massimo giovannozzihalo '03, may 21 20031 dynamic aperture for single-particle motion: overview...

Massimo GiovanMassimo Giovannozzinozzi

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Dynamic Aperture for Single-Dynamic Aperture for Single-ParticleParticle Motion: Overview of Motion: Overview of

TheoreticalTheoretical Background, Numerical Background, Numerical

Predictions and Experimental Predictions and Experimental ResultsResults

M. GiovannozziM. Giovannozzi

CERN AB/ABPCERN AB/ABP

Summary:Summary:

DefinitionDefinition

Dynamic aperture computationDynamic aperture computation

Time-dependent effectsTime-dependent effects

Nonlinear dynamics Nonlinear dynamics experimentsexperiments

Dynamic aperture Dynamic aperture experiments: experiments: FNALFNAL - E778, - E778, DESYDESY - HERA-p, - HERA-p, CERNCERN – SPS – SPS

ConclusionsConclusions


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DefinitioDefinitionn

A reasonable definition is the followingA reasonable definition is the following

Dynamic Aperture (DA) is the volume in Dynamic Aperture (DA) is the volume in phase space of the initial conditions that phase space of the initial conditions that are stable for a given number are stable for a given number NN of turns of turns in the accelerator.in the accelerator.

From a physical point of view N is From a physical point of view N is dictated by the specific problem, i.e. dictated by the specific problem, i.e. injection process duration, storage time injection process duration, storage time etc.etc.


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DA computation DA computation 1/21/2

The DA computation can be performed by:The DA computation can be performed by:

Direct computationDirect computation: the definition is : the definition is applied, i.e. the evolution of a set of applied, i.e. the evolution of a set of initial conditions is computed and the initial conditions is computed and the stable ones are kept for further analysis.stable ones are kept for further analysis.

This approach is very This approach is very CPU-time CPU-time consumingconsuming. .

It is affordable for It is affordable for short-short- medium-termmedium-term DA computation, i.e. N not exceeding DA computation, i.e. N not exceeding 101044-10-1055 (depending on the model). (depending on the model).

Important remark: Important remark: the computation can the computation can be parallelised!!!be parallelised!!!


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DA computation DA computation 2/22/2

Indirect computationIndirect computation: the problem : the problem consists in looking for dynamics consists in looking for dynamics observables well-correlated with observables well-correlated with stability of initial conditions (stability of initial conditions (early early indicatorsindicators). ).

The computational effort should be The computational effort should be limitedlimited..

This approach is aimed particularly at This approach is aimed particularly at long-termlong-term DA computation, i.e. N DA computation, i.e. N exceeding exceeding 101066..

An example of a An example of a semi-analyticalsemi-analytical method method for 2D maps will be given.for 2D maps will be given.


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Direct DA computation Direct DA computation 1/51/5

According to the definition of DA in According to the definition of DA in terms of volume in phase space, the terms of volume in phase space, the following integral have to be computed:following integral have to be computed:

yxyx dpdydpdxpypx ),,,(The function The function equals 1 if the initial equals 1 if the initial condition (x, pcondition (x, pxx, y, p, y, pyy) is stable for N ) is stable for N turns.turns.


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If one neglects the If one neglects the disconnecteddisconnected part of part of the volume, then polar coordinates can the volume, then polar coordinates can be usedbe used

In practice the integral is approximated In practice the integral is approximated by a sum…by a sum…

21

42

0

2

0

2/

021,, )2sin(),,(

81

21

dddrA

4/1

2,,

,,21

21

2

A

rDA as radius DA as radius of of equivalent equivalent hyperspherhyperspheree

Special Special weight!weight!


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This approach allows determining the error due This approach allows determining the error due to the discretisation, i.e.to the discretisation, i.e.

Discretisation in Discretisation in -> error -> error 1/K 1/K

Discretisation in Discretisation in 11, , 22 -> error -> error 1/L 1/L

Discretisation in Discretisation in r -> error r -> error 1/J 1/J

The step is optimised by imposing comparable The step is optimised by imposing comparable errors on different variables. To have a relative errors on different variables. To have a relative error on error on DADA of of 1/(4J) -> J1/(4J) -> J44, implying that , implying that N·JN·J44 iterates are needed! iterates are needed!

