text: 10 dn¯ for each curve are displayed in Fig 6b and color-coded dt a) b) to match the corresponding data set. × − 250 ions We measure the largest heating rate of 1.4(2) 104 s 1 (black points) when the trap endcap electrodes are held 185 ions at 0 V using the high voltage power supplies responsi- ble for initial ion loading and transport. Small voltage ﬂuctuations from these power supplies as well as elec- 240 ions tromagnetic interference along the connecting cables are the likely cause of this COM heating. Upon grounding the endcaps directly to the vacuum chamber at the high- voltage feedthrough, we observe an order-of-magnitude FIG. 6: (color online) Measured COM heating rates (points × − drop in the heating rate to between 4.7(8) 102 s 1 with error bars) and corresponding linear ﬁts (solid lines) plot- (blue points) and 1.2(3) × 103 s−1 (red points). This ted with both (a) linear and (b) logarit×hmic v−ertical scales. The highest heating rate of dn¯ = 1.4(2) 104 s 1 is obtained approximate factor-of-two variation in heating rates is dt with the trap endcap electrodes grounded through the high representative of our day-to-day observations with diﬀer- voltage power supplies used for initial ion loading. The two ent ion samples, and we see no evidence of a correlation lower heating rates are measured with the endcaps shorted between total ion number and heating rate with the end- directly to the trap vacuum ground at the high voltage vac- cap electrodes grounded at the feedthrough. ∼The mean uum feedthrough. The variation between the two lowest rates of the two lowest heating rates corresponds to 30 mK/s, is representative of the range of heating rates measured thus which is near previous collisional heating estimates [48]. far under the given conditions. Additional ﬁltering of the central ring and rotating-wall electrodes outside the vacuum envelope yielded no mea- surable improvement in COM heating [? ]. trapped-ion crystal, and may be used to phase-sensitively Because the distance from the trapped-ion arrays to detect weak forces [37, 49]. Section III can be used to es- ≥ the trap electrode surfaces ( 2 cm) is large compared to timate and optimize the force detection sensitivity for the diameter of the planar array (<0.5 cm), electric ﬁeld a given N and temperature of the trapped-ion crystal. noise from trap electrode surfaces will be uniform across We estimate that the spin-motion coupling technique dis- the array and preferentially heat the COM mode. For an cussed here could improve on the force sensitivity ob- array with N ions this results in a linear dependence of tained in Ref. [37] by more than an order-of-magnitude. the COM heating rate on ion number due to uniform elec- We also present the ﬁrst measurements of ambient tric ﬁeld noise [5]. The lack of an observed N-dependence heating of a resolved mode of motion in a Penning trap. in the measured ambient heating rate indicates the source Future crystal heating measurements will include other of the heating is likely not electric ﬁeld noise. However, resolved transverse motional modes with the goal of more any potential anomalous heating must be less than the clearly distinguishing between electric ﬁeld ﬂuctuations ∼ − measured 103 s 1 ambient heating rate. Dividing this (mode-speciﬁc, N-dependent) and background gas col- ∼ limit by the number of trapped ions (N 200) gives a lisions (mode- and N-independent). Nevertheless, we limit on the anomalous heating rate for a single trapped demonstrate that low heating rates are indeed achievable ∼ − ion of 5 s 1 at a trap frequency of 795 kHz. in Penning ion traps. V. CONCLUSION Acknowledgments In summary, we have demonstrated a new technique This work was supported by the DARPA-OLE pro- for analyzing the motional state of a resolved ion crystal gram and NIST. The authors thank J. P. Home, K. R. A. mode. The methods presented here do not rely on stim- Hazzard, M. Foss-Feig, and A. M. Rey for useful discus- ulated Raman transitions or Doppler linewidth analysis, sions as well as S. Kotler and J. P. Gaebler for comments and are in principle applicable to any resolved motional on the manuscript. This manuscript is a contribution of mode at any temperature provided the Lamb-Dicke con- NIST and not subject to U.S. copyright. ﬁnement criterion is satisﬁed for the given mode. The sensitivity of our spin dephasing measurements allows for a regime of operation with negligible spin-motion entan- Appendix A: Calculating Bloch Vector Length using glement and spontaneous emission decoherence. Monte Fock States Carlo simulations based on the semiclassical description of Sec. are in excellent agreement with spin dephasing A thermal motional state is described by a density ma- measurements. Furthermore, we observe a clear distinc- trix that is a statistical mixture of Fock states. This mo- tion between coherent and thermal states of motion, de- tivates a calculation that assumes each experiment be- |↑(cid:105)| (cid:105) | (cid:105) spite the randomness of the RF drive phase relative to gins with the system in the state, n , where n is our optical dipole force. the harmonic oscillator Fock state of COM motion. The The methods and analysis presented here enable very ﬁrst π/2 pulse of the Ramsey sequence yields the qubit sensitive detection of coherently driven motion of a rotation ##LTFigure## ##LTLine## ##LTLine##

text: dn¯

text: for

text: each

text: curve

text: are

text: displayed

text: in

text: Fig

text: and

text: color-coded

text: dt

text: a)

font: default

text: b)

text: to

text: match

text: the

text: corresponding

text: data

text: set.

text: ions

text: We

text: measure

text: the

text: largest

text: heating

text: rate

text: of

text: s

text: (black

text: points)

text: when

text: the

text: trap

text: endcap

text: electrodes

text: are

text: held

text: ions

text: at

text: V

text: using

text: the

text: high

text: voltage

text: power

text: supplies

text: responsi-

text: ble

text: for

text: initial

text: ion

text: loading

text: and

text: transport.

text: Small

text: voltage

text: ﬂuctuations

text: from

text: these

text: power

text: supplies

text: as

text: well

text: as

text: elec-

text: ions

text: tromagnetic

text: interference

text: along

text: the

text: connecting

text: cables

text: are

text: the

text: likely

text: cause

text: of

text: this

text: COM

text: heating.

text: Upon

text: grounding

text: the

text: endcaps

text: directly

text: to

text: the

text: vacuum

text: chamber

text: at

text: the

text: high-

text: voltage

text: feedthrough,

text: we

text: observe

text: an

text: order-of-magnitude

text: FIG.

text: (color

text: online)

text: Measured

text: COM

text: heating

text: rates

text: (points

text: drop

text: in

text: the

text: heating

text: rate

text: to

text: between

text: s

text: with

text: error

text: bars)

text: and

text: corresponding

text: linear

text: ﬁts

text: (solid

text: lines)

text: plot-

text: (blue

text: points)

text: and

text: (red

text: points).

text: This

text: ted

text: with

text: both

text: (a)

text: linear

text: and

text: (b)

text: logarit×hmic

text: v−ertical

text: scales.

text: The

text: highest

text: heating

text: rate

text: of

text: dn¯

text: s

text: is

text: obtained

text: approximate

text: factor-of-two

text: variation

text: in

text: heating

text: rates

text: is

text: dt

text: with

text: the

text: trap

text: endcap

text: electrodes

text: grounded

text: through

text: the

text: high

text: representative

text: of

text: our

text: day-to-day

text: observations

text: with

text: diﬀer-

text: voltage

text: power

text: supplies

text: used

text: for

text: initial

text: ion

text: loading.

text: The

text: two

text: ent

text: ion

text: samples,

text: and

text: we

text: see

text: no

text: evidence

text: of

text: a

text: correlation

text: lower

text: heating

text: rates

text: are

text: measured

text: with

text: the

text: endcaps

text: shorted

text: between

text: total

text: ion

text: number

text: and

text: heating

text: rate

text: with

text: the

text: end-

text: directly

text: to

text: the

text: trap

text: vacuum

text: ground

text: at

text: the

text: high

text: voltage

text: vac-

text: cap

text: electrodes

text: grounded

text: at

text: the

text: feedthrough.

text: ∼The

text: mean

text: uum

text: feedthrough.

text: The

text: variation

text: between

text: the

text: two

text: lowest

text: rates

text: of

text: the

text: two

text: lowest

text: heating

text: rates

text: corresponds

text: to

text: mK/s,

text: is

text: representative

text: of

text: the

text: range

text: of

text: heating

text: rates

text: measured

text: thus

text: which

text: is

text: near

text: previous

text: collisional

text: heating

text: estimates

text: far

text: under

text: the

text: given

text: conditions.

text: Additional

text: ﬁltering

text: of

text: the

text: central

text: ring

text: and

text: rotating-wall

text: electrodes

text: outside

text: the

text: vacuum

text: envelope

text: yielded

text: no

text: mea-

text: surable

text: improvement

text: in

text: COM

text: heating

text: trapped-ion

text: crystal,

text: and

text: may

text: be

text: used

text: to

text: phase-sensitively

text: Because

text: the

text: distance

text: from

text: the

text: trapped-ion

text: arrays

text: to

text: detect

text: weak

text: forces

text: Section

text: III

text: can

text: be

text: used

text: to

text: es-

text: the

text: trap

text: electrode

text: surfaces

text: cm)

text: is

text: large

text: compared

text: to

text: timate

text: and

text: optimize

text: the

text: force

text: detection

text: sensitivity

text: for

text: the

text: diameter

text: of

text: the

text: planar

text: array

text: cm),

text: electric

text: ﬁeld

text: a

text: given

text: N

text: and

text: temperature

text: of

text: the

text: trapped-ion

text: crystal.

text: noise

text: from

text: trap

text: electrode

text: surfaces

text: will

text: be

text: uniform

text: across

text: We

text: estimate

text: that

text: the

text: spin-motion

text: coupling

text: technique

text: dis-

text: the

text: array

text: and

text: preferentially

text: heat

text: the

text: COM

text: mode.

text: For

text: an

text: cussed

text: here

text: could

text: improve

text: on

text: the

text: force

text: sensitivity

text: ob-

text: array

text: with

text: N

text: ions

text: this

text: results

text: in

text: a

text: linear

text: dependence

text: of

text: tained

text: in

text: Ref.

text: by

text: more

text: than

text: an

text: order-of-magnitude.

text: the

text: COM

text: heating

text: rate

text: on

text: ion

text: number

text: due

text: to

text: uniform

text: elec-

text: We

text: also

text: present

text: the

text: ﬁrst

text: measurements

text: of

text: ambient

text: tric

text: ﬁeld

text: noise

text: The

text: lack

text: of

text: an

text: observed

text: N-dependence

text: heating

text: of

text: a

text: resolved

text: mode

text: of

text: motion

text: in

text: a

text: Penning

text: trap.

text: in

text: the

text: measured

text: ambient

text: heating

text: rate

text: indicates

text: the

text: source

text: Future

text: crystal

text: heating

text: measurements

text: will

text: include

text: other

text: of

text: the

text: heating

text: is

text: likely

text: not

text: electric

text: ﬁeld

text: noise.

text: However,

text: resolved

text: transverse

text: motional

text: modes

text: with

text: the

text: goal

text: of

text: more

text: any

text: potential

text: anomalous

text: heating

text: must

text: be

text: less

text: than

text: the

text: clearly

text: distinguishing

text: between

text: electric

text: ﬁeld

text: ﬂuctuations

text: measured

text: s

text: ambient

text: heating

text: rate.

text: Dividing

text: this

text: (mode-speciﬁc,

text: N-dependent)

text: and

text: background

text: gas

text: col-

text: limit

text: by

text: the

text: number

text: of

text: trapped

text: ions

text: (N

text: gives

text: a

text: lisions

text: (mode-

text: and

text: N-independent).

text: Nevertheless,

text: we

text: limit

text: on

text: the

text: anomalous

text: heating

text: rate

text: for

text: a

text: single

text: trapped

text: demonstrate

text: that

text: low

text: heating

text: rates

text: are

text: indeed

text: achievable

text: ion

text: of

text: s

text: at

text: a

text: trap

text: frequency

text: of

text: kHz.

