uspas accelerator physics 2019 northern illinois

1 T. Satogata / January 2019 USPAS Accelerator Physics

USPAS Accelerator Physics 2019 Northern Illinois University and UT-Batelle

16: Routes to Chaos

(perhaps yet another self-referential lecture)

Todd Satogata (Jefferson Lab and ODU) / [email protected] Steve Peggs (BNL) / [email protected]

Bhawin Dhital (ODU) / [email protected] and Kiel Hock (BNL) / [email protected] http://www.toddsatogata.net/2019-USPAS

Happy birthday to Irving Langmuir (Nobel 1932) and Rudolf Mössbauer (Nobel 1961)!

Happy National Hot Chocolate Day and Hell Is Freezing Over Day!


Review: 1D Beam-Beam (Steve, yesterday)

Regular non-resonant

Regular resonant Stochastic chaos

Rapidly divergent (multipoles)

Scaled betatron amplitude x/σ

Focu

sing

kic

k Δ

x’

0 6

c.f. diverging multipole

Very nonlinear

D. Hofstadter: “... an eerie type of chaos ... just behind a facade of order – and yet, deep inside the chaos lurks an even eerier type of order.”


Review: 1D Beam-Beam

§  1D beam-beam dynamics are surprisingly tractable §  Can predict where isolated resonances are located

•  Lecture yesterday and homework §  Can predict other isolated resonance properties

•  N-turn Hamiltonians give “island tunes”, resonance widths

Tune

Q0+ξ

n/6

Q0

Scaled betatron amplitude x/σ 0 6

Isolated resonance

Can we predict the onset of chaos, even roughly?


16.1: Resonance overlap

§  We have evaluated isolated resonances using the n-turn action-angle Kobayashi Hamiltonian where

§  Here we had assumed that all other resonances either §  have small Vn (so their widths are small enough to ignore) or §  their Hamiltonian terms phase average to near zero over

many ierations of this map (over many turns)

Q0 �p

n⌧ 1

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U’’(J): detuning Vn(J): resonance driving Small-amplitude frequency


16.1: Resonance overlap

§  We have evaluated isolated resonances using the n-turn action-angle Kobayashi Hamiltonian where Q0 �

p

n⌧ 1

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U’’(J): detuning Small-amplitude frequency

Tune

Q0+ξ

n/6

Q0 Scaled betatron amplitude x/σ 0 6

“Isolated” resonances

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m/k


Fig 16.1: Resonance Overlap

§ 

Even-order beam-beam resonances


16.1: Resonance overlap: Chirikov

§  What happens when this assumption breaks down? §  Resonances approach the point of overlapping §  Separatrices are the first to interact

•  But separatrices have infinite period and therefore are infinitely vulnerable to perturbation

§  Chirikov hypothesized that chaos emerges when nearby resonance widths are large enough that they overlap §  This is calculable and verifiable §  “A Universal Instability of Many-Dimensional Oscillator

Systems” – Physics Reports 52 5, May 1979, pp. 263-379. §  5000+ citations: a “famous” paper and surprisingly readable


16.1: Chirikov Overlap and the Standard Map

“Isolated” resonances

Chaotic resonance overlap: separatrix becomes chaotic RF motion with large Qs!


Whoa, Wait a sec: ξ=0.2 could be stable?

§ 

Even-order beam-beam resonances


EIC Beam-Beam Parameters (European Stragegy Submission)


16.2: 6D Motion and Tune Modulation

§  We have been (understandably) rather naive §  This is a perfect 1D uncoupled nonlinear model

§  Reality (aside from noise) §  Dispersion couples longitudinal and transverse motion

•  Almost always have to bend the beam somewhere •  Off-center motion in quadrupoles also gives dipole “feed-down”

§  Coupling couples transverse motion •  Quadrupoles have random rotations relative to design plane

§  Sextupoles are necessary (mostly)

•  Chromaticity correction in large accelerators

§  Any coupling adds different frequencies to our system §  Tune modulation, e.g. Q0 = Q00 + q sin(2⇡Qst)

