Grigory Osipov
Head of the Department of Control Theory Professor at the Institute of Information Technologies, Mathematics and Mechanics Director at the Institute of Supercomputing Simulations, Lobachevsky State University of Nizhny Novgorod
Born: June 29, 1960 1982: Diploma in Mathematics (M.Sc.), Nizhny Novgorod State University, Nizhny Novgorod, Russia. 1989: Candidate of Science (Ph.D.). Thesis title: "Research into and Development of the System of Mathematical Modeling of Dynamics of Nonlinear Networks of Rotators ", Institute of Space Research, Russian Academy of Sciences, Moscow, Russia. 1998 – 1999: Research Associate, Boston University, USA. 2002: Researcher, Lancaster University, UK. 2004: Doctor of Physics and Mathematics (Doctor Habil.). Thesis title: “Synchronization in Heterogeneous Ensembles of Locally Diffusively Coupled Regular and Chaotic Oscillators”, Nizhny Novgorod University, Nizhny Novgorod, Russia. 2006: Visiting Professor, Potsdam University, Germany. Since 2007: Head of the Department of Control Theory, Professor at the Faculty of Computational Mathematics and Cybernetics. 2009: Visiting Researcher at the Leuven University, Belgium. 2010: Visiting Professor, Academia Sinica, Taipei, Taiwan. 2011: Visiting Professor, Hong Kong Baptist University, Hong Kong. Since 2015: Director of the Institute of Supercomputing Simulations.
Field of research: nonlinear dynamics, synchronization, mathematical modeling, controlling chaos, pattern formation, theory of bifurcations, computational neuroscience, parallel programming.
Publications:
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Smirnov L. A., Kryukov A. K., Osipov G. V., Kurths J.
Bistability of Rotational Modes in a System of Coupled Pendulums
2016, vol. 21, nos. 7-8, pp. 849-861
Abstract
The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an in-phase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the in-phase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable in-phase cycle, but also two out-of phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the out-ofphase limit regimes are discussed as well.
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Grines E. A., Osipov G. V.
On Constructing Simple Examples of Three-dimensional Flows with Multiple Heteroclinic Cycles
2015, vol. 20, no. 6, pp. 679-690
Abstract
In this work we suggest a simple method for constructing $G$-equivariant systems of ODEs in $\mathbb{R}^3$ (i.e., systems whose trajectories are invariant under the action of this group on $\mathbb{R}^3$) that possess multiple disjoint heteroclinic networks. Heteroclinic networks under consideration consist of saddle equilibria that belong to coordinate axes and one-dimensional separatrices connecting them. We require these separatrices to lie on coordinate planes. We also assume the action of $G$ on $\mathbb{R}^3$ to be generated by cyclic permutation of coordinate variables and reflection with respect to one of the coordinate planes. As an example, we provide a step-by-step construction of three-dimensional flow with two disjoint heteroclinic networks. Also, we present a sketch of global dynamics analysis for the minimal example.
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Korotkov A. G., Kazakov A. O., Osipov G. V.
Sequential Dynamics in the Motif of Excitatory Coupled Elements
2015, vol. 20, no. 6, pp. 701-715
Abstract
In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of the generalized Lotka–Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized three-dimensional Lotka–Volterra model.
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Belykh V. N., Petrov V. S., Osipov G. V.
Dynamics of the Finite-dimensional Kuramoto Model: Global and Cluster Synchronization
2015, vol. 20, no. 1, pp. 37-48
Abstract
Synchronization phenomena in networks of globally coupled non-identical oscillators have been one of the key problems in nonlinear dynamics over the years. The main model used within this framework is the Kuramoto model. This model shows three main types of behavior: global synchronization, cluster synchronization including chimera states and totally incoherent behavior. We present new sufficient conditions for phase synchronization and conditions for an asynchronous mode in the finite-size Kuramoto model. In order to find these conditions for constant and time varying frequency mismatch, we propose a simple method of comparison which allows one to obtain an explicit estimate of the phase synchronization range. Theoretical results are supported by numerical simulations.
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Korotkov A. G., Zagrebin S. Y., Kadina E. Y., Osipov G. V.
Abstract
In [1], a stable heteroclinic cycle was proposed as a mathematical image of switching
activity. Due to the stability of the heteroclinic cycle, the sequential activity of the elements
of such a network is not limited in time. In this paper, it is proposed to use an unstable
heteroclinic cycle as a mathematical image of switching activity. We propose two dynamical
systems based on the generalized Lotka – Volterra model of three excitable elements interacting
through excitatory couplings. It is shown that in the space of coupling parameters there is a
region such that, when coupling parameters in this region are chosen, the phase space of systems
contains unstable heteroclinic cycles containing three or six saddles and heteroclinic trajectories
connecting them. Depending on the initial conditions, the phase trajectory will sequentially
visit the neighborhood of saddle equilibria (possibly more than once). The described behavior
is proposed to be used to simulate time-limited switching activity in neural ensembles. Different
transients are determined by different initial conditions. The passage of the phase point of the
system near the saddle equilibria included in the heteroclinic cycle is proposed to be interpreted
as activation of the corresponding element.
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