text: in

text: Penning

text: ion

text: traps.

text: V.

text: CONCLUSION

text: Acknowledgments

text: In

text: summary,

text: we

text: have

text: demonstrated

text: a

text: new

text: technique

text: This

text: work

text: was

text: supported

text: by

text: the

text: DARPA-OLE

text: pro-

text: for

text: analyzing

text: the

text: motional

text: state

text: of

text: a

text: resolved

text: ion

text: crystal

text: gram

text: and

text: NIST.

text: The

text: authors

text: thank

text: J.

text: P.

text: Home,

text: K.

text: R.

text: A.

text: mode.

text: The

text: methods

text: presented

text: here

text: do

text: not

text: rely

text: on

text: stim-

text: Hazzard,

text: M.

text: Foss-Feig,

text: and

text: A.

text: M.

text: Rey

text: for

text: useful

text: discus-

text: ulated

text: Raman

text: transitions

text: or

text: Doppler

text: linewidth

text: analysis,

text: sions

text: as

text: well

text: as

text: S.

text: Kotler

text: and

text: J.

text: P.

text: Gaebler

text: for

text: comments

text: and

text: are

text: in

text: principle

text: applicable

text: to

text: any

text: resolved

text: motional

text: on

text: the

text: manuscript.

text: This

text: manuscript

text: is

text: a

text: contribution

text: of

text: mode

text: at

text: any

text: temperature

text: provided

text: the

text: Lamb-Dicke

text: con-

text: NIST

text: and

text: not

text: subject

text: to

text: U.S.

text: copyright.

text: ﬁnement

text: criterion

text: is

text: satisﬁed

text: for

text: the

text: given

text: mode.

text: The

text: sensitivity

text: of

text: our

text: spin

text: dephasing

text: measurements

text: allows

text: for

text: a

text: regime

text: of

text: operation

text: with

text: negligible

text: spin-motion

text: entan-

text: Appendix

text: A:

text: Calculating

text: Bloch

text: Vector

text: Length

text: using

text: glement

text: and

text: spontaneous

text: emission

text: decoherence.

text: Monte

text: Fock

text: States

text: Carlo

text: simulations

text: based

text: on

text: the

text: semiclassical

text: description

text: of

text: Sec.

text: are

text: in

text: excellent

text: agreement

text: with

text: spin

text: dephasing

text: A

text: thermal

text: motional

text: state

text: is

text: described

text: by

text: a

text: density

text: ma-

text: measurements.

text: Furthermore,

text: we

text: observe

text: a

text: clear

text: distinc-

text: trix

text: that

text: is

text: a

text: statistical

text: mixture

text: of

text: Fock

text: states.

text: This

text: mo-

text: tion

text: between

text: coherent

text: and

text: thermal

text: states

text: of

text: motion,

text: de-

text: tivates

text: a

text: calculation

text: that

text: assumes

text: each

text: experiment

text: be-

text: spite

text: the

text: randomness

text: of

text: the

text: RF

text: drive

text: phase

text: relative

text: to

text: gins

text: with

text: the

text: system

text: in

text: the

text: state,

text: n

text: where

text: n

text: is

text: our

text: optical

text: dipole

text: force.

text: the

text: harmonic

text: oscillator

text: Fock

text: state

text: of

text: COM

text: motion.

text: The

text: The

text: methods

text: and

text: analysis

text: presented

text: here

text: enable

text: very

text: ﬁrst

text: pulse

text: of

text: the

text: Ramsey

text: sequence

text: yields

text: the

text: qubit

text: sensitive

text: detection

text: of

text: coherently

text: driven

text: motion

text: of

text: a

text: rotation

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: I. INTRODUCTION It has been long known, since the seminal work of Batchelor and Townsend [1], that spatial derivatives of a turbulent velocity ﬁeld do not behave as gaussian random variables. The current view on this still barely understood phenomenon is that the non-gaussian ﬂuctuations of the velocity gradients – the hallmark of turbulent intermittency – are likely to be related to the existence of long-lived coherent structures and to deviations from the Kolmogorov “K41” scaling, both important ingredients in the contemporary phenomenology of turbulence [2, 3]. The main notorious diﬃculties with ﬁrst-principle theories of intermittency stand on (i) the inadequacy of perturbative expansions to deal with the coupled dynamics of vorticity and the rate-of-strain tensor at high Reynolds numbers and (ii) the fact that the closed equations for the time evolution of the velocity gradient tensor are non-local in the space variables. Notwithstanding the strong coupling/non-local issues, it is actually possible to devise simpliﬁed ﬂuid dynamical models that would capture relevant qualitative features of the intermittent ﬂuctuations of the velocity gradient tensor [4]. Here, a fundamental role is played by the lagrangian framework of ﬂuid dynamics, since it leads in a natural way to reduced-dimensional systems, in the form of either ordinary or stochastic diﬀerential equations for the time evolution of the velocity gradient tensor [5–8]. The aforementioned lagrangian models have been mostly investigated by means of nu- merical integrations of the associated diﬀerential equations, which can then be compared to well-established results of alternative direct numerical simulations. There is, however, a large room for the exploration of analytical tools in the study of lagrangian models of intermittency, a direction we pursue here, joining other authors in this eﬀort [7, 9]. We focus our attention on one particularly interesting stochastic model, the Recent Fluid De- formation Closure (RFDC) model [8], and derive reasonable approximations for its velocity gradient probability distribution functions (vgPDFs). We put into practice standard sta- tistical ﬁeld-theoretical procedures for the computation of eﬀective actions through vertex renormalization [10, 11], which are carried out in the context of the Martin-Siggia-Rose func- tional formalism [12–15]. Our approach – essentially a semiclassical treatment – is general enough, so that, in principle, it can be applied to a large class of stochastic models. This paper is organized as follows. In Sec. II, we brieﬂy outline, as a ground for the subsequent discussions, the essential points of the RFDC model. In Sec. III, we rephrase 2

text: I.

label: section

text: INTRODUCTION

text: It

label: paragraph

text: has

text: been

text: long

text: known,

text: since

text: the

text: seminal

text: work

text: of

text: Batchelor

text: and

text: Townsend

text: that

text: spatial

text: derivatives

text: of

text: a

text: turbulent

text: velocity

text: ﬁeld

text: do

text: not

text: behave

text: as

text: gaussian

text: random

text: variables.

text: The

text: current

text: view

text: on

text: this

text: still

text: barely

text: understood

text: phenomenon

text: is

text: that

text: the

text: non-gaussian

text: ﬂuctuations

text: of

text: the

text: velocity

text: gradients

text: the

text: hallmark

text: of

text: turbulent

text: intermittency

text: are

text: likely

text: to

text: be

text: related

text: to

text: the

text: existence

text: of

text: long-lived

text: coherent

text: structures

text: and

text: to

text: deviations

text: from

text: the

text: Kolmogorov

text: scaling,

text: both

text: important

text: ingredients

text: in

text: the

text: contemporary

text: phenomenology

text: of

text: turbulence

text: The

text: main

text: notorious

text: diﬃculties

text: with

text: ﬁrst-principle

text: theories

text: of

text: intermittency

text: stand

text: on

text: (i)

text: the

text: inadequacy

text: of

text: perturbative

text: expansions

text: to

text: deal

text: with

text: the

text: coupled

text: dynamics

text: of

text: vorticity

text: and

text: the

text: rate-of-strain

text: tensor

text: at

text: high

text: Reynolds

text: numbers

text: and

text: (ii)

text: the

text: fact

text: that

text: the

text: closed

text: equations

text: for

text: the

text: time

text: evolution

text: of

text: the

text: velocity

text: gradient

text: tensor

text: are

text: non-local

text: in

text: the

text: space

text: variables.

text: Notwithstanding

text: the

text: strong

text: coupling/non-local

text: issues,

text: it

text: is

text: actually

text: possible

text: to

text: devise

text: simpliﬁed

text: ﬂuid

text: dynamical

text: models

text: that

text: would

text: capture

text: relevant

text: qualitative

text: features

text: of

text: the

text: intermittent

text: ﬂuctuations

text: of

text: the

text: velocity

text: gradient

text: tensor

text: Here,

text: a

text: fundamental

text: role

text: is

text: played

text: by

text: the

text: lagrangian

text: framework

text: of

text: ﬂuid

text: dynamics,

text: since

text: it

text: leads

text: in

text: a

text: natural

text: way

text: to

text: reduced-dimensional

text: systems,

text: in

text: the

text: form

text: of

text: either

text: ordinary

text: or

text: stochastic

text: diﬀerential

text: equations

text: for

text: the

text: time

text: evolution

text: of

text: the

text: velocity

text: gradient

text: tensor

text: The

text: aforementioned

text: lagrangian

text: models

text: have

text: been

text: mostly

text: investigated

text: by

text: means

text: of

text: nu-

text: merical

text: integrations

text: of

text: the

text: associated

text: diﬀerential

text: equations,

text: which

text: can

text: then

text: be

text: compared

text: to

text: well-established

text: results

text: of

text: alternative

text: direct

text: numerical

text: simulations.

text: There

text: is,

text: however,

text: a

text: large

text: room

text: for

text: the

text: exploration

text: of

text: analytical

text: tools

text: in

text: the

text: study

text: of

text: lagrangian

text: models

text: of

text: intermittency,

text: a

text: direction

text: we

text: pursue

text: here,

text: joining

text: other

text: authors

text: in

text: this

text: eﬀort

text: We

text: focus

text: our

text: attention

text: on

text: one

text: particularly

text: interesting

text: stochastic

text: model,

text: the

text: Recent

text: Fluid

text: De-

text: formation

text: Closure

text: (RFDC)

text: model

text: and

text: derive

text: reasonable

text: approximations

text: for

text: its

text: velocity

text: gradient

text: probability

text: distribution

text: functions

text: (vgPDFs).

text: We

text: put

text: into

text: practice

text: standard

text: sta-

text: tistical

text: ﬁeld-theoretical

text: procedures

text: for

text: the

text: computation

text: of

text: eﬀective

text: actions

text: through

text: vertex

text: renormalization

text: which

text: are

text: carried

text: out

text: in

text: the

text: context

text: of

text: the

text: Martin-Siggia-Rose

text: func-

text: tional

text: formalism

text: Our

text: approach

text: essentially

text: a

text: semiclassical

text: treatment

text: is

text: general

text: enough,

text: so

text: that,

text: in

text: principle,

text: it

text: can

text: be

text: applied

text: to

text: a

text: large

text: class

text: of

text: stochastic

text: models.

text: This

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: II,

text: we

text: brieﬂy

text: outline,

text: as

text: a

text: ground

text: for

text: the

text: subsequent

text: discussions,

text: the

text: essential

text: points

text: of

text: the

text: RFDC

text: model.

text: In

text: Sec.

text: III,

text: we

text: rephrase


text: Press (2004). [4] C. Meneveau, Ann. Rev. Fluid Mech. 43, 219 (2011). [5] P. Vieillefosse, Phys. A 125, 150 (1984). [6] B.J. Cantwell, Phys. Fluids A 4, 782 (1992). [7] M. Chertkov, A. Pumir and B.I. Shraiman, Phys. Fluids 11, 2394 (1999). [8] L. Chevillard and C. Meneveau, Phys. Rev. Lett. 97, 174501 (2006). [9] A. Naso, A. Pumir and M. Chertkov, J. Turb. 8, 1 (2007). [10] D. Amit, Field Theory, Renormalization, and Critical Phenomena, World Scientiﬁc Publishing Company (1984). [11] J.J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press (2002). [12] P.C. Martin, E.D. Siggia and H.A. Rose, Phys. Rev. A 8, 423 (1973). [13] H.K. Janssen, Z. Phys. 24, 113 (1976). [14] C. De Dominicis, J. Phys. C: Solid State Phys. 1, 247 (1976). [15] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996). [16] M.M. Afonso and C. Meneveau, Phys. D 239, 1241 (2010). [17] L. Chevillard and C. Meneveau, Phys. Fluids 23, 101704 (2011). [18] L. Chevillard, C. Meneveau, L. Biferale and F. Toschi, Phys. Fluids 20, 101504 (2008). [19] A well-written discussion on the perturbative diagrammatic expansions for stochastic diﬀer- ential equations can be found in A.-L. Barabasi and E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press (1995). [20] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Diﬀerential Equations, Springer, Berlin (1999). [21] D.P.Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cam- bridge University Press (2005). [22] C. Itzykson and J.-M. Drouﬀe, Statistical Field Theory v.2, Cambridge University Press (1989). 17

text: Press

label: reference

text: C.

