Probably, we are obliged to Science, more than to any other field of the human activity, for the origin of our sense that collective efforts are necessary indeed. F. Joliot-Curie The study of autowave processes is a young science. Its basic concepts and methods are still in the process of formation, and the field of its applications to various domains of natural sciences is expanding continuously. Spectacular examples of various autowave processes are observed experimentally in numerous laboratories of quite different orientations, dealing with investigations in physics, chemistry and biology. It is O1). r opinion, however, that if a history of the discovery of autowaves will he written some day its author should surely mention three fundamental phenomena which were the sources of the domain in view. "Ve mean combustion and phase transition waves, waves in chemical reactors where oxidation-reduction processes take place, and propagation of excitations in nerve fibres. The main tools of the theory of autowave processes are various methods used for investigating nonlinear discrete or distributed oscillating systems, the mathe matical theory of nonlinear parabolic differential equations, and methods of the theory of finite automata. It is noteworthy that the theory of autowave,. , has been greatly contributed to be work of brilliant mathematicians who anticipated the experimental discoveries in their abstract studies. One should mention R. Fishel' (1937), A. N. Kolmogorov, G. 1. Petrovskii, and N. S. Piskunov (1937), N. Wiener and A. Rosenbluth (1946), A. Turing (1952).
1 Autowave processes and their role in natural sciences.- 1.1 Autowaves in non-equilibrium systems.- 1.2 Mathematical model of an autowave system.- 1.3 Classification of autowave processes.- 1.4 Basic experimental data.- 2 Physical premises for the construction of basic models.- 2.1 Finite interaction velocity. Reduction of telegrapher's equations.- 2.2 Nonlinear diffusion equation. Finite diffusion velocity.- 2.3 Diffusion in multicomponent homogeneous systems.- 2.4 Integro-differential equations and their reduction to the basic model.- 2.5 Anisotropic and dispersive media.- 2.6 Examples of basic models for autowave systems.- 3 Ways of investigation of autowave systems.- 3.1 Basic stages of investigation.- 3.2 A typical qualitative analysis of stationary solutions in the phase plane.- 3.3 Study of the stability of stationary solutions.- 3.4 Small-parameter method.- 3.5 Axiomatic approach.- 3.6 Discrete models.- 3.7 Fast and slow phases of space-time processes.- 3.8 Group-theoretical approach.- 3.9 Numerical experiment.- 4 Fronts and pulses: elementary autowave structures.- 4.1 A stationary excitation front.- 4.2 A typical transient process.- 4.3 Front velocity pulsations.- 4.4 Stationary pulses.- 4.5 The formation of travelling pulses.- 4.6 Propagation of pulses in a medium with smooth inhomogeneities.- 4.7 Pulses in a medium with a nonmonotonic dependence v = v(y).- 4.8 Pulses in a trigger system.- 4.9 Discussion.- 5 Autonomous wave sources.- 5.1 Sources of echo and fissioning front types.- 5.2 Generation of a TP at a border between "slave" and "trigger" media.- 5.3 Stable leading centres.- 5.4 Standing waves.- 5.5 Reverberators: a qualitative description.- 6 Synchronization of auto-oscillations in space as a self-organization factor.- 6.1 Synchronization in homogeneous systems.- 6.2 Synchronization in inhomogeneous systems. Equidistant detuning case.- 6.3 Complex autowave regimes arising when synchronization is violated.- 6.4 A synchronous network of auto-oscillators in modern radio electronics.- 7 Spatially inhomogeneous stationary states: dissipative structures.- 7.1 Conditions of existence of stationary inhomogeneous solutions.- 7.2 Bifurcation of solutions and quasi-harmonical structures.- 7.3 Multitude of structures and their stability.- 7.4 Contrast dissipative structures.- 7.5 Dissipative structures in systems with mutual diffusion.- 7.6 Localized dissipative structures.- 7.7 Self-organization in combustion processes.- 8 Noise and autowave processes.- 8.1 Sources of noise in active kinetic systems and fundamental stochastic processes.- 8.2 Parametric and multiplicative fluctuations in local kinetic systems.- 8.3 The mean life time of the simplest ecological prey-predator system.- 8.4 Internal noise in distributed systems and spatial self-organization.- 8.5 External noise and dissipative structures - linear theory.- 8.6 Nonlinear effects - the two-box model.- 8.7 Wave propagation and phase transitions in media with distributed multiplicative noise.- 9 Autowave mechanisms of transport in living tubes.- 9.1 Autowaves in organs of the gastrointestinal tract.- 9.2 Waves in small blood-vessels with muscular walls.- 9.3 Autowave phenomena in plasmodia of Myxomycetes.- Concluding Remarks.- References.