Existence and Stabilization of Periodic Solutions in Equivariant Systems




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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.



Bifurcation theory, Combinatorial dynamics, Delay differential equations, Symmetry (Mathematics)



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