Vol 31, No 3 (2023)

May 10, 2023 marks the 70th birthday of Professor Sergey Vladimirovich Gonchenko, a leading researcher of Lobachevsky State University of Nizhny Novgorod.

Purpose of this work is the research of the dynamical processes and in particular the phenomenon of the synchronization in an ensemble of coupled chaotic economic oscillators. Methods. The research methods are the qualitative and numerical methods of the theory of nonlinear dynamical systems and the theory of the bifurcations. Results. The nonlinear model of economic oscillator as the system of automatic control are considered. Such kind of general economic models are unsuitable for getting some concrete economic estimations and recommendations. But such kind models are very useful for a development the theory of the economic cycles, theory of the generation, interactions, synchronization of the cycles and so on. Our numerical experiments demonstrated a good enough qualitative similarity of an chaotic economic oscillations in our model and real economic cycles. The phenomen of the synchronization of the chaotic oscillations in the ensemble of coupled economic oscillators are considered, however the accuracy of the synchronization depends with couplings essentially.

Purpose of this article is to study the transport of passive particles in the velocity field of a vortex tripole with a change in the parameter that determines the speed of the configuration movement. A structure consisting of a central vortex and satellite vortices rotating around it with the opposite vorticity is understood as a tripole. We employ a system of three point vortices, the most simple mathematical representation of a vortex tripole, which may be expressed as a system of nonlinear ordinary differential equations with a parameter. Consideration is limited to a particular case of a tripole with zero total vorticity. The influence of the speed values of vortex configuration movement on the processes of passive particle transport has been studied. Methods. The study was carried out numerically using algorithms based on the dynamical systems approaches including the construction of the Poincare map and the analysis of the dynamics of marker particles. Were carried out long ´ times calculations, corresponding to hundreds and thousands of turns around the tripole center. Integrators of high orders of accuracy were used to solve the Cauchy problems, which made it possible to adequacy of the calculation result control. Results. We found that transferring passive particles is fundamentally different depending on the speed of the tripole. A vast zone of chaotic dynamics forms in the neighborhood of the vortices when the velocity is low. This zone slowly shifts along with the tripole. There are subregions of active and slow mixing inside the chaos region. The possible stages of particle dynamics are: transfer from the region to the right of the tripole to the area to the left, vigorous mixing near the vortices, and slowly drifting to the region to the left of the tripole. At a high speed of vortex configuration in the entire chaotic region, the particles are strongly mixed. The vortex tripole removes particles from the vicinity of its initial position over long distances and practically does not capture new particles along its path. In intermediate situations, both processes can be realized at varying degrees. Conclusion. Non-trivial scenarios for the transport of passive particles by a vortex tripole, which can also occur in real vortex configurations of fluids, have been discovered and described.

Purpose of this work is to analyze oscillatory regimes in a system of nonlinear differential equations describing the competition of three non-antagonistic species in a spatially homogeneous domain. Methods. Using the theory of cosymmetry, we establish a connection between the destruction of a two-parameter family of equilibria and the emergence of a continuous family of periodic regimes. With the help of a computational experiment in MATLAB, a search for limit cycles and an analysis of multistability were carried out. Results. We studied dynamic scenarios for a system of three competing species for different coefficients of growth and interaction. For several combinations of parameters in a computational experiment, new continuous families of limit cycles (extreme multistability) are found. We establish bistability: the coexistence of isolated limit cycles, as well as a stationary solution and an oscillatory regime. Conclusion. We found two scenarios for locating a family of limit cycles regarding a plane passing through three equilibria corresponding to the existence of only one species. Besides cycles lying in this plane, a family is possible with cycles intersecting this plane at two points. We can consider this case as an example of periodic processes leading to overpopulation and a subsequent decline in numbers. These results will further serve as the basis for the analysis of systems of competing populations in spatially heterogeneous areas.

The purpose of this study is to analyze the influence of the shape of the daily illumination profile on the synchronization of rhythms in the sleep–wake state switching model. Normally, the alternation of sleep and wakefulness of a person is synchronized with his circadian rhythm and with the 24-hour rhythm of illumination. There is, however, a lot of experimental evidence of a violation of this synchronism, both in the form of phase failures (for example, during air travel) and in the form of long-term mismatch of rhythms (for example, during shift work in production). Mathematical models of the process of switching between sleep and wakefulness also demonstrate the desynchronization of rhythms and are successfully used to optimize work schedules. At the same time, the influence of a number of factors on this process has not been sufficiently studied, including the nature of changes in illumination during the day. Methods. An analysis of the six-dimensional model under study shows that, in terms of nonlinear dynamics, the problem is reduced to finding and interpreting resonance regions on a three-dimensional torus. For the specific purposes of our work, it turned out to be convenient to estimate the ratio of three periods (24 hours, the circadian period, and the current duration of the sleep–wake cycle) by numerically integrating the model equations on a grid of parameter values using parallel computing technology. The main result of our work is that the presence of round-the-clock low-intensity illumination (that is, the addition of a zero-frequency signal to the daily light cycle) causes the circadian rhythm to desynchronize with respect to the daily one in a significant range of parameters. We have proposed an explanation of this effect based on the structure of the mathematical model. Conclusion. Our results raise at least two serious questions, the first of which is related to the physiological interpretation of one of the main variables of the model, sleep homeostasis, and the second is to refine the assumptions that are used in the model description of the photoreceptor response. In any case, there are interesting prospects for further research.

On June 14, 2023, could be celebrated the 85-th anniversary of the founder of the journal “Applied Nonlinear Dynamics”, the Corresponding Member of the Russian Academy of Sciences, the Honored Scientist of the Russian Federation, the laureate of the Presidential Prize in Education, the Rector of Saratov University (1994–2003), the head of the department electronics, oscillations and waves (1981–2020) Dmitry Ivanovich Trubetskov. Published reflections of colleagues and friends of D. I. Trubetskov reveal his contribution to the development of science, education and the significance of his multifaceted personality for the history of Saratov University.