Mathematics and Its Applications

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.