text: Meneveau,

text: Ann.

text: Rev.

text: Fluid

text: Mech.

text: P.

text: Vieillefosse,

text: Phys.

text: A

text: B.J.

text: Cantwell,

text: Phys.

text: Fluids

text: A

text: M.

text: Chertkov,

text: A.

text: Pumir

text: and

text: B.I.

text: Shraiman,

text: Phys.

text: Fluids

text: L.

text: Chevillard

text: and

text: C.

text: Meneveau,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Naso,

text: A.

text: Pumir

text: and

text: M.

text: Chertkov,

text: J.

text: Turb.

text: D.

text: Amit,

text: Field

text: Theory,

text: Renormalization,

text: and

text: Critical

text: Phenomena,

text: World

text: Scientiﬁc

text: Publishing

text: Company

text: J.J.

text: Zinn-Justin,

text: Quantum

text: Field

text: Theory

text: and

text: Critical

text: Phenomena,

text: Oxford

text: University

text: Press

text: P.C.

text: Martin,

text: E.D.

text: Siggia

text: and

text: H.A.

text: Rose,

text: Phys.

text: Rev.

text: A

text: H.K.

text: Janssen,

text: Z.

text: Phys.

text: C.

text: De

text: Dominicis,

text: J.

text: Phys.

text: C:

text: Solid

text: State

text: Phys.

text: J.

text: Cardy,

text: Scaling

text: and

text: Renormalization

text: in

text: Statistical

text: Physics,

text: Cambridge

text: University

text: Press

text: M.M.

text: Afonso

text: and

text: C.

text: Meneveau,

text: Phys.

text: D

text: L.

text: Chevillard

text: and

text: C.

text: Meneveau,

text: Phys.

text: Fluids

text: L.

text: Chevillard,

text: C.

text: Meneveau,

text: L.

text: Biferale

text: and

text: F.

text: Toschi,

text: Phys.

text: Fluids

text: A

text: well-written

text: discussion

text: on

text: the

text: perturbative

text: diagrammatic

text: expansions

text: for

text: stochastic

text: diﬀer-

text: ential

text: equations

text: can

text: be

text: found

text: in

text: A.-L.

text: Barabasi

text: and

text: E.

text: Stanley,

text: Fractal

text: Concepts

text: in

text: Surface

text: Growth,

text: Cambridge

text: University

text: Press

text: P.E.

text: Kloeden

text: and

text: E.

text: Platen,

text: Numerical

text: Solution

text: of

text: Stochastic

text: Diﬀerential

text: Equations,

text: Springer,

text: Berlin

text: D.P.Landau

text: and

text: K.

text: Binder,

text: A

text: Guide

text: to

text: Monte

text: Carlo

text: Simulations

text: in

text: Statistical

text: Physics,

text: Cam-

text: bridge

text: University

text: Press

text: C.

text: Itzykson

text: and

text: J.-M.

text: Drouﬀe,

text: Statistical

text: Field

text: Theory

text: Cambridge

text: University

text: Press


text: ∝ Reynolds number is, thus, R (T/τ)2). It is then assumed that the lagrangian evolution e A of (t) is associated, for small time scales, to the approximate Cauchy-Green tensor C A A = exp[τ ] exp[τ T] , (5) A A so that the functional V [ ] in Eq. (1) gets replaced by a local function of , C− A C− A −A 1Tr( 2) − Tr( 1)A V ( ) = 2 + C− . (6) Tr( 1) 3T A We end up, therefore, with a closed and much simpler time evolution equation for : C− A C− A A F −A 1Tr( 2) − Tr( 1)A F ˙ = V ( ) + g = 2 + C− + g . (7) Tr( 1) 3T We refer the reader to Ref. [8] for a more detailed account on the conceptual and technical aspects of the RFDC model. It is convenient to set T = 1 (without loss of generality) and perform an expansion of A V ( ) up to some arbitrary power of τ in (6). A previous extensive numerical study shows that even the ﬁrst order expansion is enough to grasp the physical content of the model [16]. A We work, throughout the paper, with second(cid:88)order expansions of V ( ), which we write as 4 A A V ( ) = V ( ) , (8) p p=1 where A −A V ( ) = , (9) 1 (cid:18) I (cid:19) A −A A 2τ A A V ( ) = 2 + Tr( 2) + Tr( ) , (10) 2 3 3 I A −τ A A − 2 A A − τ2 A A A − τ2 A A V ( ) = + T Tr( ) Tr( 2) Tr( T ) Tr( 2) , (11) 3 3 3 3 3 I I A − A A A − A τ2 A A A V ( ) = τ2Tr( T )Tr( 2) τ2[Tr( 2)]2 + T Tr( 2) + 4 9 9 3 τ2 A A A + ( 2 + 2T)Tr( 2) . (12) 6 A A A It is important to note that V ( ) comprises all the contributions of O( p) to V ( ). The p RFDC model yields a promising stage for further improvements, insofar its vgPDFs as well as its geometrical statistical properties related to the coupling between the vorticity and the rate-of-strain tensor share several qualitative features in common with the ones observed in experiments and direct numerical simulations of turbulence [17, 18]. 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Reynolds

text: number

text: is,

text: thus,

text: R

text: It

text: is

text: then

text: assumed

text: that

text: the

text: lagrangian

text: evolution

text: e

text: A

text: of

text: (t)

text: is

text: associated,

text: for

text: small

text: time

text: scales,

text: to

text: the

text: approximate

text: Cauchy-Green

text: tensor

text: C

text: A

text: A

text: exp[τ

text: exp[τ

text: T]

text: A

text: A

text: so

text: that

text: the

text: functional

text: V

text: in

text: Eq.

text: gets

text: replaced

text: by

text: a

text: local

text: function

text: of

text: C−

text: A

text: C−

text: A

text: −A

text: Tr(

text: V

text: C−

text: Tr(

text: A

text: We

text: end

text: up,

text: therefore,

text: with

text: a

text: closed

text: and

text: much

text: simpler

text: time

text: evolution

text: equation

text: for

text: C−

text: A

text: C−

text: A

text: A

text: F

text: −A

text: Tr(

text: F

text: V

text: g

text: C−

text: g

text: Tr(

text: We

text: refer

text: the

text: reader

text: to

text: Ref.

text: for

text: a

text: more

text: detailed

text: account

text: on

text: the

text: conceptual

text: and

text: technical

text: aspects

text: of

text: the

text: RFDC

text: model.

text: It

text: is

text: convenient

text: to

text: set

text: T

text: (without

text: loss

text: of

text: generality)

text: and

text: perform

text: an

text: expansion

text: of

text: A

text: V

text: up

text: to

text: some

text: arbitrary

text: power

text: of

text: τ

text: in

text: A

text: previous

text: extensive

text: numerical

text: study

text: shows

text: that

text: even

text: the

text: ﬁrst

text: order

text: expansion

text: is

text: enough

text: to

text: grasp

text: the

text: physical

text: content

text: of

text: the

text: model

text: A

text: We

text: work,

text: throughout

text: the

text: paper,

text: with

text: expansions

text: of

text: V

text: which

text: we

text: write

text: as

text: A

text: A

text: V

text: V

text: p

text: where

text: A

text: −A

text: V

text: I

text: A

text: −A

text: A

text: 2τ

text: A

text: A

text: V

text: Tr(

text: Tr(

text: I

text: A

text: −τ

text: A

text: A

text: A

text: A

text: τ2

text: A

text: A

text: A

text: τ2

text: A

text: A

text: V

text: T

text: Tr(

text: Tr(

text: Tr(

text: T

text: Tr(

text: I

text: I

text: A

text: A

text: A

text: A

text: A

text: τ2

text: A

text: A

text: A

text: V

text: T

text: )Tr(

text: T

text: Tr(

text: τ2

text: A

text: A

text: A

text: A

text: A

text: A

text: It

text: is

text: important

text: to

text: note

text: that

text: V

text: comprises

text: all

text: the

text: contributions

text: of

text: O(

text: p)

text: to

text: V

text: The

text: p

text: RFDC

text: model

text: yields

text: a

text: promising

text: stage

text: for

text: further

text: improvements,

text: insofar

text: its

text: vgPDFs

text: as

text: well

text: as

text: its

text: geometrical

text: statistical

text: properties

text: related

text: to

text: the

text: coupling

text: between

text: the

text: vorticity

text: and

text: the

text: rate-of-strain

text: tensor

text: share

text: several

text: qualitative

text: features

text: in

text: common

text: with

text: the

text: ones

text: observed

text: in

text: experiments

text: and

text: direct

text: numerical

text: simulations

text: of

text: turbulence

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: Quasi-periodic motions in . . . equations with dissipative eﬀects . . . detection methods 19 in the decomposition of the Fourier spectrum are listed in Table 6.2 of [62], where the √ numerical results deﬁnitely show that the invariant attractor is the golden torus related ∗ − − to the frequency ω = 2 φ = (3 5)/2 . 1 5 Semi-analytic approach constructing the normal form for invariant tori In order to describe the procedure constructing explicitly the Kolmogorov’s normal form related to an invariant quasi-periodic attractor, it is convenient to reformulate the pseudo-Hamiltonian model of the dissipative forced pendulum (deﬁned by the equa- tions (4)–(6)) in a suitably more general context. For this purpose, let us introduce three non-negative integer numbers n , n and K ; among them, both n and K are strictly 1 2 1 positive. Let n = n + n be the number of degrees of freedom of the system described 1 2(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) by the following equations of motion: V − − p˙, q˙ = H H(0) η p Ω(0) , 0 , (20) ∈ R where Ω(0) n is the external forcing fre(cid:88)que(cid:88)ncy vector and the Hamiltonian part ∞ ∞ · H(0)(p, q) = ω p + f(0,s)(p, q) . (21) l l=0 s=0 ∈ R In the equation above, as usual, ω n must be regarded as a ﬁxed frequency vector, ∈ ∀ ≥ ≥ while f(0,s) P l 0 and s 0 , where we denote P the class of functions l l,sK l,sK which are homogeneous polynomials of degree l in p , . . . , p , do not depend on the n 1 n1 2 actions p , . . . , p and are trigonometric polynomials of degree sK with respect to n1+1∈ T n the angles q n . Let us remark that also the Hamiltonian of the forced pendulum H ε deﬁned in (4) can be expressed in the form (21) with the following values of the integer parameters: n = 1 , n = 1 and K = 2 . In fact, after having performed a translation3 1 2 of the action coordinate p so that p = p + ω , it is enough to put ω = (ω , 1) , − − 1 1 1 1 − 1 Ω(0) = Ω (ω , 0) = (Ω ω , 0) , f(0,1) = ε[cos q +cos(q q )] , f(0,0) = 1p2 and f(0,s) = 0 ∀ (cid:54) 1 1 0 1 (cid:0) 1 (cid:1)2 2 2 1 l (l, s) = (0, 1), (2, 0) . Our goal is to determine an accurate approximation of a canonical transformation ∞ ∞ − ψ( ), such that in the new coordinates (P , Q) = ψ( ) 1(p, q) the equations of mo- (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) tion (20) are transformed to the following form: V ∞ − P˙ , Q˙ = H H( ) η P , 0 , (22) ∞ where the new Hamiltonian H( ) is in Kolmogorov’s normal form, i.e., (cid:55)→ 3Let us recall that (p , p , q , q ) (p + ω , p , q , q ) is a canonical transformation. Moreover, we 1 2 1 2 1 1 2 1 2 emphasize that in this special case the deﬁnition of H(0)(p, q) here is given by avoiding the introduction of a new symbol instead of p , by abuse of notation. 1 ##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: Quasi-periodic

text: motions

text: in

text: equations

text: with

text: dissipative

text: eﬀects

text: detection

text: methods

text: in

text: the

text: decomposition

text: of

text: the

text: Fourier

text: spectrum

text: are

text: listed

text: in

text: Table

text: of

text: where

text: the

text: numerical

text: results

text: deﬁnitely

text: show

text: that

text: the

text: invariant

text: attractor

text: is

text: the

text: golden

text: torus

text: related

text: to

text: the

text: frequency

text: ω

text: φ

text: Semi-analytic

text: approach

text: constructing

text: the

text: normal

text: form

text: for

text: invariant

text: tori

text: In

text: order

text: to

text: describe

text: the

text: procedure

text: constructing

text: explicitly

text: the

text: Kolmogorov’s

text: normal

text: form

text: related

text: to

text: an

text: invariant

text: quasi-periodic

text: attractor,

text: it

text: is

text: convenient

text: to

text: reformulate

text: the

text: pseudo-Hamiltonian

text: model

text: of

text: the

text: dissipative

text: forced

text: pendulum

text: (deﬁned

text: by

text: the

text: equa-

text: tions

text: in

text: a

text: suitably

text: more

text: general

text: context.