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0L

(B⇢)x

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16.2: Tune modulation

§  The isolated resonance Kobayashi Hamiltonian was

§  Modulation of the tune looks like a time-dependent driving term §  Poincare and periodicity asides, large-N-turn maps

§  To first order, the phase modulation also appears in the resonance driving term! §  Phase-modulated pendulum: Mathieu equation §  Todd’s dissertation: http://www.toddsatogata.net/Thesis/ §  Sidebands and sideband overlap leading to chaotic motion

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Beam-beam tune modulation diagram

q = 0.001 Qs=0.00515 Q0 = Q00 + q sin(2⇡Qst)<latexit sha1_base64="Zt5hcZzR5DcUbQ0uDf5ZX5U/q0Q=">AAACH3icbVDLSgNBEJz1GeMr6tHLYBAUIcwGwSAIAS8eDRgNZMMyO+mYwdnZdaY3EJbc/Q1/wKv+gTfx6g/4HU5iDmosaCiquunuilIlLTL24c3NLywuLRdWiqtr6xubpa3ta5tkRkBTJCoxrYhbUFJDEyUqaKUGeBwpuInuzsf+zQCMlYm+wmEKnZjfatmTgqOTwtJeI2T0jDbCnLERPaL3wWlgpT6oBql0qqV4GJbKrMImoLPEn5IymeIyLH0G3URkMWgUilvb9lmKnZwblELBqBhkFlIu7vgttB3VPAbbySe/jOi+U7q0lxhXGulE/TmR89jaYRy5zphj3/71xuJ/XjvDXq2TS51mCFp8L+plimJCx8HQrjQgUA0d4cJIdysVfW64QBffry19UAPAUdEF4/+NYZZcVys+q/iN43K9No2oQHbJHjkgPjkhdXJBLkmTCPJAnsgzefEevVfvzXv/bp3zpjM75Be8jy8qyqB4</latexit><latexit sha1_base64="Zt5hcZzR5DcUbQ0uDf5ZX5U/q0Q=">AAACH3icbVDLSgNBEJz1GeMr6tHLYBAUIcwGwSAIAS8eDRgNZMMyO+mYwdnZdaY3EJbc/Q1/wKv+gTfx6g/4HU5iDmosaCiquunuilIlLTL24c3NLywuLRdWiqtr6xubpa3ta5tkRkBTJCoxrYhbUFJDEyUqaKUGeBwpuInuzsf+zQCMlYm+wmEKnZjfatmTgqOTwtJeI2T0jDbCnLERPaL3wWlgpT6oBql0qqV4GJbKrMImoLPEn5IymeIyLH0G3URkMWgUilvb9lmKnZwblELBqBhkFlIu7vgttB3VPAbbySe/jOi+U7q0lxhXGulE/TmR89jaYRy5zphj3/71xuJ/XjvDXq2TS51mCFp8L+plimJCx8HQrjQgUA0d4cJIdysVfW64QBffry19UAPAUdEF4/+NYZZcVys+q/iN43K9No2oQHbJHjkgPjkhdXJBLkmTCPJAnsgzefEevVfvzXv/bp3zpjM75Be8jy8qyqB4</latexit><latexit sha1_base64="Zt5hcZzR5DcUbQ0uDf5ZX5U/q0Q=">AAACH3icbVDLSgNBEJz1GeMr6tHLYBAUIcwGwSAIAS8eDRgNZMMyO+mYwdnZdaY3EJbc/Q1/wKv+gTfx6g/4HU5iDmosaCiquunuilIlLTL24c3NLywuLRdWiqtr6xubpa3ta5tkRkBTJCoxrYhbUFJDEyUqaKUGeBwpuInuzsf+zQCMlYm+wmEKnZjfatmTgqOTwtJeI2T0jDbCnLERPaL3wWlgpT6oBql0qqV4GJbKrMImoLPEn5IymeIyLH0G3URkMWgUilvb9lmKnZwblELBqBhkFlIu7vgttB3VPAbbySe/jOi+U7q0lxhXGulE/TmR89jaYRy5zphj3/71xuJ/XjvDXq2TS51mCFp8L+plimJCx8HQrjQgUA0d4cJIdysVfW64QBffry19UAPAUdEF4/+NYZZcVys+q/iN43K9No2oQHbJHjkgPjkhdXJBLkmTCPJAnsgzefEevVfvzXv/bp3zpjM75Be8jy8qyqB4</latexit><latexit sha1_base64="Zt5hcZzR5DcUbQ0uDf5ZX5U/q0Q=">AAACH3icbVDLSgNBEJz1GeMr6tHLYBAUIcwGwSAIAS8eDRgNZMMyO+mYwdnZdaY3EJbc/Q1/wKv+gTfx6g/4HU5iDmosaCiquunuilIlLTL24c3NLywuLRdWiqtr6xubpa3ta5tkRkBTJCoxrYhbUFJDEyUqaKUGeBwpuInuzsf+zQCMlYm+wmEKnZjfatmTgqOTwtJeI2T0jDbCnLERPaL3wWlgpT6oBql0qqV4GJbKrMImoLPEn5IymeIyLH0G3URkMWgUilvb9lmKnZwblELBqBhkFlIu7vgttB3VPAbbySe/jOi+U7q0lxhXGulE/TmR89jaYRy5zphj3/71xuJ/XjvDXq2TS51mCFp8L+plimJCx8HQrjQgUA0d4cJIdysVfW64QBffry19UAPAUdEF4/+NYZZcVys+q/iN43K9No2oQHbJHjkgPjkhdXJBLkmTCPJAnsgzefEevVfvzXv/bp3zpjM75Be8jy8qyqB4</latexit>


