G. Servizi

Normal forms for Hamiltonian maps and nonlinear effects in a particle accelerator

SummaryWe describe the motion of a particle in the lattice of a hadron accelerator using the formalism of symplectic maps. We revisit the Courant-Snyder’s theory and we stress that the reduction to normal form of a symplectic map is just the natural generalization of the linear theory. We show that a simple FODO cell (formed by linear elements and a sextupole) can be reduced to a quadratic map, for which some results are presented. We show also that it is possible to recover a continuum limit from a discrete description of the lattice. Finally a study of a one-dimensional model for LHC is made. The normal forms are used to compute the tune shifts and smear in various configurations. A comparison is made with the tracking results and an excellent agreement is found in a region whose boundary is close to the dynamical aperture. This suggests that the two-dimensional extension of the method is well suited to treat the coupled betatron nonlinear oscillations.RiassuntoSi descrive il moto di una particella in un acceleratore di adroni, usando il formalismo delle mappe simplettiche. Viene rivista la teoria di Courant-Snyder e si mostra che la riduzione in forma normale di una mappa simplettica è la naturale generalizzazione del caso lineare. Si prova che una cella FODO semplice (composta da elementi lineari e da un sestupolo) può essere ricondotta a una mappa quadratica, per la quale si presentano alcuni risultati e si mostra che è possibile recuperare il limite continuo partendo da una descrizione discreta del sistema. Infine si effettua uno studio di un modello unidimensionale per LHC; si usano le forme normali per calcolare il «tune shift» e lo «smear» in varie configurazioni, si fa un confronto con i risultati del «tracking» e si riscontra un eccellente accordo in una zona il cui limite è non lontano dall’apertura dinamica. Questo suggerisce che l’estensione del metodo a due dimensioni potrà essere un valido strumento per trattare le oscillazioni nonlineari accoppiate di betatrone.РезюмеМы описьваем движение частицы в решетке адронного ускорителя, используя формализм симплексных отображений. В связи с этим мы заново рассматриваем теорию Куранта-Снайдера. Мы подчеркиваем, что преобразование к нормальной форме симплексного отображения, в нерезонансном случае, представляет естественное обобщение теории Куранта-Снайдера, чтобы включить сдвига настройки и для размывания квадратичного обображения. Мы показываем, что можно получить непрерывный предел из дискетного описания решетки. Исследется одномерная модель для большого адронного коллайдера. Мы вычисляем сдвиг настройки и размывание в различных ситуациях, используя метод нормальных сдвиг настройки и размывание в различных ситуациях, используя метод нормальных форм в программе. Мы сравниваем результаты для нормальных форм в различных порядках возмущений с результатами, полученными с использованием модели тонких линз для той же решетки.

Resonant normal forms, interpolating Hamiltonians and stability analysis of area preserving maps

Abstract The geometrical and dynamical properties of area preserving maps in the neighborhood of an elliptic fixed point are analyzed in the framework of resonant normal forms. The interpolating flow is not obtained from a map tangent to the identity, but from the normal form of the given map and a time independent interpolating Hamiltonian H is introduced. On this Hamiltonian the local stability properties of the fixed point and the geometric structure of the orbits are transparent. Numerical agreement between the level lines of H and the orbits of the map suggests that the perturbative expansion of H is asymptotic. This is confirmed by a rigorous error analysis, based on majorant series: the error for the normal form expansion grows as n! while the truncation error for H also has a factorial growth and in a disc of radius r can be made exponentially small with 1/r. The boundary of the global stability domain is considered; for the quadratic map the identification with the inner envelope of the homoclinic tangle of the hyperbolic fixed point is strongly suggested by numerical evidence.

