scholarly journals Operator of invariant differentiation and its application for integrating systems of ordinary differential equations

2017 ◽  
Vol 9 (4) ◽  
pp. 12-21 ◽  
Author(s):  
Rafail Kavyevich Gazizov ◽  
Aliya Gainetdinova
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Invariant Differentiation

scholarly journals Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160461 ◽  
Author(s):  
A. A. Gainetdinova ◽  
R. K. Gazizov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebras ◽  
Transformation Group ◽  
Second Order ◽  
Differential Invariants ◽  
Invariant Representation ◽  
Invariant Differentiation

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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