# Recent thesis

## The chaotic behaviour of geodesics in non-homogeneous vacuum pp-wave solutions

### Department of Physics, Eastern Mediterranean University

#### Supervisor: Professor Mustafa HALILSOY

##### Abstract

We demonstrate chaotic behaviour of time-like, space like and null geodesics of free test particle in non-homogeneous vacuum plane fronted gravitational waves with asymptotic analytical and numerical methods. Geodesic motions that we derived from pp-waves metric, generate deterministic chaotic Hamiltonian flow on plane of real coordinates and corresponding phase-spaces in case of n greater than or equal to 3, which belongs to the famous Henon-Heiles family. Analysing the integrability of this type of Hamiltonian flow, show that even simple and well-know dynamics of pp-waves could be evolve in complex fashion.

The fulltext of this thesis is available at: http://www.geocities.com/nonlinearphysics/

Source: Mehmet Suzen

## Mode interactions in three-dimensional convection

### DAMTP, University of Cambridge

#### Thesis summary

The focus of this thesis is the influence of rotation or a magnetic field on the dynamics of pattern formation by thermal convection in a plane layer (the Rayleigh-B\'enard problem).

Three-dimensional solutions to the equations of motion for an incompressible fluid are examined using the Boussinesq approximation. One fundamental effect of the addition of either rotation or a vertical magnetic field to the Rayleigh-B\'enard problem is the possibility that the conduction state may lose stability via a Hopf bifurcation rather than through a steady-state bifurcation. Near the boundaries between regions of parameter space where these Hopf or steady-state bifurcations occur, the dynamics are affected by both steady and oscillatory modes of behaviour. In this thesis new interactions between marginal steady and oscillatory modes with distinct wavenumbers are considered, where the three-dimensionality of the problem plays a crucial role.

The first chapter of the thesis provides a short introduction to the theory of equivariant steady-state and Hopf bifurcations on doubly-periodic planar lattices. Known results and new observations for Hopf bifurcations on non-rotating and rotating square lattices are summarised. There is also a detailed presentation of new stability results for the Hopf bifurcation on a square superlattice.

In chapter 2 the effect of an imposed vertical magnetic field on convection is discussed. This leads to a detailed analysis of the simplest three-dimensional Hopf/steady-state mode interaction, where the ratio of the critical wavenumbers for oscillatory and steady convection is $1:\sqrt{2}$.

The remaining chapters contain results on pattern selection in rotating Rayleigh-B\'enard convection, specifically for low Prandtl number fluids. Calculations in the regime where the onset of convection is oscillatory are performed to determine possible forms of convection at onset. The transition (with increasing Prandtl number) from patterns involving oscillatory rolls to those involving steady rolls is influenced by a second $1: \sqrt{2}$ mode interaction. The resulting amplitude equations contain heteroclinic cycles and bursting behaviour in marked contrast to the magnetoconvection case. Finally, a new asymptotic regime of low Prandtl number and rapid rotation is explored. The relevance of this regime to various sets of experimental results is discussed.

J.H.P.Dawes@damtp.cam.ac.uk

Source Jonathan Dawes

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