|dc.description.abstract||This dissertation is divided into two parts. The first part, which is composed of Chapters
2 - 4, present new sufficient conditions for the occurrence of the Hopf bifurcation together
with explicit description of the symmetric properties of the bifurcating branches. Chapter 2
treats a general setting in ordinary differential equations. The main novelty in these results
is that they treat cases where eigenvalues “slide” along the imaginary axis.
Chapter 3 considers the so-called interval systems which can be used to describe models with
uncertain coefficients. These systems which are considered in the field of robust control, have
been studied from the point of view of stability. The most famous of these stability results
is the Kharitonov theorm. We formulate conditions on an interval system which allows us
to guarantee the occurrence of the Hopf bifurcation in all the selectors of an interval system
using the abstract theorem proven in Chapter 2.
Chapter 4 treats a high dimensional system with a large symmetry group. It presents
results describing the existence, multiplicity, stability and symmetric properties of periodic
solutions. We use this as an example how the general theory can be applied by computing
the equivariant spectral data. This involves computing the isotypical decomposition of the
space, the isotypical multiplicities and crossing numbers of the eigenvalues and the maximal isotropies which appear in the respective irreducible representations. These computations
are then used to verify that the conditions of the equivariant Hopf bifurcation result are
satisfied. We prove the existence of branches of small cycles bifurcating from the trivial
equilibrium and describe their symmetric properties. Further using the Hirano-Rybicki type
results we prove the existence of (not necessarily small) periodic solutions with prescribed
period and symmetry.
The second part of this dissertation, which is composed of Chapters 5 and 6, deals with
the problem of stabilization of unstable periodic orbits by use of time delayed feedback.
Chapter 5 discusses a modification of the classical Pyragas control scheme to treat the case
of equivariant systems. The main results of this chapter are to extend necessary conditions
for stabilization presented in my previous work to the equivariant setting. Examples where
stabilization has been achieved are used to illustrate these necessary conditions.
In Chapter 6 a combination of modified Pyragas controls are implemented to stabilize periodic
solutions with various spatio-temporal symmetries. This chapter follows the basic
methodology introduced by Fiedler where a control based on a single group element was
used. In our case we deal with a much larger group than in previous work by other people.
This non-trivial group means that group theoretic restrictions discussed in the literature come
into effect and make stabilization via a control based on a single group element impossible.
However as was suggested, a control based on several group eleents can be implemented and
this may help to overcome these restrictions. We confirm, for the first time, that this is
a viable control strategy. Furthermore, as opposed to previous work, we treat individual
oscillators which are not in the normal form.||