Tune Modulation Frequency Scale

§  Remember, tune modulation is really “just” modulating a pendulum §  We know kicking around a pendulum near its natural

frequency produces excitement §  What is the natural frequency of resonant motion? §  Remember these are topologically equivalent to pendula §  “Island” tune: QI<<Q00

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Slow Tune Modulation: Amplitude Modulation

§  If modulation frequency is much lower than island tune

§  then the modulation really looks like a slow variation of Q0 §  Nearly adiabatic with respect to the resonance “island” motion §  Amplitudes of resonances “breathe” up and down §  Island widths also vary because their amplitudes are changing

§  Conversely, modulation frequency >> island tune...

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Qs ⌧ QI<latexit sha1_base64="ONhgVdv3LIFpvQJV/yWu5W8fKYo=">AAACBXicbVDLTgJBEOzFF+IL9ehlIjHxRHaNiRxJvOgNEgEjbMjs0MCE2dnNzCwJ2XD2B7zqH3gzXv0Of8DvcIA9CFhJJ5Wq7nR3BbHg2rjut5Pb2Nza3snvFvb2Dw6PiscnTR0limGDRSJSjwHVKLjEhuFG4GOskIaBwFYwup35rTEqzSP5YCYx+iEdSN7njBorPdW7uiMEqXfvu8WSW3bnIOvEy0gJMtS6xZ9OL2JJiNIwQbVue25s/JQqw5nAaaGTaIwpG9EBti2VNETtp/OLp+TCKj3Sj5Qtachc/TuR0lDrSRjYzpCaoV71ZuJ/Xjsx/YqfchknBiVbLOongpiIzN4nPa6QGTGxhDLF7a2EDamizNiQlrYMUYzRTAs2GG81hnXSvCp7btmrX5eqlSyiPJzBOVyCBzdQhTuoQQMYSHiBV3hznp1358P5XLTmnGzmFJbgfP0CGqiYuw==</latexit><latexit sha1_base64="ONhgVdv3LIFpvQJV/yWu5W8fKYo=">AAACBXicbVDLTgJBEOzFF+IL9ehlIjHxRHaNiRxJvOgNEgEjbMjs0MCE2dnNzCwJ2XD2B7zqH3gzXv0Of8DvcIA9CFhJJ5Wq7nR3BbHg2rjut5Pb2Nza3snvFvb2Dw6PiscnTR0limGDRSJSjwHVKLjEhuFG4GOskIaBwFYwup35rTEqzSP5YCYx+iEdSN7njBorPdW7uiMEqXfvu8WSW3bnIOvEy0gJMtS6xZ9OL2JJiNIwQbVue25s/JQqw5nAaaGTaIwpG9EBti2VNETtp/OLp+TCKj3Sj5Qtachc/TuR0lDrSRjYzpCaoV71ZuJ/Xjsx/YqfchknBiVbLOongpiIzN4nPa6QGTGxhDLF7a2EDamizNiQlrYMUYzRTAs2GG81hnXSvCp7btmrX5eqlSyiPJzBOVyCBzdQhTuoQQMYSHiBV3hznp1358P5XLTmnGzmFJbgfP0CGqiYuw==</latexit><latexit