ELSEVIER Computer Physics Communications 106 (1997) 169-180

Computer Physics Communications

NERO: a code for the nonlinear evaluation of resonances in one-turn mappings*

E. Todesco a M. Gemini a M. Giovannozzi b a Dipartimenw di Fisica, Universitd Bologna. Via irnerio 46./-40126 Bologna. Italy

b PS Divisitm, CERN. CH.1211 Geneva 23. Switzerland

Received 6 August 1996

Abstract

We describe a code that evaluates the stability, the position and the width of resonances in four-dimensional symplectic mappings. The code is 0ased on the computation of the resonant perturbative series through the program ARES, and on the analysis of the resonant orbits of the interpolating Hamiltonian. The code is dedicated to the study and to the comparison of the nonlinear behaviour in one-turn betatronic maps. © 1997 Elsevier Science B.V.

Keywords: Symplectic maps; One-turn betatronic maps: Normal forms; Resonances

P R O G R A M S U M M A R Y PerqJhe.':r: ,~sed: Storage for input and output files

Title of program: NERO

Camh~gue identifier: ADGR

Program obtainable from: CPC Program Library, Queen's Univer- sity of Belfast, N Ireland, or: INFN, Sezione di Bologna; e-mail: etodesco @ bo .in fn.it

Licensing provisions: none

Comlmters: AIphavax 3000/500, HP 755

Operating systems: VMS. Unix

Programming language used: Fortran 77

No. of bytes m dtsaributed program, including test data. etc.: 53784

Distribution format: ASCII

l~eywords: symplectic maps, one-lure be~alronic maps, nomml forms, resonances

Nature t~f the physical problem This code allews one to reconstruct the dynamics in phase space of a four-dimensional symplectic mapping using the perturbative approach. The position of the network of resonances and their strength are evaluated v~a analytical methods. This approach is complementary to numerical methods for the ana|ysis of the global dynamics in phase space. It can be useful for evaluating the rel- evance of the nonlinear effects in betatronic motion of hadron accelerators.

High speed storage required: i.2 Mb at pertutbative order 10

* Work partially supported by EC Human Capital and Mobility Contract No. ERBCHRXCT940480.

0010-4655/97/$17.00 (~) 1997 Elsevier Science B.V. All rights re.~rved. PI! SO010-465 5 ( 97 ) 0009 ! - X

170 E. ?bdesco et aL/Computer Physics Communications 106 (1997) 169-180

Method of solution Given a sympiectic map in the neighbourhood of an elliptic fixed point, the code evaluates both the nonresonant and the resonant perturbative series (normal forms). For each resonant normal foml. the interpolating Hamiltonian is computed and the posi- tion and the stability of the resonant orbits and the width of the islands are evaluated; this analysis is carded out through the direct inspection of the coefficients of the interpolating Hamiltonian. All

the computations are carried out at an arbitrary order that can be specified by the user; the first significant perturbative order is taken as a first guess, and then a Newton method is used to eval- uate the higher orders effect.

Typical running time 30 seconds for the test run (phase space analysis up to order 8, with all resonances up to order 9) on an Alphavax 3000/500.

LONG WRITE-UP

1. I n t r o d u c t i o n

The dynamics of single particles in a magnetic lattice under the effect of nonlinear fields has become a crucial issue tbr the construction of large circular accelerators such as the Large Hadron Collider [ I ]. These problems are modelized through tbur-dimensional (4D) symplectic mappings that describe the behaviour of the single particles over one turn of the machine. The comprehension of the relation between resonances, nonlinearities, tuneshifts and the stability domain (dynamic aperture) in these models is a rather difficult task. The dynamic aperture is usually determined through numerical integration based on tracking [ 2,3]. The perturbative theory based either on Hamiltonian flows [4-6], or on symplectic mappings [7-101, provides a lot of analytical information on the detuning and on the resonance parameters.

In the case of unstable resonances, the dynamic aperture is usually determined by the hyperbolic resonant orbits (fixed lines) [ 11-13]. The stable resonances, on the other hand, feature families of islands that do not limit the stability domain, and therefore there is no direct relation with the dynamic aperture. Several studies have shown, however, that analytical indicators (Quality Factors, QFs) extracted through perturbative tools can be well correlated with the dynamic aperture 16,14,151.

