Physics Letters A 355 (2006) 104–109

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Chaotic phase oscillation of a proton beam in a synchrotron

Fei Li a,∗, Wenhua Hai b, Zhongzhou Ren a, Weixing Shu a

a Department of Physics, Nanjing University, Nanjing 210093, Chinab Department of Physics, Hunan Normal University, Changsha 410081, China

Received 18 March 2005; received in revised form 8 November 2005; accepted 9 February 2006

Available online 17 February 2006

Communicated by A.P. Fordy

Abstract

We investigate the chaotic phase oscillation of a proton beam in a cooler synchrotron. By using direct perturbation method, we construct thegeneral solution of the 1st-order equation. It is demonstrated that the general solution is bounded under some initial and parameter conditions.From these conditions, we get a Melnikov function which predicts the existence of Smale-horseshoe chaos iff it has simple zeros. Our result underthe 1st-order approximation is in good agreement with that in [H. Huang et al., Phys. Rev. E 48 (1993) 4678]. When the perturbation method isnot suitable for the system, numerical simulation shows the system may present transient chaos before it goes into periodical oscillation; changingthe damping parameter can result in or suppress stationary chaos.© 2006 Elsevier B.V. All rights reserved.

PACS: 29.27.-a; 05.45.Ac; 46.15.Ff

Keywords: Phase oscillation; Chaos; General solution; Melnikov function

1. Introduction

Ever since the first accelerator was invented, more than70 years have passed and accelerators have always been oneof the most important apparatus in nuclear and particle physicsresearches. In accelerators different electromagnetic fields gen-erated by induced currents in metallic envelopes are used toaccelerate charged particles up to very high speed. Many artifi-cial nuclides have been found by colliding diverse atomic nucleiwith high-speed particles, which greatly promotes the devel-opment of nuclear and particle physics. Today, acceleratorshave got an extensive application in high energy, high-intensitybeams, high luminosity, condensed-matter physics, radioactivesanitization and radioactive treatment of tumor, etc., [1]. As hasbeen pointed out, high energy provides new territory for poten-tial discoveries and indeed new energy frontiers usually lead tothe discovery of new physics [1]. However, during the course

* Corresponding author.E-mail address: wiselfhhy@hotmail.com (F. Li).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.02.012

of acceleration, the charged particle beams exhibit various non-linearities due to diverse factors, such as interactions betweenparticles and nonuniformity of electromagnetic field. Nonlin-earities can badly baffle the development of accelerator scienceand technology. Therefore, in recent years, nonlinear dynam-ics in accelerators has attracted more and more interests [2–8].Among the nonlinearities, a hot topic is regular and chaoticdynamics. For high-intensity accelerators halo-chaos have be-come a serious barrier that must be conquered. Halo-chaoscan result in excessive radioactivity, intolerable beam losses,radio-frequency breakdown and emittance growth in the accel-erator [9].

In this Letter, we investigate the chaotic phase oscillation ofa proton beam in a cooler synchrotron [2–4]. Here, the coolersynchrotron is actually used as a proton storage ring and it’smain purpose is to store protons moving at high speed in thesynchrotron. Moving particle beams in cooler synchrotrons ex-hibit various nonlinear dynamics. Researches on nonlinear dy-namics of particle beam is of great importance in designingfuture accelerators and colliders. It may also lead to a bettercontrol of the time-varying components for emittance dilution,

F. Li et al. / Physics Letters A 355 (2006) 104–109 105

super slow beam exaction, controlled phase space manipula-tions, etc., [4]. Firstly, by using direct perturbation method,we get the general solution of the 1st-order perturbed equa-tion. From the general solution, we reach a Melnikov functionwhich denotes the existence of Smale-horseshoe chaos iff it hassimple zeros. Our result derived from direct perturbation tech-nique is in good agreement with that in Ref. [4]. And then, witha strong effective phase modulation leading the perturbationmethod to be invalid, we perform numerical calculations andfind that there exists transient chaos [10,11] in the system be-fore it goes into regular and periodical oscillation; changing thedamping parameter can result in or suppress stationary chaos.

2. Theoretical analysis under perturbed approximation

In experiment a single bunch of protons is injected into thesynchrotron. And then the proton beam is electron-cooled andtransversely modulated simultaneously by a small dipole. Theelectron cooling imposes a damping force on the beam. In thestudies of longitudinal motion in a synchrotron, one usuallycharacterizes the longitudinal motion through the phase Φ ofa proton relative to the rf cavity voltage.

