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

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

The Myth of Autism

This thesis is a study of autism in and intimations thereof in various narrative works of different types and from different time periods. My interpretation of autism, upon which my literary analysis is based, is in accordance with the insights of a field of psychological theory known as metabletics, which studies various conditions (autism, for example) in light of the wider social, cultural, and historical contexts to which they belong. I interpret autism as an expression of a comprehensive cultural condition that affects Modern society (especially in the Western world) as a whole and finds its historical roots in the advent of linear perspective vision, which occurred in the 15th century. As I examine intimations and/or expressions of autism in the various narrative works I explore, I elaborate upon the ways in which autistic symptoms (as well as the narratives in question) are connected to the traits of linear-perspective-based consciousness. My goal is to inspire a more robust and sensitive understanding of and approach to autism than the traditional medical/diagnostic approach. After supplying the necessary background information about linear perspective vision and its effects upon Modern, Western society as a whole and then briefly attending to the implications of what some have interpreted as intimations of autism in pre-Modern narratives, I explore the autistic impulses within popular narrative works from different stages of Modern history, beginning with Shakespeare\u27s Hamlet and concluding with David Fincher\u27s 1999 film, Fight Club. In so doing, I trace the development of the autism phenomenon from within the concurrent (and enveloping) development of the condition of Modern man. After this, I devote a chapter to the portrayal of autism in contemporary children\u27s fiction, which I argue to be our key to understanding and approaching the autism phenomenon in a manner that is beneficial to autistic children and to society as a whole

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]

Shooting up illicit drugs with God and the State: the legal–spatial constitution of Sydney's Medically Supervised Injecting Centre as a sanctuary

© 2015 Institute of Australian Geographers In 1999, the Uniting Church opened a Medically Supervised Injecting Centre (MSIC) at the Wayside Chapel in the inner Sydney suburb of Kings Cross. The Uniting Church justified this overt act of civil disobedience against the State's prohibitionist model of drug usage by invoking the ancient right of sanctuary. This invocation sought to produce a specific sort of spatialisation wherein the meaning of the line constituting sanctuary effects a protected ‘inside’ governed by God's word – civitas dei – ‘outside’ the jurisdiction of state power in civitas terrena. Sanctuary claims a territory exempt from other jurisdictions. The modern assertion of sanctuary enacts in physical space the relationship between state and religious authorities and the integration and intersections of civitas terrena and civitas dei. This article draws upon conceptions of sanctuary at the intersection of the Catholic Christianity tradition and the State since medieval times to analyse the contemporary space of sanctuary in the MSIC, exploring the shifting and ambiguous boundaries in material, legislative, and symbolic spaces. We argue that even though the MSIC has now been incorporated into civitas terrena, it remains and enacts a space of sanctuary

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