4/1

1 1 1

4212,,

1 22121

)2sin(),,(2

K

k

L

l

L

lkllkr

LKr


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This approach is not the unique solution to This approach is not the unique solution to the problem of computing the volume the problem of computing the volume integral:integral:

The scan in The scan in 11, , 22 (and the averaging step) is (and the averaging step) is replaced by a replaced by a time averagetime average. The underlying . The underlying assumption is that the motion is assumption is that the motion is ergodic (it ergodic (it might fail near low-order resonances)might fail near low-order resonances). The . The power power 44 in the scaling law for the error is in the scaling law for the error is replaced by a replaced by a 22!!

Normal formsNormal forms are used to compute the are used to compute the nonlinear invariant. This method has nonlinear invariant. This method has samesame behaviour (behaviour (errorerror) as previous one. ) as previous one. Precautions should be taken for Precautions should be taken for convergenceconvergence issues issues ((near low-order near low-order resonancesresonances).).


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Tracking examples: Tracking examples: regular casesregular cases

SPS model SPS model with strong with strong sextupoles. sextupoles. xx=26.637, =26.637, yy=26.533.=26.533.

N=1000N=1000

xx=26.605, =26.605, yy=26.538.=26.538.

N=1000N=1000

Henon map with Henon map with octupoles and octupoles and xx=0.28=0.28, , yy=0.31=0.31. Initial . Initial coordinates (x,0,y,0). coordinates (x,0,y,0). N=1000N=1000


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Tracking examples: Tracking examples: pathological casespathological cases

Henon map with Henon map with octupoles and octupoles and

xx=0.25=0.25, , yy=0.61803=0.61803. . Initial coordinates Initial coordinates (x,0,y,0). N=1000(x,0,y,0). N=1000

Henon map with Henon map with octupoles and octupoles and

xx=0.25=0.25, , yy=0.61803=0.61803. . Initial coordinates Initial coordinates (x,0,y,0). N=50000(x,0,y,0). N=50000


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HowHow do we perform the tracking of initial do we perform the tracking of initial conditions?conditions?

The The HamiltonianHamiltonian character of the dynamics (for character of the dynamics (for protons) makes it necessary to preserve protons) makes it necessary to preserve symplecticitysymplecticity..

Two approaches can be outlined:Two approaches can be outlined:

Element-by-element trackingElement-by-element tracking: the : the thickthick nonlinear nonlinear elements are replaced with elements are replaced with thinthin lens kicks. The lens kicks. The tracking is tracking is exactly symplecticexactly symplectic, but the solution is an , but the solution is an approximate one. However, approximate one. However, errorerror can be can be controlledcontrolled exactly and exactly and high-orderhigh-order symplectic integrators are symplectic integrators are available.available.

Map trackingMap tracking: the polynomial map obtained by : the polynomial map obtained by composition of the elements’ map cannot be used composition of the elements’ map cannot be used directly (due to truncation errors). Hence, methods directly (due to truncation errors). Hence, methods have been proposed to restore symplecticity, have been proposed to restore symplecticity, however, it is not clear whether however, it is not clear whether new physics is new physics is addedadded……


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Semi-analytical Semi-analytical computation of DA (2D) computation of DA (2D) 1/41/4

A semi-analytical method A semi-analytical method is based on is based on invariant invariant manifoldsmanifolds..

From each hyperbolic From each hyperbolic fixed point emanates fixed point emanates invariant manifolds invariant manifolds tangent to the tangent to the eigenvectors of the eigenvectors of the linearised map.linearised map.

These manifolds can be These manifolds can be constructed numerically constructed numerically by iterating initial by iterating initial conditions on a small conditions on a small segment along the segment along the eigenvectors.eigenvectors.

Blue arrows: Blue arrows: eigenvectorseigenvectors


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Invariant manifolds emanating from different fixed Invariant manifolds emanating from different fixed points may intersect creating a complex structure points may intersect creating a complex structure ((homoclinic/heteroclinic tanglehomoclinic/heteroclinic tangle).).

