Efficient detection of periodic orbits in chaotic systems by stabilising transformations

An algorithm for detecting periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp.~6172--6175], which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp.~4733--4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional systems. The difficulty in applying the algorithm to higher-dimensional systems is mainly due to the fact that the number of the stabilising transformations grows extremely fast with increasing system dimension. Here we analyse the properties of stabilising transformations and propose an alternative approach for constructing a smaller set of transformations. The performance of the new approach is illustrated on the four-dimentional kicked double rotor map and the six-dimensional system of three coupled Henon maps

Efficient method for detection of periodic orbits in chaotic maps and flows

An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional system. The difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of stabilising transformations grows extremely fast with increasing system dimension. In this thesis, we construct stabilising transformations based on the knowledge of the stability matrices of already detected periodic orbits (used as seeds). The advantage of our approach is in a substantial reduction of the number of transformations, which increases the efficiency of the detection algorithm, especially in the case of high-dimensional systems. The performance of the new approach is illustrated by its application to the four-dimensional kicked double rotor map, a six-dimensional system of three coupled H\'enon maps and to the Kuramoto-Sivashinsky system in the weakly turbulent regime.Comment: PhD thesis, 119 pages. Due to restrictions on the size of files uploaded, some of the figures are of rather poor quality. If necessary a quality copy may be obtained (approximately 1MB in pdf) by emailing me at [email protected]

Analytical modelling of multi-spacecraft reconnection layer measurements at the magnetopause

An approach to determine and analyse the structure of Petschek-type magnetic reconnection is developed. This is achieved by extending an analytical model based on the Rankine-Hugoniot wave equation for shock jump conditions and is described in terms of its past applications and limitations. The model is applied to data from the CLUSTER multi-spacecraft mission using a boundary condition method optimised by two interlinked genetic algorithms. Case studies for a range of locations within the magnetopause region and local conditions are described and subjected to fluid and particle analyses to confirm the presence of reconnective signatures. Genetic algorithms are used to optimise the fit of the model, by modifying the boundary condition selection and internal structure parameters. This information then facilitates the construction of a more accurate modelled layer structure for each event. The calculated values for state variables within these layers are compared quantitatively and qualitatively to the magnetopause boundary crossings present in the CLUSTER data. Case study results are summarised and compared before being compiled into quantitative statistics for describing the local and possibly global applicability of the model. The fast application of these methods by means of an automatic process to a large set of data is described, as are the wider possibilities arising from this and the limitations of model, methods and data. These results are used to support several assertions. Firstly, that this model is indeed applicable, within its limitations, to the study of reconnection events within the magnetospheric environment. It can also facilitate deeper studies of individual reconnection events, in addition to being employed as a basis to classify wider statistical trends in spatio-temporal structures

On the use of stabilising transformations for detecting unstable periodic orbits in the Kuramoto-Sivashinsky equation

In this paper we develop further a method for detecting unstable periodic orbits (UPOs) by stabilising transformations, where the strategy is to transform the system of interest in such a way that the orbits become stable. The main difficulty of using this method is that the number of transformations, which were used in the past, becomes overwhelming as we move to higher dimensions (Davidchack and Lai 1999; Schmelcher et al. 1997, 1998). We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs (Crofts and Davidchack 2006). The main benefit of using the proposed set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. In a typical situation the dimension of the unstable manifold is much smaller than the dimension of the system so the number of transformations is much smaller. Here we extend this approach to high-dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system -- the Kuramoto-Sivashinsky equation. A comparison is made between the performance of this new method against the competing methods of Newton-Armijo (NA) and Levernberg-Marquardt (LM). In the latter case, we take advantage of the fact that the LM algorithm is able to solve under-determined systems of equations, thus eliminating the need for any additional constraints

Spreading dynamics on spatially constrained complex brain networks

The study of dynamical systems defined on complex networks provides a natural framework with which to investigate myriad features of neural dynamics and has been widely undertaken. Typically, however, networks employed in theoretical studies bear little relation to the spatial embedding or connectivity of the neural networks that they attempt to replicate. Here, we employ detailed neuroimaging data to define a network whose spatial embedding represents accurately the folded structure of the cortical surface of a rat brain and investigate the propagation of activity over this network under simple spreading and connectivity rules. By comparison with standard network models with the same coarse statistics, we show that the cortical geometry influences profoundly the speed of propagation of activation through the network. Our conclusions are of high relevance to the theoretical modelling of epileptic seizure events and indicate that such studies which omit physiological network structure risk simplifying the dynamics in a potentially significant way

Spreading dynamics on spatially constrained complex brain networks

The study of dynamical systems defined on complex networks provides a natural framework with which to investigate myriad features of neural dynamics, and has been widely undertaken. Typically, however, networks employed in theoretical studies bear little relation to the spatial embedding or connectivity of the neural networks that they attempt to replicate. Here, we employ detailed neuroimaging data to define a network whose spatial embedding represents accurately the folded structure of the cortical surface of a rat and investigate the propagation of activity over this network under simple spreading and connectivity rules. By comparison with standard network models with the same coarse statistics, we show that the cortical geometry influences profoundly the speed propagation of activation through the network. Our conclusions are of high relevance to the theoretical modelling of epileptic seizure events, and indicate that such studies which omit physiological network structure risk simplifying the dynamics in a potentially significant way

Phosphoramidite derivatives of macrocycles

The invention is directed to phosphoramidite derivatives of macrocycles, such as porphyrins and expanded porphyrins, including sapphyrins and texaphyrins. The phosphoramidite derivatives are useful as intermediates in the preparation of macrocycle-oligonucleotide conjugates.Board of Regents, University of Texas Syste