Spatiotemporal patterns in the Turing-Takens-Bogdanov scenario

Many spatial dinamical systems exhibit the coexistence of diffusion drive instabilities (Turing bifurcation) and Homoclinic of the homogeneous solution. However, the interaction between these bifurcation has not been deeply studied 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 bifurcation and their interactions. These bifurcations curves converge in a new high codimension point, the called Turin-Takens-Bogdanov point. The system has shown a wide variety of stable solutions such as steady patterns, homogeneous oscilatory states , different more complex spatiotemporal periodic solution, pseudo-periodic states and turbulent regimes.