It has been shown that DA can be accurately It has been shown that DA can be accurately computed using the following approach:computed using the following approach:

Find the Find the lowest periodlowest period ( (oneone or or twotwo) hyperbolic fixed ) hyperbolic fixed point.point.

Determine Determine eigenvectorseigenvectors of the linearised map. of the linearised map.

ConstructConstruct invariantinvariant manifoldsmanifolds emanating from that emanating from that fixed point.fixed point.

Compute the Compute the minimum distanceminimum distance from the origin: this from the origin: this will give the radius of the connected part of DA.will give the radius of the connected part of DA.

Essentially, intersections between the invariant Essentially, intersections between the invariant manifolds from different unstable fixed points, will manifolds from different unstable fixed points, will make the complex structure approaching the border make the complex structure approaching the border of stability from outside.of stability from outside.


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Semi-analytical Semi-analytical computation of DA (2D) computation of DA (2D) 3/43/4 Invariant manifolds form the Invariant manifolds form the

hyperbolic points of period one hyperbolic points of period one for the cubic map (for the cubic map (xx=0.34). The =0.34). The stability domain is also shown.stability domain is also shown.

Global behaviour of the Global behaviour of the dynamic aperture for the dynamic aperture for the cubic polynomial mapcubic polynomial map


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Orbits of a Orbits of a nonlinear nonlinear polynomial polynomial mapmap

Invariant Invariant manifoldsmanifolds

Stability Stability domaindomain


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Early indicators 1/4Early indicators 1/4

The goal is to find The goal is to find dynamical observablesdynamical observables, , to be computed easily, over a to be computed easily, over a limited limited number of turnsnumber of turns, showing a , showing a goodgood correlationcorrelation with with long-termlong-term stability! stability!

Outstanding issuesOutstanding issuesIntermittencyIntermittency: particles showing a stable : particles showing a stable behaviour over behaviour over 101033-10-1044 turns, suddenly turns, suddenly escape to infinity. No remedy seems to be escape to infinity. No remedy seems to be available to deal with such a situation.available to deal with such a situation.

StableStable chaoschaos: initial conditions showing a : initial conditions showing a highlyhighly chaoticchaotic dynamics are dynamics are not lostnot lost over a over a large numberlarge number of turns. No result is of turns. No result is available to prove that chaotic particles will available to prove that chaotic particles will be lost anyway…be lost anyway…


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Bounds on invariantsBounds on invariants

The approach is based on an accurate The approach is based on an accurate computation of the invariants.computation of the invariants.

Based on this, the maximum variation Based on this, the maximum variation of the invariant of the invariant JJ in the accessible in the accessible region of phase space is computed region of phase space is computed (Monte Carlo) for a given number of (Monte Carlo) for a given number of turns turns nn00..

Stability times are derived via (Stability times are derived via (J is a J is a parameter)parameter)

0nJJ

N


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Lyapunov exponentLyapunov exponent

A threshold is defined using numerical A threshold is defined using numerical simulations of a simple model (Henon simulations of a simple model (Henon map)map)

Particles satisfyingParticles satisfying

are are unstableunstable

1

|ˆ|log

1)(

NN xxN

N

15.0log1

)( ANAN

N

)()( NN


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Tune variationTune variation

A threshold is defined using numerical A threshold is defined using numerical simulations of a simple model (Henon simulations of a simple model (Henon map)map)

Particles satisfyingParticles satisfying

are are unstableunstable

2,

:12/2/:121

)(

yxi

ii NNNN

2.0)(

ANA

N

)()( NN


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Inverse Logarithm Interpolation Inverse Logarithm Interpolation 1/41/4

Results of numerical Results of numerical evaluation of DA can evaluation of DA can be presented in be presented in different forms.different forms.

It is customary to use It is customary to use the so-called the so-called survival survival plotsplots: DA is plotted : DA is plotted against the number of against the number of turns.turns.

If DA is computed If DA is computed using too few angles, using too few angles, and/or without and/or without applying the proposed applying the proposed averaging, the survival averaging, the survival plots do not convey all plots do not convey all the possible the possible information.information.


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If the DA, If the DA, computed using computed using the approach the approach presented before, presented before, is plotted against is plotted against the number of the number of turns a smooth turns a smooth curve appears!curve appears!

