- Bifurcation Properties of Dynamic Urban Models by Thomas Buchendorfer (IERC, Cranfield University).
- Generalised Markov Coarse Graining and the Observables of Chaos by Donal MacKernan (Universite Libre de Bruxelles).
- Semilinear problems and Spectral Theory by Ms. Wenying Feng (University of Glasgow).
- Renormalisation for Siegel Discs by Andrew D. Burbanks (Loughborough University).

**Supervisor**: Professor Peter Allen.

Structural change in cities is an important phenomenon in urban development which is also of current interest in the context of the sustainability debate. Urban models can help to address issues related to structural change and the group of dynamic urban models offers an interesting approach to this task. A significant subset of these models makes use of the concept of bifurcation as a generator of structural change.

An examination of the literature shows that there is a fuzzy understanding of bifurcations amongst researchers in the field of urban modelling. To clarify the concept a range of situations that can arise in urban models is discussed using armchair experiments. A methodological gap is then identified: what is missing is a reliable way to identify bifurcations in dynamic urban models.

Such a technique is then proposed and validated. It can produce plots that are in essence bifurcation diagrams from which the unstable branches have been omitted.

A series of experiments is then conducted to provide empirical evidence for the theoretical argument presented so far. A first phase consists of a re-evaluation of earlier work with the proposed technique for bifurcations analysis using an abstract city as an experimental setup. A second phase follows showing in principle how the bifurcation analysis of a real case can be conducted. Here the object of study is an imaginary city based on Brussels.

The results prove that structural change is an unreliable base from which to conclude bifurcations and that bifurcations can occur without any structural change being observable. Structural change and bifurcations are thus related but neither of them follows conclusively from the other. Some implications of bifurcations for policy making are also outlined.

**Source**:
Thomas Buchendorfer
(`
T.Buchendorfer@cranfield.ac.uk`.

**Supervisor**: Professor Gregoire Nicolis.

Chaotic systems are characterised by the exponentially
rapid separation of initially close-by phase space trajectories. As realistic
initial conditions are never known precisely, after a short time one cannot
specify the current state of the system. It is therefore
more natural
to adopt a statistical perspective. One approach is to specify the initial
condition as a sharply peaked probability distribution, and to describe
its subsequent evolution; another is to examine the systems long time average
properties. Each approach entails an analysis of the properties of the
Frobenius-Perron operator **U **which evolves
probability densities,
residing in an `a priori`
infinite dimensional function space. The main thesis
on which the present dissertation is based is that a large class of
deterministic
chaotic systems can in fact be described statistically as an essentially
finite dimensional process, through a generalisation of traditional Markov
coarse graining to a function analytic setting. This possibility arises
because typically evolving probability densities exhibit recurrent patterns,
which when systematically accounted for, can be used to obtain essentially
finite matrix representations of **U**.
We thus describe local statistical
behaviour which enables one to characterise the evolution of point-like
initial conditions, and global behaviour, which primarily is concerned
with the properties of time-correlations of observables (such as position)
which are continuous over the system's entire phase space, and of Liapunov
exponents.

This thesis is organised as follows. In chapter 2 the principal results in the literature on one-dimensional maps are given, and Markov coarse graining is generalised for piecewise linear Markov maps, and equivalent repellers. The connection between our representations and the Zeta-function formalism is established. In so doing, we point out that for piecewise linear Markov maps, where the minimal Markov partition consists of more than one cell, the dynamical zeta functions cannot be expected to give all the decay rates of time correlation functions of analytical observables. This is due to a subtle counting error, which fortunately can be easily corrected. We then show that local statistical properties can be described through an appropriate basis related to the systems periodic orbits, and derive a formula for the corresponding spectra.

The previous results are generalised to any nonlinear
Markov analytic map (and equivalent repellers) in chapter 3.
We show rigorously,
and constructively how spectral decompositions of
**U** can be obtained.

The case of non-hyperbolic, but chaotic maps (of the S-unimodal) type) is treated in chapter 4; in particular this includes the logistic map. Singular, but integrable, probability densities are generic to these systems, necessitating the construction of a special orthogonal basis. We establish that these systems are statistically isomorphic to Markov analytic maps. We argue heuristically that these techniques can also be applied to non-Markov maps, and provide numerical results remarkably consistent with this claim. Finally, a formula for the dependence of the spectral gap, as a function of the control parameter, is obtained.

In chapter 5 classes of two-dimensional chaotic systems both dissipative and conservative are considered. All the results of chapter 2 are generalised. In particular, we show that the results of any statistical measurement (involving piecewise analytic observables) can be calculated through an essentially finite dimensional matrix. Thus an effective spectral decomposition can be obtained, where the eigenvalues lie within the unit disc (with the exception of eigenvalues corresponding to invariant states). We close the chapter by commenting on the origins of macroscopic irreversibility.

In the final chapter we discuss applications, and partial results on: intermittency, non-Markov maps, stochastic resonance-like phenomena and sensitive dependence of chaotic systems to time dependent perturbations.

**Source**: Donal MacKernan
(`
dmack@ulb.ac.be`).

**Supervisor**: Professor J.R.L. Webb.

Department of Mathematics,

University of Glasgow, July 1997.

**Source**: Jeff Webb
(`
jrlw@maths.gla.ac.uk`)

**Supervisor**: A.H. Osbaldestin.

This thesis investigates the existence of Siegel discs for iterated complex maps and looks at the properties of their boundary curves for golden mean rotation number. The key tool used is the idea of a renormalisation operator acting on a space of functions. Firstly, a computer-assisted proof is discussed and verified, which establishes the existence of a fixed point of the relevant renormalisation operator. In particular, the proof yields a ball of functions around an approximate fixed point that is guaranteed to contain the true fixed point. The rigorous computational techniques which allow computers to be used for this purpose are then discussed. Given the existence of the fixed point, we verify certain topological conditions, known as the necklace hypotheses, on the action of the maps making up the fixed point. This proves the existence of a Siegel disc having a Holder continuous (invariant) boundary curve for all maps attracted to the fixed point. Further, it is shown that the motion on the boundary is conjugate to a pure rotation, that the boundary curve passes through a critical point of the map, and that the conjugator is not differentiable on a dense set of points. Finally, by viewing the invariant curve as the limit set of an iterated function system (IFS), a further investigation is made to get rigorous bounds on the fractal dimension of the Siegel disc boundary. This involves calculating bounds on the contractivities and coercivities of the maps of the IFS and solving corresponding partition equations. In particular, a rigorous upper bound on the dimension of 1.16 is obtained. For a different IFS a tighter upper bound of 1.08523 is obtained, although this second bound relies on an unproved assumption (namely, that the second IFS satisfies an open set condition). These bounds are found to be in good agreement with earlier numerical estimates.

**Source**: A.H. Osbaldestin
(`
A.H.Osbaldestin@Lboro.ac.uk`).

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Last Updated: 25th July 1997.