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A new autosoliton view is developed for the seismic process. In physical terms, faults correspond to stationary autosolitons, and inter- and intrafault deformation disturbances are traveling autosolitons. Slow dynamics reveals itself only on large time scales because slow autosoliton disturbances, as a rule, have velocities 4–7 orders of magnitude lower than the sound velocity. It is shown that, in the loaded strong medium, slow autowave and autosoliton disturbances are generated by short dynamic actions (pulses) at interfaces. In real geomaterials, these are block boundaries and various-scale faults. Dynamic movements of structural elements cause the deformation autowaves and autosolitons to propagate from the interfaces into blocks and along faults. Velocities of such deformation autowaves and autosolitons are low and proportional to velocities of the related movements of structural elements in the geomedium. Propagating in structural elements that are in a certain stress-strain state, deformation autowaves and autosolitons can be taken as small disturbances of the existing fields of the stress-strain state. A mathematical model is represented for the geomaterial treated as a nonequilibrium randomly inhomogeneous medium. Special features of the generation and propagation of deformation autosolitons in such media are studied.

We consider the regularities of a plastic flow in materials with strain-induced phase transformation in titanium nickelide and trip steel as examples. The experimental analysis of plastic deformation processes has been carried out using the technique of digital correction of speckle images, which makes it possible to describe quantitatively the behavior of plasticity fronts associated with phase transformation in the given materials. The mechanisms of formation of the Luders fronts as well as the Portevin–Le Chatelier fronts at different stages of the plastic flow are considered.

Addition of a 20-μL drop of a mixture containing 1 mM Fe2+ + 0.5 M glutathione to the surface of 0.5 M S-nitrosoglutathione solution, which is expected to form dinitrosyl iron complexes with glutathione, generated short-lived (0.2 s) concentric ring-shaped structures in the S-nitrosoglutathione solution. The number of the ring-like structures exceeded the number of concentric circles spread across the water surface (mechanical waves) when droplets of water fall into a body of an aqueous solution. It appears likely that the ring-shaped structures observed in the S-nitrosoglutathione solution are associated with an autowave distribution of products generated during the formation of dinitrosyl iron complexes with glutathione.

The behavior of localized plastic flow autowaves in a Fe–Ni–Cr alloy at temperatures 143 K ≤ T ≤ 420 K is considered. The temperature variation of the autowave propagation velocity is studied. It is found that, for the region of low temperatures, the autowave velocity is inversely proportional to the work hardening coefficient and the quadratic dispersion law takes place. The elastoplastic strain invariant does not depend on temperature.

An approach and a model of the birth and development of deformation bands and their fronts under plastic deformation due to phase transformation are proposed. A solid body is considered as an open nonequilibrium system the relaxation of which is determined both by vibrational and by electronic degrees of freedom. Electron transitions between different states of the system of nuclei and electrons under deformation initiate excitation of the unstable vibrational mode of atomic displacements into positions peculiar to the martensitic phase.

The behavior of characteristics of localized plastic flow autowaves in a Fe–Cr–Ni alloy in the temperature interval of 143 ≤ T ≤ 420 K (0.34 ≤ T/ΘD ≤ 1), where ΘD = 420 K is the Debye temperature (parameter), is considered. It is shown that the product of the length and propagation rate of the autowave λVaw in this interval decreases with temperature. At the same time, the ratio of this quantity to viscosity of the photon gas in the metal in the investigated temperature interval remains almost constant. This indicates that the elastoplastic strain invariant does not depend on temperature.

Topical problems of the formation of slow deformation waves and their connection with recent geodynamic (deformational) processes are discussed. It is shown that the term “diffusion of stresses (displacements, strains)” is ill defined from the standpoint of physics of transfer phenomena because in the case of diffusion, mass transfer takes place, whereas the wave processes transfer energy. It is noted that the existing models describing the “diffusion of stresses” are solved based on the mathematical formalism of heat conduction theory which relies on the phenomenon of energy transfer. It is demonstrated that applying the term “wave” to the “stress diffusion” processes is untenable because in the classical sense, wave processes describe undamped (sustained, continuous) oscillations propagating in a homogeneous medium at constant velocity. The processes describing the “diffusion of stresses” form strongly damped oscillations whose propagation velocity substantially decreases with time. As a mechanism corresponding to the wave canonical concepts, a model of autowave deformation processes is proposed that implement the relay-race transfer and successive re-initiation of deformation activity from a fault to a fault or from one activated segment of a fault to another segment. Problematic issues of identifying the slow deformation waves are discussed, and the recommendations are proposed for constructing a network of observation points for the in situ measurements of spatiotemporal migration of the Earth’s surface deformations. It is substantiated that the existence of slow deformation waves does not explain the entire observed spatiotemporal spectrum of recent movements of the Earth’s surface.

