: positive

text: In

text: the

text: controlled

text: case

text: the

text: µ

text: ε

text: Kepler

text: problem

text: for

text: the

text: energy

text: cost

text: has

text: been

text: shown

text: to

text: be

text: integrable

text: (and

text: geodesic)

text: when

text: suitably

text: averaged

text: (see

text: for

text: a

text: survey).

text: The

text: aim

text: of

text: this

text: paper

text: is

text: to

text: study

text: the

text: integrability

text: properties

text: of

text: the

text: Kepler

text: problem

text: for

text: time

text: minimization.

text: The

text: pioneering

text: work

text: of

text: Ziglin

text: in

text: the

text: followed

text: by

text: the

text: modern

text: formu-

text: lation

text: of

text: diﬀerential

text: Galois

text: theory

text: in

text: the

text: late

text: by

text: Moralès,

text: Ramis

text: and

text: Simó

text: have

text: led

text: to

text: a

text: very

text: diverse

text: literature

text: on

text: the

text: integrability

text: of

text: Hamiltonian

text: systems.

text: According

text: to

text: Pontrjagin’s

text: Maximum

text: principle,

text: one

text: can

text: turn

text: general

text: op-

text: timization

text: problems

text: with

text: dynamical

text: constraints

text: into

text: Hamiltonian

text: systems,

text: which

text: are

text: generally

text: not

text: everywhere

text: diﬀerentiable.

text: Optimal

text: control

text: theory

text: thus

text: provides

text: an

text: abundant

text: class

text: of

text: dynamical

text: systems,

text: for

text: which

text: integrability

text: naturally

text: is

text: a

text: cen-

text: tral

text: question.

text: Yet,

text: diﬀerential

text: Galois

text: theory

text: has

text: not

text: so

text: often

text: been

text: applied

text: in

text: this

text: context,

text: in

text: part

text: because

text: of

text: the

text: diﬃculty

text: brought

text: by

text: the

text: singularities.

text: Notwith-

text: standing

text: theses

text: singularities

text: (vanishing

text: of

text: the

text: adjoint

text: variable

text: here),

text: we

text: show

text: p

text: v

text: how

text: to

text: apply

text: these

text: ideas

text: to

text: the

text: system

text: Setting

text: We

text: ﬁrst

text: recall

text: some

text: classical

text: facts

text: on

text: optimal

text: control.

text: We

text: refer

text: for

text: example

text: to

text: the

text: book

text: of

text: Agrachev

text: for

text: more

text: details.

text: Let

text: be

text: an

text: -dimensional

text: smooth

text: M

text: n

text: R

text: manifold

text: and

text: U

text: an

text: arbitrary

text: subset

text: of

text: m.

text: A

text: controlled

text: dynamical

text: system

text: is

text: a

text: smooth

text: family

text: of

text: vector

text: ﬁelds

text: f

text: M

text: U

text: T

text: M

text: (weaker

text: smoothness

text: assumptions

text: may

text: also

text: be

text: made).

text: Let

text: U

text: R

text: u

text: t

text: u(t)

text: L

text: U)

text: loc

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CONTROLLED KEPLER PROBLEM 5 By the Maximum Principle, the extremals of (6) must(cid:113)be projections on of M solutions of the maximized Hamiltonian system − p x + p x H(x, p) = p x + p x 3 1 4 2 + p2 + p2 1 3 2 4 (x2 + x2)3/2 3 4 1 2 ∗ on , while the control is given(cid:112)by T M 1 u = (p , p ). 3 4 p2 + p2 3 4 Now, let M { ∈ C × C } = (x, p, r) 8 2∗, r2 = x2 + x2, r2 = p2 + p2 1 1 2 2 3 4 be the Riemann surface of . It is a complex symplectic manifold (with local H Darboux coordinates outside the singular hypersurface ), over which (x, p) r r = 0 1 2 extends meromorphically, and even rationally, since H − p x + p x (7) 3 1 4 2 H(x, p, r) = p x + p x + r . 1 3 2 4 r3 2 1 The Hamiltonian has four degrees of freedom, hence (see [2]) the meromorphic H M Liouville integrability of over would mean that there would exist three H M independent ﬁrst integrals, in addition to itself, almost everywhere in . This H does not happen. Theorem 2. The minimum time Kepler problem is not meromorphically Liouville M integrable on . It is well known that the classical Kepler problem is integrable, and even super in- tegrable (since there are more ﬁrst integrals than degrees of freedom, which relates to Kepler’s ﬁrst law and the dynamical degeneracy of the Newtonian potential see [8], for example). But the three-body problem is not, as Poincaré, Julliard- Tosel, Tsygvintsev or Combot have proved in various settings [7, 9, 13, 16]. Simi- larly, the above theorem asserts that lifting the Kepler problem to the cotangent bundle and introducing the singular control term break integrability. r 2 This result prevents the existence of enough real analytic (and even meromor- M phic) ﬁrst integrals to ensure integrability over . Or course, it does not prevent the existence of an additional ﬁrst integral which would have a natural frontier outside asymptotic to the real domain and which would thus not extend to the complex plane. Future work may be dedicated to extend this result to real ﬁrst integrals: The Hamiltonian is invariant by two rotations, and homogeneous with respect to several coordinates. If it is real Liouville integrable, one can ﬁnd real ﬁrst integrals have the same symmetries. This may allow us to extend them to the complex domain, creating a contradiction with theorem 2. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CONTROLLED

