Pavel Kuptsov
Professor at Saratov State Technical University
19891994: Student of physical Faculty of Saratov State University
19941997: Postgraduate student of Saratov State University
1998 Candidate of Science from Saratov State University (Supervisor Prof. S.P. Kuznetsov)
19972003, 20052010: Lecturer, then docent in Department of Informatics of Saratov State Law Academy
2004 Postdoc in Centre for Mathematical Science, City University London (United Kingdom)
20102022: 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 RadioEngineering and Electronics of RAS
19891994: Student of physical Faculty of Saratov State University
19941997: Postgraduate student of Saratov State University
1998 Candidate of Science from Saratov State University (Supervisor Prof. S.P. Kuznetsov)
19972003, 20052010: Lecturer, then docent in Department of Informatics of Saratov State Law Academy
2004 Postdoc in Centre for Mathematical Science, City University London (United Kingdom)
20102022: 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 RadioEngineering and Electronics of RAS
Publications:
Kuptsov P. V.
Synchronization and Bistability of Two Uniaxial SpinTransfer Oscillators with Field Coupling
2022, vol. 27, no. 6, pp. 697712
Abstract
A spintransfer oscillator is a nanoscale device demonstrating selfsustained precession
of its magnetization vector whose length is preserved. Thus, the phase space of this
dynamical system is limited by a threedimensional 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 higherorder synchronization tongues except
the main one are observed.

Kuptsov P. V., Kuznetsov S. P.
Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction
2018, vol. 23, nos. 78, pp. 908932
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 wellknown
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 finitetime 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 finitetime 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.

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. 551565
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 quasiperiodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds.
