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Grigory Osipov

Nizhny Novgorod State University
Nizhny Novgorod State University


Smirnov L. A., Kryukov A. K., Osipov G. V., Kurths J.
Bistability of Rotational Modes in a System of Coupled Pendulums
2016, vol. 21, no. 7-8, pp.  849-861
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.
Keywords: coupled elements, bifurcation, multistability
Citation: Smirnov L. A., Kryukov A. K., Osipov G. V., Kurths J.,  Bistability of Rotational Modes in a System of Coupled Pendulums, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 849-861
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
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.
Keywords: heteroclinic cycle, heteroclinic network
Citation: Grines E. A., Osipov G. V.,  On Constructing Simple Examples of Three-dimensional Flows with Multiple Heteroclinic Cycles, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 679-690
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
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.
Keywords: Neuronal motifs, Lotka–Volterra model, heteroclinic cycle, period-doubling bifurcation, Feigenbaum scenario, strange attractor, Lyapunov exponents
Citation: Korotkov A. G., Kazakov A. O., Osipov G. V.,  Sequential Dynamics in the Motif of Excitatory Coupled Elements, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 701-715
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
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.
Keywords: phase oscillators, Kuramoto model, global synchronization, existence and stability conditions, asynchronous mode
Citation: Belykh V. N., Petrov V. S., Osipov G. V.,  Dynamics of the Finite-dimensional Kuramoto Model: Global and Cluster Synchronization, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 37-48

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