Many chaotic and complicated systems cannot be analysed by traditional methods. In 1987 P.Bak, C.Tang, and K.A.Wiesenfeld developed a new concept called Self-Organised Criticality (SOC) to explain the behaviour of composite systems containing a large number of elements that interact over a short range. In general this theory applies to complex systems that naturally evolve to a critical state in which a minor event starts a chain reaction that can affect any number of elements in the system. It was later shown that many complex phenomena such as flux pinning in superconductors, dynamics of granular systems, earthquakes, droplet formation and biological evolution show signs of SOC. The dynamics of complex systems in nature often occurs in terms of punctuation, or avalanches rather than following a smooth, gradual path. Extremal dynamics is used to model the temporal evolution of many different complex systems. Specifically the Bak-Sneppen evolution model, the Sneppen interface depinning model, the Zaitsev flux creep model, invasion percolation, and several other depinning models. This thesis considers extremal dynamics at constant flux where M>1 smallest barriers are simultaneously updated as opposed to models in the limit of zero flux where only the smallest barrier is updated. For concreteness, we study the Bak-Sneppen (BS) evolution model [Phys. Rev. Lett. 71, 4083 (1993)]. M=1 corresponds to the original BS model. The aim of the present work is to understand analytically through mean field theory the random neighbour version of the generalised BS model and verify the results against the computer simulations. This is done in order to scrutinise the trustworthiness of our numerical simulations. The computer simulations are found to be identical with results obtained from the analytical approach. Due to this agreement, we know that our simulations will produce reliable results for the nearest neighbour version of the generalised BS model. Since the nearest neighbour version of the generalised BS model cannot be solved analytically, we have to rely on simulations. We investigate the critical behaviour of both versions of the model using the scaling theory. We look at various distributions and their scaling properties, and also measure the critical exponents accurately verifying whether the scaling relations holds. The effect of increasing from M=1 to M>1 is surprising with dramatic decrease in size of the scaling regime.

Large-eddy simulation is rapidly becoming the preferred method for calculations involving turbulent phenomena. However, filtering equations as performed in traditional LES procedures leads to significant problems. In this work we present some key components in the construction of a novel LES solver for compressible turbulent flow, designed to overcome most of the problems faced by traditional LES procedures. We describe the construction of the large-scale algorithm, which employs fairly standard numerical techniques to solve the Navier{Stokes equations. We validate the algorithm for both transonic and supersonic ow scenarios. We further explicitly show that the solver is capable of capturing boundary layer effects. We present a detailed derivation of the chaotic map termed the \compressible poor man's Navier{Stokes (CPMNS) equation" starting from the Navier{Stokes equations themselves via a Galerkin procedure, which we propose to use as the fluctuating component in the SGS model. We provide computational results to show that the chaotic map can produce a wide range of temporal behaviors when the bifurcation parameters are varied over their ranges of stable behaviors. Investigations of the overall dynamics of the CPMNS equation demonstrates that its use increases the potential realism of the corresponding SGS model.

L'objectif de cette thèse est de modéliser la transmission du paludisme dans la région montagneuse de l'Ouest du Kenya, en se servant des outils de systèmes dynamiques. Nous considérons deux modèles mathématiques. Le premier prend en compte une susceptibilité et une infectivité différentielle dans les métapopulations, et le second un taux de saturation des repas sanguins dans la population des moustiques.

The theory of nonlinear mass-spring chains has a history stretching back to the now famous numerical simulations of Fermi, Pasta and Ulam. The unexpected results of that experiment have led to many new fields of study. Despite this, the mathematics of the lattice equations have proved sufficiently rich to attract continued attention to the present day. This work is concerned with the motions of an infinite one dimensional lattice with nearest-neighbour interactions governed by a generic potential. The Hamiltonian of such a system may be written $H = \sum_{i=-\infty}^{\infty} \, \Bigl(\frac{1}{2}p_i^2 + V(q_{i+1}-q_i)\Bigr)$, in terms of the momenta $p_i$ and the displacements $q_i$ of the lattice sites. All sites are assumed to be of equal mass. Certain generic conditions are placed on the potential $V$. Of particular interest are the solitary wave solutions which are known to exist upon such lattices. The KdV equation has long been known to emerge in a formal manner from the lattice equations as a continuum limit. More recently, the lattice's localized nonlinear modes have been rigorously approximated by the KdV's well-studied soliton solution, in the lattice's long wavelength regime. To date, however, little is known about how, and to what extent, lattice solitary waves differ from KdV solitons. It is proved in this work that a solution (which we prove to be unique) to a particular linear ordinary differential equation provides a correction to the KdV approximation. This gives, in an explicit way, the lowest order effect of lattice discreteness upon lattice solitary waves. It is also shown how such discreteness effects are propagated along the lattice both in isolation (single soliton case), and in the presence of another soliton correction (the bisoliton case). In the latter case their interaction is studied and the impact of lattice discreteness upon lattice solitary wave interactions is observed. This is possible by virtue of the discovery of an evolution equation for discreteness effects on the lattice. This equation is proved to have appropriate unique solutions and is found to be strikingly similar to corresponding equations known in both the theories of shallow water waves and ion-acoustic waves.

