scholarly journals Ulam’s type stability of impulsive ordinary differential equations

2012 ◽  
Vol 395 (1) ◽  
pp. 258-264 ◽  
Author(s):  
JinRong Wang ◽  
Michal Fec˘kan ◽  
Yong Zhou
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Ulam’S Type Stability

scholarly journals On the Stability of Nonautonomous Linear Impulsive Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
JinRong Wang ◽  
Xuezhu Li
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Impulsive Differential Equations ◽  
The Stability ◽  
Ulam’S Type Stability ◽  
Unbounded Intervals ◽  
Stability Results ◽  
Rassias Stability ◽  
Stability Concepts

We introduce two Ulam's type stability concepts for nonautonomous linear impulsive ordinary differential equations. Ulam-Hyers and Ulam-Hyers-Rassias stability results on compact and unbounded intervals are presented, respectively.

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.

Sign in / Sign up

Export Citation Format

Share Document