text: For

text: this

text: purpose,

text: let

text: us

text: introduce

text: three

text: non-negative

text: integer

text: numbers

text: n

text: n

text: and

text: K

text: among

text: them,

text: both

text: n

text: and

text: K

text: are

text: strictly

text: positive.

text: Let

text: n

text: n

text: n

text: be

text: the

text: number

text: of

text: degrees

text: of

text: freedom

text: of

text: the

text: system

text: described

text: by

text: the

text: following

text: equations

text: of

text: motion:

text: V

text: p˙,

text: q˙

text: H

text: η

text: p

text: R

text: where

text: n

text: is

text: the

text: external

text: forcing

text: vector

text: and

text: the

text: Hamiltonian

text: part

text: q)

text: ω

text: p

text: q)

text: l

text: R

text: In

text: the

text: equation

text: above,

text: as

text: usual,

text: ω

text: n

text: must

text: be

text: regarded

text: as

text: a

text: ﬁxed

text: frequency

text: vector,

text: while

text: P

text: l

text: and

text: s

text: where

text: we

text: denote

text: P

text: the

text: class

text: of

text: functions

text: l

text: l,sK

text: l,sK

text: which

text: are

text: homogeneous

text: polynomials

text: of

text: degree

text: l

text: in

text: p

text: p

text: do

text: not

text: depend

text: on

text: the

text: n

text: actions

text: p

text: p

text: and

text: are

text: trigonometric

text: polynomials

text: of

text: degree

text: sK

text: with

text: respect

text: to

text: T

text: n

text: the

text: angles

text: q

text: n

text: Let

text: us

text: remark

text: that

text: also

text: the

text: Hamiltonian

text: of

text: the

text: forced

text: pendulum

text: H

text: ε

text: deﬁned

text: in

text: can

text: be

text: expressed

text: in

text: the

text: form

text: with

text: the

text: following

text: values

text: of

text: the

text: integer

text: parameters:

text: n

text: n

text: and

text: K

text: In

text: fact,

text: after

text: having

text: performed

text: a

text: of

text: the

text: action

text: coordinate

text: p

text: so

text: that

text: p

text: p

text: ω

text: it

text: is

text: enough

text: to

text: put

text: ω

text: (ω

text: Ω

text: (ω

text: (Ω

text: ω

text: ε[cos

text: q

text: +cos(q

text: q

text: and

text: l

text: (l,

text: s)

text: Our

text: goal

text: is

text: to

text: determine

text: an

text: accurate

text: approximation

text: of

text: a

text: canonical

text: transformation

text: ψ(

text: such

text: that

text: in

text: the

text: new

text: coordinates

text: (P

text: Q)

text: ψ(

text: q)

text: the

text: equations

text: of

text: mo-

text: tion

text: are

text: transformed

text: to

text: the

text: following

text: form:

text: V

text: P˙

text: Q˙

text: H

text: H(

text: η

text: P

text: where

text: the

text: new

text: Hamiltonian

text: H(

text: is

text: in

text: Kolmogorov’s

text: normal

text: form,

text: i.e.,

text: us

text: recall

text: that

text: (p

text: p

text: q

text: q

text: (p

text: ω

text: p

text: q

text: q

text: is

text: a

text: canonical

text: transformation.

text: Moreover,

text: we

text: emphasize

text: that

text: in

text: this

text: special

text: case

text: the

text: deﬁnition

text: of

text: q)

text: here

text: is

text: given

text: by

text: avoiding

text: the

text: introduction

text: of

text: a

text: new

text: symbol

text: instead

text: of

text: p

text: by

text: abuse

text: of

text: notation.

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Quasi-periodic motions in . . . equations with dissipative eﬀects . . . detection methods 25 Figure 5: Explicit construction of the Kolmogorov’s normal form for the dissipative forced pendulum, deﬁned by the equations (4)–(6), with ε = 0.03 and η = 0.1 . Norms of the generating functions X(r), ξ(r), χ(r) and of the external forcing frequency vectors Ω(r) as 2 a function of the normalization step r . Actually, in the ﬁgure above, their values have been plotted by using the symbols + , (cid:63) , ♦ , ⊕, respectively. See the text for more details. 5.2 Semi-analytic results 5.2.1 Checking the explicit construction of the Kolmogorov’s normal form It could be astonishing that both the equations of motion (36) and the Hamiltonian part (43) deﬁned at the r-th normalization step have exactly the same structure as those introduced by the previous step, which are written in (24) and (25), respectively. Indeed, performing the algorithm described in the previous subsection is advantageous, because the unwanted Hamiltonian terms of degree 0 and 1 in the actions get smaller and smaller as r increases (under the usual KAM hypotheses). This implies that the algorithm is successful if and only if also the generating functions decrease with r (recall the equa- (cid:8) (cid:9) tions (29) and (39)); this remark can be easily translated in a numerical test concerning the construction of the normal form. The behavior of the sequence of the external forcing frequency vectors Ω(r) ≥ de- r 0 serves a particular discussion. Let us recall that the main theorem in [63] actually proves the existence of a pair of objects: an initial frequency vector Ω(0) and a canonical trans- ∞ formation ψ( ) such that the equations of motion (20) are conjugated to those in (22), where the Hamiltonian part is in the Kolmogorov’s normal form (23). Thus, the proof scheme determines Ω(0) a posteriori, i.e., as a result of the normalization procedure (ac- tually, here we followed the approach originally designed in [18]). Of course, this is unpractical when we focus on comparisons with numerical results, because any integrator of the equations of motion (20) requires that Ω(0) must be known in advance. This is the main reason why, in subsection we developed a Newton method based on the frequency analysis, so to provide a good approximation of the initial Ω(0), corresponding to the ﬁxed angular velocity vector ω of the quasi-periodic motion on the wanted invariant torus. If such a vector Ω(0) would be perfectly determined, then the algorithm constructing the → conjugacy to (22) (where any external frequency is not appearing) requires that Ω(r) 0 → ∞ for r . In the following, we will test numerically this condition. From a practical point of view, let us proceed to a further study of the equations of motion (6), where the Hamiltonian part H is given in (4) and the values of the parameters ε are ﬁxed so that ε = 0.03 , η = 0.1 , Ω = Ω˜(5) = 0.3870821721708347 . (49) Let us recall that for this system the frequency analysis results of subsection clearly √ show the existence of an attracting invariant torus, characterized by a quasi-periodic ∗ − − motion related to the golden frequency ω = 2 φ = (3 5)/2 . In our opinion, 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Quasi-periodic

text: motions

text: in

text: equations

text: with

text: dissipative

text: eﬀects

text: detection

text: methods

text: Figure

text: Explicit

label: caption

text: construction

text: of

text: the

text: Kolmogorov’s

text: normal

text: form

text: for

text: the

text: dissipative

text: forced

text: pendulum,

text: deﬁned

text: by

text: the

text: equations

text: with

text: ε

text: and

text: η

text: Norms

text: of

text: the

text: generating

text: functions

text: X(r),

text: ξ(r),

text: χ(r)

text: and

text: of

text: the

text: external

text: forcing

text: frequency

text: vectors

text: Ω(r)

text: as

text: a

text: function

text: of

text: the

text: normalization

text: step

text: r

text: Actually,

text: in

text: the

text: ﬁgure

text: above,

text: their

text: values

text: have

text: been

text: plotted

text: by

text: using

text: the

text: symbols

text: respectively.

text: See

text: the

text: text

text: for

text: more

text: details.

text: Semi-analytic

text: results

text: Checking

text: the

text: explicit

text: construction

text: of

text: the

text: Kolmogorov’s

text: normal

text: form

text: It

text: could

text: be

text: astonishing

text: that

text: both

text: the

text: equations

text: of

text: motion

text: and

text: the

text: Hamiltonian

text: part

text: deﬁned

text: at

text: the

text: r-th

text: normalization

text: step

text: have

text: exactly

text: the

text: same

text: structure

text: as

text: those

text: introduced

text: by

text: the

text: previous

text: step,

text: which

text: are

text: written

text: in

text: and

text: respectively.

text: Indeed,

text: performing

text: the

text: algorithm

text: described

text: in

text: the

text: previous

text: subsection

text: is

text: advantageous,

text: because

text: the

text: unwanted

text: Hamiltonian

text: terms

text: of

text: degree

text: and

text: in

text: the

text: actions

text: get

text: smaller

text: and

text: smaller

text: as

text: r

text: increases

text: (under

text: the

text: usual

text: KAM

text: hypotheses).

text: This

text: implies

text: that

text: the

text: algorithm

text: is

text: successful

text: if

text: and

text: only

text: if

text: also

text: the

text: generating

text: functions

text: decrease

text: with

text: r

text: (recall

text: the

text: equa-

text: tions

text: and

text: this

text: remark

text: can

text: be

text: easily

text: translated

text: in

text: a

text: numerical

text: test

text: concerning

text: the

text: construction

text: of

text: the

text: normal

text: form.

text: The

text: behavior

text: of

text: the

text: sequence

text: of

text: the

text: external

text: forcing

text: frequency

text: vectors

text: Ω(r)

text: de-

text: r

text: serves

text: a

text: particular

text: discussion.

text: Let

text: us

text: recall

text: that

text: the

text: main

text: theorem

text: in

text: actually

text: proves

text: the

text: existence

text: of

text: a

text: pair

text: of

text: objects:

text: an

text: initial

text: frequency

text: vector

text: and

text: a

text: canonical

text: trans-

text: formation

text: ψ(

text: such

text: that

text: the

text: equations

text: of

text: motion

text: are

text: conjugated

text: to

text: those

text: in

text: where

text: the

text: Hamiltonian

text: part

text: is

text: in

text: the

text: Kolmogorov’s

text: normal

text: form

text: Thus,

text: the

text: proof

text: scheme

text: determines

text: a

text: posteriori,

text: i.e.,

text: as

text: a

text: result

text: of

text: the

text: normalization

text: procedure

text: (ac-

text: tually,

text: here

text: we

text: followed

text: the

text: approach

text: originally

text: designed

text: in

text: Of

text: course,

text: this

text: is

text: unpractical

text: when

text: we

text: focus

text: on

text: comparisons

text: with

text: numerical

text: results,

text: because

text: any

text: integrator

text: of

text: the

text: equations

text: of

text: motion

text: requires

text: that

text: must

text: be

text: known

text: in

text: advance.

