Complexity and Evolution of Dissipative Systems

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