*H. Mais* `mais@mail.desy.de`

DESY3mm

In this short contribution I want to describe some of the activities and
goals of the European collaboration **Nonlinear Problems in
Beam Dynamics and Transport**. This collaboration is part of the
Human Capital Mobility program of the European Community and was
approved in 1994 by the European Commission. The program will run
for 39 months starting from September 1994.
The participating universities and
laboratories (together with the local coordinators) are:

- University of Bologna (G. Turchetti)
- University of Madrid (L. Vazquez)
- University of Patras (T. Bountis)
- CERN, Geneva (W. Scandale)
- DESY, Hamburg (H. Mais)
- ENEA, Frascati (G. Dattoli)
- GSI, Darmstadt (I. Hofmann)
- LNL, Legnaro (A. Dainelli)

The coordinator of the whole net is G. Turchetti from Bologna. The financial support of 150 000 ECU ( or approximately 300 000 DM) is mainly used to pay postdocs and doctoral students working at Bologna, DESY and GSI. Altogether, about 30 scientists are involved in this collaboration.

The scientific program can be divided into three main parts:

- stability and structure of phase space
- diffusion and transport
- high intensity and collective phenomena

- dynamic aperture calculations and scaling laws; application to the LHC lattice
- geometry of 4D maps
- resonance network in action space from Fourier and normal forms
- diffusion of action in stochastically perturbed 4D maps; comparison of tracking and Fokker-Planck solutions for a model of the SPS
- diffusion of action due to periodic modulation; description by adiabatic theory
- modelling of the sorting problem for random magnetic errors
- development of numerical schemes to solve the Fokker-Planck equation with 3 or 4 independent variables; application to stochastic beam dynamics (rf-noise, beam-beam interaction)
- development of symplectic schemes for transport equations in FEL
- symplectic high order algorithms to compute transfer maps
- studies of the nonlinearities in the SIS lattice at GSI; dynamic aperture with random and systematic sextupoles
- development of the beam halo with space charge forces and lattice nonlinearities
- algorithms for tracking on parallel architectures

The results of these investigations are published in joint publications some of which are listed at the end of this report.

This European net has now been in existence for over two years and has proven to be very efficient. The regular exchange of ideas and the frequent visits of the researchers involved to the various partner institutions not only strengthens the collaboration between universities and laboratories but also provides an excellent transfer of knowledge and expertise. Furthermore, this collaboration is ideally suited for training and educating young scientists in the field of beam dynamics.

- G. Mahmoud, T. Bountis, G. Turchetti "Synchronization in
parametrically driven Hamiltonian systems" Il Nuovo Cimento
**B110**, 1311 (1995) - A. Bazzani, M. Giovannozzi, E. Todesco "A program to compute
Birkhoff normal forms of symplectic maps in "
Comp. Phys. Comm.
**86**, 199 (1995) - M. Giovannozzi, R. Grassi, W. Scandale, E. Todesco "Sorting
approach to magnetic random errors" Phys. Rev.
**E52**, 3093 (1995) - G. Servizi, D. Bortolotti, E. Todesco, M. Giovannozzi, M.N.
Vrahatis "GIOTTO: a code for the nonlinear analysis of area
preserving mappings" Int. J. Mod. Phys
**C6**, 651 (1995) - E. Todesco, M. Giovannozzi "Dynamic aperture estimates
and phase space distortions in nonlinear betatronic motion"
Phys. Rev.
**E53**, 4067 (1996) - A. Bazzani, R. Bartolini, M. Giovannozzi, W. Scandale, E. Todesco
"Tune evaluation in simulations and experiments" Part. Accel.
**52**, 147 (1996) - M. Giovannozzi, E. Todesco, A. Bazzani, R. Bartolini "PLATO: a program library for the analysis of nonlinear betatronic motion" Nucl. Instrum. and Meth. in press (1996)
- G. Dattoli, P.L. Ottaviani, A. Torre, L. Vazquez "Evolution of operator equations: integration with algebraic and finite difference methods, application to physical problems" La rivista del Nuovo Cimento to be published (1996)
- E. Todesco, M. Giovannozzi, W. Scandale "Fast indicators of long-term stability" Part. Accel. to appear (1996)
- M. Giovannozzi, E. Todesco "Numerical methods to estimate the dynamic aperture" Part. Accel. to appear (1996)
- R. Bartolini, A. Bazzani, M. Giovannozzi, W. Scandale, E. Todesco "Precise measurement of the betatron tune" Part. Accel. to appear (1996)