Soil amplification of plane seismic waves

Abstract Amplitudes of surface particle velocities are calculated when time-harmonic seismic waves of uniform amplitude are incident upon an arbitrary stratified elastic soil layer from the underlying bedrock. Whereas previous workers have mainly treated normally incident S waves, we allow the waves to be of SV, P, or SH types and to have arbitrary angles of incidence. Following standard practice the problem is set up as a matrix differential system, but in such a way that incident SV and P waves may be treated together (the system for SH decoupling). Though complicated, the 4 × 4 SVP system has considerable structure which is elucidated in Appendices 1 and 2. These results, though not altogether new, are of independent interest, and are gathered together in concise form for reference. The theory for low- and for high-frequency approximations is given. The main results of the work are illustrated by two numerical examples: Model 1 where the soil layer is homogeneous; and Model 2 where the soil layer has a linear velocity profile.

Resonances and asymptotic behavior of Birkhoff series

Abstract For hamiltonian systems with two degrees of freedom a mechanism accounting for the divergence of perturbation series and the asymptotic relation between true and formal dynamics is proposed. In the special case of conservative quadratic maps numerical and analytical support is given for a piecewise geometric structure of the Birkhoff series, that is a sequence of pseudoconvergence radii is found which decreases to zero and is associated with the resonances approaching the rotation angle of the linear map.

Topological degree theory and local analysis of area preserving maps

We consider methods based on the topological degree theory to compute periodic orbits of area preserving maps. Numerical approximations to the Kronecker integral give the number of fixed points of the map provided that the integration step is “small enough.” Since in any neighborhood of a fixed point the map gets four different combinations of its algebraic signs we use points on a lattice to detect the candidate fixed points by selecting boxes whose corners show all combinations of signs. This method and the Kronecker integral can be applied to bounded continuous maps such as the beam–beam map. On the other hand, they cannot be applied to maps defined on the torus, such as the standard map which has discontinuity curves propagating by iteration. Although the use of the characteristic bisection method is, in some cases, unable to detect all fixed points up to a given order, their distribution gives us a clear picture of the dynamics of the map.

A chronotopic model of mobility in urban spaces

In this paper, we propose an urban mobility model based on individual stochastic dynamics driven by the chronotopic action with a deterministic public transportation network. Such a model is inspired by a new approach to the problem of urban mobility that focuses the attention to the individuals and considers the presence of random components and attractive areas (chronotopoi), an essential ingredient to understand the citizens dynamics in the modern cities. The computer simulation of the model allows virtual experiments on urban spaces that describe the mobility as the evolution of a non-equilibrium system. In the absence of chronotopoi the relaxation to a stationary state is studied by the mean-field equations. When the chronotopoi are switched on the different classes of people feel an attraction toward the chronotopic areas proportional to a power law of the distance. In such a case, a theoretical description of the average evolution is obtained by using two diffusion equations coupled by local mean-field equations.

Mellin transforms and correlation dimensions

We consider the Mellin transform of the correlation integrals and show that the divergence abscissa is the correlation dimension. The analytic structure of the Mellin transform is explicitly described for some Julia and Cantor sets. The existence of oscillations in the correlation integral for the Cantor sets is proved. Extensions of the results to the order d correlation integrals are discussed.

Generalized dynamical variables and measures for the Julia sets

SummaryWe give rigorous estimates of the dimensions, entropies, characteristic exponents and scaling function of hyperbolic Julia sets, for any Gibbs measure, by the direct computations of the topological pressure.RiassuntoTramite il calcolo diretto della pressione topologica sono fornite stime rigorose su dimensioni generalizzate, entropie, indici di Lyapunov e funzioni di scala di Julia sets iperbolici per una misura di Gibbs qualsiasi.РезюмеИспользуя непосредс твенные вычисления топологического дав ления, мы приводим строгие о ценки размеров, энтро пии, характерных показат елей экспонент и функции п одобия для гиперболи ческих систем Джулиа, для про извольной меры Гиббса. Затем пре дложенный метод обоб щается на случай негиперболич еских систем Джулиа.

