scholarly journals ALGORITHM FOR FINDING EXPLICIT SOLUTIONS OF OVERDETERMINED SYSTEMS OF DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 12 (4) ◽  
pp. 5-18
Author(s):  
M.L. Zaytsev ◽  
◽  
V.B. Akkerman ◽  
Keyword(s):  
Differential Equations ◽  
Explicit Solutions ◽  
Polynomial Equations ◽  
Hydrodynamic Equations ◽  
Scientific Interest ◽  
Huge Number ◽  
Overdetermined Systems ◽  
Systems Of Differential Equations ◽  
Pde Systems ◽  
General Method

Previously, the authors proposed a general method for finding particular solutions for overdetermined systems of partial differential equations (PDE), where the number of equations is greater than the number of unknown functions. In this paper, an algorithm for finding solutions for overdetermined PDE systems is proposed, where the authors use a method for finding an explicit solution for overdetermined algebraic (polynomial) equations. Using this algorithm, some overdetermined PDE systems can be solved in explicit form. The main difficulty of this algorithm is the huge number of polynomial equations that arise, which need to be investigated and solved numerically or explicitly. For example, the overdetermined hydrodynamic equations obtained earlier by the authors give at least 10 million such equations. However, if the equations are solved explicitly, then it is possible to write out the solution of the hydrodynamic equations in a general form, which is of great scientific interest.

scholarly journals General Method of Construction of Implicit Discrete Maps Generating Chaos in 3D Quadratic Systems of Differential Equations

2014 ◽  
Vol 24 (02) ◽  
pp. 1450025 ◽  
Author(s):  
Vasiliy Ye. Belozyorov
Keyword(s):  
Differential Equations ◽  
Quadratic Differential ◽  
Chaotic Dynamics ◽  
Autonomous Systems ◽  
2D Systems ◽  
Lambert Function ◽  
Quadratic Systems ◽  
Discrete Maps ◽  
Systems Of Differential Equations ◽  
General Method

A method allowing to study the dynamics of 3D systems of quadratic differential equations by the reduction of these systems to the special 2D systems is presented. The mentioned 2D systems are used for the construction of new types of discrete maps generating the chaotic dynamics in some 3D autonomous systems of quadratic differential equations. Strong simplification of all results gives an introduction of the Lambert function. Due to this function some implicit discrete maps become explicit. Examples are given.

SOLUTION OF SOME SYSTEMS OF DIFFERENTIAL EQUATIONS IN MATHEMATICAL SYSTEM MAPLE

2013 ◽  
Vol 1 (05) ◽  
pp. 58-65
Author(s):  
Yunona Rinatovna Krakhmaleva ◽  
◽  
Gulzhan Kadyrkhanovna Dzhanabayeva ◽  
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations ◽  
Mathematical System

On decomposability of countable systems of differential equations

1993 ◽  
Vol 45 (10) ◽  
pp. 1598-1608
Author(s):  
A. M. Samoilenko ◽  
Yu. V. Teplinskii
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations

scholarly journals Differential Games for an Infinite 2-Systems of Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1467
Author(s):  
Muminjon Tukhtasinov ◽  
Gafurjan Ibragimov ◽  
Sarvinoz Kuchkarova ◽  
Risman Mat Hasim
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Differential Games ◽  
Differential Game ◽  
Infinite System ◽  
The State ◽  
Geometric Constraints ◽  
Control Parameters ◽  
Systems Of Differential Equations ◽  
Pursuit And Evasion

A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found.

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

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