S. Vakulenko, Petersburg State University of Technology and Design, Russian Academy of Sciences, Saint Petersburg.

• Complexity and evolution of spatially extended systems: analytical approachChapter 1: Introduction Dynamical systems Attractors Strange attractors Neural and genetic networks Reaction diffusion systems Systems with random perturbations and Gromov-Carbone problemChapter 2: Method to control dynamics: Invariant manifolds, realization of vector fields Invariant manifolds Method of realization of vector fields Control of attractor and inertial dynamics for neural networksChapter 3: Complexity of patterns and attractors in genetic networks Centralized networks and attractor complexity in such network A connection with computational problems, Turing machines and finite automatons Graph theory, graph growth and computational power of neural and genetical networks Mathematical model that shows how positional information can be transformed into body plan of multicellular organism Applications to TF- microRNA networks. Bifurcation complexity in networksChapter 4: Viability problem, Robustness under noise and evolution Here we consider neural and genetic networks under large random perturbations Viability problem We show that network should evolve to be viable, and network complexity should increase A connection with graph growth theory (Erdos-Renyi, Albert-Barabasi) Relation between robustness, attractor complexity and functioning speed Why Stalin and Putin's empires fall (as a simple illustration) The Kolmogorov complexity of multicellular organisms and genetic codes: nontrivial connections Robustness of multicellular organisms (Drosophila as an example) A connection with the Hopfield systemChapter 5: Complexity of attractors for reaction diffusion systems and systems with convection Existence of chemical waves with complex fronts Existence of complicated attractors for reaction diffusion systems Applications to Ginzburg Landau systems and natural computing Existence of complicated attractors for Navier Stokes equations