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Tangency bifurcations of global Poincaré maps

Osinga, H (Bristol)
Thursday 02 July 2009, 16:30-17:30

Seminar Room 1, Newton Institute


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 (abstract or preprint)
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 (abstract or preprint)
Bernd Krauskopf, Clare M. Lee & Hinke M. Osinga
Nonlinearity 22(5): 1091-1121, 2009.

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