It turns out the it It turns out the it can be fitted by can be fitted by the following law:the following law:

N

bDND

log1)(


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Some comments:Some comments:

The scaling law depends on The scaling law depends on twotwo parametersparameters only!only!

They have a They have a clearclear physicalphysical meaningmeaning::

DD: it represent the DA at “: it represent the DA at “infinite timeinfinite time”. It ”. It is the is the limiting valuelimiting value of DA. of DA.

bb: gives the : gives the relative importancerelative importance between between sort-termsort-term and and long-termlong-term DA. DA.

The scaling law has been tested also for The scaling law has been tested also for 6D6D motion in realistic models (LHC lattice).motion in realistic models (LHC lattice).

N

bDND

log1)(


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There is an interesting connection with fundamental There is an interesting connection with fundamental theorems such as theorems such as KAMKAM and and NekhoroshevNekhoroshev..

By inverting the relation stating the dependence of the By inverting the relation stating the dependence of the stability time on the amplitude (stability time on the amplitude (NekhoroshevNekhoroshev) one ) one obtainsobtains

The The KAMKAM theorem is applied to support the statement theorem is applied to support the statement that that diffusiondiffusion is practically is practically negligiblenegligible, at least , at least not too not too far from the originfar from the origin, thus ensuring the existence of a , thus ensuring the existence of a stable region of radius stable region of radius rr, then particles beyond , then particles beyond rr

escape according toescape according to

0/1 /log)(

NN

RNr A

0/1 /log)(

NN

RrNr A


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Comparison of DA Comparison of DA calculationscalculations

ExtrapolationExtrapolationParticleParticle

LossLossLyapunLyapun

ovovTuneTune

-19 %-19 %-5 %-5 %0 %0 %101055

-17 %-17 %-4 %-4 %-6 %-6 %-15 %-15 %6 %6 %81928192

-21 %-21 %-6 %-6 %-4 %-4 %-13 %-13 %11 %11 %20482048

-6 %-6 %3 %3 %0 %0 %-12 %-12 %17 %17 %512512

-4 %-4 %4 %4 %24 %24 %128128

101077

NN

LH

C 4

DLH

C 4

D

-31 %-31 %-7 %-7 %0 %0 %101055

-21 %-21 %-2 %-2 %-7 %-7 %-6 %-6 %13 %13 %81928192

-21 %-21 %-2 %-2 %-4 %-4 %0 %0 %21 %21 %20482048

-20 %-20 %-1 %-1 %9 %9 %29 %29 %512512

12 %12 %39 %39 %128128

LH

C 6

DLH

C 6

D


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Time-dependent effects Time-dependent effects 1/41/4

Real machines are plagued by Real machines are plagued by time-time-dependentdependent effects, e.g. effects, e.g. rippleripple in the in the power supplies, power supplies, persistentpersistent currentscurrents (superconducting machines).(superconducting machines).

This generates This generates tunetune modulationmodulation. .

Tune can be modulated also by Tune can be modulated also by synchro-betatronsynchro-betatron coupling. coupling.

A complex phenomenology arises: A complex phenomenology arises: phase space topology depends on phase space topology depends on mutualmutual interactioninteraction of parameters, i.e. of parameters, i.e. linear tunes, amplitude and frequency linear tunes, amplitude and frequency of tune modulation.of tune modulation.


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In general the In general the performanceperformance of of earlyearly indicatorsindicators get worse in presence of get worse in presence of tune modulation.tune modulation.

Lyapunov Lyapunov exponent provides very exponent provides very pessimistic estimates of pessimistic estimates of DADA due to the due to the large scalelarge scale chaotic motion. chaotic motion.

How does the inverse logarithm How does the inverse logarithm interpolation behave in presence of interpolation behave in presence of tune modulation?tune modulation?


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The following form of the interpolating The following form of the interpolating function was chosenfunction was chosen

Strictly speaking, in presence of tune Strictly speaking, in presence of tune modulation Nekhoroshev theorem does not modulation Nekhoroshev theorem does not hold (hold (modulation is not a weak perturbationmodulation is not a weak perturbation).).

From a phenomenological point of view tune From a phenomenological point of view tune modulation induces strong long-term effects.modulation induces strong long-term effects.

