Spatiotemporal patterns in the Turing-Takens-Bogdanov scenario

Abstract

[eng] Some spatial dinamical systems exhibit, for close values of the parameter, diffusion drive instability (Turing bifurcation) and a Homoclinic bifurcation of the homogeneous solution. However, the interaction between these bifurcations has not been studied in detail in the literature. In this thesis we explore the interaction between a Turing and a Homoclinic bifurcation in a Reaction-Diffusion system. For this purpose we incorporate a diffusion term to the normal form for the Cusp Takens-Bogdanov codimension-3 point, in such a way that a Turing instability might occur. We analyse the spatio-temporal bifurcations and their interactions. These bifurcation curves converge in a new high codimension point, that we call Turin-Takens-Bogdanov point. The system shows a wide variety of stable solutions such as steady patterns, homogeneous oscilatory states , different more complex spatio-temporal periodic solution, pseudo-periodic states and turbulent regimes.