An analytical method for determining the amplitude and velocity of an autowave pulse emerging in isentropically unstable heat-releasing media is presented. The method is suitable for any values of acoustic dispersion and growth rate and requires the knowledge of heat-loss function only without the need for a numerical solution of a complete system of gas-dynamic equations.

Basic concepts of the autowave model of development of a localized plastic flow of solids of different natures are considered. It is shown that plastic deformation develops in a localized (at the macroscale level) way throughout the process. The form of the observed localization patterns is related to stages of strain hardening of the material. The patterns are projections of different modes of the localized plastic flow autowave on the observation surface. An elastoplastic strain invariant is introduced, which is considered as the main equation of the autowave plasticity model, and its physical nature is discussed.

The paper investigates a model simulating crustal fault dynamics and strain wave generation in a fault block geological medium, the parameters determining sliding regimes in faults, and the physics of transitions between different deformation regimes. The model comprises the most important mechanisms responsible for the interaction of fault walls: friction, geometric irregularities (roughness and asperities on the fault surface), and external load, which govern sliding along the fault. The results of field and laboratory studies of deformation migration on the macro/mesoscale are consistent with the concept of localized deformation propagation in the form of solitary waves (kinks, solitons) and autowaves. The conditions are defined which make possible the transition from the model simulating solitary waves in a conservative medium with low “friction” (soliton-like behavior of the system) toward the model of solitary waves in an active medium with diffusion (autowave-like behavior of the system). Two possible deformation regimes of the fault block structure in the high-friction limit are considered. The fault wall displacement is stopped due to this friction, but the adjacent blocks move relative to each other in the core of the fault. It is shown that in the high-friction limit a perturbed sine-Gordon equation applied for fault dynamics modeling is reduced to a reaction-diffusion equation, whereas the system goes from the soliton regime to the autowave regime. In the case of high friction and a lack of energy supply to the fault from an external source, the transfer of localized deformation is changed by a diffusive dissipation of stress.

The relationship between the spatial parameter of the development of a localized plastic flow (the length of the localized plasticity autowave) and the length of the deformed sample is studied. According to experiments carried out on polycrystalline samples of a zirconium alloy and commercial purity aluminum, these quantities are related to each other by a logarithmic law valid at the stage of parabolic strain hardening.

The infrared thermography method was used to study changes in the temperature of austenitic stainless steel in the process of tensile deformation. It was revealed that the temperature of the sample stops growing in the transition region between the linear stages of strain hardening. This phenomenon can be connected with a change in the entropy of the deformed medium as a result of a reconstruction of the ordered autowave structures. In the region of true deformation exceeding 0.4, a phenomenon of discontinuous flow (Portevin–Le Chậtelier effect) is observed, which is accompanied by the motion of the single fronts of localized deformation. The observed fronts are heat sources, and the value of the temperature at the arbitrary point of the sample is determined by its position relative to the front of deformation.

A diffusion neuron model representing a system of $$m,$$ $$m \geqslant 2$$ , identical nonlinear delay differential equations coupled by linear diffusion terms is considered. It is shown that, with a suitable choice of the diffusion coefficient, the system has a set of $$m$$ stable relaxation cycles.

TiAl-based composite materials have been produced by spin-casting self-propagating high-temperature synthesis metallurgy methods using thermite-type mixtures, general relationships in the formation of their composition and structure have been investigated, the synthesis conditions have been optimized, a method for obtaining large ingots has been proposed, and their composition and structure have been determined. We have obtained cast Nb- and Cr-doped TiAl-based composite materials using mixtures of Ti, Nb, Cr, and Ca oxides with a combined reducing agent (Al and Ca) under the effect of an overload above 200g. It has been shown that partial Ca substitution for Al in the starting mixture ensures complete reduction of the TiO2, and the resultant metallic phase contains calculated amounts of Ti and Al. Mixtures of optimized composition make it possible to produce a cast composite material similar in composition to alloy 4822.