text: This

text: is

text: the

text: main

text: reason

text: why,

text: in

text: subsection

text: we

text: developed

text: a

text: Newton

text: method

text: based

text: on

text: the

text: frequency

text: analysis,

text: so

text: to

text: provide

text: a

text: good

text: approximation

text: of

text: the

text: initial

text: corresponding

text: to

text: the

text: ﬁxed

text: angular

text: velocity

text: vector

text: ω

text: of

text: the

text: quasi-periodic

text: motion

text: on

text: the

text: wanted

text: invariant

text: torus.

text: If

text: such

text: a

text: vector

text: would

text: be

text: perfectly

text: determined,

text: then

text: the

text: algorithm

text: constructing

text: the

text: conjugacy

text: to

text: (where

text: any

text: external

text: frequency

text: is

text: not

text: appearing)

text: requires

text: that

text: Ω(r)

text: for

text: r

text: In

text: the

text: following,

text: we

text: will

text: test

text: numerically

text: this

text: condition.

text: From

text: a

text: practical

text: point

text: of

text: view,

text: let

text: us

text: proceed

text: to

text: a

text: further

text: study

text: of

text: the

text: equations

text: of

text: motion

text: where

text: the

text: Hamiltonian

text: part

text: H

text: is

text: given

text: in

text: and

text: the

text: values

text: of

text: the

text: parameters

text: ε

text: are

text: ﬁxed

text: so

text: that

text: ε

text: η

text: Ω

text: Let

text: us

text: recall

text: that

text: for

text: this

text: system

text: the

text: frequency

text: analysis

text: results

text: of

text: subsection

text: clearly

text: show

text: the

text: existence

text: of

text: an

text: attracting

text: invariant

text: torus,

text: characterized

text: by

text: a

text: quasi-periodic

text: motion

text: related

text: to

text: the

text: golden

text: frequency

text: ω

text: φ

text: In

text: our

text: opinion,

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: 26 U. Locatelli, L. Stefanelli the explicit construction of the normal form related to that torus is rather challenging, because the the perturbing parameter ε is larger than the breakdown threshold value for (cid:39) the conservative case (i.e., ε¯ 0.0275856) and at the same time it is not so far from c (cid:0) (cid:1) the breakdown threshold corresponding to the chosen friction coeﬃcient η = 0.1 (i.e., (cid:39)(cid:0) (cid:1) (cid:0) (cid:1) ε 0.0372 , see the table appearing in Figure 4). c Of course, it is convenient to reformulate the equations of motion in the form p˙, q˙ = V − − H H(0) η p Ω(0) , 0 where all the terms appearing in the expansion of the Hamilto- (cid:32) (cid:33) (cid:32) (cid:33) nian H(0) are determined as in the discussion following formula (21). In particular, we have √ √ − − 3 5 − 3 5 ω = , 1 , Ω(0) = Ω˜(5) , 0 = (0.0047704825942882482 , 0) . 2 2 Starting from these settings, we explicitly performed 60 steps of the normalization pro- cedure described in subsection , by using the software package X(cid:37)´oνoς, that is designed for making computer algebra, with a special care to its possible applications to Celestial Mechanics problems (see [32] for an introduction to its main concepts). Since none of the canonical transformations prescribed by the algorithm increases the polynomial degree, all the expansions of the Hamiltonian parts are rather compact, because they are at most quadratic in the actions as the initial H(0). Figure 5 shows the norms of the generating ∈ functions X(r), ξ(r) and χ(r) with the index r [1, 60] ; more precisely, we have calculated 2 the sum of the absolute values of the coeﬃcients appearing in (30), (29) and (42), respect- (cid:80) ively. Actually, the precision of the computation is aﬀected by the truncation rules on the expansions, that we arranged so to neglect all the terms having a Fourier harmonic k | | |(cid:12) | (cid:12) with l –norm k = 2 (cid:12)k >(cid:12) 122 . In particular, this implies also that the contributions 1 i=1 i due to many relevant terms independent from the angles are not taken into account for r > 30 ; thus, the plot of ξ(r) has been stopped when the values corresponding to some indexes r have begun to be unrealistically small with respect to the previous ones. Let us emphasize that the geometrical decrease of the generating functions looks quite sharp in the semi–log scale of Figure 5; this behavior is in agreement with the analytical estimates on the algorithm constructing the Kolmogorov’s normal form, when it is reformulated according a classical formal scheme (see [30]). Furthermore, also the l –norm of the ex- (cid:12) (cid:12) 1 ternal forcing frequen(cid:12) cy v(cid:12)ector Ω(r) is reported in Figure 5. In this case, the geometrical − decrease is rather sharp until a “saturation threshold value”, that is of order 10 16; for r > 30 the value of Ω(r) is approximately constant. This unpleasant phenomenon can be easily explained, by taking into account that the numerical determination of the initial Ω(0) is aﬀected by the unavoidable round-oﬀ errors. Such an uncertainity (due to the ap- (cid:12) (cid:12) plication of the frequenc(cid:12)y an(cid:12)alysis numerical method) is propagated to all the sequence of the external forcing frequency vectors by the recursive deﬁnitions (27) and (37). Thus, the (cid:55)→ computed plot of r Ω(r) agrees with the expectation that lim →∞ Ω(r) = 0 . Finally, r we can conclude that Figure 5 makes evident that the constructing procedure is conver- ging to the Kolmogorov’s normal form for dissipative systems, which is characterized by equations (22) and (23). We now perform another test, checking the accuracy of the conjugacy canonical trans- K formation (r) which is provided after having carried out the r–th normalization step ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: U.

text: Locatelli,

text: L.

text: Stefanelli

text: the

text: explicit

text: construction

text: of

text: the

text: normal

text: form

text: related

text: to

text: that

text: torus

text: is

text: rather

text: challenging,

text: because

text: the

text: the

text: perturbing

text: parameter

text: ε

text: is

text: larger

text: than

text: the

text: breakdown

text: threshold

text: value

text: for

text: the

text: conservative

text: case

text: (i.e.,

text: ε¯

text: and

text: at

text: the

text: same

text: time

text: it

text: is

text: not

text: so

text: far

text: from

text: c

text: the

text: breakdown

text: threshold

text: corresponding

text: to

text: the

text: chosen

text: friction

text: coeﬃcient

text: η

text: (i.e.,

text: ε

text: see

text: the

text: table

text: appearing

text: in

text: Figure

text: c

text: Of

text: course,

text: it

text: is

text: convenient

text: to

text: reformulate

text: the

text: equations

text: of

text: motion

text: in

text: the

text: form

text: p˙,

text: q˙

text: V

text: H

text: η

text: p

text: where

text: all

text: the

text: terms

text: appearing

text: in

text: the

text: expansion

text: of

text: the

text: Hamilto-

text: nian

text: are

text: determined

text: as

text: in

text: the

text: discussion

text: following

text: formula

text: In

text: particular,

text: we

text: have

text: ω

text: Starting

text: from

text: these

text: settings,

text: we

text: explicitly

text: performed

text: steps

text: of

text: the

text: normalization

text: pro-

text: cedure

text: described

text: in

text: subsection

text: by

text: using

text: the

text: software

text: package

text: that

text: is

text: designed

text: for

text: making

text: computer

text: algebra,

text: with

text: a

text: special

text: care

text: to

text: its

text: possible

text: applications

text: to

text: Celestial

text: Mechanics

text: problems

text: (see

text: for

text: an

text: introduction

text: to

text: its

text: main

text: concepts).

text: Since

text: of

text: the

text: canonical

text: transformations

text: prescribed

text: by

text: the

text: algorithm

text: increases

text: the

text: polynomial

text: degree,

text: all

text: the

text: expansions

text: of

text: the

text: Hamiltonian

text: parts

text: are

text: rather

text: compact,

text: because

text: they

text: are

text: at

text: most

text: quadratic

text: in

text: the

text: actions

text: as

text: the

text: initial

text: Figure

text: shows

text: the

text: norms

text: of

text: the

text: generating

text: functions

text: X(r),

text: ξ(r)

text: and

text: χ(r)

text: with

text: the

text: index

text: r

text: more

text: precisely,

text: we

text: have

text: calculated

text: the

text: sum

text: of

text: the

text: absolute

text: values

text: of

text: the

text: coeﬃcients

text: appearing

text: in

text: and

text: respect-

text: ively.

text: Actually,

text: the

text: precision

text: of

text: the

text: computation

text: is

text: aﬀected

text: by

text: the

text: truncation

text: rules

text: on

text: the

text: expansions,

text: that

text: we

text: arranged

text: so

text: to

text: neglect

text: all

text: the

text: terms

text: having

text: a

text: Fourier

text: harmonic

text: k

text: with

text: l

text: –norm

text: k

text: In

text: particular,

text: this

text: implies

text: also

text: that

text: the

text: contributions

text: i

text: due

text: to

text: many

text: relevant

text: terms

text: independent

text: from

text: the

text: angles

text: are

text: not

text: taken

text: into

text: account

text: for

text: r

text: thus,

text: the

text: plot

text: of

text: ξ(r)

text: has

text: been

text: stopped

text: when

text: the

text: values

text: corresponding

text: to

text: some

text: indexes

text: r

text: have

text: begun

text: to

text: be

text: unrealistically

text: small

text: with

text: respect

text: to

text: the

text: previous

text: ones.

text: Let

text: us

text: emphasize

text: that

text: the

text: geometrical

text: decrease

text: of

text: the

text: generating

text: functions

text: looks

text: quite

text: sharp

text: in

text: the

text: semi–log

text: scale

text: of

text: Figure

text: this

text: behavior

text: is

text: in

text: agreement

text: with

text: the

text: analytical

text: estimates

text: on

text: the

text: algorithm

text: constructing

text: the

text: Kolmogorov’s

text: normal

text: form,

text: when

text: it

text: is

text: reformulated

text: according

text: a

text: classical

text: formal

text: scheme

text: (see

text: Furthermore,

text: also

text: the

text: l

text: –norm

text: of

text: the

text: ex-

text: ternal

text: forcing

text: cy

text: Ω(r)

text: is

text: reported

text: in

text: Figure

text: In

text: this

text: case,

text: the

text: geometrical

text: decrease

text: is

text: rather

text: sharp

text: until

text: a

text: “saturation

text: threshold

text: value”,

text: that

text: is

text: of

text: order

text: for

text: r

text: the

text: value

text: of

text: Ω(r)

text: is

text: approximately

text: constant.

text: This

text: unpleasant

text: phenomenon

text: can

text: be

text: easily

text: explained,

text: by

text: taking

text: into

text: account

text: that

text: the

text: numerical

text: determination

text: of

text: the

text: initial

text: is

text: aﬀected

text: by

text: the

text: unavoidable

text: round-oﬀ

text: errors.