text: KEPLER

text: PROBLEM

text: By

text: the

text: Maximum

text: Principle,

text: the

text: extremals

text: of

text: projections

text: on

text: of

text: M

text: solutions

text: of

text: the

text: maximized

text: Hamiltonian

text: system

text: p

text: x

text: p

text: x

text: H(x,

text: p)

text: p

text: x

text: p

text: x

text: on

text: while

text: the

text: control

text: is

text: T

text: M

text: u

text: (p

text: p

text: Now,

text: let

text: M

text: C

text: C

text: (x,

text: p,

text: r)

text: be

text: the

text: Riemann

text: surface

text: of

text: It

text: is

text: a

text: complex

text: symplectic

text: manifold

text: (with

text: local

text: H

text: Darboux

text: coordinates

text: outside

text: the

text: singular

text: hypersurface

text: over

text: which

text: (x,

text: p)

text: r

text: r

text: extends

text: meromorphically,

text: and

text: even

text: rationally,

text: since

text: H

text: p

text: x

text: p

text: x

text: H(x,

text: p,

text: r)

text: p

text: x

text: p

text: x

text: r

text: The

text: Hamiltonian

text: has

text: four

text: degrees

text: of

text: freedom,

text: hence

text: (see

text: the

text: meromorphic

text: H

text: M

text: Liouville

text: integrability

text: of

text: over

text: would

text: mean

text: that

text: there

text: would

text: exist

text: three

text: H

text: M

text: independent

text: ﬁrst

text: integrals,

text: in

text: addition

text: to

text: itself,

text: almost

text: everywhere

text: in

text: This

text: H

text: does

text: not

text: happen.

text: Theorem

text: The

text: minimum

text: time

text: Kepler

text: problem

text: is

text: not

text: meromorphically

text: Liouville

text: M

text: integrable

text: on

text: It

text: is

text: well

text: known

text: that

text: the

text: classical

text: Kepler

text: problem

text: is

text: integrable,

text: and

text: even

text: super

text: in-

text: tegrable

text: (since

text: there

text: are

text: more

text: ﬁrst

text: integrals

text: than

text: degrees

text: of

text: freedom,

text: which

text: relates

text: to

text: Kepler’s

text: ﬁrst

text: law

text: and

text: the

text: dynamical

text: degeneracy

text: of

text: the

text: Newtonian

text: potential

text: see

text: for

text: example).

text: But

text: the

text: three-body

text: problem

text: is

text: not,

text: as

text: Poincaré,

text: Julliard-

text: Tosel,

text: Tsygvintsev

text: or

text: Combot

text: have

text: proved

text: in

text: various

text: settings

text: Simi-

text: larly,

text: the

text: above

text: theorem

text: asserts

text: that

text: lifting

text: the

text: Kepler

text: problem

text: to

text: the

text: cotangent

text: bundle

text: and

text: introducing

text: the

text: singular

text: control

text: term

text: break

text: integrability.

text: r

text: This

text: result

text: prevents

text: the

text: existence

text: of

text: enough

text: real

text: analytic

text: (and

text: even

text: meromor-

text: M

text: phic)

text: ﬁrst

text: integrals

text: to

text: ensure

text: integrability

text: over

text: Or

text: course,

text: it

text: does

text: not

text: prevent

text: the

text: existence

text: of

text: an

text: additional

text: ﬁrst

text: integral

text: which

text: would

text: have

text: a

text: natural

text: frontier

text: outside

text: asymptotic

text: to

text: the

text: real

text: domain

text: and

text: which

text: would

text: thus

text: not

text: extend

text: to

text: the

text: complex

text: plane.

text: Future

text: work

text: may

text: be

text: dedicated

text: to

text: extend

text: this

text: result

text: to

text: real

text: ﬁrst

text: integrals:

text: The

text: Hamiltonian

text: is

text: invariant

text: by

text: two

text: rotations,

text: and

text: homogeneous

text: with

text: respect

text: to

text: several

text: coordinates.

text: If

text: it

text: is

text: real

text: Liouville

text: integrable,

text: one

text: can

text: ﬁnd

text: real

text: ﬁrst

text: integrals

text: have

text: the

text: same

text: symmetries.

text: This

text: may

text: allow

text: us

text: to

text: extend

text: them

text: to

text: the

text: complex

text: domain,

text: creating

text: a

text: contradiction

text: with

text: theorem

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##