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Renormalization in a variety of quasiperiodically forced systems

Student thesis: Doctoral Thesis

This thesis is based on the application of renormalization group techniques to examine a variety of characteristics of quasiperiodically forced systems. The initial focus of the work is symmetric barrier billiards, a pseudo-integrable system consisting of a particle moving at constant speed in a rectangular chamber with a partial barrier placed in the centre. A renormalization analysis of the autocorrelation function (ACF) is presented for a class of quadratic irrational trajectories, and depending on the nature of the barrier, this can lead to either self-similar or chaotic behaviour of the correlations. In the case of the golden mean trajectory, this is then explained by constructing a map which identifies the action of the renormalization operator with a subshift of finite type, and it is shown that orbits of the renormalization operator in a space of pairs of piecewise constant functions explore a specified attracting set. A projection of the function pairs in this set obtained by averaging them (to yield the correlations) gives rise to the presence of invariant sets embedded in three-dimensional space on which the correlations lie. We extend this work by giving a renormalization analysis of correlations in a quasiperiodically forced two-level quantum system in a time dependent magnetic field, which consists of periodic kicks whose amplitude is determined by a general class of discontinuous modulation function.

Another additional application of renormalization techniques occurs in the study of strange non-chaotic attractors (SNAs). We investigate the non-smooth pitchfork bifurcation route to SNA in systems of \pinched skew-product" type and give conditions for self-similar behaviour of the attractor at the critical point of transition. In addition, we describe how the attractor scales as we approach a bifurcation curve. To conclude, we study the box-counting dimension of strange non-chaotic attractors (SNAs) created by this bifurcation. We provide compelling evidence that a non-critical SNA has dimension 2. The method we adopt becomes more accurate in the study of piecewise linear SNAs. We also provide numerical evidence that the dimension of a critical SNA is not necessarily equal to 2, but can lie between 1 and 2.
Original languageEnglish
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Award dateSep 2015

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