A 4D quadratic map can be used to represent the transfer map of a FODO cell with a sextupolar nonlinearity in the kick approximation. This map describes the transverse betatronic motion of a charged p article in a circular accelerator. The dynamic aperture of such a mapping is analysed, i.e., the domain in phase space where stable motion occurs, as a function of the linear tunes. Starting from the study of the stability properties of the fixed points of low period (one or two), it is shown that the dynamic aperture is related to the invariant manifolds emanating from unstalbe points. This repre nsets a generalisation of a similar result obtained for generic two-dimensional sympletic maps

Giovannozzi, Massimo

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STUDY OF THE DYNAMIC APERTURE OF THE 4D QUADRATIC MAPUSING INVARIANT MANIFOLDSM. Giovannozzi, CERN, Geneva, SwitzerlandAbstractA 4D quadratic map can be used to represent the transfermap of a FODO cell with a sextupolar nonlinearity in thekick approximation. This map describes the transverse be-tatronic motion of a charged particle in a circular accelera-tor. The dynamic aperture of such a mapping is analysed,i.e. the domain in phase space where stable motion occurs,as a function of the linear tunes. Starting from the studyof the stability properties of the fixed points of low period(one or two), it is shown that the dynamic aperture is re-lated to the invariant manifolds emanating from unstablepoints. This represents a generalisation of a similar resultobtained for generic two-dimensional symplectic maps.1 INTRODUCTIONThe evaluation of the dynamic aperture (DA), which is thevolume in phase space where stable motion occurs, of four-dimensional dynamical systems, is based on CPU-time in-tense simulations which do produce numerical informa-tion, but cannot provide a deep insight into the phenomenawhich determine the DA.In this paper, a new method to estimate the stability do-main of a four-dimensional system is presented. It is basedon the properties of low-order unstable fixed points and it isa generalisation of a technique successfully applied to two-dimensional Hamiltonian systems [1, 2, 3]. The method isbased on the construction of the invariant manifolds ema-nating from the unstable fixed points (such manifolds areequivalent to the separatrices for continuous-time Hamilto-nian systems) of low period. Thanks to the phenomenonof homoclinic/heteroclinic intersections, these manifoldsform a dense network in phase space which extends fromthe outer, unstable part of phase space to the DA. Hencefrom the knowledge of the unstable fixed points and theinvariant manifolds it is possible to deduce the stability do-main of the dynamical system.In the paper a proof-of-principle of this method is pre-sented using a 4D quadratic map, the generalisation of thewell-known 2D area-preserving He´non map. The model isanalysed and its DA is evaluated using the standard numeri-cal methods [4]. Using a simplified technique, the invariantmanifolds are numerically constructed and it is shown howthe DA can be deduced.2 THE MODELThe starting point of the analysis is the transfer map Mof a FODO cell with a sextupole magnet represented by asextupolar kick. In case of absence of linear coupling, thelinear part L of the transfer function can be written as ablock diagonal matrixL =Mx00 My; (1)while the nonlinear kick can be expressed as a polynomialfunction of degree two, namelyx1= x0x01= x00+K2(x20  y20)y1= y0y01= y00  2K2x0y0;where K2is the integrated sextupolar gradient, given byK2=ZL1B00@2By@x2(0;0;s)d s: (2)The global transfer map is given by the composition of Land the kick0BB@x1x01y1y011CCA=Mx00 My0BB@x0x00+K2(x20  y20)y0y00  2K2x0y01CCA:(3)Eq. (3) can be cast in a different form by means of the stan-dard transformation to normalised coordinates (Courant-Snyder variables), together with a rescaling of the physicalvariables by the quantity 3=2xK20BB@x1x01y1y011CCA= R(!1; !2)0BB@x0x00+ (x20   y20)y0y00+ 2 x0y01CCA: (4)In the previous equation, R(!1; !2) is a 4D block diago-nal matrix representing a 2D rotation in each phase spaceplane by an angle !i= 2i, while the dimensionless pa-rameter  represents the ratio of the vertical to the horizon-tal -functions at the centre of the sextupole. The trans-fer function (4) is a symplectic quadratic polynomial mapH and it represents the natural generalisation of the well-known 2D He´non map [5]. It has two remarkable proper-ties:P1: In the limit  ! 