UK Nonlinear News, August 1999

Recent thesis


Follicle Selection Dynamics in the Mammalian Ovary

M.A. Ch'avez-Ross

Centre for Nonlinear Dynamics and its Applications, University College London

Supervisor: Jaroslav Stark

Abstract:

The main objective of this thesis is to develop and analyse mathematical models of the regulation of the ovulation cycle in mammals. Specifically, we are interested in understanding the mechanisms that control the number of follicles ovulated in each cycle. In humans, a failure of such control mechanisms can lead to Polycystic Ovary Syndrome (PCOS), which accounts for a substantial fraction of all cases of anovulatory infertility found in women of reproductive age. Although treatment is available, it is highly desirable to improve it. Thus, a better understanding of the selection process of the ovulatory follicle is still required.

The thesis begins with a biological description of the terminal phase of the ovarian cycle. This provides the necessary background for the understanding and formulation of the mathematical models presented in later chapters. Next, a review of existing literature models is given and their relevance to the regulation process is analysed. Of these, the one due to Lacker (also referred as the symmetric model) is the best understood in terms of the control of ovulation and PCO. It is given by a system of nonlinear differential equations and assumes the same growth rate for each follicle. This assumption is biologically implausible and leads the model to exhibit unrealistic behaviour in some cases. A non-symmetric generalisation is therefore developed and Lacker's analysis of the symmetric model is extended to this case. The non-symmetric model exhibits behaviour which more closely reflects that observed in PCO. The thesis then goes on to present a theoretical and numerical analysis of another version of the symmetric model which has been proposed by Mariana et al. This incorporates a variable representing the ageing of the follicle in the same framework as that of Lacker's original model. Finally, all of the above models use a somewhat arbitrary function to describe a follicle's sensitivity to hormonal stimulation. In order to provide a more biologically motivated basis for our analysis we therefore develop a model in terms of the gonadotropic receptors of follicular cells. It is believed that the degree of sensitivity of a follicle to pituitary hormones is one of the factors determining its selection. This model is studied using numerical techniques, since its mathematical structure is too complicated to allow a theoretical analysis.

Tentative conclusions underlying the mechanisms that select the ovulatory follicle are given in terms of the models described in this thesis. Some of these are rather speculative due to the greatly simplified nature of the models in comparison to the real biological system. Nevertheless, since the behaviour of the models is qualitatively consistent with the results obtained from experimental data, they provide useful insights into the mechanisms that control the ovulation number in mammals.

Source: Alexandra-Chavez Ross ( alex@pims.ubc.ca).


The Forced Vibration of a Partially Delaminated Beam

Roger P. Menday

Mathematical Sciences, Loughborough University

Supervisors: Martin Harrison (Loughborough) and Rhian Green (Leicester)

Abstract

The forced vibration of a partially delaminated structure such as an aircraft wing can result in catastrophic crack growth. In order to look at the underlying mechanism of the dynamics and failure of the material, a simplified model of a cantilever beam with a single delamination at its free end is considered. We investigate a number of aspects of this system, using mathematical models to gain insight into its behaviour. This first model makes a number of broad, general assumptions. The model produced is known mathematically as an impact oscillator, a non-linear system where the nonlinearity is caused by the discontinuity of an impact between the main beam and its delamination. In the earlier chapters of the thesis, we examine both numerically and analytically, the resulting motions from the model subject to a harmonic forcing of the main beam. The model displays several types of motion; periodic, chattering-periodic and chaotic motion. Experimental evidence, a sample of which is provided, also shows some of these features. In the later chapters of the thesis, we consider the impact between the main beam and its delamination in more detail. In order to study the detailed nature of an impact between the two bodies, we construct a series of models, of increasing complexity. This serves to establish the validity of the assumptions made when deriving the initial model of the system.

A copy of the thesis is available at http://www.lboro.ac.uk/departments/ma/research/ncsg/papers/index.html .

Source: Andy Osbaldestin


POD-Galerkin Modelling of the Martian Atmosphere

Stephen G. Whitehouse

Mathematics Institute, University of Oxford

Abstract

The aim of this thesis is to seek a low-dimensional description of baroclinic instability in general, and of the Martian atmosphere in particular, where both forcing and spatial resonance are relevant to the dynamics of the system being analysed.

The Proper Orthogonal Decomposition (POD) is used to determine a basis for the modal decomposition of climatic simulations of Mars, obtained by using two General Circulation Models (GCMs): (a) a simple GCM, which is an idealised model in which the meteorological primitive equations are solved on a sphere with simplified physical parameters and (b) the Martian GCM, a more realistic model in which a comprehensive range of the relevant Martian physical parameters and topography are represented. Results of these analyses are presented for a range of Martian seasons and climatic conditions. The effects of using different forms of energy norm in performing the analysis is considered, with the objective of providing analyses which represents the physically most significant components of the circulation, with optimal efficiency.

Reduced low-dimensional models that replicated the full simple GCM stream function simulations are formulated by projecting the spherical quasi-geostrophic equations onto the PODs of the large-scale calculations. The resulting models are analysed by using a combination of solution continuation and numerical integration methods. A thorough analysis of the models reveals that a 6-D POD model is capable of reproducing the amplitude, frequency and behaviour of the leading oscillatory structures of the simple GCM, to within a 1% error. Such an excellent reproduction of the original system is shown to be due to (1) an accurate vertical formulation scheme, (2) the use of the correct norm, (3) a sufficiently high level of truncation and (4) the fact that the original system is a steady wave flow.

The behaviour of the various regimes observed in the low-order models are comparable with observations from studies of large-scale waves and instabilities in planetary atmospheres, including a range of hydrodynamical experiments on baroclinic wave interactions of a stratified fluid in cylindrical containers.

Source: Peter Read ( read@atm.ox.ac.uk).


Qualitative Approximation to Study of Nonlinear Dynamic Applications to Time Series Analysis: A Simulation

Ana María López Jiménez

Experimental Department of the University of Seville (Spain)

Index:

  1. Introduction
    1. The complexity in the mechanistic paradigm: A historical revision.
    2. Characteristics more relevants of complexity paradigm.
    3. Objects of this job.

FIRST PART. QUALITATIVE APPROXIMATION TO STUDY OF NONLINEAR DYNAMIC.

  1. Mathematical models from study to change: preliminary concepts.
  2. Qualitative theory of discrete dynamic systems (DDS): fixed points and stability.
  3. Qualitative theory of DDS: bifurcations.

SECOND PART. QUALITATIVE THEORY OF DISCRETE DYNAMIC SYSTEMS (DDS). APPLICATIONS TO TIME SERIES ANALYSIS: A SIMULATION STUDY.

  1. Some indices to behaviour of a dynamic system.
  2. Robustness of the methods from the estimation of Lyapunov`s exponent from SDD in short time series and with additive noise: a simulation study.

CONCLUSIONS
REFERENCES
APPENDIX

Ana is currently working on the nonlinear identification with neural networks

Source: Ana María López Jiménez.



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