text: Such

text: an

text: uncertainity

text: (due

text: to

text: the

text: ap-

text: plication

text: of

text: the

text: numerical

text: method)

text: is

text: propagated

text: to

text: all

text: the

text: sequence

text: of

text: the

text: external

text: forcing

text: frequency

text: vectors

text: by

text: the

text: recursive

text: deﬁnitions

text: and

text: Thus,

text: the

text: computed

text: plot

text: of

text: r

text: Ω(r)

text: agrees

text: with

text: the

text: expectation

text: that

text: lim

text: Ω(r)

text: Finally,

text: r

text: we

text: can

text: conclude

text: that

text: Figure

text: makes

text: evident

text: that

text: the

text: constructing

text: procedure

text: is

text: conver-

text: ging

text: to

text: the

text: Kolmogorov’s

text: normal

text: form

text: for

text: dissipative

text: systems,

text: which

text: is

text: characterized

text: by

text: equations

text: and

text: We

text: now

text: perform

text: another

text: test,

text: checking

text: the

text: accuracy

text: of

text: the

text: conjugacy

text: canonical

text: trans-

text: K

text: formation

text: (r)

text: which

text: is

text: provided

text: after

text: having

text: carried

text: out

text: the

text: r–th

text: normalization

text: step

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Quasi-periodic motions in . . . equations with dissipative eﬀects . . . detection methods 7 equations of motion (6), using the Taylor1 method (see [41]). In our tests, such a software package is able to numerically integrate the ﬂow Φ2π , performing less than 30 steps; ε,η ,Ω each step is aﬀected by an uncertainity not greater than the round-oﬀ error on double type variables of the C programming language. The precision of the integration scheme could be further improved, by using long double type variables or multiple precision arithmetic, that can be very well performed also by using the TIDES software package (see [1] for an introduction). We consider that our numerical results should be very slightly modiﬁed by such a further improvement and, so, it has not been implemented. An important feature of the dissipative systems is that they need a relaxation time before converging to the invariant attractor. From the computational point of view, this means that a certain number W of preliminary iterations is necessary, in addition to those required by the frequency map analysis. In order to provide a criterion for the choice of the value of W , let us consider the unperturbed case of the dissipative standard map (2) and assume the initial value of the ordinate is y ; then, one can easily check that the 0 − − ∀ ∈ N sequence of the iterated points is such that y = (1 η)n(y Ω) + Ω , n . In a n 0 numerical experiment, the value of y is determined so that the initial point (x , y ) is 0 −0 0 rather close to the wanted invariant attractor. Therefore, we deﬁne W so that (1 η)n is ∀ ≥ at most of the order of the machine precision n W , i.e., (cid:100)− − (cid:101) W = (15 log 10)/ log(1 η) , (8) (cid:100) (cid:101) ∈ R being α the smallest integer greater than or equal to α . 3 Adapting the frequency map analysis to dissipative systems 3.1 Frequency map analysis for Hamiltonian systems: a short overview Since our investigation approach for dissipative systems is strongly reminiscent of the method designed by Laskar to study conservative systems (see, e.g., [43] and [44]), we think that it is convenient to recall some of its features in the present subsection. This will allow us to introduce our adaptation for dissipative systems in a more natural way. Let us consider an n–d.o.f. quasi-integrable system, described by an analytic Hamilto- nian H(I, θ) = h(I) + εf(I, θ) , (9) ∈ G×T G ⊂ R where (I, θ) n (being n an open set) are action–angle variables. According to KAM theory (see, e.g., [59]), if the following conditions are satisﬁed: 1A software package implementing the numerical integration of the ordinary diﬀerential equations by means of the Taylor method is publicly available at the following website: ∼ http://www.maia.ub.es/ angel/soft.html ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Quasi-periodic

text: motions

text: in

text: equations

text: with

text: dissipative

text: eﬀects

text: detection

text: methods

text: equations

text: of

text: motion

text: using

text: the

text: method

text: (see

text: In

text: our

text: tests,

text: such

text: a

text: software

text: package

text: is

text: able

text: to

text: numerically

text: integrate

text: the

text: ﬂow

text: Φ2π

text: performing

text: less

text: than

text: steps;

text: ε,η

text: ,Ω

text: each

text: step

text: is

text: aﬀected

text: by

text: an

text: uncertainity

text: not

text: greater

text: than

text: the

text: round-oﬀ

text: error

text: on

text: double

text: type

text: variables

text: of

text: the

text: C

text: programming

text: language.

text: The

text: precision

text: of

text: the

text: integration

text: scheme

text: could

text: be

text: further

text: improved,

text: by

text: using

text: long

text: double

text: type

text: variables

text: or

text: multiple

text: precision

text: arithmetic,

text: that

text: can

text: be

text: very

text: well

text: performed

text: also

text: by

text: using

text: the

text: TIDES

text: software

text: package

text: (see

text: for

text: an

text: introduction).

text: We

text: consider

text: that

text: our

text: numerical

text: results

text: should

text: be

text: very

text: slightly

text: modiﬁed

text: by

text: such

text: a

text: further

text: improvement

text: and,

text: so,

text: it

text: has

text: not

text: been

text: implemented.

text: An

text: important

text: feature

text: of

text: the

text: dissipative

text: systems

text: is

text: that

text: they

text: need

text: a

text: relaxation

text: time

text: before

text: converging

text: to

text: the

text: invariant

text: attractor.

text: From

text: the

text: computational

text: point

text: of

text: view,

text: this

text: means

text: that

text: a

text: certain

text: number

text: W

text: of

text: preliminary

text: iterations

text: is

text: necessary,

text: in

text: addition

text: to

text: those

text: required

text: by

text: the

text: frequency

text: map

text: analysis.

text: In

text: order

text: to

text: provide

text: a

text: criterion

text: for

text: the

text: choice

text: of

text: the

text: value

text: of

text: W

text: let

text: us

text: consider

text: the

text: unperturbed

text: case

text: of

text: the

text: dissipative

text: standard

text: map

text: and

text: assume

text: the

text: initial

text: value

text: of

text: the

text: ordinate

text: is

text: y

text: then,

text: one

text: can

text: easily

text: check

text: that

text: the

text: N

text: sequence

text: of

text: the

text: iterated

text: points

text: is

text: such

text: that

text: y

text: η)n(y

text: Ω)

text: Ω

text: n

text: In

text: a

text: n

text: numerical

text: experiment,

text: the

text: value

text: of

text: y

text: is

text: determined

text: so

text: that

text: the

text: initial

text: point

text: (x

text: y

text: is

text: −0

text: rather

text: close

text: to

text: the

text: wanted

text: invariant

text: attractor.

text: Therefore,

text: we

text: deﬁne

text: W

text: so

text: that

text: η)n

text: is

text: at

text: most

text: of

text: the

text: order

text: of

text: the

text: machine

text: precision

text: n

text: W

text: i.e.,

text: W

text: log

text: η)

text: R

text: being

text: α

text: the

text: smallest

text: integer

text: greater

text: than

text: or

text: equal

text: to

text: α

text: Adapting

text: the

text: frequency

text: map

text: analysis

text: to

text: dissipative

text: systems

text: Frequency

text: map

text: analysis

text: for

text: Hamiltonian

text: systems:

text: a

text: short

text: overview

text: Since

text: our

text: investigation

text: approach

text: for

text: dissipative

text: systems

text: is

text: strongly

text: reminiscent

text: of

text: the

text: method

text: designed

text: by

text: Laskar

text: to

text: study

text: conservative

text: systems

text: (see,

text: e.g.,

text: and

text: we

text: think

text: that

text: it

text: is

text: convenient

text: to

text: recall

text: some

text: of

text: its

text: features

text: in

text: the

text: present

text: subsection.

text: This

text: will

text: allow

text: us

text: to

text: introduce

text: our

text: adaptation

text: for

text: dissipative

text: systems

text: in

text: a

text: more

text: natural

text: way.

text: Let

text: us

text: consider

text: an

text: n–d.o.f.

text: quasi-integrable

text: system,

text: described

text: by

text: an

text: analytic

text: Hamilto-

text: nian

text: H(I,

text: θ)

text: h(I)

text: εf(I,

text: θ)

text: G×T

text: G

text: R

text: where

text: (I,

text: θ)

text: n

text: (being

text: n

text: an

text: open

text: set)

text: are

text: action–angle

text: variables.

text: According

text: to

text: KAM

text: theory

text: (see,

text: e.g.,

text: if

text: the

text: following

text: conditions

text: are

text: satisﬁed:

text: software

text: package

text: implementing

text: the

text: numerical

text: integration

text: of

text: the

text: ordinary

text: diﬀerential

text: equations

text: by

text: means

text: of

text: the

text: Taylor

text: method

text: is

text: publicly

text: available

text: at

text: the

text: following

text: website:

text: angel/soft.html

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Quasi-periodic motions in . . . equations with dissipative eﬀects . . . detection methods 9 w . In all our numerical experiments (described below), we used the Hanning window ﬁlter w(u) = 1 + cos(πu) ; here, we just recall that with this kind of weight function, the diﬀerence between the computed value of the frequency vector ω and the true one is O → ∞ (1/T 4) for T . A ﬁrst test to check if an orbit lies on an invariant KAM torus can be made by controlling that the Fourier decomposition (10) of a corresponding signal holds true, within the limitations due to the unavoidable numerical errors. Let us remark that the frequency vector ω is not given a priori, but its detection is often not so diﬃcult, in (cid:0) (cid:1) practical applications. For instance, let us consider a quasi-integrable Hamiltonian of the ∀ type (9) and satisfying the conditions (A) and (B); j = 1, . . . , n , let us study the signal z(t) = I (t) exp iθ (t) ; then the point σ¯ corresponding to the absolute maximum of j j T the function (11) gives an approximation of ω , that gets more and more accurate for → ∞ j T . Another natural numerical investigation concerns the local regularity and invertibility (cid:55)→ − of the action-frequency map I ω(I) such that ω(I) = Ψ 1(I, θ) for any ﬁxed value ∈ T of θ n . The frequency map analysis mainly aims to obtain directly, in a numerical (cid:55)→ manner, the map I ω(I) . The procedure, can be summarized as follows: we ﬁrst (cid:0) (cid:1) arbitrarily ﬁx the initial values of the angles θ ; we pick up the initial actions I from J 0 0 a regular grid of values. For each initial condition, we consider the corresponding (cid:55)→ motion law t (I(t), θ(t)) and we analyze the n signals z (t) = I (t) exp iθ (t) with ∈ − ∀ j j j t [t T, t + T ], j = 1, . . . , n ; for each signal, we ﬁnd the value of ω corresponding 0 0 j to the absolute maximum of the function (11). Following this procedure, we can then ∈ J calculate the frequency ω = ω(I ) for all the initial values of the actions I . 0 0 The analysis of the frequency map can distinguish among three diﬀerent dynamical situations: (a) when the values of the initial conditions (I , θ ) are such that the corresponding ∀ ∈ J (cid:55)→0 0 motions are chaotic I , then the map I ω(I ) looks highly irregular; 0 0 0 (b) when the initial conditions are such that the orbits are on the regular mani- folds inside a resonant region (these are the so called “librational” maximal tori in the neighborhood of a stable equilibrium point or an elliptic lower dimensional torus), some components ω of the frequency vector are constant while I is changed; j 0 (c) when the initial conditions are in a region (nearly) ﬁlled by KAM tori, a thin enough enlargement of the frequency map highlights a quasi-linear and invertible behavior. This is in agreement with the property (I) (that is described above and proved in [59]) joined with the approach described in, e.g., [54]: the signature of the existence of a KAM torus (cid:55)→ is the local regularity and invertibility of the action–frequency map I ω(I) such that − ∈ T ω(I) = Ψ 1(I, θ) for any ﬁxed value of θ n . These three diﬀerent regimes can be sharply highlighted with some numerical exper- S iments on symplectic maps. For instance, let us consider the standard map , as it is ε deﬁned by formula (2) when η = 0 . The results plotted in Figure 1 are obtained by ana- (cid:55)→ lyzing the signal n z(n) with z(n) = y exp(ix ) , where the pair (y , x ) is obtained by S n n n n n iterations of , starting from the initial condition (y , x ) . Let us remark that we are ε 0 0 assuming that the signal is sampled in a trivial way, so that the “elapsed time” between an iteration of the standard map and the next one is ∆ = 1 . By looking at the deﬁnitions ##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: Quasi-periodic

text: motions

text: in

text: equations

text: with

text: dissipative

text: eﬀects

text: detection

text: methods

text: w

text: In

text: all

text: our

text: numerical

text: experiments

text: (described

text: below),

text: we

text: used

text: the

text: Hanning

text: window

text: ﬁlter

text: w(u)

text: cos(πu)

text: here,

text: we

text: just

text: recall

text: that

text: with

text: this

text: kind

text: of

text: weight

text: function,

text: the

text: diﬀerence

text: between

text: the

text: computed

text: value

text: of

text: the

text: frequency

text: vector

text: ω

text: and

text: the

text: true

text: one

text: is

text: O

text: for

text: T

text: A

text: ﬁrst

text: test

text: to

text: check

text: if

text: an

text: orbit

text: lies

text: on

text: an

text: invariant

text: KAM

text: torus

text: can

text: be

text: made

text: by

text: controlling

text: that

text: the

text: Fourier

text: decomposition

text: of

text: a

text: corresponding

text: signal

text: holds

text: true,

text: within

text: the

text: limitations

text: due

text: to

text: the

text: unavoidable

text: numerical

text: errors.