0 the two phase planes (x; x0) and(y; y0) are decoupled. In fact, under this condition, the mapreduces to a 2D He´non map in the (x; x0) plane and a linearrotation in the (y; y0) plane;P2: the plane (x; x0) is invariant under the action of the 4Dmap (4).3 FIXED POINTSA fixed point of a 4D map G is a root of the polynomialequationG(x) = x; x 2 R4: (5)The stability properties of the solutions of Eq. (5), whichis actually a system of four nonlinear coupled equations,derive from the linearisation GLof the map G around thefixed point. More precisely, the eigenvalues of GLdeter-mine whether the fixed point is unstable (also called hyper-bolic) or stable (also called elliptic). The classification ofthe fixed points is similar to the one used for the 2D maps,although the number of possible cases is bigger.Furthermore, it is possible to define higher-order fixedpoints (also called cycles or periodic orbits) by replacing Gin Eq. (5) with the nth power of the polynomial mapGn(x) = x; x 2 R4: (6)As far as the fixed points of the map H with n = 1; 2 areconcerned, it is possible to show the following results [6]R1: Four fixed points exist. One is the origin (0; 0; 0; 0)and the others arex2D= 2 tan!12x02D=  x2Dtan!12y2D= 0 y02D= 0andx=  1tan!22x0=  xtan!12y= 112tan2!22+22tan!12tan!221=2y0=  ytan!22:By definition, the origin is always stable. The second fixedpoint x2Dis real for all the values of the linear tunes !1; !2.It coincides with the unique hyperbolic fixed point of the2D He´non map [5]: this is a consequence of the propertyP2. In the (x; x0) plane this fixed point is always hyper-bolic, while in the (y; y0) its stability type changes fromelliptic to hyperbolic, according to the value of the lineartunes.The third and fourth fixed points are real provided thefollowing condition is satisfied12tan2!22+22tan!12tan!22 0: (7)They have the same stability type which varies as a functionof the linear tunes and ;R2: No real fixed point of period two exists for the 4DHe´non map.4 INVARIANT MANIFOLDSFor a hyperbolic fixed point xhypof a 4D map, the eigen-vectors of the linearisation of the map GLaround xhypde-fine two 2D planes in the 4D phase space, where the motioninduced by the linearised map has an expanding or a con-tracting behaviour.We can extend these sets to the original non-linear mapG, i.e. it is possible to define two manifolds emanating fromthe unstable fixed point, called Wu(xhyp) and Ws(xhyp),having the same expanding (superscript u) or contracting(superscript s) behaviour. The eigenvectors of GLare tan-gential toWu;s(xhyp) at the fixed point.The invariant manifolds have at least the hyperbolic fixedpoint as intersection. An additional intersection, xhom,is either called homoclinic or heteroclinic depending onwhether the two intersecting manifolds emanate from thesame hyperbolic fixed point. Provided the two manifoldsare non-tangential at the point xhom, they will oscillatearound each other generating a 2D object which fills mostof the phase space of the system (at least the unstable partof the phase space).It is not a trivial task to numerically construct such amanifold. The problem has been completely solved for 1Dinvariant manifolds and efficient algorithms have been de-veloped. However, in this case even the straightforwardmethod, based on the iteration of a set of initial conditionschosen on the eigenvectors of the linearised map, providesexcellent results [1, 2, 3].The straightforward generalisation