Perturbative expansions for area-preserving maps

SummaryThe structure of perturbation series for area-preserving maps is investigated. A basically different behaviour is found between the Birkhoff series, which formally conjugate with circles all the orbits in a neighbourhood of the origin, and the series which map into circles the individual invariant curves with fixed diophantine winding number. The former series exhibit an asymptotic character, the latter a convergent one, as one should expect from the KAM theorem. The source of this difference is found to be the different way in which the contributions of the relevant resonances propagate. In the first case, if ε is the size of the divisor associated to a resonanceM/N, then at each ordern> N an ε−1 contribution occurs, in the second case subtle cancellations provide a new ε−1 only when a harmonic (that isn =pN) is reached. This precise asymptotic statement and the properties of the relevant resonances obtained from the continued fraction expansion allow us, in the case of quadratic irrationals, to understand the limit process which leads to divergence or convergence. In the divergent case the asymptotic properties of the series are exhaustively described.RiassuntoSi studia la struttura délie serie perturbative per mappe conservative e si riscontra una sostanziale diversità di comportamento tra le série di Birkhoff, che coniugano formalmente con cerchi tutte le orbite in un intorno dell’origine, e le serie che trasformano in cerchi le singole curve invariant! con un fissato numero di rotazione diofantino ; le prime hanno carattere asintotico, le seconde sono convergenti, in accordo col teorema KAM. La causa di tale diversità risiede nel diverso modo di propagazione dei contributi delle risonanze: nel primo caso, dettoε l’ordine di grandezza del denominatore associato alla risonanzaM/N, ad ogni ordinen > N si ha un contributo di ordine ε-1 nel secondo caso si hanno cancellazioni tali da fornire termini di ordine ε-1 solo quando si raggiunge un ordine multiplo della risonanza (cioèn =pN). Ciò, congiuntamente alle proprietà delle risonanze connesse con lo sviluppo in frazione continua, consente di comprendere, nel caso di numeri irrazionali quadratici, il processo che porta a serie convergenti o divergenti, di cui si descrivono ampiamente le proprietà asintotiche.РезюмеИсследуется структу ра пертурбационных рядов для отображени й, сохраняющих площадь. Обнаружено существе нное различие в поведении рядов Биркхоффа, которые фо рмально сопряжены с к ругами всех орбит в окресности на чала отсчета, и рядов, Котор ые отображают в круги отдельные инвариантные кривые с фиксированным числ ом диофантных вращен ий. Первые ряды имеют асимптоти ческий характер, а вторые ряд ы являются сходящими ся, согласно теореме КАМ. Причина такого разли чия связана с различн ым распространением вк ладов соответствующ их резонансов. В первом с лучае, если е есть поря док величины знаменател я, связанного с резона нсом M/N, то в каждом поря дке п > N возникает вкла д ε-1; каждом порядке п > N воз никает вклад ε-1; во втором случае тонк ая взаимная компенса ция дает новый вклад ε-1, только когда достигается кратный резонанс (т.е. если от = pN). Точное асимптотическое пов едение и свойства соответст вующих резонансов, по лученные из разложения непрерыв ной дроби, позволяют, в слу чае квадратичных иррациональностей, п онять предельный процесс, который прив одит к сходимости или расходимости. В расхо дящемся случае подробно опис ываются асимптотиче ские свойства рядов.

Bifurcations of beam-beam like maps

The bifurcations of a class of mappings including the beam-beam map are examined. These maps are asymptotically linear at infinity where they exhibit invariant curves and elliptic periodic points. The dynamical behaviour is radically different with respect to the Henon-like polynomial maps whose stability boundary (dynamic aperture) is at a finite distance. Rather than the period-doubling bifurcations exhibited by the Henon-like maps, we observe a systematic appearance of tangent bifurcations and in phase space one observes the disappearance of chains of islands born from the origin and coming from infinity. This behaviour has relevant consequences on the transport process.