text: Let

text: us

text: remark

text: that

text: the

text: frequency

text: vector

text: ω

text: is

text: not

text: given

text: a

text: priori,

text: but

text: its

text: detection

text: is

text: often

text: not

text: so

text: diﬃcult,

text: in

text: practical

text: applications.

text: For

text: instance,

text: let

text: us

text: consider

text: a

text: quasi-integrable

text: Hamiltonian

text: of

text: the

text: type

text: and

text: satisfying

text: the

text: conditions

text: (A)

text: and

text: (B);

text: j

text: n

text: let

text: us

text: study

text: the

text: signal

text: z(t)

text: I

text: (t)

text: exp

text: iθ

text: (t)

text: then

text: the

text: point

text: σ¯

text: corresponding

text: to

text: the

text: absolute

text: maximum

text: of

text: j

text: j

text: T

text: the

text: function

text: gives

text: an

text: approximation

text: of

text: ω

text: that

text: gets

text: more

text: and

text: more

text: accurate

text: for

text: j

text: T

text: Another

text: natural

text: numerical

text: investigation

text: concerns

text: the

text: local

text: regularity

text: and

text: invertibility

text: of

text: the

text: action-frequency

text: map

text: I

text: ω(I)

text: such

text: that

text: ω(I)

text: Ψ

text: θ)

text: for

text: any

text: ﬁxed

text: value

text: T

text: of

text: θ

text: n

text: The

text: frequency

text: map

text: analysis

text: mainly

text: aims

text: to

text: obtain

text: directly,

text: in

text: a

text: numerical

text: manner,

text: the

text: map

text: I

text: ω(I)

text: The

text: procedure,

text: can

text: be

text: summarized

text: as

text: follows:

text: we

text: ﬁrst

text: arbitrarily

text: ﬁx

text: the

text: initial

text: values

text: of

text: the

text: angles

text: θ

text: we

text: pick

text: up

text: the

text: initial

text: actions

text: I

text: from

text: J

text: a

text: regular

text: grid

text: of

text: values.

text: For

text: each

text: initial

text: condition,

text: we

text: consider

text: the

text: corresponding

text: motion

text: law

text: t

text: (I(t),

text: θ(t))

text: and

text: we

text: analyze

text: the

text: n

text: signals

text: z

text: (t)

text: I

text: (t)

text: exp

text: iθ

text: (t)

text: with

text: j

text: j

text: j

text: t

text: [t

text: T,

text: t

text: T

text: j

text: n

text: for

text: each

text: signal,

text: we

text: ﬁnd

text: the

text: value

text: of

text: ω

text: corresponding

text: j

text: to

text: the

text: absolute

text: maximum

text: of

text: the

text: function

text: Following

text: this

text: procedure,

text: we

text: can

text: then

text: J

text: calculate

text: the

text: frequency

text: ω

text: ω(I

text: for

text: all

text: the

text: initial

text: values

text: of

text: the

text: actions

text: I

text: The

text: analysis

text: of

text: the

text: frequency

text: map

text: can

text: distinguish

text: among

text: three

text: diﬀerent

text: dynamical

text: situations:

text: (a)

text: when

text: the

text: values

text: of

text: the

text: initial

text: conditions

text: (I

text: θ

text: are

text: such

text: that

text: the

text: corresponding

text: J

text: motions

text: are

text: chaotic

text: I

text: then

text: the

text: map

text: I

text: ω(I

text: looks

text: highly

text: irregular;

text: (b)

text: when

text: the

text: initial

text: conditions

text: are

text: such

text: that

text: the

text: orbits

text: are

text: on

text: the

text: regular

text: mani-

text: folds

text: inside

text: a

text: resonant

text: region

text: (these

text: are

text: the

text: so

text: called

text: “librational”

text: maximal

text: tori

text: in

text: the

text: neighborhood

text: of

text: a

text: stable

text: equilibrium

text: point

text: or

text: an

text: elliptic

text: lower

text: dimensional

text: torus),

text: some

text: components

text: ω

text: of

text: the

text: frequency

text: vector

text: are

text: constant

text: while

text: I

text: is

text: changed;

text: j

text: (c)

text: when

text: the

text: initial

text: conditions

text: are

text: in

text: a

text: region

text: (nearly)

text: ﬁlled

text: by

text: KAM

text: tori,

text: a

text: thin

text: enough

text: enlargement

text: of

text: the

text: frequency

text: map

text: highlights

text: a

text: quasi-linear

text: and

text: invertible

text: behavior.

text: This

text: is

text: in

text: agreement

text: with

text: the

text: property

text: (I)

text: (that

text: is

text: described

text: above

text: and

text: proved

text: in

text: joined

text: with

text: the

text: approach

text: described

text: in,

text: e.g.,

text: the

text: signature

text: of

text: the

text: existence

text: of

text: a

text: KAM

text: torus

text: is

text: the

text: local

text: regularity

text: and

text: invertibility

text: of

text: the

text: action–frequency

text: map

text: I

text: ω(I)

text: such

text: that

text: T

text: ω(I)

text: Ψ

text: θ)

text: for

text: any

text: ﬁxed

text: value

text: of

text: θ

text: n

text: These

text: three

text: diﬀerent

text: regimes

text: can

text: be

text: sharply

text: highlighted

text: with

text: some

text: numerical

text: exper-

text: S

text: iments

text: on

text: symplectic

text: maps.

text: For

text: instance,

text: let

text: us

text: consider

text: the

text: standard

text: map

text: as

text: it

text: is

text: ε

text: deﬁned

text: by

text: formula

text: when

text: η

text: The

text: results

text: plotted

text: in

text: Figure

text: are

text: obtained

text: by

text: ana-

text: lyzing

text: the

text: signal

text: n

text: z(n)

text: with

text: z(n)

text: y

text: exp(ix

text: where

text: the

text: pair

text: (y

text: x

text: is

text: obtained

text: by

text: S

text: n

text: n

text: n

text: n

text: n

text: iterations

text: of

text: starting

text: from

text: the

text: initial

text: condition

text: (y

text: x

text: Let

text: us

text: remark

text: that

text: we

text: are

text: ε

text: assuming

text: that

text: the

text: signal

text: is

text: sampled

text: in

text: a

text: trivial

text: way,

text: so

text: that

text: the

text: “elapsed

text: time”

text: between

text: an

text: iteration

text: of

text: the

text: standard

text: map

text: and

text: the

text: next

text: one

text: is

text: By

text: looking

text: at

text: the

text: deﬁnitions

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text: The model of a level crossing with a Coulomb band 10 (cid:34) (cid:40) (cid:41) (cid:35) (cid:18) (cid:19) (cid:88) (cid:89) − − − − − m 1 m 1 − N Γ(1 + h ξ ) γ = (2)m 1/2(i)m (Γ(h )) 1 (h 1) r − s . (42) m,r r r j j Γ(1 + h h ) r s j=1 s=1 (cid:40) (cid:41) The Stirling numbers of the(cid:88)second kind satisfy the relation m m − (x) = xm 1, (43) j j j=1 (cid:18) (cid:19) which simpliﬁes (42) to (cid:89) − − − − − N Γ(1 + h ξ ) γ = (2)m 1/2(i)m (Γ(h )) 1 (h 1)m 1 r − s . (44) m,r r r Γ(1 + h h ) r s s=1     Using notation from (23), with b as in (38), 0         g a (0) γ 1 1 1,r . − . .. = V 1 .. . (45) N g a (0) γ N N N,r Setting ζ (a vector) equal to the right-hand side of (45), we let c be the qth column r   q   of      · · ·   −  ... ...  1 g1 0 · · · 0  · · · 0 g 0 2 ζ.1 ζ.N ... ... ... ... , (46) .. .. · · · 0 0 g N   and denote the components of c by q     c 1,q . cq = .. . (47) c N,q If we put (cid:88) (cid:16) (cid:12) (cid:17) (cid:12) (cid:12) N b = c tikr/2G1,N ξN βit , (48) 0 r,q N,N+1 hr,0,hNr r=1 · · · this corresponds to the vector of states where b (0) = a (0) = = a − (0) = · · · 0 1 q 1 a (0) = = a (0) = 0, and a (0) = 1. q+1 N q Again applying the asymptotics from (A.8) provides an asymptotic for P (v): q0 ##LTLine## ##LTLine##

text: The

text: model

text: of

text: a

text: level

text: crossing

text: with

text: a

text: Coulomb

text: band

label: equation

text: m

text: m

text: N

text: h

text: ξ

text: γ

text: (Γ(h

text: (h

text: r

text: s

text: m,r

text: r

text: r

text: j

text: j

text: h

text: h

text: r

text: s

text: The

text: Stirling

text: numbers

text: of

text: kind

text: satisfy

text: the

text: relation

text: m

text: m

text: (x)

text: xm

text: j

text: j

text: which

text: simpliﬁes

text: to

text: N

text: h

text: ξ

text: γ

text: (Γ(h

text: (h

text: r

text: s

text: m,r

text: r

text: r

text: h

text: h

text: r

text: s

text: Using

text: notation

text: from

text: with

text: b

text: as

text: in

text: g

text: a

text: γ

text: V

text: N

text: g

text: a

text: γ

text: N

text: N

text: N,r

text: Setting

text: ζ

text: (a

text: vector)

text: equal

text: to

text: the

text: right-hand

text: side

text: of

text: we

text: let

text: c

text: be

text: the

text: qth

text: column

text: r

text: q

text: of

text: g1

text: g

text: ζ.N

text: g

text: N

text: and

text: denote

text: the

text: components

text: of

text: c

text: by

text: q

text: c

text: cq

text: c

text: N,q

text: If

text: we

text: put

text: N

text: b

text: c

text: ξN

text: βit

text: r,q

text: this

text: corresponds

text: to

text: the

text: vector

text: of

text: states

text: where

text: b

text: a

text: a

text: q

text: a

text: a

text: and

text: a

text: N

text: q

text: Again

text: applying

text: the

text: asymptotics

text: from

text: provides

text: an

text: asymptotic

text: for

text: P

text: (v):

text: ##LTLine##

text: ##LTLine##


text: 5 Examples In this section we consider two examples to illustrate how the assumptions of Theorem 6 can be veriﬁed in practice. We preface the examples by a technical lemma, which will be very useful for checking the assumptions. Z Z ∈ Lemma 10. Let (Z, ) be a measurable set and ξ be a measure on (Z, ). Let α (0, 1) and let ϕ, ψ and q be nonnegative mea(cid:90)surable functions, such that (cid:90) ∞ ψ(z)ϕ(z)ξ(dz) < , (29) − ∞ ϕ 1−αα (z)q(z)ξ(dz) < . (30) (cid:90) Then, − ∞ ψα(z)q1 α(z)ξ(dz) < . (31) (cid:90) (cid:90) (cid:2) (cid:3) Proof. The result follows from H¨older’s inequality: ψα(z)q1−α(z)ξ(dz) = (cid:18)(cid:90)[ψ(z)ϕ(z)]α ϕ−(cid:19)1−αα(cid:18)(z(cid:90))q(z) 1−α ξ(dz) (cid:19) − ≤ α − 1 α ψ(z)ϕ(z)ξ(dz) ϕ 1−αα (z)q(z)ξ(dz) . 5.1 A nonlinear model with additive measurement noise We consider ﬁrst a class of nonlinear state space models where the latent process is observed in additive noise, X = hξ(X ) + σ W (32) t+1 t W t+1 Y = φX + σ U (33) t t U t { ∈ N} { ∈ N} where W , t and U , t are two independent sequences of i.i.d. standard t {t ∈ } Gaussian random variables and hξ, ξ Ξ is a parametric family of measurable real-valued functions, where Ξ is a compact subset of a Euclidean space. We denote by θ = (ξ, φ, σ , σ ) the ∈ U× W∞ parameters of the model. It is assumed that θ Θ, where Θ is a compact subset of Ξ (0, )3. ∈ (cid:55)→ | | | | We assume that for all ξ Ξ, x hξ(x) is continuous and sup ∈ lim sup →∞ hξ(x) / x < 1. | | ξ Ξ x ∈ For any δ > 0, we set V (x) = eδ x . It is easily seen that there exist constants λ (0, 1) and ∞ δ δ b < such that δ E ≤ sup θ [V (X )] λ V (x) + b . (34) ∈ x δ 1 δ δ δ θ Θ 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Examples

text: In

text: this

text: section

text: we

text: consider

text: two

text: examples

text: to

text: illustrate

text: how

text: the

text: assumptions

text: of

text: Theorem

text: can

text: be

text: veriﬁed

text: in

text: practice.

