Complexity and evolution of dissipative systems. An analytical approach
This book deals with dynamics and stability of dissipative systems arising in biology. The existence of strange attractors of arbitrary high dimension is proved analytically for Hopfield neural networks, genetic circuits and basic systems of phase transition theory. Existence of chaotic dynamics for classes of reaction-diffusion systems, coupled oscillators and population dynamics systems is investigated in the book. The author studies the chaotic behavior of unbounded complexity leading to attractor control algorithms. An explanation of the increasing complexity of dynamical systems is presented. Rigorous mathematical approaches for viability, pattern complexity, evolution rate and feasibility are provided based on the attractor theory, the Kolmogorov complexity and new methods for combinatorial problems. Moreover, a connection between the attractor complexity problem and the viability of biological systems is found. Hopfield neural networks are studied and an explicit algorithm is presented in order to construct networks with a prescribed dynamics. Lotka-Volterra systems with \( n \) resources are considered which can generate all structurally stable dynamics. Moreover, a standard ecological model is investigated which also exhibits all kinds of structurally stable dynamics and can be reduced to the Lotka-Volterra system. Genetic networks producing any spatiotemporal patterns are presented. It is shown that centralized networks can be applied in order to organize multi-cellular organism. An asymptotical variational principle is formulated in order to explain the Drosophila segmentation patterns. Wave solutions of more complicated structure are found for reaction-diffusion systems. Two-component reaction diffusion systems are studied from the point of view of chaotic theory.
scientific article; zbMATH DE number 6161396