During the past years, arbitrary-order codes have been developed to compute nonresonant perturbative series (normal forms) of a generic truncated one-turn map 17,8,10]. More recently, a code has been developed to evaluate also the resonant normal forms (code ARES, see Ref. [ 161), In this paper we present a new program that includes ARES and postprocesses its output in order to provide the following quantities: - It allows the automatic evaluation of quality factors defined in previous papers [ 15 l, such as the norm of the

map, the norm of the tuneshifi, and the norm of the resonances. - It allows the reconstruction of the phase space dynamics: the network of resonances involved in the nonlinear

motion is evaluated and the position and the width of the islands are computed. Furthermore, another quality factor, defined as the fraction of the phase space volume that is locked on resonances, is evaluated.

- it can automatically analyse several lattices at the same time and produce correlation plots of the QFs with the dynamic aperture obtained through standard tracking. This feature is relevant, for instance, in the analysis of the effect of random errors [ 151, and in general for all the optimization procedures.

This code can be useful not only for studying the stability of betatronic motion, but also to obtain an analytical picture of the global dynamics of generic 4D symplectic mappings in the neighbourhood of an elliptic fixed point. This approach is complementary to the powerful numerical methods based on frequency analysis [ 17,18].

The plan of the paper is the following: in Section 2 we describe the theory of single-resonance normal forms. In Section 3 we o,tline the algorithms used in the code for evaluating the position and the stability of resonant orbits. In Section 4 we describe the input and in Section 5 the output of the code. A test run is given in Section 6.

E. Todesco el al./Computer Physics Communications 106 t!997) 169-180 171

2. Theory

We consider betatronic one-turn maps that model the transverse oscillations of charged particles in a magnetic lattice [7,8,10,19]. We assume that the synchrotron radiation is negligible, and that the transverse motion and the longitudinal motion are uncoupled. Let ( x , p , , y,p:, ) be the physical coordinates and the conjugate momenta at a given point of the machine. The one-turn map is the nonlinear function giving the phase space coordinates and momenta as a function of the same quantities at the previous turn and at the same section of the machine. It can be expanded as a "lhylor series truncated at order M,

M

n=! jl +j2+j.~+j4=n

M

n=! ji+j2+j~+J4 =n

M

yl = ~

n=l

M

S, +J~ +j~ +j4=n

• , j ~ j4 & :j, O:.j.~44x s' p.{~ y P i . .

F2: j. .j:,)~ 44 xh P,{" YJ~ p ~4 ,

F3 ;j~ , ]2. j~ ..J.~ Xjl P.{~" YJ;~ PJ~ "

" nJ21,J; nJ4 . { ] ) F4:fi.j:..i~,j4xslrx ~ r r

,;~;i jl +j'~ +j~+J4 =n

The truncation order M is limited by the available storage memory and by the CPU lime; for a large machine such as the LHC, M can be chosen between I0 and 15. We denote by (~,/~x,.v,/~y ) the Courant-Snyder coordinates where the linear part of the map is the direct product of two 2D rotations of angles Q~ and Q:. (the linear tunes). We also denote by zl = ~ - i/~.~ and z2 = ~ - ipy the complex Courant-Snyder coordinates that diagonalize the linear part of the motion; i = x/'-zT is the imaginary unit.

Given an initial condition in the 4D phase space, its or'bit is t:haracteri~ed by two nol~linear frequencies v,

and v:. that can be extracted through Fourier analysis [ 17,20]. When the amplitude of the initial condition tends to zero, the nonlinear frequencies (t,,, v, ) tend to the linear ones (Q.,, Q,. ). We define the tune footprint as the region in the frequency plane that corresponds to stable orbits of the one-turn map [ 15,18].

2.1. Normal f o rms

The perturbative theory of normal forms consists in transforming the one-turn map to a simpler, more symmetric map U (the normal tbrm) that explicitly shows the motion invariants and the geometry of the orbits [7,8,10]. The normal form U is then written as the Lie series of an interpolating Hamiltonian h. We denote by g" the transformation from (zl, z2) to (( l , (2) such that the original map written in the new coordinates is reduced to its normal form U. The Hamiltonian is more easily expressed in terms of the amplitude-angles coordinates (01,02, pl , p2), given by (i = v / ~ d°' and (2 = x / ~ euj2. Therefore, pl and p2 are the generalization of the emittances to the nonlinear case, i.e., at first order they agree with ,~2 + I~ and 92 +/~2, respectively.

