Fractal formations in the Lattice Limit Cycle model

We examine the fractal patterns arising in the Lattice Limit Cycle model, when it is restricted on square and fractal lattices. We show that, for processes taking place on regular 2d substrates, the fractal dimensions depend on the kinetic constants and we have observed a sharp phase-transition from uniform 2d spatial distributions (d<Subscript>f</Subscript>=2) when the kinetic parameters are near the Hopf bifurcation point, to a <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$d_f \not =2$</EquationSource> </InlineEquation> inside the limit cycle region. For processes taking place on substrates which contain inactive sites, we observe nucleation of homologous species around inactive regions leading to poisoning, when the active sites are distributed in a fractal manner on the substrate. This is less frequent in cases where the active sites are distributed uniformly and randomly on the lattice leading, normally, to non-trivial steady states. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006