Sturman, Robert John; (2001) Strange nonchaotic attractors in quasiperiodically forced systems. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

We study a class of dynamical system known as a quasiperiodically forced system. This is a mechanism characterised by a driving system based on incommensurate frequencies. A feature which appears common to many quasiperiodically forced systems is the strange nonchaotic attractor (SNA). We apply both analytical and numerical methods to study the structure and creation of SNAs. One of the few rigorous established results is that an SNA of a quasiperiodically forced system cannot be the graph of a continuous function. This is shown by deriving uniform estimates for growth rates from non-uniform hypotheses in uniquely ergodic systems. We show how conditions on growth rates with respect to all the invariant measures of a broader range of system can be used to derive one-sided uniform convergence in both the Birkhoff and the sub-additive ergodic theorems. We apply the latter to show that any strange compact nonchaotic invariant set of a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Lyapunov exponent; in other words, it must contain some 'non-attracting' orbits. This was already known for the few examples of SNAs that have been rigorously been proved to exist. In the light of the theoretical results we investigate numerically a paradigm example which highlights the complexity of the structure of the SNA, and discuss some apparent contradictions between the analysis and the numerics. We also describe a type of intermittency present in the SNA. This has a similar scaling behaviour to the intermittency found in an attractor-merging crisis of chaotic attractors. By studying rational approximations to the irrational forcing we present a reasoning behind this scahng, which also provides insight into the mechanism which creates the SNA. Finally, renormalisation techniques are applied to a similar system, which also exhibits an SNA. This procedure makes it possible to study numerical approximations to quasiperiodically forced systems, whilst retaining a mathematically rigorous method of relating the approximations. The renormalisation scenario is used to discuss various aspects of the SNA, including its fine-scale structure and the logarithmic growth of orbits.

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