The plastic flow macrolocalization in single crystals and polycrystalline alloys at various stages of strain hardening is considered. The macrolocalized plastic flow at these stages is shown to be an autowave self-organization process. The ultrasound velocity is found to be an informative parameter of autowave plastic deformation.

The method of frontal polymerization (FP) has been presented repeatedly in different publications but, taking into account the aim of this review, we intend to focus detailed and oriented attention on the process specificity that causes its unexpected advantages. This work has disclosed the ways to detect, reveal, and investigate the foundations of the frontal polymerization method and their use in different areas of science and technology. As researchers gain closer insight into this nontypical method of polymerization, they reveal new advantages and possibilities for its use to solve many problems both in the synthesis of polymer nanocomposites with retention of nanoscale dimensions, uniform distribution, and the optimal number of nanoparticles and in the synthesis of intercalated polymeric high-temperature superconductors, multifunctional gradient materials, hydrogels with prescribed properties, and other materials. It has been shown that the models and revealed features for the transition of linear autowaves into nonlinear ones are of interest and can be used similarly to the autowave processes that occur by the Belousov–Zhabotinsky mechanism in biology and human organism. With allowance for that noted above, this paper also describes the kinetic features of frontal polymerization that cause us to reconsider certain approaches and principles of formal kinetics. The ways and methods to pass from formal (isothermal) kinetics to nonisothermal kinetics (by the example of adiabatic polymerization) with finding the new principles of kinetic computations under nonisothermal conditions in application to frontal polymerization are given.

The evolution of gas-dynamic perturbations in a heat-releasing medium was investigated in two-dimensional geometry under conditions of isentropic instability. Decomposition of the initial perturbation of “step” type, possessing transversal modulation (ripples) of front, into a sequence of flat autowave fronts predicted by a generalized nonlinear acoustic equation is shown. Evolution of localized perturbation results in decomposition into sequence of concentric autowaves. The front of two-dimensional structures is resistant to transversal modulation. A quasi-regular cellular structure is formed behind the fronts.

An interpretation of the nature of the relation between elastic and plastic strains, called the elastic–plastic strain invariant, is proposed which takes into account the change in the entropy of the system during autowave generation at the stage of linear strain hardening. It is shown that this approach consistently explains the nature of the invariant and its role in the description of plasticity.

The features of the macroscopic inhomogeneity of plastic deformation in the form of autowaves with a pulsating amplitude are analyzed, and data on the localization of sources of acoustic emission at different stages of plastic flow in the stretching of fcc mono- and polycrystals are presented. The relationship between the local components of the plastic distortion tensor in the strain localization zone is traced. The role of acoustic phenomena accompanying the localization of plastic strain in the development of the process of plastic deformation is considered.

We consider the autowave mechanism of evolution of a localized plastic deformation of crystalline solids of different origins. It is found that localization of the plastic flow is determined by the relation between elastic and plastic phenomena in deforming materials. It is shown that the main parameter of deformation processes is the elastic–plastic deformation invariant, viz., a dimensionless quantity connecting quantitatively the parameters of elastic waves and self-sustained waves (autowaves) of localized plastic deformation. The correctness of this statement is verified for metals, alkali-halide crystals, and rocks. The physical origin of the invariant is explained on the basis of thermodynamic considerations.

The paper considers active wave processes in changes in the lumina of cylindrical hollow organs. A mathematical model is proposed for autowave transport of the internal contents of an organ based on mechanochemical interactions. The self-organization of changes in the shapes of organs are discussed in application to lymphatic vessels of living organisms.

A singularly perturbed parabolic system of nonlinear reaction–diffusion equations is studied. Systems of this class are used to simulate autowave processes in chemical kinetics, biophysics, and ecology. A detailed algorithm for constructing an asymptotic approximation of a travelling front solution is proposed. In addition, methods for constructing an upper and a lower solution based on the asymptotics are described. According to the method of differential inequalities, the existence of an upper and a lower solution guarantees the existence of a solution to the problem under consideration. These methods can be used for asymptotic analysis of model systems in applications. The results can also be used to develop and justify difference schemes for solving problems with moving fronts.