As a first step, one can build a nonresonant normal form: in this case the Hamiltonian only depends on the

amplitudes pl and p2,

k~ .k2

The phase space described by this Hamiltonian is given by 2D KAM tori whose nonlinear frequencies are 111e

partial derivatives of h with respect to pl and p2.

172 E. Todesco et al./Computer Physics Communications 106 f!997) 169-180

When the linear tunes are close to a resonance (q ,p) , where q E [','t and p E Z, it may happen that the resonant condition on the nonlinear frequencies,

q u a + p ~ ' y = m + e , e<< I , m E Z , (3)

is exactly satisfied (i.e., c = O) for some positive amplitudes. In this specific case, the general topology of the 2D invariant KAM tori breaks down, and a family of islands arises. Such islands can be described by the single-resonance normal form theory that consists in retaining in the normal form U and in the interpolating Hamiltonian, the resonant combinations of the phases qO! + p02. Thus h re~ds

Z kt+tq/2 t~, h(pl ,P2,01,02) = hk~.t~,..t Pl p2 "+tlt'l/2 cos(i(qOt + p02) + tPk,,t,.:.l) . kt .k_~ ,i

(4)

2.2. Topology of single resonances

In order to analyse the dynamics of the above Hamiltonian, one can perform the canonical transformation

rl = p l , ,;hi = 01 + ~ 02, q

r2 = -P- Pl + P2 ~b2 = 02, q

such that the new Hamiltonian reads

h(qbl , r l ; r2)= Z h ~ , . k , . t r k l " + t q / Z ( P r l + r 2 ) k~ ,k2.t q

(5)

k: +tit, I~2 cos( lq$j + tPk,.k,.t ) • (6)

Since there is no dependence on ~b2,, the quantity r2 is invariant ! and the problem is reduced to analysing a 2D Hamiltonian with a parametric dependence on r2~ For each value of r2 one can compute the fixed points, the eigenvalues of the linear map about those fixed points, and the position of the separatrices. Finally, one has to check that the obtained solutions rl and r2 correspond to positive values of the original amplitudes Pl and P2.

We define the excitation order lexc as twice the power in the amplitudes of the first nonzero resonant coefficient in the interpolating Hamiltonian. In generic cases, where all the resonant coefficients are different from zero, the excitation order is equal to the resonance order q + IPl. If there are some symmetries, sorae resonant coefficients can be equal to zero and therefore the excitation order can be greater than the resonance order. One has to distinguish two cases.

2.2. I. Excitation order equal to three or four When one considers a resonance (q,p) such that qp < 0 (also called difference resonance), the second

invariant is r2 ~ -PPl + qP2 and therefore the motion is always bounded. In general one can have islands, even though the geometry of these structures is more complicated and each resonance needs an appropriate analysis. For this reason, the code NERO does not perform the phase space analysis for these resonances.

In the case of sum resonances (qp > 0) with excitation order three, the resonance is dominant over the detuning terms and therefore one has hyperbolic fixed points whose related separatrices go to infinity. The cede can evaluate the position of these separatrices that are the limits of the stability boundary.

In the case of sum resonances with excitation order four, one can have different cases according to the relative strength of the resonant and of the detuning terms. One can have either separatrices that limit the stability boundary or stable islands. Also in this case the analysis is rather involved, and it has not been implemented in the code for the phase space analysis.

t As the Hamiltonian is always an invariant of motion. Ihe quantiiy r2 is also called the second invariant.

E. Todesco et aL/Computer Phy.,~ics Communications 106 (!997) 169-180 173

2.2.2. £xcitation order greater than four In this case there is always a detuning term (i.e. a nonresonant coefficient) of order lower than the resonance.

One can show that in general there exist two families of fixed points, one elliptic and one hyperbolic, in agreement with the Poinear6-Birkhoff theorem. The separatrices that pass through the hyperbolic fixed points are the border of the islands. The code NERO automatically evaluates the position of the fixed points, of the separatrices, and the area of the islands.

