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2013
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# Pavel Kuptsov

ul. Zelenaya 38, Saratov, 410019, Russia
IRE RAS Saratov branch
Professor at Saratov State Technical University

1989-1994: Student of physical Faculty of Saratov State University
1994-1997: Postgraduate student of Saratov State University
1998 Candidate of Science from Saratov State University (Supervisor Prof. S.P. Kuznetsov)
1997-2003, 2005-2010: Lecturer, then docent in Department of Informatics of Saratov State Law Academy
2004 Postdoc in Centre for Mathematical Science, City University London (United Kingdom)
2010-2022: Docent, then professor in Saratov State Technical University
2013 Doctor of Sciences from Saratov State Technical University (Scientific advisor Prof. S.P. Kuznetsov)
Since 2021 Head of the Laboratory of Theoretical nonlinear dynamics in Saratov branch of Kotelnikov Institute of Radio-Engineering and Electronics of RAS

## Publications:

 Kuptsov P. V. Synchronization and Bistability of Two Uniaxial Spin-Transfer Oscillators with Field Coupling 2022, vol. 27, no. 6, pp.  697-712 Abstract A spin-transfer oscillator is a nanoscale device demonstrating self-sustained precession of its magnetization vector whose length is preserved. Thus, the phase space of this dynamical system is limited by a three-dimensional sphere. A generic oscillator is described by the Landau – Lifshitz – Gilbert – Slonczewski equation, and we consider a particular case of uniaxial symmetry when the equation yet experimentally relevant is reduced to a dramatically simple form. The established regime of a single oscillator is a purely sinusoidal limit cycle coinciding with a circle of sphere latitude (assuming that points where the symmetry axis passes through the sphere are the poles). On the limit cycle the governing equations become linear in two oscillating magnetization vector components orthogonal to the axis, while the third one along the axis remains constant. In this paper we analyze how this effective linearity manifests itself when two such oscillators are mutually coupled via their magnetic fields. Using the phase approximation approach, we reveal that the system can exhibit bistability between synchronized and nonsynchronized oscillations. For the synchronized one the Adler equation is derived, and the estimates for the boundaries of the bistability area are obtained. The twodimensional slices of the basins of attraction of the two coexisting solutions are considered. They are found to be embedded in each other, forming a series of parallel stripes. Charts of regimes and charts of Lyapunov exponents are computed numerically. Due to the effective linearity the overall structure of the charts is very simple; no higher-order synchronization tongues except the main one are observed. Keywords: uniaxial spin-transfer oscillators, mutual synchronization, bistability Citation: Kuptsov P. V.,  Synchronization and Bistability of Two Uniaxial Spin-Transfer Oscillators with Field Coupling, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 697-712 DOI:10.1134/S1560354722060077
 Kuptsov P. V., Kuznetsov S. P. Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction 2018, vol. 23, no. 7-8, pp.  908-932 Abstract Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in the sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if it occurs, is weaker than contractions in the second one. In this paper we analyze the local structure of several chaotic attractors recently suggested in the literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also, we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well-known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite-time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite-time Lyapunov exponents, we demonstrate that average expanding and contracting properties specific to pseudohyperbolicity are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace. Keywords: chaotic attractor, strange pseudohyperbolic attractor, method of angles, hyperbolic isolation, Lyapunov exponents, finite-time Lyapunov exponents, instant Lyapunov exponents, covariant Lyapunov vectors Citation: Kuptsov P. V., Kuznetsov S. P.,  Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 908-932 DOI:10.1134/S1560354718070079
 Kuptsov P. V., Kuznetsov S. P., Stankevich N. V. A Family of Models with Blue Sky Catastrophes of Different Classes 2017, vol. 22, no. 5, pp.  551-565 Abstract A generalized model with bifurcations associated with blue sky catastrophes is introduced. Depending on an integer index $m$, different kinds of attractors arise, including those associated with quasi-periodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds. Keywords: dynamical system, blue sky catastrophe, quasi-periodic oscillations, hyperbolic chaos, Smale–Williams solenoid Citation: Kuptsov P. V., Kuznetsov S. P., Stankevich N. V.,  A Family of Models with Blue Sky Catastrophes of Different Classes, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 551-565 DOI:10.1134/S1560354717050069