Mironenko reflecting function
Encyclopedia
The reflecting function 
of a dynamical system
connects the past state
of it with the future state 
of it by the formula 
The concept of the reflecting function was introduсed by Vladimir Ivanovich Mironenko.
with the general solution 
in Cauchy form Reflecting Function is defined by formula 
is periodic in 
with respect to 
, then 
is a transformation (Poincaré map
) periodic in
of the differential system 
Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial date 
of periodic solutions of the differential system and investigate the stability
of those solutions.
For the Reflecting Function
of the system 
the basic relation
is holding.
Therefore we have an opportunity sometimes to find Poincaré map of the non-integrable in quadrature
systems even in elementary functions.
of a dynamical systemDynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
connects the past state
of it with the future state of it by the formula The concept of the reflecting function was introduсed by Vladimir Ivanovich Mironenko.Definition
For the differential system with the general solution in Cauchy form Reflecting Function is defined by formula Application
If a vector-function is periodic in with respect to , then is a transformation (Poincaré mapPoincaré 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...
) periodic in
of the differential system Therefore the knowledge of the Reflecting Function give us the opportunity to find out the initial date of periodic solutions of the differential system and investigate the stabilityLyapunov stability
Various types of stability may be discussed for the solutions of differential 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 Lyapunov...
of those solutions.
For the Reflecting Function
of the system the basic relation
is holding.
Therefore we have an opportunity sometimes to find Poincaré map of the non-integrable in quadrature
Quadrature
Quadrature may refer to:In signal processing:*Quadrature amplitude modulation , a modulation method of using both an carrier wave and a 'quadrature' carrier wave that is 90° out of phase with the main, or in-phase, carrier...
systems even in elementary functions.
Literature
- Мироненко В. И. Отражающая функция и периодические решения дифференциальных уравнений. — Минск, Университетское, 1986. — 76 с.
- Мироненко В. И. Отражающая функция и исследование многомерных дифференциальных систем. — Гомель: Мин. образов. РБ, ГГУ им. Ф. Скорины, 2004. — 196 с.