Results of measurements of the coefficient of microwave reflection from a block of freshwater ice subjected to plastic deformation are presented. Strong fluctuations in the phase of reflection coefficient are revealed, with the extremum at an electromagnetic radiation wavelength of 1.35 cm. This wavelength corresponds to the region of localized plastic flow autowaves in crystalline media.

The production of hafnium aluminides from mixtures of hafnium or its oxide with aluminum in the combustion mode is impossible because of a near-zero thermal effect. The present study is aimed at stimulating autowave chemical transformation with the help of energetic formulations and producing cast hafnium aluminides by means of combustion methods. The characteristics of the autowave synthesis and the conditions for producing cast hafnium aluminides are determined, and their composition and structure are investigated.

Composite materials based on Nb with functional (Si, C, B) and alloying dopants (Hf, Ti, Al, etc.) are promising materials for aircraft engine applications. Our previous studies have shown that such composites can be synthesized in the autowave mode (combustion mode) using highly exothermic mixtures of Nb2O5 with Al, Si, Hf, and Ti. It has been found that in the combustion wave, Hf actively participates in the reduction of Nb2O5, which complicates its introduction into the composite material and leads to an excess of Al in the alloy. In the present study, we investigated the possibility of replacing Hf in the starting mixture by less active HfAl3 and also determined the effect of the grain size of HfAl3 on the Hf content in the composite material.

Isolated Helianthus tuberosus L. polysaccharide (HTLP) with a molecular weight of 1–2 MDa has a wide range of biological activities, including the ability to switch transformed cells to an apoptotic pathway. A qualitative explanation for this effect has previously been proposed on the basis of the interactions among components of intracellular signaling pathways associated with the Dectin-1, TLR-6, and TNFR1 receptors and intracellular factors involved in regulating apoptosis. In this work, an autowave model is proposed to explain the switch between alternative modes of cell behavior in response to polysaccharide. The model is based on the modified FitzHugh–Nagumo system of equations. The parameters of the model were estimated numerically.

Conditions have been considered for the formation of autowaves of localized plastic flow in the deformed metals upon the propagation of Luders bands in the case of the Portevin–Le Chatelier effect, and upon the formation of a neck with taking into account the differences in the microscopic mechanisms of plastic deformation in the case of these phenomena. The laws that govern the development of the localized plastic flow of metals and the role of these laws in the development of the above effects have been investigated. It has been established that the main features of the deformation characteristic of these phenomena are determined by the differences in the properties of the active media that are formed in the material upon plastic deformation. The mechanisms of the generation of different autowave modes of the localized plastic flow upon the Luders deformation, Portevin–Le Chatelier effect, and the formation of a neck in active media of different nature during deformation have been considered.

It is shown that, in a wide class of physicochemical systems described by a system of conservation equations, including filtration combustion, the autowave mode of the process is impossible if the thermodynamic forces and flows are linearly related and the boundary conditions are stationary.

The development of macroscopic plastic flow heterogeneities in metals in the form of Chernov–Luders bands and the Portevin–Le Chatelier effect is studied. The main laws of deformation development in these two cases are found, and the kinetics of the motion of Chernov–Luders band fronts and the serrated deformation during the Portevin–Le Chatelier effect is investigated. It is shown that both versions of heterogeneities can be considered as macroscopic autowave processes of switching and excitation in deformable media.

Here we provide a review of research on slow motions and strain waves in the Earth and propose a substantiated hypothesis that all stress-strain perturbations in the form of slow waves propagating in solids and geomedia, including plastic waves in metals and waves in faults of different scales, are of common physical nature. Loaded solids and geomedia are active hierarchically organized multiscale systems that display nonlinear dynamics and lose their stability when disturbed by any dynamic processes at block boundaries, e.g., displacements in fault zones. Such a medium cooperatively responds to parametric excitation by generating slow strain waves (autowaves) as a way of its self-organization. In support of the proposed concept, a consistent mathematical model is suggested for describing the evolution of stress-strain states and slow strain autowaves in an unstable elastoplastic medium, and examples of simulations are presented for strain autowaves in ductile materials under tension and quasi-brittle materials and geomedia with a fault zone under compression.

