# DIS

## Seminar

### Tangency bifurcations of global Poincaré maps

Seminar Room 1, Newton Institute

#### Abstract

A common tool for analyzing the qualitative behaviour of a periodic
orbit of a vector field in *R^n* is to consider the
Poincaré return map to an *(n-1)*-dimensional section.
Poincaré used this technique to show instabilities in the solar
system and Birkhoff continued these ideas to find a Poincaré
map that gives information aobut the entire dynamics in the context of
Hamiltonian systems. For general vector fields, particularly in
experiments, people often choose an unbounded
*(n-1)*-dimensional section of *R^n* and
assume that the Poincaré map gives all the information about
the dynamics. However, for such choices there will typically be points
where the flow is tangent to the section. Such tangencies cause
bifurcations of the Poincaré return map if the section is
moved, even when there are no bifurcations in the underlying vector
field. This talk discusses the interactions of invariant manifolds
with the tangency loci on the section. Using tools from singularity
theory and flowbox theory, we present normal forms of codimension-one
tangency bifurcations in the neighbourhood of a tangency point. The
study of these bifurcations is motivated by and illustrated with
examples arising in applications.

This is joint work with Clare Lee (University of Strathclyde), Bernd Krauskopf (University of Bristol) and Pieter Collins (CWI, Amsterdam).

Tangency bifurcations of global Poincaré maps

Clare M. Lee, Pieter J. Collins, Bernd Krauskopf & Hinke M. Osinga

*SIAM Journal on Applied Dynamical Systems* **7**(3): 712-754,
2008.

Codimension-one tangency bifurcations of global Poincaré maps
of four-dimensional vector fields

Bernd Krauskopf, Clare M. Lee & Hinke M. Osinga

*Nonlinearity* **22**(5): 1091-1121, 2009.

Dr Hinke Osinga Home Page: www.enm.bris.ac.uk/staff/hinke