`UK Nonlinear News`,
`August 2001`

## Alona Ben-Tal

### Department of Mathematics, University of Auckland, New Zealand.

### Supervisors: Vivien Kirk, Geoff Nicholls.

**Abstract**: In this thesis we study a class of symmetric forced
oscillators modeled by non-linear ordinary differential equations. Solutions for
this class of systems can be symmetric or non-symmetric. When a symmetric
periodic solution loses its stability as a physical parameter is varied, and two
non-symmetric periodic solutions appear, this is called a symmetry breaking
bifurcation. In a symmetry increasing bifurcation two conjugate chaotic
attractors (i.e., attractors which are related to each other by the symmetry)
collide and form a larger symmetric chaotic attractor. Symmetry can also be
restored via explosions where, as a physical parameter is varied, two conjugate
attractors (chaotic or periodic) which do not intersect are suddenly embedded in
one symmetric attractor.

In this thesis we show that all these apparently distinct bifurcations can be
realized by a single mechanism in which two conjugate attractors collide with a
symmetric limit set. The same mechanism seems to operate for at least some
bifurcations involving non-attracting limit sets. We illustrate this point with
examples of symmetry restoration in attracting and non-attracting sets found in
the forced Duffing oscillator and in a power system. Symmetry restoration in the
power system is associated with a phenomenon known as ferroresonance.

The study of the ferroresonance phenomenon motivated this thesis. Part of
this thesis is devoted to studying one aspect of the ferroresonance phenomenon;
the appearance of a strange attractor with a band-like structure. This attractor
was called previously a `pseudo-periodic' attractor.

Some methods for analyzing the non-autonomous systems under study are shown.
We construct three different maps which highlight different features of symmetry
restoring bifurcations. One map in particular captures the symmetry of a
solution by sampling it every half the period of the forcing. We describe a
numerical method to construct a bifurcation diagram of periodic solutions and
present a non-standard approach for converting the forced oscillator to an
autonomous system.

A copy of the thesis is available from http://www.math.auckland.ac.nz/deptdb/staff_view.php?upi=aben031

**Source**: Alona
Ben-Tal

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