Pavel Vladimirovich Kuptsov

Saratov State Technical University · Instrumentation Engineering

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 partic...

We consider Hodgkin-Huxley-type model that is a stiff ODE system with two fast and one slow variables. For the parameter ranges under consideration the original version of the model has unstable fixed point and the oscillating attractor that demonstrates bifurcation from bursting to spiking dynamics. Also a modified version is considered where the...

A system of three non-identical Josephson junctions connected via an RLC circuit is considered. The method of Lyapunov exponents charts is used, which makes it possible to identify the main types of dynamics of the system and to analyze the dependence of its properties on parameters. The possibility of both two and three-frequency invariant tori is...

We suggest a universal map capable to recover a behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations ar...

The purpose of this review is to present in a unified manner the latest results on mathematical modeling of rough hyperbolic chaos in systems of various physical nature. Main research Methods are the numerical solution of systems of differential equations and partial differential equations, numerical extraction of the phase of oscillatory processes...

We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variatio...

We consider a self-oscillator whose excitation parameter is varied. Frequency of the variation is much smaller then the natural frequency of the oscillator so that oscillations in the system are periodically excited and decay. Also a time delay as added such that when the oscillations start to grow at a new excitation stage they are influenced via...

The dynamics of a nonlinear numerical model of a nonlinear optical interaction in the semiconductor disk laser resonator under influence of the time delay is investigated. The conditions of self-excitation, stationary generation modes and their stability are studied. The analysis of stationary generation stability was performed with DDEBiftool pack...

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...

Dimension of an inertial manifold for a chaotic attractor of spatially distributed system is estimated using autoencoder neural network. The inertial manifold is a low dimensional manifold where the chaotic attractor is embedded. The autoencoder maps system state vectors onto themselves letting them pass through an inner state with a reduced dimens...

In this paper we analyze local structure of several chaotic attractors recently suggested in 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 a...

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...

We develop an extension of the fast method of angles for hyperbolicity verification in chaotic systems with an arbitrary number of time-delay feedback loops. The adopted method is based on the theory of covariant Lyapunov vectors and provides an efficient algorithm applicable for systems with high-dimensional phase space. Three particular examples...

We consider extended starlike networks where the hub node is coupled with several chains of nodes representing star rays. Assuming that nodes of the network are occupied by nonidentical self-oscillators we study various forms of their cluster synchronization. Radial cluster emerges when the nodes are synchronized along a ray, while circular cluster...

We develop the numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes the computation of angle distribution between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is...

In this paper we categorize dynamical regimes demonstrated by starlike networks with chaotic nodes. This analysis is done in view of further studying of chaotic scale-free networks, since a starlike structure is the main motif of them. We analyze starlike networks of Hénon maps. They are found to demonstrate a huge diversity of regimes. Varying the...

A generalized model of starlike network is suggested that takes into account non-additive coupling and nonlinear transformation of coupling variables. For this model a method of analysis of synchronized cluster stability is developed. Using this method three starlike networks based on Ikeda, predator-prey and H\'enon maps are studied.

In this paper we categorize dynamical regimes demonstrated by star-like networks with chaotic nodes. This analysis is important for further studying of chaotic scale-free networks, since a star-like structure is the main motif of them. We analyze star-like networks of Henon maps. They are found to demonstrate a huge diversity of regimes. Varying th...

Covariant Lyapunov vectors for scale-free networks of Henon maps are highly localized. We revealed two mechanisms of the localization related to full and phase cluster synchronization of network nodes. In both cases the localization nodes remain unaltered in course of the dynamics, i.e., the localization is nonwandering. Moreover this is predictabl...

Broadband synchronization of two coupled spin torque nano-oscillators is reported. The frequency detuning is controlled by the spin current densities, the coupling is introduced via magnetic fields generated by the oscillators, and the coupling strength corresponds to their distance. The fields are computed in a simplified form based on a dipole ap...