The phase oscillation of a proton beam in a synchrotron isgoverned by the equation [1,3]

d2Φ

dt2+ 2α

dΦ

dt+ ω2

s sinΦ

= ωmωsa cosωmt + 2αωsa sinωmt

with α being the damping coefficient, a the effective phasemodulation amplitude parameter, ωs the small amplitude syn-chrotron angular frequency and ωm the modulation angular fre-quency of the rf voltage.

Due to the condition ωmωsa � 2αωsa [2–4], we omit thelast term of the right-hand side in the above equation

(1)d2Φ

dt2+ 2α

dΦ

dt+ ω2

s sinΦ = ωmωsa cosωmt.

It is obvious that our system has the similar form of a two-dimensional forced pendulum with damping which has beenstudied extensively. They may possess similar nonlinear fea-tures.

Setting τ = ωst , and we have

d2Φ

dτ 2+ sinΦ = ε,

(2)ε = −2α

ωs

dΦ

dτ+ aωm

ωs

cos

(ωm

ωs

τ

).

According to the experimental parameters [2–4], the follow-ing inequality can be easily satisfied

(3)2α ∼ aωm � ωs.

So, ε can be regarded as a nonautonomous and dissipative per-turbation to a system described by the following unperturbedequation:

(4)d2Φ0

dτ 2+ sinΦ0 = 0.

Through integrating Eq. (4), we reach the unperturbed solutionΦ0 by

(5)∫ [

2(B + cosΦ0)]−1/2

dΦ0 = ±(τ − τ0),

where B = 12 [ dΦ0

dτ|τ0 ]2 − cosΦ0(τ0) is an integral constant de-

termined by the initial conditions. Assuming B = 1, we arriveat the well-known heteroclinic solution of Eq. (4)

(6)Φ0 = ±2 arctan[sinh(τ − D)

]with the initial conditions

D = τ0 ∓ Ar sinh

[tan

(Φ0(τ0)

2

)],

(7)Φ̇0(τ0) = ±2 cos

[Φ0(τ0)

2

].

The dot here represents the differentiation with respect to τ . Tothe best of our knowledge, Eq. (6) is just the separatrix solutionof the system.

We expand the solution close to the separatrix as follows:

(8)Φ = Φ0 + Φ1, |Φ0| � |Φ1|.Substituting Eq. (8) into Eq. (2), we obtain the 1st-order

equations [12]

(9)Φ̈1 + (cosΦ0)Φ1 = ε1,

(10)ε1 = −2α

ωs

dΦ0

dτ+ aωm

ωs

cos

(ωm

ωs

τ

).

Inserting Eqs. (6) and (7) into Eqs. (9) and (10), we have thelinearized equation [13]

Φ̈1 + [2 sech2(τ − D) − 1

]Φ1 = ε1,

(11)ε1 = −4α

ωs

sech(τ − D) + aωm

ωs

cos

(ωm

ωs

τ

).

When ε1 = 0, Eq. (11) has two linearly independent solutions

Φ〈1〉1 = sech(τ − D),

(12)Φ〈2〉1 = 1

2

[sinh(τ − D) + (τ − D) sech(τ − D)

].

Evidently, Φ〈2〉1 is unbounded and will exponentially increase

with the growth of τ due to its including of sinh(τ − D).By using Eq. (12) and the variation of constants, we get the

general solution of Eq. (11) as [12–17]

(13)Φ1 = Φ〈2〉1

τ∫E

Φ〈1〉1 ε1 dτ − Φ

〈1〉1

τ∫F

Φ〈2〉1 ε1 dτ.

Here E and F are arbitrary constants adjusted by the initialconditions.

Applying Eqs. (10) and (12) to Eq. (13) and calculating thelimits for τ → ∞, we find

(14)limτ→±∞Φ1 → ±∞.

Eq. (14) indicates function Φ1(τ ) is unbounded due to the ex-ponentially increasing of Φ

〈2〉 to infinity as τ → ±∞. But what

1

106 F. Li et al. / Physics Letters A 355 (2006) 104–109

we are interested in is the bounded orbit. To this aim, we findthe necessary and sufficient conditions for the boundedness ofthe orbit as

(15)G± = limτ→±∞

τ∫E

Φ〈1〉1 ε1 dτ = 0.

