Towards characterisation of chaotic attractors in terms of embedded coherent structures
thesisposted on 30.08.2017, 14:44 by Daniel Lewis Crane
The central theme of this thesis is the development of general methods for the modelling of the dynamics on chaotic attractors by a coarse-grained representation constructed through the use of embedded periodic orbits & other coherent structures. Our aim is to develop tools for constructing two types of reduced representations of chaotic attractors: Markov-type models, and symbolic dynamics. For Markov models, we present construction of a minimal cover of chaotic attractors of maps and high-dimensional flows by embedded coherent structures such as periodic orbits from which a Markov chain of the dynamics can be constructed. For the symbolic dynamics, we investigate the utility of unstable periodic orbits for the construction of an approximate generating partition of a chaotic attractor. In the first section of Part 1 we present an original method by which chaotic attractors of discrete-time dynamical systems can be covered using a small set of unstable periodic orbits (UPOs) following an iterative selection algorithm that only chooses those UPOs that provide additional covering of the attractor to be included into the cover. We then show how this representation can be used to represent trajectories in the system as a series of transition between cover elements, using which as a basis for the construction of a Markov chain representation of the dynamics. In the second section we extend this method to continuous-time dynamical systems, introducing methods by which covers of high-dimensional attractors can be constructed in low dimensional projections with as little information loss as possible, and also giving an example of how group symmetries of the system can be dealt with. In Part 2 we change our focus to the construction of symbolic dynamics of discrete-time systems, presenting an extension to an existing method for the computational construction of approximate generating partitions that increases the applicability of the method to a wider range of systems, and also significantly improving the results for more complex maps.