2.3. Quality factor:

Previous studies [6,14,15] have shown the importance of analytical quality factors (QF) to understand the nonlinear dynamics. These QF provide useful information on the one-turn map, such as the relative strength of the different orders of the map, of the tuneshift, or of a resonance. They are useful to evaluate in a very short CPU time the goodness or the badness of a magnetic lattice with nonlinearities, and can be used in order to carry out analytical or numerical optimizations of the lattice. Therefore we have implemented the computation of the such quantities in NERO. All the QF depend on an amplitude A that is necessary to weight the different orders that contribute to the QE The quantity A represents the maximum amplitude where we want to analyse the nonlinear motion, and therefore it is interesting to choose it close to the expected dynamic aperture. - Norm of the map Qt; it is the sum of the absolute values of the map coefficients weighted with the powers

of the maximum amplitude A,

Q, ( A ) - A" I I . (7 ) n=2 jt +j2 +)3-~-)4-n i=1,4

- Norm of the tuneshift Q2; it is the average of the square root of the sum of the squares of the amplitude- dependent t.uneshifts divided by twg. The average is carried out over the border of the analysed phase space Pt + P2 = A. The tuneshift is evaluated through nonresonant normal forms truncated at the specified order N.

- Norm of the resonance Q3(q,p); it is the sum of the absolute values of the resonant coefficients of the interpolating Hamiltonian weighted at the amplitude A,

Q3(A; q,p) = E E hk"tq'tAtq+k:+t~'t+lt'l)/2 (8) 14,0 km .k:

- Hypervolume of the resonance Q4(q, p); if the ~esonance is stable, it is the ratio between the hypervolume in 4D of the initial conditions that are locked on ~,he resonance (q ,p) , and the total phase space hypervolume analysed (that is limited by the amplitude A and therefore corresponds to 2¢r2A~). If the resonance is unstable, it is the ratio between the 4D hypervolume of initial conditions that lie outside the unstable separatrix, and the total hypervolume.

3. A l g o r i t h m s

NERO incorporates the code ARES that evaluates the normal form series: the related algorithms have been described in a previous paper [ 16]. The main part of the code ,,T~nr,,.,,,.,,j performs the analysis of the interpolating Hamiltonian in order to evai~:ate the existence, the position and the stability of the resonanl orbits. We will restrict ourselves to outline the algorithms used for the case of excitation order greater than four. A similar strategy is used for the case of excitation order equal to three.

We consider a resonance (q,p) and the related interpolating Hamiltonian h. First of all, a scan over the tune values that lie on the resonant line qv., + pv,, = m is carried out. A Newton method is used to compute the values of the amplitudes p~ and P2 that correspond to that detuning. It makes use of the nonresonanl part of

174 E. ~ulesco et al./Computer Physics Communications 106 (I 997) 169-180

the Hamiltonian up to the truncation order. To obtain an initial guess for the Newton method, one analytically computes the values of the amplitudes pi and P2 assuming only first-order detuning. The final result of the Newton corresponds to the average amplitudes pO and p0 of the resonance.

If the obtained amplitudes are positive, the second invariant is evaluated. One then has to analyse the reduced Hamiltonian (6) in order to find out the position of the fixed points that are the solutions of the system,

ah arl (¢~1' ri; r2 ) = 0,

ah a~l (q / i , r l , r2 ) O. (9)

Also ia this case one has to use a Newton method (with two variables rl and eq ). The amplitude can be initialized using p0. The initial guess for the angle can be worked out by considering the first-order truncation of the second equation of system (9): if there is only one sine te~m, the guess can be computed analytically. Otherwise, if there is more than one term (i.e., the resonance excitation is greater than the resonance order), one can make a numerical scan over [0, 2¢r1 in order to find out a good guess. In both cases two guesses are needed; they must correspond to the elliptic and to the hyperbolic solution. The stability is evaluated through the computation of the Hessian of the Hamiltonian at the fixed points.

Once the fixed points have been found, one has to evaluate the equation of the separatrix. Also in this case we use a first-order analytical guess for the position of the inner and outer separatrix for the angle ~bl,~ corresponding to the elliptic fixed point. Then, each guess is used to start a Newton method that solves the separa!rix equation at arbitrary order,

h( t~le, rl ; r2) = Eh =-- h( t~h, rih; r2) , (10)

where (~h,r lh) denote the position of the hyperbolic fixed point, and therefore Eh is the Hamiitonian value on the hyperbolic fixed point. The obtained values r~ ~in and r~ nax are than transformed back to the original amplitudes to have the minimum and maximum amplitude of the separatrix (p~ain p~in) and (p~ax, p~nax).

