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Mironenko Reflecting Function
In applied mathematics, the reflecting function \,F(t,x) of a differential system \dot x=X(t,x) connects the past state \,x(-t) of the system with the future state \,x(t) of the system by the formula \,x(-t)=F(t,x(t)). The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka. Definition For the differential system \dot x=X(t,x) with the general solution \varphi(t;t_0,x) in Cauchy form, the Reflecting Function of the system is defined by the formula F(t,x)=\varphi(-t;t,x). Application If a vector-function X(t,x) is \,2\omega-periodic with respect to \,t, then \,F(-\omega,x) is the in-period \, \omega;\omega/math> transformation (Poincaré map) of the differential system \dot x=X(t,x). Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial dates \,(\omega,x_0) of periodic solutions of the differential system \dot x=X(t,x) and investigate the stability of those solutions. For the Reflecting Function \,F(t,x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential System
In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential equations. Linear systems of differential equations A first-order linear system of ODEs is a system in which every equation is first order and depends on the unknown functions linearly. Here we consider systems with an equal number of unknown functions and equations. These may be written as \frac = a_(t) x_1 + \ldots + a_(t)x_n + g_(t), \qquad j=1,\ldots,n where n is a positive integer, and a_(t),g_(t) are arbitrary functions of the independent variable t. A first-order linear system of ODEs may be written in matrix form: \frac \begin x_1 \\ x_2 \\ \vdots \\ x_n \end = \begin a_ & \ldots & a_ \\ a_ & \ldots & a_ \\ \vdots & \ldots & \vdots \\ a_ & & a_ \end \begin x_1 \\ x_2 \\ \vdots \\ x_n \end + \begin g_1 \\ g_2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Mironenko
Uladzimir Ivanavich Mironenka (; born 9 February 1942, Bych, Homel region, Belarus) is a Belarusian mathematician. Biography In 1964 Uladzimir Ivanavich graduated from Mogilev State University, Faculty of Physics and Mathematics. In 1970 he defended his PhD thesis on the topic "Embeddable Systems". In 1975 approved in the rank of associate professor and in 1992 in the rank of professor. Professor Mironenka has introduced in the theory of the differential equations motions of ''φ''-solution, embeddable system and reflecting function. He applied these concepts to study of the existence of periodic solutions and solutions boundary value problems of differential systems, and to study of stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Exponential stability ** Linear stability **Lyapunov stability ** Marginal s ... of these solutions. References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Solution
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. Types of solution A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis), pioneered the field complex analysis, and the study of permutation groups in abstract algebra. Cauchy also contributed to a number of topics in mathematical physics, notably continuum mechanics. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: : "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific worker; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. Biography Youth and education Cauchy was the son of Loui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Map
In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system. More precisely, one considers a periodic orbit with initial conditions within a section of the space, which leaves that section afterwards, and observes the point at which this orbit first returns to the section. One then creates a map to send the first point to the second, hence the name ''first recurrence map''. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it. A Poincaré map can be interpreted as a discrete dynamical system with a state space that is one dimension smaller than the original continuous dynamical system. Because it preserves many properties of periodic an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Solution
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called ''aperiodic''. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x_e stay near x_e forever, then x_e is Lyapunov stable. More strongly, if x_e is Lyapunov stable and all solutions that start out near x_e converge to x_e, then x_e is said to be ''asymptotically stable'' (see asymptotic analysis). The notion of '' exponential stability'' guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ('' quadrature'' or ''squaring''), as in the quadrature of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Functions
In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addition, sums, multiplication, products, algebraic function, roots and composition of functions, compositions of finite set, finitely many Polynomial#Polynomial functions, polynomial, Rational function, rational, Trigonometric functions, trigonometric, Hyperbolic functions, hyperbolic, and Exponential function, exponential functions, and their Inverse function, inverses (e.g., Inverse trigonometric functions, arcsin, Natural logarithm, log, or ''x''1/''n''). All elementary functions are continuous on their Domain of a function, domains. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An abstract algebra, algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