Necessity of Eq. (15) is obvious. So, under the condition (15),using the l’Hospital rule and substituting Eq. (12) into Eq. (13),we can prove that Φ1 and Φ̇1 have finite inferior and super lim-its for τ � τ0 as follows:

limτ→±∞Φ1

= limτ→±∞Φ̇1

= limτ→±∞

[Φ

〈2〉1

τ∫E

Φ〈1〉1 ε1 dτ − Φ

〈1〉1

τ∫F

Φ〈2〉1 ε1 dτ

]

� limτ→±∞Φ

〈1〉1 ε1

(1/Φ〈2〉1 )τ

+ limτ→±∞Φ

〈2〉1 ε1

(1/Φ〈1〉1 )τ

= 3aωm

8ωs

,

limτ→±∞Φ1

= limτ→±∞Φ̇1

= limτ→±∞

[Φ

〈2〉1

τ∫E

Φ〈1〉1 ε1 dτ − Φ

〈1〉1

τ∫F

Φ〈2〉1 ε1 dτ

]

� limτ→±∞Φ

〈1〉1 ε1

(1/Φ〈2〉1 )τ

− limτ→±∞Φ

〈2〉1 ε1

(1/Φ〈1〉1 )τ

(16)= −3aωm

8ωs

.

Eq. (16) is the proof for the sufficiency of the boundedness con-dition (15).

Given Eq. (15), we have a Melnikov function of the 1st-orderEq. (11) through a straightforward integration

M(τ0) = G+ − G−

=+∞∫

−∞Φ

〈1〉1 ε1 dτ

=+∞∫

−∞sech(τ − D)

[−4α

ωs

sech(τ − D)

+ aωm

ωs

cos

(ωm

ωs

τ

)]dτ

= −8α

ωs

+ aπωm

ωs

cos

(ωm

ωs

D

)sech

(πωm

2ωs

)(17)= 0.

Melnikov function is also called Melnikov distance between thestable and unstable manifolds in the Poincaré section at τ0. It iswell known that if the Melnikov function M(τ0) has simple ze-ros, the stable and unstable manifolds intersect transversely atthe point corresponding to τ = τ0, which shows the existence

of Smale-horseshoe chaos in our system for the orbit (13). Ow-ing to this, we call the orbit (13) a chaotic one. To guaranteethe Melnikov function M(τ0) has simple zeros, the followingequation

(18)dM

dD= aπω2

m

ω2s

sin

(ωm

ωs

D

)sech

(πωm

2ωs

)�= 0

must be satisfied, i.e., sin(ωm

ωsD) �= 0, which leads to

(19)cos

(ωm

ωs

D

)�= ±1.

Combining Eqs. (17) and (19), we arrive at an inequality

(20)1 >

∣∣∣∣ 8α

aπωm sech(πωm

2ωs)

∣∣∣∣.According to the experimental parameters [1–3], a > 0, α > 0,therefore, Eq. (20) can be changed into

(21)a >8α

πωm

cosh

(πωm

2ωs

)> 0.

Setting

(22)ac = 8α

πωm

cosh

(πωm

2ωs

)where ac is the threshold critical value resulting in chaos. So, itis clear that if a > ac, the Melnikov function M(τ0) has simplezeros, which leads to the onset of Smale-horseshoe chaos.

Our result (21) obtained through direct perturbation methodis in good agreement with that in Ref. [4]. As had pointed outin Ref. [4], the parameters used in experiments did satisfy thecondition (21) well. When the values of parameters are set suchthat a > ac, the longitudinal motion of protons gets into confu-sion and then chaotic phase oscillation will occur. For example,in the presence of a weak damping force, condition (21) canbe easily satisfied, which results in protons being damped in-coherently toward islands in succession. These islands lead thelongitudinal beam profile to be split [4]. We also notice that, inRef. [4], when chaos occurs, the values of parameters and con-stants do satisfy condition a > ac. In practical experiments theelectron cooling course, the modulation amplitude parameter a,the modulation angular frequency ωm and the small amplitudesynchrotron ωs are controllable; regulating the electron coolingcourse, one can control the damping coefficient α. It is obvi-ous that, through adjusting the values of α, a, ωs , and ωm, onecan successfully manipulate the phase dynamics of the protonbeam.

From Eq. (21), we plot the phase modulation amplitude pa-rameter a verse the modulation angular frequency ωm (Fig. 1:ωs/ωm = 3, α = 0.5 Hz). Fig. 1 shows that the region abovethe line corresponds to chaotic oscillations and that below cor-responds to regular oscillations. Generally speaking, chaos ina physical system is undesirable. In order to avoid this kind ofsituation we should set the parameters in the region under theline.

