E-Book Overview
The authors consider applications of singularity theory and computer algebra to bifurcations of Hamiltonian dynamical systems. They restrict themselves to the case were the following simplification is possible. Near the equilibrium or (quasi-) periodic solution under consideration the linear part allows approximation by a normalized Hamiltonian system with a torus symmetry. It is assumed that reduction by this symmetry leads to a system with one degree of freedom. The volume focuses on two such reduction methods, the planar reduction (or polar coordinates) method and the reduction by the energy momentum mapping. The one-degree-of-freedom system then is tackled by singularity theory, where computer algebra, in particular, Gröbner basis techniques, are applied. The readership addressed consists of advanced graduate students and researchers in dynamical systems.
E-Book Content
Preface “How can anything a computer produces have to do with chaos? I thought computers were based on logic.” Inspector Morse, in response to an explanation of the Mandelbrot fractal. This book deals with nonlinear Hamiltonian systems, depending on parameters. Such systems occur for example in the modeling of frictionless mechanics and optics. The general goal is to understand their dynamics in a qualitative, and if possible, also quantitative way. The dynamical behavior generally is expressed in terms of equilibria, periodic and quasi periodic solutions as well as corresponding homo– and heteroclinic connections between those. Such connections often are accompanied by chaotic dynamics. In many important cases, it is possible to reduce a skeleton of the dynamics to lower dimensions, sometimes leading to a Hamiltonian system in one degree of freedom. Such reduced systems allow a singularity theory or catastrophe theory approach which gives rise to transparent, in a sense polynomial, nor