Complexity and Evolution of Dissipative Systems

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

Complexity and evolution of spatially extended systems: analytical approach

Chapter 1: Introduction

• Dynamical systems
• Attractors
• Strange attractors
• Neural and genetic networks
• Reaction diffusion systems
• Systems with random perturbations and Gromov-Carbone problem

Chapter 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 networks

Chapter 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 networks

Chapter 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 system

Chapter 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