We arbitrarily select 8 points in the (a, ωm) plain, 4 abovethe curve and 4 blow the curve, to numerically solve Eq. (1).For the points above the curve chaotic dynamics are demon-

F. Li et al. / Physics Letters A 355 (2006) 104–109 107

strated in the phase space of (Φ , Φ̇) (Fig. 2: (a) ωm =1500 Hz, a = 0.086 rad > ac = 0.00097 rad, (b) ωm =1400 Hz, a = 0.0073 rad > ac = 0.001 rad, (c) ωm = 2000 Hz,a = 0.0008 rad > ac = 0.0007 rad, and (d) ωm = 2500 Hz, a =0.0006 rad > ac = 0.00058 rad). Contrary to these situations itis the other way round for the points below the curve (Fig. 3:(a) ωm = 2500 Hz, a = 0.00050 rad < ac = 0.00058 rad,(b) ωm = 2000 Hz, a = 0.00048 rad < ac = 0.00073 rad,(c) ωm = 1700 Hz, a = 0.00081 rad < ac = 0.00085 rad, and(d) ωm = 900 Hz, a = 0.0012 rad < ac = 0.0016 rad). After atransient process which is unavoidable from the beginning ofthe injection, all the trajectories go into regular and periodicalattractors, which denotes the regularity of the phase oscillationof the proton beam.

The including of the parameter D in Eq. (17) exhibits thatthe stability of the system is sensitively dependent on the ini-

Fig. 1. The regions of regular and chaotic oscillations in plane of (a,ωm) forωs/ωm = 3, α = 0.5 Hz.

tial conditions, which is one of the leading features of strangechaotic attractors. To see clearly this kind of dependence, weplot the stability curves in Fig. 4 from Eq. (17) under a set ofparameters. Fig. 4 shows that, with the increasing of the valueof D, the curves became more and more and denser and denser.If D → ∞, the number of curves will also tends to infinity; thisdenotes the existence of chaos in the system.

3. Unperturbed numerical computations

If the system’s phase oscillation is far away from the separa-trix or the phase modulation amplitude is larger than 0.087 rad,the above perturbative expansion will be invalid [4]. Accord-ingly, in this section, we will numerically investigate the chaoticproperty of the phase oscillation of the beam. Due to the damp-ing force, the system is a dissipative one, so the volume inphase space may decrease during the process of transient evolu-tion. By using MATHEMATICA, we solve Eq. (1) numericallyand plot a sequence of phase space pictures of (Φ,dΦ/dt )in Figs. 5–7. From Fig. 5(a), for a set of system parameters:α = 4 Hz, a = 0.090 rad, ωm = 1500 Hz, ωs = 3ωm, we can seetypical phase space structure of dissipative systems; the phasespace volume contracts continuously till all trajectories are at-tracted into a stable periodical attractor shown in Fig. 5(b). Thatis to say, the phase oscillation will become regular and period-ical after a transient process during which chaotic behaviorssometimes may exist. This kind of chaotic behaviors is alsocalled transient chaos. We can observe transient chaos clearlyin Fig. 6 for α = 2.0 Hz, a = 0.17 rad, ωm = 1200 Hz, ωs =1.2ωm. Fig. 6(a) exhibit a transient chaotic attractor formed inthe transient chaotic process; Fig. 6(b) is the corresponding fi-nal periodical attractor. As has been pointed out by Y. Wang

Fig. 2. The phase portrait in (Φ,Φ̇) plain with α = 0.5 Hz, ωs/ωm = 3, and a > ac .

108 F. Li et al. / Physics Letters A 355 (2006) 104–109

Fig. 3. The phase portrait in (Φ,Φ̇) plain with α = 0.5 Hz, ωs/ωm = 3, and a < ac .

Fig. 4. Curves for different initial conditions with α = 2.5 Hz and a = 0.025 rad.

et al. [4], the damping force can be very small. So we keep theother parameters of Fig. 6 unchanged and decrease the valueof the damping parameter α to 0.003 Hz, we find that the sys-tem goes into a stationary chaotic state (Fig. 7). Thus it can beseen that, for a set of fixed parameters, the damping parameteris very important. Changing the damping parameter can resultin or suppress stationary chaos.