of the previousmethod to the 4D case, could lead to some problems asthe different expansion/contraction rates could stretch onedirection more than the others, thus collapsing the recon-structed invariant manifold to an-almost-1D object. Re-cently, some new methods have been proposed to pro-vide an accurate reconstruction of the 2D invariant mani-folds [7, 8].For the proof-of-principle of the method presented in thispaper, the invariant manifolds have been constructed bysimply computing the evolution of a set of initial conditionsdistributed in a 4D neighbourhood of the unstable fixedpoint of the He´non map: those initial conditions which donot belong to the invariant manifolds are lost due to the hy-perbolic dynamics, while the others allow the reconstruc-tion the manifold.5 SIMULATION RESULTSThe starting point of this study is the numerical evaluationof the DA of the 4D He´non map as a function of the lin-ear tunes 1; 2and the parameter . The stability domainD(N) in the transverse four-dimensional phase space isgiven by the volume containing all the initial conditionsthat are stable at least for N turns. It has been shown [4]that D(N) can be computed by iterating a set of initial con-ditions (x; 0; y; 0) withx = r cos ; y = r sin ; r 2 [0; R];  2 [0; =4[using the following formulaD(N) = Z=20[r(;N)]4sin 2d!1=4(8)where r(;N) is the average amplitude along the last sta-ble orbit for a given  [4]. In Fig. 5 the stability domain isshown for the He´non map computed using N = 1000 and = 1 as a function of the linear tunes. One can clearlyFigure 1: Dynamic aperture D(N) for the 4D He´non mapas a function of the linear tunes 1; 2for N = 1000 and = 1.see the unstable resonances (third order), where the motionis totally unstable and the DA is zero. Other resonancesare also visible, but they have a smaller impact on the DA.In Fig. 5 the DA for N = 1000 and  = 0:1 is shown.The scenario is rather different now: most of the harmfulFigure 2: Dynamic aperture D(N) for the 4D He´non mapas a function of the linear tunes 1; 2for N = 1000 and = 0:1.resonances have disappeared and D(N) shows only twovertical and two diagonal third order resonances. Further-more, D(N) show almost no sensitivity to !2: this is aconsequence of the property P2 as in this case the nonlin-ear coupling between the two planes is weaker.Finally Fig. 5 shows the minimum distance of the unsta-ble invariant manifold emanating from the fixed point x2Dfrom the origin ( = 0:1). According to the assumptionsmade, the invariant manifolds should surround the regionwhere stable motion occurs. Hence, the minimum distanceof the manifold from the origin should give an estimate ofthe DA. In Fig. 5 one clearly detects some features similarFigure 3: Minimum distance from the origin of Wu(x2D)as a function of the linear tunes 1; 2for  = 0:1. Theinitial conditions are iterated 1000 times.to those shown by the DA, namely the unstable third orderresonance (vertical lines) and also a trace of the coupledthird order resonance (diagonal lines).6 CONCLUSIONSThe numerical simulations showed that the invariant man-ifolds allow to reproduce the main features of the dynamicaperture. Although the numerical values of the DA and theminimum distance of the invariant manifolds agree within30-40 %, we are confident that this can be improved by us-ing more sophisticated methods to reconstruct the invariantmanifolds.7 REFERENCES[1] M. Giovannozzi, Phys. Lett. A 182, (1993) 255.[2] M. Giovannozzi, Phys. Rev. E 53, (1996) 6403.[3] M. Giovannozzi, Cel. Mech. 68, (1997) 177.[4] E. Todesco, M. Giovannozzi, Phys. Rev. E 53, (1996) 4067.[5] M. He´non, Q. Appl. Math. 27, (1969) 291.[6] M. Giovannozzi, in preparation , (1998) .[7] M. Dellnitz, A. Hohmann, Numer. Math. 75, (1997) 293.[8] H. W. Broer, H. M. Osinga and G. Vegter, Z. angew. Math.Phys. 48, (1997) 480.

Study of the dynamic aperture of the 4D quadratic map using invariant manifolds

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