text: We

text: preface

text: the

text: examples

text: by

text: a

text: technical

text: lemma,

text: which

text: will

text: be

text: very

text: useful

text: for

text: checking

text: the

text: assumptions.

text: Z

text: Z

text: Lemma

text: Let

text: (Z,

text: be

text: a

text: measurable

text: set

text: and

text: ξ

text: be

text: a

text: measure

text: on

text: (Z,

text: Let

text: α

text: and

text: let

text: ϕ,

text: ψ

text: and

text: q

text: be

text: nonnegative

text: functions,

text: such

text: that

text: ψ(z)ϕ(z)ξ(dz)

text: ϕ

text: 1−αα

text: (z)q(z)ξ(dz)

text: Then,

text: α(z)ξ(dz)

text: Proof.

text: The

text: result

text: follows

text: from

text: H¨older’s

text: inequality:

text: 1−α

text: ξ(dz)

text: α

text: α

text: ψ(z)ϕ(z)ξ(dz)

text: ϕ

text: 1−αα

text: (z)q(z)ξ(dz)

text: A

text: nonlinear

text: model

text: with

text: additive

text: measurement

text: noise

text: We

text: consider

text: ﬁrst

text: a

text: class

text: of

text: nonlinear

text: state

text: space

text: models

text: where

text: the

text: latent

text: process

text: is

text: observed

text: in

text: additive

text: noise,

text: X

text: hξ(X

text: σ

text: W

text: t

text: W

text: Y

text: φX

text: σ

text: U

text: t

text: t

text: U

text: t

text: N}

text: N}

text: where

text: W

text: t

text: and

text: U

text: t

text: are

text: two

text: independent

text: sequences

text: of

text: i.i.d.

text: standard

text: t

text: {t

text: Gaussian

text: random

text: variables

text: and

text: hξ,

text: ξ

text: Ξ

text: is

text: a

text: parametric

text: family

text: of

text: measurable

text: real-valued

text: functions,

text: where

text: Ξ

text: is

text: a

text: compact

text: subset

text: of

text: a

text: Euclidean

text: space.

text: We

text: denote

text: by

text: θ

text: (ξ,

text: φ,

text: σ

text: σ

text: the

text: U×

text: W∞

text: parameters

text: of

text: the

text: model.

text: It

text: is

text: assumed

text: that

text: θ

text: Θ,

text: where

text: Θ

text: is

text: a

text: compact

text: subset

text: of

text: Ξ

text: We

text: assume

text: that

text: for

text: all

text: ξ

text: Ξ,

text: x

text: hξ(x)

text: is

text: continuous

text: and

text: sup

text: lim

text: sup

text: hξ(x)

text: x

text: ξ

text: Ξ

text: x

text: For

text: any

text: δ

text: we

text: set

text: V

text: (x)

text: eδ

text: x

text: It

text: is

text: easily

text: seen

text: that

text: there

text: exist

text: constants

text: λ

text: and

text: δ

text: δ

text: b

text: such

text: that

text: δ

text: E

text: sup

text: θ

text: [V

text: (X

text: λ

text: V

text: (x)

text: b

text: x

text: δ

text: δ

text: δ

text: δ

text: θ

text: Θ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:104) (cid:105) (cid:101) Using (57), (58) and (53) veriﬁes the second condition in (30). Lemma 10 can thus be used E (cid:104) (cid:105) ∞ (cid:104)∈ (cid:105) ∈ N to conclude that θ (Bθ Y )α < for all α (0, 1). Since this holds for any t and ∈ µ t t (cid:101) θ Θ, we establish the ﬁrst part of (51). ∈ E (cid:104) (cid:105) ∞ We will now check that, for all α (0, 1), θ (Cθ Y )α < . Using (49) and (50), we µ t t:t+1 get (cid:34) (cid:0)(cid:82) (cid:1) (cid:35) (cid:104) (cid:105) (cid:101) | (cid:104) (cid:105)| Eθ (Cθ(cid:104)Y (cid:105))α = E(cid:90)(cid:90)θ wθ Yt α∞(cid:18)(cid:90) gθ(xt+1, Yt+1)dxt+1 (cid:19)α µ t t:t+1 (pθ (Y ))α µ,t t:t+1 | (cid:90)(cid:104)(cid:90) (cid:105)| α − = wθ y α∞ gθ(x , y )dx (pθ (y ))1 αdy , t t+1 t+1 t+1 µ,t t:t+1 t:t+1 ≤ | |− − DαDα y y α(pθ (y ))1 αdy . (59) 1 2 t t+1 µ,t t:t+1 t:t+1 We use again Lemma 10 with | |− | |− ψ(y , y ) = y 1 y 1 (60) 0 1 0 1 and | | | | y γ y γ ϕ(y , y ) = ∨0 1 ∨ , (61) 0 1 (y2 1) (y2 1) 0 1 (cid:90)(cid:90) − ∈ (cid:90)(cid:90) (cid:8) (cid:9) with γα/(1 α) < 1 and γ (0, 1). Note ﬁrst that | || | − ∨ ∨ − ∞ ψ(y , y )ϕ(y , y )dy = ( y y )1 γ(y2 1)(y2 1) 1 dy < . (62) 0 1 0 1 0:1 0 1 0 1 0:1 (cid:90)H(cid:90)ence, (29) is satisﬁed. We ﬁnally check (30). Using the conditional independence of the (cid:2) (cid:2) (cid:12) (cid:3)(cid:3) observations given the states and (58), (cid:12) ϕ(cid:34)−1−αα(cid:34)(y(cid:18)t:t+1)pθµ,t(yt:t+(cid:19)1)dyt:(cid:35)t+1 = E(cid:34)θµ(cid:18)Eθµ ϕ−1−αα (Yt(cid:19):t+1) (cid:35)X(cid:35)t:t+1 (cid:104) (cid:105) E E β2eX0U2 ∨ 1 1−αα E β2eX0U2 ∨ 1 1−αα ≤ E | | | | = θ θ | | θ | | D2 θ eδ Xt eδ Xt+1 . µ Xt βeγX0/2 U γ Xt+1 βeγX0/2 U γ 3 µ (cid:104) (cid:105) (cid:16) (cid:104) (cid:105) (cid:104) (cid:105)(cid:17) Using (52), we get, from the Cauchy-Schwarz inequality, E | | | | ≤ E | | E | | 1/2 θ eδ Xt eδ Xt+1 θ e2δ Xt θ e2δ Xt+1 , µ µ µ Applying (53) with δ replaced by 2δ yields (30). Using Lemma 10 thus establishes (51) and √ thereby, holds. Provided that θ converges to θ at a rate 1/ T (see Remark 7), we may therefore apply T (cid:63) ∈ { − } Theorem 6 which shows that for any γ (0, 1), (cid:15) 1 (θ ) ≥ is tight with N = T1/γ. T,NT T T 1 T 6 Proof of Theorem 1 We will now turn to the proof of the minorization condition in Theorem 1. As in the statement of the theorem, we will not explicitly indicate any possible dependence on unknown 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Using

text: and

text: veriﬁes

text: the

text: second

text: condition

text: in

text: Lemma

text: can

text: thus

text: be

text: used

text: E

text: N

text: to

text: conclude

text: that

text: θ

text: (Bθ

text: Y

text: )α

text: for

text: all

text: α

text: Since

text: this

text: holds

text: for

text: any

text: t

text: and

text: µ

text: t

text: t

text: θ

text: Θ,

text: we

text: establish

text: the

text: ﬁrst

text: part

text: of

text: E

text: We

text: will

text: now

text: check

text: that,

text: for

text: all

text: α

text: θ

text: (Cθ

text: Y

text: )α

text: Using

text: and

text: we

text: µ

text: t

text: get

text: Eθ

text: wθ

text: Yt

text: µ

text: t

text: (pθ

text: (Y

text: ))α

text: µ,t

text: α

text: wθ

text: y

text: α∞

text: gθ(x

text: y

text: )dx

text: (pθ

text: (y

text: αdy

text: t

text: µ,t

text: DαDα

text: y

text: y

text: α(pθ

text: (y

text: αdy

text: t

text: µ,t

text: We

text: use

text: again

text: Lemma

text: with

text: ψ(y

text: y

text: y

text: y

text: and

text: y

text: γ

text: y

text: γ

text: ϕ(y

text: y

text: ∨0

text: with

text: α)

text: and

text: γ

text: Note

text: ﬁrst

text: that

text: ψ(y

text: y

text: )ϕ(y

text: y

text: )dy

text: y

text: y

text: dy

text: is

text: satisﬁed.

text: We

text: ﬁnally

text: check

text: Using

text: the

text: conditional

text: independence

text: of

text: the

text: observations

text: given

text: the

text: states

text: and

text: ϕ−1−αα

text: E

text: E

text: 1−αα

text: E

text: 1−αα

text: E

text: θ

text: θ

text: θ

text: θ

text: eδ

text: Xt

text: eδ

text: µ

text: Xt

text: U

text: γ

text: U

text: γ

text: µ

text: Using

text: we

text: get,

text: from

text: the

text: Cauchy-Schwarz

text: inequality,

text: E

text: E

text: E

text: θ

text: eδ

text: Xt

text: eδ

text: θ

text: Xt

text: θ

text: µ

text: µ

text: µ

text: Applying

text: with

text: δ

text: replaced

text: by

text: 2δ

text: yields

text: Using

text: Lemma

text: thus

text: establishes

text: and

text: thereby,

text: holds.

text: Provided

text: that

text: θ

text: converges

text: to

text: θ

text: at

text: a

text: rate

text: T

text: (see

text: Remark

text: we

text: may

text: therefore

text: apply

text: T

text: Theorem

text: which

text: shows

text: that

text: for

text: any

text: γ

text: (θ

text: is

text: tight

text: with

text: N

text: T,NT

text: T

text: T

text: T

text: Proof

text: of

text: Theorem

text: We

text: will

text: now

text: turn

text: to

text: the

text: proof

text: of

text: the

text: minorization

text: condition

text: in

text: Theorem

text: As

text: in

text: the

text: statement

text: of

text: the

text: theorem,

text: we

text: will

text: not

text: explicitly

text: indicate

text: any

text: possible

text: dependence

text: on

text: unknown

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##