In this thesis we study the coalescence of invariant curves in certain codimension-two bifurcations of families of mappings. This phenomenon has been described in considerable detail by A Chenciner [Bifurcations de points fixes elliptiques: I - Courbes invariants, Publ. Math. I.H.E.S. 61, p67-127, 1985] in families of mappings which exhibit the Hopf bifurcation with higher order degeneracy. We give a review of some of his reviews in Chapter 1. In Chapter 2 we study the coalescence of invariant curves in two-parameter families of mappings which exhibit the Hopf bifurcation with 1:2 strong resonance. That is, families of the form <I>f<SUB>μ,v</SUB></I> : <I>U </I>→ R<SUP>2</SUP>, where <I>U</I> is an open subset of R<SUP>2</SUP> containing the origin, such that <I>f<SUB>μ, v</SUB></I> (0) = 0 and <I>Df<SUB>0,0</SUB></I> (0) has -1 as a double eigenvalue. We obtain a first return map for <I>f<SUB>μ,v</SUB></I> on a suitably defined region which shows that the coalescence of invariant curves in this case occurs in much the same way as described by Chenciner. The derivation of the first return map is dependent on the value of a complex valued functional χ which depends analytically on the entire Taylor expansion of f<SUB>0,0</SUB>. We require χ<I> </I>(f<SUB>0,0</SUB>) - 0. In Chapter 3 we describe some of the properties of χ and develop numerical methods for estimating χ(<I>g</I>), where <I>g </I>is a polynomial of degree ≤ 2 with linear part (<SUP>-1</SUP><SUB>0 </SUB><SUP>-1</SUP><SUB>-1</SUB>). Finally we use the numerical estimation methods to investigate the set {<I>g : </I>χ<I>(g) </I>= 0}.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:643296 |
| Date | January 1999 |
| Creators | Cockburn, James Keith |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/11996 |
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