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Non-linear dynamics Yannis PAPAPHILIPPOU

CERN

United States Particle Accelerator School,University of California - Santa-Cruz, Santa Rosa, CA

14th – 18th January 2008

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Summary

Driven oscillators and resonance condition Field imperfections and normalized field errors Perturbation treatment for a sextupole Poincaré section Chaotic motion Singe-particle diffusion

Dynamics aperture Frequency maps

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Damped and driven harmonic oscillator Damped harmonic oscillator:

Q is the damping coefficient (amplitude decreases withtime)

ω0 is the Eigenfrequency of the harmonic oscillator An external force can pump energy into the system

General solution with

ω the frequency of the driven oscillation Amplitude U(ω) can become large for certain

frequencies

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Resonance effect

excitation by strong wind on theeigenfrequencies

Tacoma Narrow bridge 1940

0!! =

U(ω)

ωω

α(ω)

ω0ω0

π/2

Q>1/2

Q<1/2

Q>1/2

Q<1/2

• Without or with weak damping aresonance condition occurs for• Infamous example:

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Perturbation in Hills equations Hill’s equations with driven harmonic force

where the F is the Lorentz force from perturbing fields Linear magnet imperfections: derivation from the design dipole

and quadrupole fields due to powering and alignment errors Time varying fields: feedback systems (damper) and wake fields

due to collective effects (wall currents) Non-linear magnets: sextupole magnets for chromaticity correction

and octupole magnets for Landau damping Beam-beam interactions: strongly non-linear field Space charge effects: very important for high intensity beams non-linear magnetic field imperfections: particularly difficult to

control for super conducting magnets where the field quality isentirely determined by the coil winding accuracy

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Localized Perturbation

0

1{)( 0 =! ssL"

for ‘s’ = s0

otherwiseand 1)( 0 =!" dsss

L#

•Periodic delta function

•Equation of motion for a single perturbation in the storage ring

•Expanding in Fourier series the delta function

•Infinite number of driving frequencies!!!•Recall that the driving force can be the result of any multi-pole

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•Equations of motion (u = x or y) including all multi-pole errors

•Solved with perturbation theory approachwith

•At first order

Resonance condition

Resonance conditions and tune diagram

There are resonance lines everywhere !!!

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•Regions with fewresonances:

•Avoid low orderresonances•< 12th order for a protonbeam without damping•< 3rd 5th order forelectron beams withdamping•Close to couplingresonances: regionswithout low orderresonances but relativelysmall!

Choice of the working point

Qy

Qx

7th11th4th & 8th9th

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Single Sextupole Perturbation•Consider a thin sextupole perturbation•Equations of motion

•With•The equation is written

•Resonance conditions integer third integer

•No exact solution•Need numerical tools to integrate equations of motion

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•Record the particle coordinates at onelocation (BPM)•Unperturbed motion lies on a circle (simplerotation)•Resonance is described by fixed points•For a sextupole

•The particle does not lie on a circle!•The change of tune per turn is

Poincaré Section

x

0/!x"

R!

Poincaré Section:

y

x

s

0/ !x"

x

0/!x"

turn!"

3

12x

0/!x"

R+ΔR

02 Q!

x

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x

0/!x"

3

2

1

Rfp

Q < r/3

Topology of a sextupole resonance•Small amplitude, regular motion•Large amplitude, instability, chaoticmotion and particle loss•Separatrix: barrier between stableand unstable motion (location ofunstable fixed points)

x

x’

x

Octupole

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Sextupole effects

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Optimization of Dynamic aperture Keep chromaticity sextupole strength low Include sextupoles in quadrupoles for more flexibility Try an interleaved sextupole scheme (-I transformer) Choose working point far from systematic resonancesIterate between linear and non-linear lattice

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Frequency Map analysis Oscillating electrons in storage ring generally obey “quasi-harmonic”motion close to the origin for a “good working point” Large amplitudes sample more non-linear fields and motion becomeschaotic - i.e., the frequency of oscillation (tune) changes with turn number. Motion close to a resonance also exhibits diffusion Frequency map analysis examines dynamics in frequency space ratherthan configuration space. Regular or quasi-regular periodic motion is associated to unique tunevalues in frequency space Irregular motion exhibits diffusion in frequency space, i.e. tunes change The mapping of configuration space (x and y) to frequency space (Qx andQy) is regular for regular motion and irregular for chaotic motion. Numerically integrate the equations of motion for a set of initialconditions (x, y, x’,y’) and compute the frequencies as a function of time Small amplitude, regular motion Large amplitude, chaotic motion and particle loss

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NAFF algorithmQuasi-periodic approximation through NAFFalgorithm

of a complex phase space function

for each degree of freedom with

defined over

Advantages of NAFF:

with high precision

b) Determination of frequency vector

a) Very accurate representation of the “signal” (if quasi-periodic)and thus of the amplitudes

for Hanning Filter

and

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Aspects of frequency map analysis• Construction of frequency map

and construction of diffusion map

• Determination of tune diffusion vector

LHC Simulations ALS MeasurementsPapaphilippou PAC99 Robin et al. PRL2000

• Determination of resonance drivingterms associated with amplitudes Bengtsson PhD thesis CERN88-05

LHC SimulationsPapaphilippou PAC99

SPS Measurements Bartolini et al. PAC99

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Building the frequency map Choose coordinates (xi, yi) with px and py=0 Numerically integrate the phase trajectories through the lattice for sufficientnumber of turns Compute through NAFF Qx and Qy after sufficient number of turnsPlot them in the tune diagram

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Frequency Map for the ESRFAll dynamics represented in thesetwo plots Regular motion represented byblue colors (close to zero amplitudeparticles or working point)

Resonances appear as distortedlines in frequency space (or curvesin initial condition space Chaotic motion is represented byred scattered particles and definesdynamic aperture of the machine

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ReferencesO. Bruning, Non-linear dynamics, JUAS

courses, 2006.