A posterioriA posteriori: the : the fit quality is always very fit quality is always very goodgood!!

In presence of strong modulation In presence of strong modulation or Aor A becomes negativebecomes negative: the entire phase space is : the entire phase space is unstable.unstable.

NB

AND log)(


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DA DA simulations simulations for Henon for Henon modulated modulated mapmap


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Nonlinear dynamics Nonlinear dynamics experimentsexperiments

On the experimental front, On the experimental front, considerable efforts were devoted to considerable efforts were devoted to test some outstanding issues related to test some outstanding issues related to nonlinear beam dynamics.nonlinear beam dynamics.

These measurements were aimed at These measurements were aimed at determining quantities such as determining quantities such as detuningdetuning with amplitude, with amplitude, smearsmear, , islands’islands’ parameters (size, position). parameters (size, position).

In some cases In some cases DADA measurements were measurements were attempted, including its parametric attempted, including its parametric dependence (on ripple, for instance).dependence (on ripple, for instance).


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Phase space reconstruction Phase space reconstruction 1/21/2

Extraction Extraction lineline

Slow Slow bumpbump

Extraction Extraction septumseptum

Sextupole Sextupole magnetsmagnets

Octupole Octupole magnetsmagnets

Kicker Kicker magnet magnet

Phase space Phase space measuremenmeasuremen

ttUsed to Used to generate generate

stable stable islandsislands


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Phase space reconstruction Phase space reconstruction 2/22/2

Regular Regular motion motion near the near the origin of origin of phase phase spacespace

Motion Motion inside inside islandsislands

Motion Motion beyond beyond islands’ islands’ separatriseparatrixx

Motion Motion inside inside islandsislands

Measurement performed at the CERN PS in Measurement performed at the CERN PS in 20022002


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Nonlinear dynamics Nonlinear dynamics experiments: experiments: FNAL E778FNAL E778

Machine conditions:Machine conditions:Tevatron at injection.Tevatron at injection.

Dedicated Dedicated sextupolessextupoles are used to create are used to create nonlinear effects.nonlinear effects.

The tune is near The tune is near xx=2/5=2/5..

Measurements:Measurements:SmearSmear..

DetuningDetuning with amplitude. with amplitude.

EvolutionEvolution of beam of beam profilesprofiles..

Islands properties (including trapping efficiency Islands properties (including trapping efficiency vs. tune modulation parameters).vs. tune modulation parameters).

DADA: reasonable agreement with tracking (: reasonable agreement with tracking (20 %20 %).).


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Nonlinear dynamics Nonlinear dynamics experiments: experiments: DESY HERA-pDESY HERA-p

Machine conditions:Machine conditions:HERA-p at injection (after decay of HERA-p at injection (after decay of persistent currents).persistent currents).

Nonlinear effects are the natural ones due Nonlinear effects are the natural ones due to superconducting magnets.to superconducting magnets.

Measurements:Measurements:DetuningDetuning with amplitude (used to cross- with amplitude (used to cross-check the model).check the model).

Tune ripple Tune ripple (including the possibility of (including the possibility of correcting the ripple).correcting the ripple).

DADA: it was measured using two different : it was measured using two different methods.methods.


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Nonlinear dynamics Nonlinear dynamics experiments: experiments: DESY HERA-pDESY HERA-p

Beam profiles: Beam profiles: 1)1) the beam is kicked; the beam is kicked; 2)2) the the width of the beam profile is measured after width of the beam profile is measured after some time.some time.

Beam losses: Beam losses: 1)1) the beam is scraped; the beam is scraped; 2)2) scrapers scrapers are retracted; are retracted; 3)3) the beam is kicked to sample a the beam is kicked to sample a given amplitude.given amplitude.

Observations:Observations:The control of experimental conditions in the The control of experimental conditions in the verticalvertical planeplane is rather is rather difficultdifficult..

At At injectioninjection the influence of the influence of rippleripple is is negligible.negligible.

At At top-energytop-energy rippleripple effects are effects are relevantrelevant..

ScraperScraper measurements measurements agreeagree with a with a diffusiondiffusion model at model at top-energytop-energy, but , but notnot at at injection.injection.

