Dynamics of ensembles of coupled oscillators

Modeling tumor growth

Investigation of the dynamics and structure of combustion waves

For more than ten years we have been actively studying nonlinear wave processes associated with combustion. Our research is aimed at analyzing the stability, structure, and dynamics of laminar flames in mixtures and diffusion flames. In addition, we have extensive experience in studying nonlinear wave phenomena in active media and reaction-diffusion systems. Based on the results of our work, more than 50 articles have been published.

We first used the Evans function method to analyze the diffusion-thermal stability of premixed and diffusion flames. In particular, we first investigated the stability of combustion waves in the Zel'dovich-Linian and Zeldovich-Barenblatt models with a two-stage chain reaction mechanism, which are the basic fundamental representation of hydrocarbon flames with a chain reaction mechanism. In the parameter space, the critical values ​​of the parameters for the loss of stability were found, the types of bifurcations that led to the appearance of complex dynamic modes of propagation of combustion waves were established, and the properties of these regimes were investigated. The analysis of flame extinguishing scenarios is carried out. It has been established that there are three scenarios for suppression of combustion waves: the standard folding-related folding bifurcation; The case when the flame velocity decreases continuously to zero at finite values ​​of the parameters; dynamic scenario. The latter is associated with the creation of a chaotic flame propagation regime and the crisis of a strange attractor in the scenario of transition chaos.

Later, we applied, the knowledge gained for the analysis of the stability of rich hydrogen-air flames within the models with reduced and detailed kinetics. New results were obtained in the field of studying the critical phenomena of loss of stability and quenching of deflagration waves in mixtures with a chain kinetic mechanism of the reaction, and a new method for verifying the kinetic mechanisms of combustion reactions in such systems was proposed.

The main areas of current research include:
-Investigation of the dynamics and structure of combustion waves and diffusion flames
-Development of new methods of reduction and verification of kinetic mechanisms
-Investigation of processes of inhibition and interaction of a flame with walls
-Examination of the propagation of combustion waves in composite solid energy materials
-Investigations of the fundamentals of energy-efficient hydrocarbon fuel combustion technologies for practical applications
-Investigation of combustion processes in microgravity and simulation of experiments
-Development of low-dimensional kinetic schemes of combustion of hydrocarbon fuels, study of inhibition processes and dynamics of flames in the framework of these models

Partial synchronization states in neural networks

One of the new directions of the laboratory is the study of dynamic modes realized in different network configurations of coupled neurons.

Systems of interacting oscillators of various natures (phase, dynamical, chemical, optical, and biological) are capable of demonstrating a rich variety of possible dynamic manifestations and properties. Active study of the dynamics of point elements, as well as network structures, led to the discovery of a new, largely counterintuitive dynamical phenomenon, which was soon given the name “chimera”. Суть химерных структур заключается в согласованном сосуществовании порядка и хаоса (пространственной когерентности и инкогерентности) в системах идентичных осцилляторов и обуславливается согласованием внутренней динамики точечных подсистем с их формообразующей сетевой структурой. Not surprisingly, chimeric states have taken a special place in the tasks of neuroscience . Now, chimeras are identified with empirically recorded phenomena arising in the cerebral cortex and responsible for information processing, unihemispheric sleep (occurring in some mammalian and avian species), and some pathological conditions.

In our research, a mathematical model based on a system of fractional-differential reaction-superdiffusion equations was proposed, implying the possibility of cluster activation of elements. The proposed model is compared with a locally coupled system organized on the basis of classical reaction-diffusion equations. Moreover, the possibility of building network structures with variable and adaptive form of connections based on the above method was demonstrated. Further development and study of the proposed model led to the discovery of chimeric states as one of its possible dynamic manifestations. As in most nonlinear problems, various combinations of parameters responsible for dynamic properties were analyzed in detail. In particular, regions of synchronization, developed incoherence, and chimeric states were found in the space of exponents of the fractional Laplace operator (which determine the network properties of the proposed system). Finally, the relationship between the parameters responsible for the activation of point subsystems (neurons) and the parameters characterizing the network features of their aggregates (neural networks) was demonstrated in terms of chimera dynamics and the development of complete synchronization and incoherence.

We have analyzed in detail synchronization processes in neural networks, the configuration of which is set based on a superdiffusive type of interaction. Network configurations are demonstrated, the weak variation of which leads to significant (heterogeneous synchronization transition), as well as to insignificant (homogeneous synchronization transition) changing the dynamics of the system.