Bifurcation of invariant tori in a neighborhood of an equilibrium of a system of ordinary differential equations

2000 ◽  
Vol 36 (4) ◽  
pp. 639-641
Author(s):  
Yu. V. Usachev
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Invariant Tori

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF INVARIANT TORI IN MULTIDIMENSIONAL DYNAMICAL SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 681-689
Author(s):  
V. S. AFRAIMOVICH ◽  
A. L. ZHELEZNYAK ◽  
I. L. ZHELEZNYAK
Keyword(s):  
Mathematical Model ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Strange Attractors

A method for analyzing the existence of a multidimensional torus for certain systems of ordinary differential equations is proposed. Using this method, the existence of a multidimensional torus in one of such systems is analyzed. This system is a mathematical model of the dynamics of interacting structures within the drift hydrodynamical systems. The behavior of trajectories on the multidimensional torus is numerically investigated. The existence of two- and three-dimensional tori as well as strange attractors are considered.

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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