For scale-free networks of Henon maps we show that the first covariant Lyapunov vectors demonstrate high nonwandering localization. The nodes of localization are not synchronized with others, and the distributions of square deviations of dynamical variables from their neighborhood have identical power law shapes for all of such nodes. The revealed...

We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes posit...

We study an ensemble of identical noisy phase oscillators with a blinking mean-field coupling, where one-cluster and two-cluster synchronous states alternate. In the thermodynamic limit the population is described by a nonlinear Fokker-Planck equation. We show that the dynamics of the order parameters demonstrates hyperbolic chaos. The chaoticity m...

An effective numerical method for testing the hyperbolicity of chaotic dynamics is suggested. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicates a tangency between expan...

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the pat...

In this Letter, we show that the analysis of Lyapunov-exponent fluctuations contributes to deepen our understanding of high-dimensional chaos. This is achieved by introducing a gaussian approximation for the large-deviation function that quantifies the fluctuation probability. More precisely, a diffusion matrix D (a dynamical invariant itself) is m...

In this Letter we show that the analysis of Lyapunov-exponents fluctuations contributes to deepen our understanding of high-dimensional chaos. This is achieved by introducing a Gaussian approximation for the entropy function that quantifies the fluctuation probability. More precisely, a diffusion matrix D (a dynamical invariant itself) is measured...

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically...

In the tangent space of some spatially extended dissipative systems one can observe "physical" modes which are highly involved in the dynamics and are decoupled from the remaining set of hyperbolically "isolated" degrees of freedom representing strongly decaying perturbations. This mode splitting is studied for the Ginzburg-Landau equation at diffe...

Departing from a system of two nonautonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a one-dimensional medium as an ensemble of such local elements introducing spatial coupling via diffusion. When length of the medium is small, all spatial cells oscillate synchronously, reproducing the local hyperbolic dynamics....

Flow and diffusion distributed structures (FDS) are stationary spatially periodic patterns that can be observed in reaction-diffusion-advection systems. These structures arise when the flow rate exceeds a certain bifurcation point provided that concentrations of interacting species at the inlet differ from steady-state values and the concentrations...

Recently, a system with uniformly hyperbolic attractor of Smale-Williams type has been suggested [Kuznetsov, Phys. Rev. Lett., 95, 144101, 2005]. This system consists of two coupled non-autonomous van der Pol oscillators and admits simple physical realization. In present paper we introduce amplitude equations for this system and prove that the attr...

We study stationary patterns arising from a combination of flow and diffusion in a two-dimensional (2D) reaction-diffusion system in a channel with Poiseuille flow. Both transverse and longitudinal modes are investigated and compared with numerical computations.

Stationary flow- and diffusion-distributed structures (FDS) patterns appear in a reaction-diffusion-advection system when a constant forcing is applied at the inlet of the reactor. We show that if the forcing is subject to noise, the FDS can be destroyed via the noise-induced Hopf instability. However, the FDS patterns are restored if the flow rate...

Stationary flow and diffusion distributed structure (FDS) is known to appear in a reaction–diffusion system with open flow when the constant perturbation is applied at the inlet. Usually, the FDS is considered in the oscillatory Hopf domain when the instability of the Hopf mode is convective. This paper focuses on the formation of the FDS in presen...

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We consider the interaction of a small moving particle with a stationary space-periodic pattern in a chemical reaction diffusion system with a flow. The pattern is produced by a one-dimensional Brusselator model that is perturbed by a constant displacement from the equilibrium state at the inlet. By partially blocking the flow, the particle gives r...

We consider reaction–diffusion instabilities in a flow reactor whose cross-section slowly expands with increasing longitudinal coordinate (cone shaped reactor). Due to deceleration of the flow in this reactor, the instability is convective near the inlet to the reactor and absolute at the downstream end. In sustained regimes the two regions are sep...

We consider the system, exhibiting pitch-fork bifurcation, forced by the external perturbation with fractal properties. Developing a renormalization group approach, we show that the situation is characterized by nonclassical critical exponents. These exponents appear to depend on external influence intensity and we get the analytical expressions fo...