USPAS Accelerator Physics 2019 Northern Illinois ...
Transcript of 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!
2 T. Satogata / January 2019 USPAS Accelerator Physics
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.”
3 T. Satogata / January 2019 USPAS Accelerator Physics
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?
4 T. Satogata / January 2019 USPAS Accelerator Physics
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
5 T. Satogata / January 2019 USPAS Accelerator Physics
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
6 T. Satogata / January 2019 USPAS Accelerator Physics
Fig 16.1: Resonance Overlap
§
Even-order beam-beam resonances
7 T. Satogata / January 2019 USPAS Accelerator Physics
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
8 T. Satogata / January 2019 USPAS Accelerator Physics
16.1: Chirikov Overlap and the Standard Map
“Isolated” resonances
Chaotic resonance overlap: separatrix becomes chaotic RF motion with large Qs!
9 T. Satogata / January 2019 USPAS Accelerator Physics
Whoa, Wait a sec: ξ=0.2 could be stable?
§
Even-order beam-beam resonances
10 T. Satogata / January 2019 USPAS Accelerator Physics
EIC Beam-Beam Parameters (European Stragegy Submission)
11 T. Satogata / January 2019 USPAS Accelerator Physics
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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(B⇢)x
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12 T. Satogata / January 2019 USPAS Accelerator Physics
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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13 T. Satogata / January 2019 USPAS Accelerator Physics
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>
14 T. Satogata / January 2019 USPAS Accelerator Physics
15 T. Satogata / January 2019 USPAS Accelerator Physics
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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16 T. Satogata / January 2019 USPAS Accelerator Physics
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>
17 T. Satogata / January 2019 USPAS Accelerator Physics
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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18 T. Satogata / January 2019 USPAS Accelerator Physics
E778 Persistant Signal
19 T. Satogata / January 2019 USPAS Accelerator Physics
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
20 T. Satogata / January 2019 USPAS Accelerator Physics
Lichtenberg and Lieberman, Jose and Saletan