Drift and Meander of Spiral Waves

TL;DR

This thesis develops an asymptotic theory for the drift and meander of spiral waves in reaction-diffusion systems under symmetry-breaking perturbations, combining previous theories and providing numerical methods for analysis.

Contribution

It extends existing spiral wave theories by integrating drift and meander phenomena using symmetry quotient systems and Floquet theory, with practical numerical implementations.

Findings

The new theory successfully models spiral wave drift and meander.

Numerical methods demonstrate convergence and effectiveness in complex calculations.

Linking theory to response functions enhances understanding of spiral wave dynamics.

Abstract

In this thesis, we are concerned with the dynamics of spiral wave solutions to Reaction-Diffsion systems of equations, and how they behave when subject to symmetry breaking perturbations. We present an asymptotic theory of the study of meandering (quasiperiodic spiral wave solutions) spiral waves which are drifting due to symmetry breaking perturbations. This theory is based on earlier theories: the 1995 Biktashev et al theory of drift of rigidly rotating spirals, and the 1996 Biktashev et al theory of meander of spirals in unperturbed systems. We combine the two theories by first rewriting the 1995 drift theory using the symmetry quotient system method of the 1996 meander theory, and then go on to extend the approach to meandering spirals by considering Floquet theory and using a singular perturbation method. We demonstrate the work of the newly developed theory on simple examples. We also develop a numerical implementation of the quotient system method, demonstrate its numerical convergence and its use in calculations which would be difficult to do by the standard methods, and also link this study to the problem of calculation of response functions of spiral waves.