Patterns of Sources and Sinks in Oscillatory Systems
In oscillatory systems, a fundamental spatiotemporal pattern is wavetrains, which are spatially periodic solutions moving with constant speed (also known as periodic travelling waves). In many numerical simulations, one observes finite bands of wavetrains, separated by sharp interfaces known as "sources" and "sinks". This talk is concerned with patterns of sources and sinks in the complex Ginzburg-Landau equation with zero linear dispersion; in this case the CGLE is a reaction-diffusion system. I will show that patterns with large source-sink separations occur in a discrete family, due to a constraint of phase-locking type on the distance between a source and its neighbouring sinks. I will then consider the changes in source-sink patterns in response to very slow increases in the coefficient of nonlinear dispersion. I will present numerical results showing a cascade of splittings of sources into sink-source-sink triplets, culminating in spatiotemporal chao s at a parameter value that matches the change in absolute stability of the underlying periodic travelling wave. In this case the gradual change in pattern form represents an ordered, structured transition from a periodic solution to spatiotemporal chaos. The work that I will present was done in collaboration with Matthew Smith (Microsoft Research. Cambridge) and Jens Rademacher (CWI, Amsterdam).