Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386902
Title: The classification of bifurcations in maps with symmetry
Author: Brown, Anthony Graham
ISNI:       0000 0001 3484 8817
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1992
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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.
Supervisor: Not available Sponsor: Science and Engineering Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.386902  DOI: Not available
Keywords: QA Mathematics
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