DADA: a : a 20 %20 % agreement with tracking was agreement with tracking was achieved.achieved.


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Nonlinear dynamics Nonlinear dynamics experiments: experiments: CERN SPSCERN SPS

Machine conditions:Machine conditions:SPS at 120 GeV (the machine is very linear).SPS at 120 GeV (the machine is very linear).

Eight sextupoles are powered to excite Eight sextupoles are powered to excite nonlinear resonances (but not the third-order).nonlinear resonances (but not the third-order).

Octupoles are used to reduce the detuning (by Octupoles are used to reduce the detuning (by product of these measurements: the correction product of these measurements: the correction of the tuneshift is beneficial for the dynamics).of the tuneshift is beneficial for the dynamics).

Measurements:Measurements:DetuningDetuning with amplitude (used to cross-check with amplitude (used to cross-check the model).the model).

Tune ripple.Tune ripple.

DADA: it was measured using : it was measured using beam losses.beam losses.


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Nonlinear dynamics Nonlinear dynamics experiments: experiments: CERN SPSCERN SPS

Observations:Observations:The lack of a real pencil beam is believed to The lack of a real pencil beam is believed to be the major obstacle (and limiting factor) be the major obstacle (and limiting factor) for this measurement.for this measurement.

No dependence of DA on ripple (No dependence of DA on ripple (artificially artificially introducedintroduced) frequency emerged from the ) frequency emerged from the measurements.measurements.

The presence of a second frequency has a The presence of a second frequency has a negative impact on the beam stability.negative impact on the beam stability.

DADA: a : a 20 %20 % agreement with tracking was agreement with tracking was achieved.achieved.


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Conclusions Conclusions 1/21/2

Although a definitive solution to the Although a definitive solution to the problem of computing efficiently and problem of computing efficiently and correctly DA is not at hand, yet, a number correctly DA is not at hand, yet, a number of results have been obtained:of results have been obtained:

VariousVarious approaches have been proposed to approaches have been proposed to compute the connected volume of stable compute the connected volume of stable initial conditions in phase space (initial conditions in phase space (DADA).).

The The dependencedependence of the of the errorerror on on computedcomputed DA is DA is knownknown, thus allowing for an , thus allowing for an optimaloptimal choicechoice of the of the grid stepgrid step..

Early indicators have been defined, and Early indicators have been defined, and their properties studied in details.their properties studied in details.

An An interpolationinterpolation lawlaw has been worked out: has been worked out: it allows a reliable it allows a reliable extrapolationextrapolation of tracking of tracking results.results.


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Conclusions Conclusions 2/22/2

• Nonlinear dynamics experiments have been Nonlinear dynamics experiments have been performed at performed at FNALFNAL ( (E778E778), ), DESYDESY ( (HERA-pHERA-p), ), CERNCERN ( (SPSSPS). ). • A A goodgood agreement with tracking results was agreement with tracking results was

found for found for detuningdetuning with with amplitudeamplitude, , smearsmear, and , and islands propertiesislands properties..

Whenever the experimental conditions are well Whenever the experimental conditions are well under control, the agreement between under control, the agreement between DADA measurements and tracking is of the order of measurements and tracking is of the order of 20 %.20 %.


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AcknowledgemeAcknowledgementsnts

I am particularly in debt with I am particularly in debt with G. TurchettiG. Turchetti who introduced who introduced me in this field. He was the first to point out the deep me in this field. He was the first to point out the deep implications of the logarithm interpolation.implications of the logarithm interpolation.

Most of the results on computation of DA represent a Most of the results on computation of DA represent a joint effort with joint effort with E. TodescoE. Todesco and and W. ScandaleW. Scandale..

W. FischerW. Fischer and and F. SchmidtF. Schmidt are gratefully acknowledged are gratefully acknowledged for the collaboration during the SPS experiment for the collaboration during the SPS experiment (tracking studies, long night-shifts in control room).(tracking studies, long night-shifts in control room).

F. ZimmermannF. Zimmermann pointed out a number of references pointed out a number of references concerning nonlinear dynamics, and DA measurements.concerning nonlinear dynamics, and DA measurements.

