Bifurcations

This book collects several works, giving an overview of some models and theories, which are useful for the study and analysis of complex biological and physical systems. It can provide a good guidance for physicists with interest in biology, applied research scientists and postgraduate students.

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 subject of differential and difference equations is an old and much-honored chapter in science, one which germinated in applied fields such as celestial mechanics, nonlinear oscillations, and fluid dynamics. In recent years, due primarily to the proliferation of computers, dynamical systems has once more turned to its roots in applications with perhaps a more mature look.

The material in this book is based predominantly on my recent work. It is the first monograph on the subject, though some support material may overlap other monographs. The investigation of wave packets and their bi­ furcations is very interesting, and useful theoretically and in practice, not only in geophysical fluid dynamics, which is the field to which the theory is being applied here, but also in other fields in mathematics and the natural sciences.