An example of physical system with hyperbolic attractor of Smale - Williams type

A simple and transparent example of a non-autonomous flow system, with hyperbolic strange attractor is suggested. The system is constructed on a basis of two coupled van der Pol oscillators, the characteristic frequencies differ twice, and the parameters controlling generation in both oscillators undergo a slow periodic counter-phase variation in time. In terms of stroboscopic Poincar\'{e} section, the respective four-dimensional mapping has a hyperbolic strange attractor of Smale - Williams type. Qualitative reasoning and quantitative data of numerical computations are presented and discussed, e.g. Lyapunov exponents and their parameter dependencies. A special test for hyperbolicity based on statistical analysis of distributions of angles between stable and unstable subspaces of a chaotic trajectory has been performed. Perspectives of further comparative studies of hyperbolic and non-hyperbolic chaotic dynamics in physical aspect are outlined.Comment: 7 pages, 4 figure

A Note on the Real Fermionic and Bosonic quadratic forms: Their Diagonalization and Topological Interpreation

We explain in this note how real fermionic and bosonic quadratic forms can be effectively diagonalized. Nothing like that exists for the general complex hermitian forms. Looks like this observation was missed in the Quantum Field theoretical literature. The present author observed it for the case of fermions in 1986 making some topological work dedicated to the problem: how to construct Morse-type inequalities for the generic real vector fields? This idea also is presented in the note.Comment: 9pages, Late

On one example of a Nikishin system

The paper puts forward an example of a~Markov function $f=\operatorname{const}+\widehat{\sigma}$ such that the three functions $f,f^2$ and $f^3$ form a Nikishin system. A conjecture is proposed that there exists a~Markov function $f$ such that, for each $n\in\mathbb N$, the system $f,f^2,\dots,f^n$ constitutes a~Nikishin system. Bibliography:~20~titles

The existence of a smooth interface in the evolutionary elliptic Muskat--Verigin problem with nonlinear source

We study the two-phase Muskat--Verigin free-boundary problem for elliptic equations with nonlinear sources. The existence of a smooth solution and a smooth free boundary is proved locally in time by applying the parabolic regularization of a condition on the free boundary.Comment: This is an English translation of the article S.P. Degtyarev, The existence of a smooth interface in the Muskat-Verigin elliptic evolution problem with a nonlinear source, http://www.ams.org/mathscinet-getitem?mr=280906

Analysis and interpretation of the X-ray properties of black hole candidates

The thesis is mainly devoted to the study of spectral and timing characteristics of the X-ray emission from Galactic black hole candidates GRS 1915+105, GX 339-4, 4U 1630-47, XTE J1748-288, GRS 1739-278, KS/GRS 1730-312 and GRS 1737-31. The main results include: observation of the correlated evolution of spectral and timing parameters of these sources and its interpretation in the framework of two-phase model of the accretion flow near the compact objects; development of the quantitative model for the evolution of GRS 1915+105 during flaring state providing direct estimate of the disk accretion rate in the system; restrictions on the parameters of the spatial distribution and luminosity function of the Galactic hard X-ray transient sources; application of the the bulk motion comptonization model to the analytic approximation of the broad--band energy spectra of Galactic black hole candidates in the high/very high state obtained with RXTE.Comment: The summary of PhD thesis, 12 pages, 9 figure

Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study

A system of $N$ rotators is investigated with a constraint given by a condition of vanishing sum of the cosines of the rotation angles. Equations of the dynamics are formulated and results of numerical simulation for the cases $N$=3, 4, and 5 are presented relating to the geodesic flows on a two-dimensional, three-dimensional, and four-dimensional manifold, respectively, in a compact region (due to the periodicity of the configuration space in angular variables). It is shown that a system of three rotators demonstrates chaos, characterized by one positive Lyapunov exponent, and for systems of four and five elements there are, respectively, two and three positive exponents (`hyperchaos'). An algorithm has been implemented that allows calculating the sectional curvature of a manifold in the course of numerical simulation of the dynamics at points of a trajectory. In the case of $N$=3, curvature of the two-dimensional manifold is negative (except for a finite number of points where it is zero), and Anosov's geodesic flow is realized. For $N$=4 and 5, the computations show that the condition of negative sectional curvature is not fulfilled. Also the methodology is explained and applied for testing hyperbolicity based on numerical analysis of the angles between the subspaces of small perturbation vectors; in the case of $N$=3, the hyperbolicity is confirmed, and for $N$=4 and 5 the hyperbolicity does not take place.Comment: 20 pages, 5 figure

Interval regularization for imprecise linear algebraic equations

In this paper, we consider the solution of ill-conditioned systems of linear algebraic equations that can be determined imprecisely. To improve the stability of the solution process, we "immerse" the original imprecise linear system in an interval system of linear algebraic equations of the same structure and then consider its tolerable solution set. As the result, the "intervalized" matrix of the system acquires close and better conditioned matrices for which the solution of the corresponding equation system is more stable. As a pseudo-solution of the original linear equation system, we take a point from the tolerable solution set of the intervalized linear system or a point that provides the largest tolerable compatibility (consistency). We propose several computational recipes to find such pseudo-solutions.Comment: The work presented at the International Conference "Computational and Applied Mathematics 2017" (CAM 2017), June 25-30, 2017, Akademgorodok, Novosibirsk, Russia (http://conf.ict.nsc.ru/cam17

Numerical computation of formal solutions to interval linear systems of equations

The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the classical interval arithmetic algebraically. The need to solve these problems naturally arises, for example, in inner and outer estimation of various solution sets to interval linear systems of equations. The work develops two approaches to the construction of stationary iterative methods for computing the formal solutions that are based on splitting the matrix of the system. We consider their convergence and implementation issues, compare with the other approaches to computing formal solutions

On rational definite summation

We present a partial proof of van Hoeij-Abramov conjecture about the algorithmic possibility of computation of finite sums of rational functions. The theoretical results proved in this paper provide an algorithm for computation of a large class of sums $S(n) = \sum_{k=0}^{n-1}R(k,n)$.Comment: LaTeX 2.09, 7 pages, submitted to "Programming & Computer Software

From geodesic flow on a surface of negative curvature to electronic generator of robust chaos

Departing from the geodesic flow on a surface of negative curvature as a classic example of the hyperbolic chaotic dynamics, we propose an electronic circuit operating as a generator of rough chaos. Circuit simulation in NI Multisim software package and numerical integration of the model equations are provided. Results of computations (phase trajectories, time dependences of variables, Lyapunov exponents and Fourier spectra) show good correspondence between the chaotic dynamics on the attractor of the proposed system and of the Anosov dynamics for the original geodesic flow.Comment: 6 pages, 6 figure