Finally, the equation for the separatrix (10) is solved also for all the values of ~bl that are between ~bl,. and ~lh; in this way one can integrate the area of the islands. Also in this case a Newton method is used, initialized by the values of the preceding angle ~bi.

4. Program input

Map input

Three options are available: (0) The map can be given as a file of coefficients in Courant-Snyder coordinates. ( I ) The map can be given as a file of coefficients in physical coordinates. (2) One can analyse a generalized form of the H6non map, which is already built in the code. The analysed

map is

f x ' ~

yl 0 R(Qy)

Px + Ke 2.,,,=,,. k, ( x + iy

Y

py - Im 5 -`"2 n=l ! I

k , ( x + iy)" /

( i l )

E. Todesco et al. / Computer Phvsic~ Communications 106 (1997) 169-180 175

where R ( Q i ) is the 2D rotation matrix of an angle 2~rQ;; this is the map of a linear lattice with a multipole in the kick approximation, where nl and n2 are the minimum and the maximum multipolar order, and k,, are the muitipole coefficients. For nl = n2 = 2 and k2 = I, one recovers the 4D H6non mapping [ 10]. In the input file one has to include also the values of the parameters

Qx, Qy,

tll , 112 ,

k , , . . . . , k, ,: .

In the first two cases, it is possible to analyse up to 99 files relative to different maps. Their names are read from a file NAME.DAT (one for each record). The map coefficients are stored in the format provided by the DA code [3,19], each map in one separate file. The truncation order of the map is automatically read from the fi!es. To each map a two-digits integer number n n , starting from one, is associated; such a number is used to distinguish the output files.

Resonances to analyse

The following menu of four options is prompted: ( 1 ) Analysis of only one resonance (q, p). (2) Analysis of all the resonances (q, p) whose order q + IPl satisfies nmin < q + IPl < nma,,. (3) Analysis of all the resonances (q, p) whose order q + iP] is greater or equal to nmin and less or equal to

nmax, and such Wat the euclidean distance of the resonant line from the linear tunes in the tune diagram is less than ~.

(4) Ana|ysis of all the resonances specified in the input file RESON.DAT. The file is in free lormat, and each line contains two integer numbers q, and p: that specify the single-resonance.

For the selected resonances, the code evaluates the normal form, the interpolating Hamiltonian, and the .quality factors Qi, Q2 and Q3 (see next section, file HAMInn.DAT).

Phase space analysis

The following menu of three options is prompted: (0) No phase space analysis is performed. ( I ) In this case the phase space analysis is c ~ i e d out. ~ le position and the width of the islands o f the

resonances that lie inside the analysed domain are evaluated. In order to find out which resonances lie in the selected phase space domain, the footprint through nonresonant normal form is evaluated and then only the resonances that fall inside the footprint are selected. This method can fail whenever the linear tunes are close to a low order resonance: this produces a divergent behaviour in the nonresonant normal form at low orders, and a wrong footprint. Moreover, it can exclude resonances that are close to the edge of the analysed phase space domain.

(2) In this case the phase space analysis is carried out, but, to avoid the difficulties of the automatic detection of resonances, one can provide the set of resonances to analyse in file RPHSP.DAT.

In both cases ( I ) and (2), the reconstruction of the global dynamics in phase space is provided in tile PHSPnnoDAT. The total fraction of phase space occupied ~y resonances (quality factor Q~) is evaluated and

written in file QEDAT.

Normal form order N

It is tile order (power of coordinates) up to which the normal form is computed. Tile interpolating Hamiltonian of the normal form is of order N + 1 in the coordinates. The normal tbrm order N must be smaller or equal to

176 E. Todesco et al./ Computer Physics Communication,~ 106 (1997) 160-t~0

the truncation order M of the one-turn map. A normal form of order N neglects all the nonlinearities higher than the 2N + 2 multipole, and all the resonances of order higher than N 4- I.