The development of macroscopic inhomogeneities in the form of Chernov–Luders bands and serrated deformation bands (Portevin–Le Chatelier effect) in plastic metal flow is studied. For these two cases, regularities in the development of deformation inhomogeneity were established and the kinetics of motion of the fronts of Chernov–Luders bands and serrated deformation bands was studied. It is shown that the Chernov–Luders fronts and the serrated Portevin–Le Chatelier deformation can be considered as macroscopic autowave switching and excitation processes, respectively, in deformable media of different nature.

We have studied the regularities in the evolution of macroscopic nonuniformities in the plastic flow of metals in the form of the Chernov–Luders bands and the Portevin—Le Chatelier effect. The regularities in the evolution of strain inhomogeneity in these two cases have been established, and the kinetics of motion of the Chernov–Luders fronts and abrupt deformation in the Portevin–Le Chatelier effect has been analyzed. It has been shown that the Chernov–Luders fronts and Portevin–Le Chatelier jumpwise straining can be treated as macroscopic autowave processes of switching and excitation, respectively, in deformed media of various origins.

The correlation between lattice characteristics of a deformed medium and parameters of autowaves of a localized plastic flow in metals at the stage of linear strain hardening is considered. Such correlation testifies for the existence of a close connection between elastic and plastic deformations of the medium and, more generally, between characteristics of plasticity and electronic structure of elements.

Under study is the process of heating a moving medium by high-frequency electromagnetic radiation in presence of heat exchange with the environment, in the approximation of a thermally thin layer. It is shown that, due to the competitive nature of the processes of heat emission in the heated media and heat exchange with the environment, the temperature profiles are realized in the form of autowaves. Inspecting analytical and numerical solutions, we determine the basic regularities of the dynamics of temperature waves.

An experimental and theoretical study of plastic shear localization mechanisms observed under dynamic deformation using the shear–compression scheme on a Hopkinson–Kolsky bar has been carried out using specimens of AMg6 alloy. The mechanisms of plastic shear instability are associated with collective effects in the microshear ensemble in spatially localized areas. The lateral surface of the specimens was photographed in the real-time mode using a CEDIP Silver 450M high-speed infrared camera. The temperature distribution obtained at different times allowed us to trace the evolution of the localization of the plastic strain. Based on the equations that describe the effect of nonequilibrium transitions on the mechanisms of structural relaxation and plastic flow, numerical simulation of plastic shear localization has been performed. A numerical experiment relevant to the specimen-loading scheme was carried out using a system of constitutive equations that reflect the part of the structural relaxation mechanisms caused by the collective behavior of microshears with the autowave modes of the evolution of the localized plastic flow. Upon completion of the experiment, the specimens were subjected to microstructure analysis using a New View-5010 optical microscope–interferometer. After the dynamic deformation, the constancy of the Hurst exponent, which reflects the relationship between the behavior of defects and roughness induced by the defects on the surfaces of the specimens is observed in a wider range of spatial scales. These investigations revealed the distinctive features in the localization of the deformation followed by destruction to the script of the adiabatic shear. These features may be caused by the collective multiscale behavior of defects, which leads to a sharp decrease in the stress-relaxation time and, consequently, a localized plastic flow and generation of fracture nuclei in the form of adiabatic shear. Infrared scanning of the localization zone of the plastic strain in situ and the subsequent study of the defect structure corroborated the hypothesis about the decisive role of non-equilibrium transitions in defect ensembles during the evolution of a localized plastic flow.

The regularities for the autowave synthesis (SHS) of a cast oxide composite material (Al2O3–Cr2O3 · ZrO2) were studied by example of combustion of thermite-type systems under an inert gas pressure. The ratio of starting reagents with optimal zirconia content was determined for the synthesis of the desired product on the basis of analysis of published data and thermodynamic calculations (Thermo program). In the experiments, it was shown that combustion temperature and sample mass are the most important parameters influencing the composition and microstructure of the final product. Optimal conditions for the process control were defined. The final product includes solid solution Al2O3–Cr2O3 grains located on the boundaries of ZrO2 particles. The material is promising for production of machine parts working in conditions of intense wear and shock loads in corrosive environments at elevated temperatures (for example, sintering of ceramic cutting tools).

