Brown, Anthony Graham (1992) The classification of bifurcations in maps with symmetry. PhD thesis, University of Warwick.

Preview

PDF WRAP_Theses_Brown_1992.pdf - Unspecified Version - Requires a PDF viewer. Download (3920Kb) | Preview

Request Changes to record.

Abstract

The aim of this work is to classify the generic codimension 1 bifurcations of a map with symmetry.

Many of the results proved here are analogous to results already proved for vector fields, in fact many of the proofs given are straightforward adaptations. However the nature of mapping allows for more complicated dynamics to be observed, dynamics which have no direct counterparts in the world of continuous flows.

We start the thesis with a review of result for non symmetric maps on R and R². We then motivate the introduction of symmetry with a system of identical coupled oscillators (simple predator prey models) which exhibit Dn symmetry, Dn is the symmetry group of an n-gon. In the next section we make the concept of symmetry more rigorous and introduce the language of groups which is a natural way to talk about symmetric systems. We also explain why symmetry effects can both complicate problems and the methods used to overcome these difficulties.

In the next chapter we prove some useful results about normal forms of symmetric mappings, these help in later chapters in the consideration of stability and bifurcation directions.

In the last two chapters we describe symmetric Hopf bifurcations, that is bifurcations to invariant circles and symmetric subharmonic bifurcations, i.e. bifurcations to periodic orbits. Using the spatial symmetries of the group action as well as the temporal symmetries which arc introduce by the existence of periodic orbits and invariant circles we can predict generically the existence of solutions.

We finish with an example of how these solutions can be calculated and interpreted for a physical system.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Bifurcation theory, Classification, Symmetry (Mathematics), Mathematical models
Official Date:	December 1992
Dates:	Date Event December 1992 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Roberts, Mark
Sponsors:	Science and Engineering Research Council (Great Britain)
Extent:	v, 128 leaves
Language:	eng

Request changes or add full text files to a record

Repository staff actions (login required)

View Item

Downloads