sha1_base64="ONhgVdv3LIFpvQJV/yWu5W8fKYo=">AAACBXicbVDLTgJBEOzFF+IL9ehlIjHxRHaNiRxJvOgNEgEjbMjs0MCE2dnNzCwJ2XD2B7zqH3gzXv0Of8DvcIA9CFhJJ5Wq7nR3BbHg2rjut5Pb2Nza3snvFvb2Dw6PiscnTR0limGDRSJSjwHVKLjEhuFG4GOskIaBwFYwup35rTEqzSP5YCYx+iEdSN7njBorPdW7uiMEqXfvu8WSW3bnIOvEy0gJMtS6xZ9OL2JJiNIwQbVue25s/JQqw5nAaaGTaIwpG9EBti2VNETtp/OLp+TCKj3Sj5Qtachc/TuR0lDrSRjYzpCaoV71ZuJ/Xjsx/YqfchknBiVbLOongpiIzN4nPa6QGTGxhDLF7a2EDamizNiQlrYMUYzRTAs2GG81hnXSvCp7btmrX5eqlSyiPJzBOVyCBzdQhTuoQQMYSHiBV3hznp1358P5XLTmnGzmFJbgfP0CGqiYuw==</latexit><latexit sha1_base64="ONhgVdv3LIFpvQJV/yWu5W8fKYo=">AAACBXicbVDLTgJBEOzFF+IL9ehlIjHxRHaNiRxJvOgNEgEjbMjs0MCE2dnNzCwJ2XD2B7zqH3gzXv0Of8DvcIA9CFhJJ5Wq7nR3BbHg2rjut5Pb2Nza3snvFvb2Dw6PiscnTR0limGDRSJSjwHVKLjEhuFG4GOskIaBwFYwup35rTEqzSP5YCYx+iEdSN7njBorPdW7uiMEqXfvu8WSW3bnIOvEy0gJMtS6xZ9OL2JJiNIwQbVue25s/JQqw5nAaaGTaIwpG9EBti2VNETtp/OLp+TCKj3Sj5Qtachc/TuR0lDrSRjYzpCaoV71ZuJ/Xjsx/YqfchknBiVbLOongpiIzN4nPa6QGTGxhDLF7a2EDamizNiQlrYMUYzRTAs2GG81hnXSvCp7btmrX5eqlSyiPJzBOVyCBzdQhTuoQQMYSHiBV3hznp1358P5XLTmnGzmFJbgfP0CGqiYuw==</latexit>


Tune Modulation Diagram

Tune modulation normalized frequency, QM/QI<latexit sha1_base64="2i9LmAM7KrJ1pOf8CUzT/D2KblE=">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</latexit><latexit sha1_base64="2i9LmAM7KrJ1pOf8CUzT/D2KblE=">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</latexit><latexit sha1_base64="2i9LmAM7KrJ1pOf8CUzT/D2KblE=">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</latexit><latexit sha1_base64="2i9LmAM7KrJ1pOf8CUzT/D2KblE=">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</latexit>

Tunemodulationnormalized

amplitude,q/Q

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E778 Persistant Signal


Arnold Diffusion and Integrability

§  This is still just 1.5 dimensions!! §  One dimension plus time

§  In more dimensions, particles can move along resonance lines and diffuse in tune space

§  A mechanism for long-term amplitude growth in tracking and perhaps even reality

§  Visualization makes my brain hurt

§  Integrability class next door


Lichtenberg and Lieberman, Jose and Saletan

uspas accelerator physics 2019 northern illinois

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