Maximum amplitude A

It is the m~imum value of the sum of the nonlinear emittances p~ + p2 that defines the region around the origin where the evaluation of the position of resonances is carried out. This value should approximately correspond to the dynamic ape.,!.ure. The crucial check of the g ~ ~_haviour of the perturbative series at !he maximum amplitude is carried out in file HAMInn.DAT

5. Program output

For each map identified by the number nn, the following files are produced:

TUNEnn.DAT: analytical tuneshifts

This file contains the coefficients of the analytical tuneshifts series, in powers of the nonlinear emittances pi and P2, worked out through nonresonant normal forms.

FOOTnn.DAT: analytical tune footprint

The triangular phase space region given by pl > 0, P2 > 0, and pz + P2 < A is divided in a rectangular grid of 2450 initial conditions. For each initial condition the amplitude-dependent tunes ~,:, and Uy are evaluated through a nonresonant normal form at the specified truncation order. The file is made up of two columns with ~,.~ and Uy, and of 2450 lines relative to each initial condition.

HAMlnn.DAT: Hamiltonian coefficients

This file contains the information about the analysed single resonance(s). The following quantities are given: - q, p: integers denoting the single resonance; - le,~c: the excitation order, that is, the lowest order of the non~,ero resonant Hamil~onian coefficients. If no

resonant coefficients are different from zero it is set to the firsv~ neglected order in the Hamiltonian N + 2. - Comments on the topology of the phase space. - The values of the Hamiitonian coefficients [ see Eq. (4) ] are given in the following format:

tT kl k2 I hk~.k,.t ~Pkl.k;.~ Ci Cr2 (12)

where the first integer n = 2k~ + 2k2 + !(q + Ipl) is the order of the coefficient, and the positive real C; is give~l by

Cl(n) = ~ hkl k, oA kl+k: (13) 2kt +2k~=n

for detuning coefficients and zero otherwise, and by

C2(kl , k2,1) =hkl.k~i , I ~ 0 (14)

for re~nant coefficients and zero otherwise. The quantities C~ give the strength of the different orders of the Hamilt¢~nian at the maximum amplitude, and they are very useful to check out whether of the normal form series are well ~haved; if they are not decreasing with the order, one has a dominance of the higher orders, and therefore the maximum amplitude must be reduced.

E. Todesco et al./Computer Physics Communications 106 (1997) 169-180 177

PHSPnn.DAT: location and width of resonances

This file contains the information about the location of stable single resonances in the space of the nonlinear emittances Pl and P2. A scan of the resonant structure along the second invariant is made, and for each value of the second invariant the following quantities are printed out:

pi 0 p0 p~ni, p~nin p~nax p~aX q p (15)

(a, , ) and are the , . ,.. ,'," (,,_,,,,,o po) d,'--'," . . . . . . . . . . . . the mverage amplitude of the fixed lines, whilst . min. pmin. '~Hlt ,,,max, pmax)_ minimum and maximum width of the oscillations inside the islands. All the phase space is analysed up to P~ + P2 < A. The last two columns denote the resonance.

In the case of unstable resonances, the code provides the minimum amplitude (p~nin, p~in) and the maximum amplitude (p~nax, p~nax) where the unstable separatrix is met. Also in this case the file is made up by eight columns:

p?m pT+. pT. 0 0 q p ( i 6)

LOGFnn.DAT: log file for phase space analysis

This file contains messages that are useYul to understand whether the code has successfully evaluated the island width or the reasons for which the computation has failed. For each resonance and for each value of the second invariant that gives rise to resonant orbits inside the selected phase space domain, the code prints out the value of the second invariant and different men, sages, if the procedure of evaluation of the resonance has been carried out, the message "Island width/Separatrix successfully computed" is printed out. On the other hand, if the procedure has been interrupted for any reason (i.e., a nonconvergent Newton algorithm, or a Newton converging to negative amplitudes, or wrong stability of the fixed points...), the cause of the mtc~uption is printed out. In any case the program does not interrupt but continues the computations with the next value of

the second invariant.

Finally, the following file containing the quality factors of all the analysed maps is produced:

QEDAT: quality factors

In this file each record corresponds to a different map. The first number is the identifier of the map. Then, one has the quality factors QI and Q2; then, the quality factors Q3 relative to all the resonances selected from the menu 'Resonances to analyse'. Finally, if the phase space analysis is active, the sum of all the quality factors Q4, which corresponds to the fraction of the 4D hypervolume occupied by all the resonances, is given.

