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Pattern formation in squares and rectangles

Jhugroo, Eric (2007). Pattern formation in squares and rectangles. (Unpublished Doctoral thesis, City University)

Abstract

This thesis considers pattern formation governed by the two dimensional Swift-Hohenberg equation in square and rectangular domains.

For the square, the dependence of the solution on the size of the square relative to the characteristic wavelength of the pattern is investigated for periodic, non-periodic (rigid) and quasi-periodic boundary conditions. Linear and weakly nonlinear analysis is used together with numerical computation to identify the bifurcation structure of steady-state solutions and to track their nonlinear development as a function of the control parameter. Nonlinear solutions arising from secondary bifurcations and fold bifurcations are also found. Time-dependent computations are also carried out in order to investigate stability, and to find certain nonlinear steady states.

The structure of solutions in the limit where the size of the square is much larger than the characteristic wavelength of the pattern is investigated using asymptotic methods.

For the rectangle, the dependence of the solution on the size of the rectangle relative to the characteristic wavelength of the pattern is investigated for non-periodic (rigid) boundary conditions. Most results are obtained for two aspect ratios, 0.75 and 0.5. Linear analysis is used together with numerical computations to identify the bifurcation structure of steady-state solutions and to track their nonlinear development. Nonlinear solutions arising from secondary bifurcations and fold bifurcations are also found, again making use of time-dependent calculations where necessary.

Finally, the structure of solutions in the limit where the size of the rectangle is much larger than the characteristic wavelength of the pattern is investigated using asymptotic methods.

The results are discussed in relation to patterns observed in physical systems such as Rayleigh-Benard convection.

Publication Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
Doctoral Theses
Doctoral Theses > School of Mathematics, Computer Science and Engineering
URI: http://openaccess.city.ac.uk/id/eprint/18271
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