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Vol 74, No 3 (2019)
- Year: 2019
- Articles: 6
- URL: https://journal-vniispk.ru/0027-1322/issue/view/10057
Article
87-97
Billiards and Integrability in Geometry and Physics. New Scope and New Potential
Abstract
Description of bifurcations and symmetries of integrable systems is an important branch of geometry that has many applications. Important results have been obtained recently in the descriptions of bifurcations of integrable billiards and in modelling of Hamiltonian systems of mechanics and dynamics by billiards. The paper contains interesting problems, as well as a research program for the near future. In the closing of the paper, the results allowing one to describe hidden symmetries of Hamiltonian bifurcations are given as an example of a work close to billiards subject.
98-107
Strengthened Ul’yanov’s Inequalities for Partial Moduli of Smoothness for Functions from Spaces with Various Metrics
Abstract
Ul’yanovs inequality connecting moduli of continuity in different metrics is well known for functions of one variable. In this paper functions of two variables are considered. Sharp Ul’yanov’s inequalities connecting partial moduli of smoothness of positive order are proved in different mixed metrics.
108-120
Reducibility of Linear Differential Systems to Linear Differential Equations
Abstract
Lyapunov reducibility of any bounded and sometimes unbounded linear homogeneous differential system to some bounded linear homogeneous differential equation is established. The preservation of the additional property of periodicity of coefficients is guaranteed, and for two-dimensional or complex systems the constancy of their coefficients is preserved. The differences in feasibility of asymptotic and generalized Lyapunov reducibility from Lyapunov one are indicated.
121-126
On Some Analytic Method for Approximate Solution of Systems of Second Order Ordinary Differential Equations
Abstract
An approach to using Chebyshev series to solve canonical second-order ordinary differential equations is described. This approach is based on the approximation of the solution to the Cauchy problem and its first and second derivatives by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using the Markov quadrature formula. It is shown that the described approach allows one to propose an approximate analytical method of solving the Cauchy problem. A number of canonical second-order ordinary differential equations are considered to represent their approximate analytical solutions in the form of partial sums of shifted Chebyshev series.
127-130
The Set of Lower Semi-Continuity Points of Topological Entropy of a Continuous One-Parametric Family of Dynamical Systems
Abstract
The description of the set of lower semi-continuity points and the set of upper semi-continuity points of the topological entropy of the systems considered as a function on some parameter is obtained for a family of dynamical systems continuously dependent on parameter.
131-133