6. Test run

We consider a test run where the H~non map with linear frequencies 0.28 and 0.31 is analysed. We choose the option of the phase space analysis, using the automatic selection of the resonances since the tunes are far

TUN,..~ .~,/~. and HAMI0 I.DAT (where from low-order resonances. In the following, we give the output files , rr~n, ~ -r only the first resonance is given for the sake of brevity). In Fig. 1 we plot the analytical footprint using the data given in file F ~ I . D A T . In Fig. 2 we plot the network of resonances in phase space, using the data

stored in file PHSP01.DAT.

1"/8 E. Todesco et al./Computer Physic,~ Communications 106 (1997) ! 69-180

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":.:,.~'r'=..=..r "'~i~ !~i)~i .." i ~'...../, . . . . . . . . . . . . . . . . . . . . . .........

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Fig. I.

0 2 5 ¢'4

0 . 2

0 . 1 5

0 . 1

0 . 0 5

C

I

0 . 3 5 ",.; " C. ~ 5 0 2 0 . 2 5 p l

Fig. 2.

Fig. I. Analytical footprint for the H6non map at (2:, = 0.28 and Qy = 0.31 through nonresonant normal fomls at order 8 (data slored in file FOOT01 .DAT).

Fig, 2. Analytical determination of the network of resonances and of their width for the HEnon map at Qx = 0.28 and Qy = 0.31 through normal forms at order 8 in the space of nonlinear invariants pt and/)2 (data stored in file PHSPOI.DAT).

I n p u t f i l e f o r t h e c o d e

2 ! (Henon map) 0.28 0.31 ! (Linear frequencies) 2 2 ! (minimum and maxJmummultipole) 1 f (sextupole coefficient) I ! (analysis of one resonance) I -4 ! (resonance to analyse)

I ! (phase space analysis with automatic resonance evaluation) 8 ! (normal form order) 0.25 t (maximum amplitude)

T U N E O I . D A T

9

T u n e x

+ 0 . 2 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D + 0 0

+ 0 . 8 3 2 0 7 6 5 2 0 9 9 4 1 0 9 4 I ) - 0 1

- 0 . 6 5 9 3 2 1 8 2 3 0 7 9 7 0 2 6 D - 0 2

+ 0 . 6 9 5 4 2 8 1 7 7 3 9 6 1 9 5 0 D - 0 1

+ 0 . 1 0 7 3 1 9 8 8 0 0 3 2 3 4 3 3 D + 0 0

- 0 . 3 2 9 0 4 2 5 8 4 7 2 0 0 1 7 7 D - 0 1

+ 0 . 6 2 8 0 5 2 8 4 8 1 6 0 2 9 6 4 D + 0 0

- 0 . 7 3 2 9 7 2 3 6 7 0 3 4 4 9 8 3 D + 0 0

+ 0 . 5 1 0 2 0 2 2 4 0 2 3 5 9 1 7 5 D + 0 0

( 0 0 ) ( 0 1 ) ( 1 0 ) ( 0 2 ) ( 1 1 ) ( 2 0 ) ( 0 3 ) ( 1 2 ) ( 2 1 )

~descoetaL/Compu~rPh~icsCommunications106(199~ 169-180

+0.3762293050242204D-02 ( 3 O)

179

Tune y

+0.3100000000000000D+00 +0.3166690041697037D-02

+0.8320765209941094D-01 +0.7952782521579365D-02

+0.1390856354792390D+00 +0.5365994001617163D-01 -0.4419383404440940D-01 +0.1884158544480889D+01 -0.7329723670344983D+00 +0.1700674134119725D+00

( o o ) ( 0 1 ) ( 1 o ) ( o 2 ) (11) (2o) (03) ( 1 2 ) ( 2 1 ) ( 3 0 )

FOOT01.DAT

0.28000 0.31000 0.28030 0.31001 0.28060 0.31002 0.28090 0.31003 0.28120 0.31005 0.28151 0.31006 0.28182 0.31007

o . . . . .

