Joint Universities Accelerator School
Non-linear effects

a) Accelerator performance parameters and non-linear effects


 b) Linear and non-linear oscillators: Integral and frequency of motion, the pendulum, damped harmonic oscillator


 c) Phase space dynamics: Fixed point analysis


 d) Non-autonomous systems: Driven (damped) harmonic oscillator, resonance conditions


 e) Linear equations with periodic coefficients – Hill’s equations: Floquet solutions and normalized coordinate


 f) Perturbation theory: Non-linear oscillator, Perturbation  by periodic function – single dipole perturbation, Application to single multipole – resonance conditions, Examples: single quadrupole, sextupole, octupole perturbation, General multi-pole perturbation, examples: linear coupling, Resonance conditions and working point choice


 g) Resonances and the path to chaos: Topology of 3rd and 4th order resonance, Path to chaos and resonance overlap, Dynamic aperture


 h) Frequency map analysis: NAFF algorithm, Aspects of frequency maps, Frequency and diffusion maps for the LHC, Frequency map for lepton rings, Working point choice, Beam-beam effect


 i) Experiments: Experimental frequency maps, Beam loss frequency maps, Space-charge frequency scan