Stability and Geometry of Third-order Resonances in Four-dimensional Symplectic Mappings

Nero: a Code for Evaluation of Nonlinear Resonances in 4d Symplectic Mappings

A code to evaluate the stability, the position and the width of nonlinear resonances in four-dimensional symplectic mappings is described. NERO is based on the computation of the resonant perturbative series through the use of Lie transformation implemented in the code ARES, and on the analysis of the resonant orbits of the interpolating Hamiltonian. The code is aimed at studying the nonlinear ...

Local Analysis of Formal Stability and Existence of Fixed Points in 4d Symplectic Mappings

OF FIXED POINTS IN 4D SYMPLECTIC MAPPINGS E. Todesco INFN, Sezione di Bologna, Via Irnerio 46, 40126, Bologna, Italy. Abstract Symplectic mappings in a four-dimensional phase space are analysed; in the neighbourhood of an elliptic xed point whose eigenvalues are close to satisfy a double resonance condition, the perturbative tools of resonant normal forms are outlined. The analysis of the inter...

A New High Order Closed Newton-Cotes Trigonometrically-fitted Formulae for the Numerical Solution of the Schrodinger Equation

In this paper, we investigate the connection between closed Newton-Cotes formulae, trigonometrically-fitted methods, symplectic integrators and efficient integration of the Schr¨odinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant lit...

Resonant Normal Forms for Four Dimensional Symplectic Mappings and Applications to Nonlinear Beam Dynamics

2 00 8 The Symplectic Geometry of Penrose Rhombus

The purpose of this article is to view Penrose rhombus tilings from the perspective of symplectic geometry. We show that each thick rhombus in such a tiling can be naturally associated to a highly singular 4–dimensional compact symplectic space MR, while each thin rhombus can be associated to another such space Mr; both spaces are invariant under the Hamiltonian action of a 2–dimensional quasit...