On the examples of frontal polymerization, tumor growth, and propagation of the nerve impulse, we examined nonlinear phenomena in chemistry and biology. The results of theoretical and experimental studies on the loss of stability of thermal waves of frontal polymerization were presented. The stationarity loss limits for the reactions of epoxide resin curing, polymerization of metal-containing acrylamide complexes, and copolymerization of methyl methacrylate and styrene in the presence of single-walled nanotubes were experimentally studied and theoretically calculated. It was shown that stability of the frontal process waves depends on the heat balance of the exothermic polymerization reaction. Nonlinear phenomena at variation of the polymerization kinetic parameters were studied. The limits of the transition from stationary to nonstationary mode were determined on the basis of experimental data and theoretical calculations. Theoretical study of a diffusion-kinetic model of tumor growth was discussed; the wave nature of the propagation of tumors was shown. The FitzHugh–Nagumo model was studied in view of the diffusion equation; the possibility of formation of the oscillatory modes during the nerve impulse propagation was demonstrated.

The propagation of plane, cylindrical, and spherical waves in a thermally unstable gas–dust medium has been simulated numerically. As applied to the photodissociation regions near O and B stars, we take into account the interaction of ultraviolet radiation with dust grains and large polycyclic aromatic hydrocarbon molecules as well as the gas cooling through the excitation of СII ions and OI atoms and the deexcitation of rotational levels of CO molecules. The instability regions have been determined. The perturbation growth times corresponding to them are ~103−105 yr. We show that wave breaking occurs irrespective of the geometry of motion, while a perturbation in the form of a single pulse gives rise to a sequence of shock waves. The post-shock gas velocity is approximately 0.1−0.5 of the sound velocity, so that the autowaves can contribute noticeably to the observed velocity dispersion of the gas near the boundaries of HII regions. Two-dimensional simulations suggest that the presence of multiple shocks in a thermally unstable medium can accelerate significantly the destruction of preexisting isolated condensations.

A correlation between the characteristics of localized plasticity autowaves and the Debye temperature of crystalline lattice in 19 metals has been analyzed. It is shown that this experimentally established linear dependence follows from the previously introduced elastoplastic invariant of deformation that relates the parameters of elastic and plastic processes in a deformable medium.

The physics of seismic process in subduction is discussed with the case study of Kamchatka region being considered. Seismic process is not attributed to either cracking or rupturing. Hydrogen degassing controls variations in voluminous-stress state of the geomedium and such parameters of boundary structures as amorphization, texturization, and destruction. Both rapid and slow dynamic processes are controlled by the medium structure and by the parameters of boundaries; these processes are induced by strain autowaves related to reversible structural transformations of the medium at ascending diffusion of hydrogen. Seismic processes are related to rapid or slow disturbance of accommodation of the medium elements relative to each other. Owing to the properties of boundaries, this process runs in a non-barrier manner and has superplasticity features. The monitoring methods for media with active inner energy sources are proposed. Difference equations of system state evolution are used; these equations are derived and corrected from the measurements of structurally sensitive parameters (saying in more precise, seismic wave velocities). Equation of system state evolution, being derived in these dimensions, reflects the effects of interaction between object and medium. As a result of the study, monitoring of phase state of the geomedium is proposed to predict small probability of the strongest earthquakes during the controlled period.

In this paper, the investigations on stochastic cellular automata models of stable oscillations and autowaves in active media are summarized to formulate a concept of stochastic cellular automaton (CA) corresponding to asynchronous CAs with probabilistic transition rules. A formal description of a stochastic CA and a stochastic CA model is given. The properties and methods of synthesis of CA models for a specified set of elementary physical and chemical transformations are described. Some possibilities of simulating autowave and oscillatory processes are demonstrated using, as an example, a carbon monoxide oxidation reaction on a platinum catalyst with surface restructuring. The CA-simulation has made it possible to reveal the range of reaction parameter values with stable oscillations of reagent concentrations and to observe autowaves on the platinum surface. Special attention is given to providing a highly efficient parallel implementation of the stochastic CA algorithm. For this, the asynchronous mode must be preliminarily transformed to a block-synchronous one. Its equivalence to the asynchronous mode is justified. This is done for the CA reaction model by a comparative statistical analysis of the simulation results.