The goal of this thesis is to investigate model imperfection in the context of forecasting. We focus on an electronic circuit built in a laboratory and then enclosed to reduce environmental effects. The non-dimensionalised model equations, obtained by applying Kirchhoff’s current and voltage laws, are the Moore-Spiegel Equations [47], but they exhibit a large disparity with the circuit. At parameter values used in the circuit, they yield a periodic trajectory whilst the circuit exhibits chaotic behaviour. Therefore, alternative models for the circuit are sought. The models we consider are local and global prediction models constructed from data. We acknowledge that all our models have errors and then seek to quantify how these errors are distributed across the circuit attractor. To this end, q-pling times of initial uncertainties are computed for the various models. A q-pling time is the time for an initial uncertainty to increase by a factor of q [67], where q is a real number. Whereas it is expected that different models should have different q-pling time distributions, it is found that the diversity in our models can be increased by constructing them in different coordinate spaces. To forecast the future dynamics of the circuit using any of the models, we make probabilistic forecasts [8]. The question of how to choose the spread of the initial ensemble is addressed by the use of skill scores [8, 9]. Finally, the diversity in our models is exploited by combining probabilistic forecasts from them so as to minimise some skill score. It is found that the skill of combined not-so-good models can be increased by combining them as discussed in this thesis.

Physical processes are often modelled using nonlinear dynamical systems. If such models are relevant then they should be capable of demonstrating behaviour observed in the physical process. In this thesis a new measure of model optimality is introduced: the distribution of ɩ-shadowing times defines the durations over which there exists a model trajectory consistent with the observations. By recognising the uncertainty present in every observation, including the initial condition, ɩ-shadowing distinguishes model sensitivity from model error; a perfect model will always be accepted as optimal. The traditional root mean square measure may confuse sensitivity and error, and rank an imperfect model over a perfect one. In a perfect model scenario a good variational assimilation technique will yield an ɩ-shadowing trajectory but this is not the case given an imperfect model; the inability of the model to ɩ-shadow provides information on model error, facilitating the definition of an alternative assimilation technique and enabling model improvement. While the ɩ-shadowing time of a model defines a limit of predictability, it does not validate the model as a predictor. Ensemble forecasting provides the preferred approach for evaluating the uncertainty in predictions, yet questions remain as to how best to construct ensembles. The formation of ensembles is contrasted in perfect and imperfect model scenarios in systems ranging from the analytically tractable to the Earth's atmosphere, thereby addressing the question of whether the apparent simplicity often observed in very high-dimensional weather models fails `even in or only in' low-dimensional chaotic systems. Simple tests of the consistency between constrained ensembles and their methods of formulation are proposed and illustrated. Specifically, the commonly held belief that initial uncertainties in the state of the atmosphere of realistic amplitude behave linearly for two days is tested in operational numerical weather prediction models and found wanting: nonlinear effects are often important on time scales of 24 hours. Through the kind consideration of the European Centre for Medium-range Weather Forecasting, the modifications suggested by this are tested in an operational model.

Discontinuous dynamical system defined by impulsive autonomous differential equation is a field that has actually been considered rarely. Also, the properties of such systems have not been discussed thoroughly in the course of mathematical researches so far. This thesis comprises two parts, elaborated with a number of examples. In the first part, some results of the previous studies on the classical dynamical system are exposed. In the second part, the definition of discontinuous dynamical system defined by impulsive autonomous differential equation is formulated, and its properties are investigated, in the view of the known results of the studies on the classical dynamical system and impulsive differential equations.

Thesis (M.S.)--University of Missouri--Rolla, 2007. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed November 28, 2007) Includes bibliographical references.

Thesis (M.S.E.C.E.)--Baylor University, 2007. / Includes bibliographical references (p. 99).

Thesis (Ph. D.)--Auburn University, 2008. / Abstract. Vita. Includes bibliographical references (p. 113).