The colleagues of the Accelerator Physics group at CERN The colleagues of the Accelerator Physics group at CERN for discussions.for discussions.


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Tracking examples: Tracking examples: regular casesregular cases

Henon map with octupoles and Henon map with octupoles and xx=0.28=0.28, , yy=0.31=0.31. Initial . Initial coordinates (x,0,y,0). N=10coordinates (x,0,y,0). N=1077


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Early indicators 3/5: Early indicators 3/5: HenonHenon

Particles lost Particles lost before before 101077 turns are turns are marked in marked in black black (Henon (Henon map).map).


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Early indicators 3/5: Early indicators 3/5: LHCLHC

Particles lost Particles lost before before 101077 turns are turns are marked in marked in black black (LHC (LHC model).model).


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Harmonic analysis of Harmonic analysis of time-seriestime-series


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Early indicators 4/5: Early indicators 4/5: HenonHenon

Particles lost Particles lost before before 101077 turns are turns are marked in marked in black black (Henon (Henon map).map).


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Early indicators 4/5: Early indicators 4/5: LHCLHC

Particles lost Particles lost before before 101077 turns are turns are marked in marked in black black (LHC (LHC model).model).


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Semi-analytical Semi-analytical computation of DA (2D) computation of DA (2D) 1/31/3 A polynomial A polynomial

map of order map of order three is used.three is used.

Dynamic Dynamic aperture aperture (radius of the (radius of the stability stability domain) vs. domain) vs. linear tune linear tune (solid line).(solid line).

Minimum Minimum distance of the distance of the inner envelope inner envelope of the of the homoclinic homoclinic tangle (dots).tangle (dots).


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Orbits of a Orbits of a polynomial polynomial map of order map of order 6.6.

Island chain Island chain of period 7 is of period 7 is clearly clearly visible.visible.

A thick A thick stochastic stochastic layer is also layer is also visible.visible.


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Detail of one Detail of one islands of period islands of period 7.7.

Higher-period Higher-period islands are islands are visible inside.visible inside.

The homoclinic The homoclinic tangle is clearly tangle is clearly visible.visible.


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Stability domain for Stability domain for the polynomial map the polynomial map of order 6. It is of order 6. It is obtained by direct obtained by direct tracking.tracking.

The invariant The invariant manifold emanating manifold emanating for the hyperbolic for the hyperbolic fixed point of period fixed point of period one is shown.one is shown.


HALO '03, May 21 2003HALO '03, May 21 2003 5050


Distribution Distribution of Lyapunov of Lyapunov exponent for exponent for modulated modulated Henon map. Henon map. Particles lost Particles lost before before 101077 turns are turns are marked in marked in blackblack


HALO '03, May 21 2003HALO '03, May 21 2003 5151


Simulation results for Simulation results for modulated Henon modulated Henon map:map:

xx=0.168, =0.168, yy=0.201=0.201

Modulation Modulation amplitude=amplitude=11

= 1.2= 1.2+0.5+0.5-0.5-0.5

A= 0.40A= 0.40+0.04+0.04-0.09-0.09

B= 0.6B= 0.6+0.2+0.2-0.1-0.1

Small stars represent Small stars represent Lyapunov estimateLyapunov estimate


HALO '03, May 21 2003HALO '03, May 21 2003 5252



xx=0.168, =0.168, yy=0.201=0.201


= 0.6= 0.6+0.5+0.5-0.4-0.4

A= 0.24A= 0.24+0.13+0.13-0.56-0.56

B= 0.6B= 0.6+0.5+0.5-0.0-0.0



HALO '03, May 21 2003HALO '03, May 21 2003 5353



xx=0.168, =0.168, yy=0.201=0.201


= 0.1= 0.1+0.4+0.4-0.5-0.5

A=A=-1.5-1.5

B= 2.3B= 2.3



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xx=0.168, =0.168, yy=0.201=0.201


==-0.5-0.5+0.4+0.4-0.3-0.3

A= 1.0A= 1.0+2.0+2.0-0.2-0.2

B=B=-0.3-0.3+0.2+0.2-2.0-2.0


massimo giovannozzihalo '03, may 21 20031 dynamic aperture for single-particle motion: overview...

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