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