4. Conclusions

The phase oscillation of a proton beam in a cooler syn-chrotron has been studied. By using the direct perturbation

technique, we construct the general solution of the 1st-orderperturbed equation. Theoretical analysis indicates the generalsolution is bounded when some initial and parameter conditionsare satisfied. From these conditions, we get a zeros Melnikovfunction which predicts the onset of Smale-horseshoe chaosnear the heteroclinic solution. According to the zero Melnikovfunction, we plot the relationship between the effective phasemodulation amplitude parameter a and the modulation angu-lar frequency of the rf voltage ωm. When the phase oscillationsare not close to the separatrix solution or the value of the ef-fective phase modulation amplitude parameter a is big enough,numerical method demonstrates there exists transient chaos in

F. Li et al. / Physics Letters A 355 (2006) 104–109 109

Fig. 5. Phase portraits of dΦ/dt vs Φ for α = 4 Hz, a = 0.090 rad, ωm = 1500 Hz, ωs = 3ωm.

Fig. 6. Phase portraits of dΦ/dt vs Φ for α = 2 Hz, a = 0.17 rad, ωm = 1200 Hz, ωs = 1.2ωm .

Fig. 7. Phase portraits of dΦ/dt vs Φ for α = 0.003 Hz, a = 0.17 rad,ωm = 1200 Hz, ωs = 1.2ωm .

the phase oscillation of the beam. And the chaotic phase oscilla-tion will ultimately become periodical motion after a transientevolution. We also find that changing the damping parametercan result in or suppress stationary chaos. These research find-ings are useful and provide a more detailed analysis for chaosin beam manipulation in cooler synchrotron phase space, whichis of great importance in designing future accelerators and col-liders. It may be also a great help to a better control of the time-varying components for emittance dilution, super slow beamexaction, controlled phase space manipulations, etc.

Acknowledgements

This work is supported by the National Natural ScienceFoundations of China (10125521 and 10535010), the 973 Na-tional Major State Basic Research and Development of China(G200007740), by the CAS Knowledge Innovation Project

No. KJCX2-SW-N02 and by the Fund of Education Ministryof China under contract 20010284036.

References

[1] S.Y. Lee, Accelerator Physics, World Scientific, Singapore, 1999.[2] M. Ellison, et al., Phys. Rev. Lett. 70 (1993) 591;

D. Li, et al., Phys. Rev. E 48 (1993) 1638;D.D. Caussyn, et al., Phys. Rev. A 46 (1992) 7942.

[3] M. Syphers, et al., Phys. Rev. Lett. 71 (1993) 719.[4] H. Huang, et al., Phys. Rev. E 48 (1993) 4678;

Y. Wang, et al., Phys. Rev. E 49 (1994) 5697;D.D. Caussyn, et al., Phys. Rev. E 51 (1995) 4947.

[5] A. Riabko, et al., Phys. Rev. E 54 (1996) 815;D. Jeon, et al., Phys. Rev. Lett. 80 (1998) 2314;C.M. Chu, et al., Phys. Rev. E 60 (1999) 6051.

[6] A.V. Fedotov, R.L. Gluckstern, Phys. Rev. ST Accel. Beam 2 (1999)014201.

[7] J.Q. Fang, Ziran 22 (2000) 63.[8] J.Q. Fang, Z.S. Wang, G.R. Chen, Commun. Theor. Phys. 42 (2004) 557.[9] Y. Fink, C. Chen, Phys. Rev. E 55 (1997) 7557.

[10] G. Chong, W. Hai, Q. Xie, Phys. Rev. E 70 (2004) 036213.[11] J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57 (1985) 617.[12] W. Hai, Y. Xiao, J. Fang, W. Haung, X. Zhang, Phys. Lett. A 265 (2000)

128.[13] W. Hai, Y. Xiao, G. Chong, Q. Xie, Phys. Lett. A 295 (2002) 220.[14] W. Hai, M. Feng, X. Zhu, L. Shi, K. Gao, X. Fang, J. Phys. A 32 (1999)

8265;W. Hai, M. Feng, X. Zhu, L. Shi, K. Gao, X. Fang, Phys. Rev. A 61 (2000)052105.

[15] F. Li, W. Hai, Acta Phys. Sinica 53 (2004) 1309.[16] F. Li, W. Hai, G. Chong, Q. Xie, Commun. Theor. Phys. 42 (2004) 599.[17] C. Lee, et al., Phys. Rev. A 64 (2001) 053604.