HAMI01.DAT

Resonance [1,-4] Resonance excited at order 5 Chain of I islands .in thn ~p:c plane and of 4 islands in the y,py plane

2 ( 0 1 O) 0,59136D-01 +0.31416D+01 2 ( 1 0 O) 0.14784D-01 +O.O0000D+O0 0.01848 4 ( 0 2 O) 0.99485D-02 -0o69748D-15 4 ( 1 1 O) 0,52281D+00 -0.20572D-15 4 ( 2 0 O) 0.20713D-01 +0.31416D+01 0.03459 5 ( 0 0 1) 0.36738D+00 +0.30159D+01 6 ( 0 3 O) 0.16656D-01 -0.69432D-16 6 ( 1 2 O) 0.43695D+00 +0.69079D-15 6 ( 2 1 O) 0.33716D+00 +0.55568D-15 6 ( 3 0 O) 0.68915D-01 -0.31416D+01 0.01343 7 ( 0 1 1) 0.95905D+00 +0.30159D+01 7 ( 1 0 1) 0.46874D+00 +0.30159D+01 8 ( 0 4 O) 0.64834D-01 -0.19673D-15 8 ( 1 3 O) 0.17981D+01 +0.3767!D-15 8 ( 2 2 O) 0.23027D+01 +0.31416D+01 8 ( 3 1 O) 0.10686D+01 +0.49352D-15 8 ( 4 0 O) 0.59098D-02 +0.56413D-15 0.02047 9 ( 0 2 1) 0.28416D+01 +0.30159D+01

0.01148

0 .00749 0 .00366

0 .00555

i 80 E. Todesco et al./C¢Jmputer Physics Communications 106 (1997) 169-180

Acknowledgements

We wish to thank G. Turchetti and W. Scandale tbr stimulating this work and for helpful discussions. We also want to acknowledge A. Bazzani for providing the code ARES that computes the normal form and the i n t ~ l a t i n g Hamiltonian series. A special thank is due to E Schmidt for valuable help in checking the code with tracking simulations for complicated lattices and for useful discussions. We want to thank the Accelerator Physics group of the CERN SL Division for financial support; the work has also been partially supported by E¢,,: Human Capital and Mobility Contract No. ERBCHRXC~40480.

References

!11 121 131 141 151 161 I71 181 191

!101 l l l l 1!21 1131 [141 1151 [16l ii7l 1181 !!91 1201

The LHC Study Group, CERN AC (LHC) 95-05 (1995). H. Grote and E C. Iselin, CERN SL (AP) 90-13 (1990). E Schmidt, CERN SL (AP) 94-56 (1994). A. Schoch, CERN 57-21 (1957). G. Guignard, CERN 76-06 (1976). E Willeke, DESY HERA 87-12 (1987). A. B ~ ' ~ i , P. Ma~.anti, G. Servizi and G. Turchetti, Nuovo Cim. B 102 (1988) 51-80. E. Forest, M. Berz and J. Irwin, Part. Accel. 24 (1989) 91-113. A. Bazzani, M. Giovannozzi, G. Servizi, E. Todesco and G. Turchetti, Physica D 64 (1993) 66-93. A. Bazzani, E. Todesco, G. Turchetti and G. Servizi, CERN 94-02 (1994). E. Todesco, Phys. Rev. E 50 (1994) R4298-301. G. H~A,ter and S. Wiggins, Physica D 90 (1996) 3i9-36.5. E. Todesco and M. Gemini, phys. Rev. E, submitted. W. Scandale, E Schmidt and E. Todesco, Part. Accel. 35 ( 1991 ) 33-88. M. Giovannozzi, R. Grassi, W. Scandale and E. Todesco, phys. Rev. E 52 (1995) 3093-101. A. Bazzani, M. Giovannozzi and E. Todesco, Compet. Phys. Comm. 86 (1995) 199-207. j. Laskar, C. Frocschh~ and A. Ceiletti, physica D 56 (1992) 253-69. J. Laskar, Physica D 67 (i992) 257-81. M. Be~, Part. Accel. 24 (1989) 109. g. Bartolini, A. Bazzani, M. Giovannozzi, W. Scandale and E. Todesco, Tune evaluation in simulations and cxpcriments, CERN SL (AP) 95-84 ( 1995): Part. Accel., in press.