The concept of active medium is used as a biophysical basis for modeling spatial and temporal self-organization in anthropogenic ecosystems that results in formation of regular dynamic structures with stable or unstable modes of development. Urban ecosystem is represented as a hierarchy of interacting active media, and their non-linearity is the result of extreme anthropogenic load and mismatch between characteristic times and scales in evolution of the natural and anthropogenic components together with the complex set of positive and negative feedbacks between the subsystems. Description of the presented model is deliberately simplified so as to use a modified Fitz-Hugh-Nagumo equation. The approach developed here is quite general and allows for systematic description of urban ecosystems as distributed dissipative systems.

The propagation of autowaves in a moving liquid excitable medium was considered. The shapes of autowave fronts in the cases of the Poiseuille and Couette flows were determined in flat capillaries within the kinematic approach. It was shown that the autowave fronts should break when the flow speed reaches a critical value. The possibility of the occurrence of a diode effect i.e., one-way capillary conductivity, was studied. The results of the computer simulations are in good agreement with theory.

We introduce a mathematical model of a continual circular chain of unidirectionally coupled oscillators. It is a nonlinear hyperbolic boundary value problem obtained from a circular chain of unidirectionally coupled ordinary differential equations in the limit as the number of equations indefinitely increases. We study the attractors of this boundary value problem. Combining analytic and numerical methods, we establish that one of the following two alternatives takes place in this problem: either the buffer phenomenon (unbounded accumulation of stable periodic motions) or chaotic attractors of arbitrarily high Lyapunov dimensions.

In the context of a numerical experiment, it is shown that the switching wave described by the reaction-diffusion equation can be delayed at a medium inhomogeneity with a thickness Δ and amplitude Δβ for a finite time τ = τ(Δβ, Δ) up to a complete stop at it (τ = ∞). Critical values Δβc and Δc corresponding to the autowave stop are found. The similarity laws \(\tau \sim (\Delta _c - \Delta )^{ - \gamma _\Delta } \) and \(\tau \sim (\Delta \beta _c - \Delta \beta )^{ - \gamma _\beta } \) are established, and the critical indices and are found. The similarity law is established for critical values of amplitude and width of the inhomogeneity corresponding to the autowave stop Δβc ∼ Δc-δ where δ ≈ 1.

Indium-oxide films are synthesized by the autowave-oxidation reaction. It is shown that, upon exposure to optical radiation, the resistance of the films sharply decreases and the maximal relative change in the resistance is 52% at room temperature. Two resistance relaxation rates after termination of the irradiation, 15 Ω s−1 during the first 30 s and 7 Ω s−1 over the remaining time, are determined. The data of infrared spectroscopy of the films show that exposure to optical radiation induces a 2.4% decrease in the transmittance at a wavelength of 6.3 μm. It is found that, after termination of the irradiation, the transmittance gradually increases with a rate of 0.006% s−1. It is suggested that photoreduction is the dominant mechanism responsible for changes in the electrical and optical properties of the In2O3 films.

A model of blood coagulation has been investigated. The model includes 25 “reaction-diffusion” equations describing the space-time dynamics of distribution of blood coagulation factor concentrations. The one-dimensional statement of the problem is considered. The autowave velocity has been estimated based on the spatial distribution of blood coagulation factors obtained by numerical calculation of the problem. The dependence of the autowave velocity on diffusion coefficients of prothrombin, AT-III, thrombin, and thrombin bound to α2-macroglobulin is estimated.

The general regular patterns in localization of deformation at pre-failure stage in rocks under compression are found and analyzed (sylvinite, marble sandstone). Applicability of speckle-photography methods to problems on rock deformation and failure is proved. The authors define the self-sustained behavior of the localized plastic deformation in rocks under compression, due to effect of various plastic micromechanisms. The autowaves in rock specimens under compression propagate at a rate of ∼10−5–10−4 m/s (0.3–3 km/yr), which is close to slow motions induced in the earth crust by an earthquake or a rockburst. The ratio of the experimental and calculated failure times is correlated with the coordinates of the failure points in the test rock specimens.

An equation describing the evolution of fast and slow magnetoacoustic waves in a heat-releasing completely ionized gas has been obtained. The shape and parameters of the magnetoacoustic pulse that is an automodel solution of this equation under conditions of magnetoacoustic instability have been determined.