On Explicit Difference Schemes for Autonomous Systems of Differential Equations on Manifolds

2019 ◽  
pp. 343-361 ◽  
Author(s):  
E. A. Ayryan ◽  
M. D. Malykh ◽  
L. A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Differential Equations ◽  
Autonomous Systems ◽  
Difference Schemes ◽  
Systems Of Differential Equations ◽  
Differential Equations On Manifolds ◽  
Explicit Difference Schemes

scholarly journals On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme

2021 ◽  
Vol 29 (1) ◽  
pp. 63-72
Author(s):  
Yu Ying ◽  
Mikhail D. Malykh
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Projective Geometry ◽  
Autonomous Systems ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Duality Principle ◽  
Integrals Of Motion ◽  
Quadratic Integral

The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as t0.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.

scholarly journals Ways for detection of the exact number of limit cycles of autonomous systems on the cylinder

2019 ◽  
Vol 55 (2) ◽  
pp. 182-194
Author(s):  
A. A. Hryn ◽  
S. V. Rudzevich
Keyword(s):  
Differential Equations ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Autonomous Systems ◽  
Type System ◽  
Transversality Condition ◽  
Second Step ◽  
Constant Sign ◽  
Exact Number ◽  
Systems Of Differential Equations

For real autonomous systems of differential equations with continuously differentiable right-hand sides, the problem of detecting the exact number and localization of the second-kind limit cycles on the cylinder is considered. To solve this problem in the absence of equilibria of the system on the cylinder, we have developed our previously proposed ways consisting in a sequential two-step application of the Dulac – Cherkas test or the Dulac test. Additionally, a new way has been worked out using the generalization of the Dulac – Cherkas or Dulac test at the second step, where the requirement of constant sign for divergence is replaced by the transversality condition of the curves on which the divergence vanishes. With the help of the developed ways, closed transversal curves are found that divide the cylinder into subdomains surrounding it, in each of which the system has exactly one second-kind limit cycle.The practical efficiency of the mentioned ways is demonstrated by the example of a pendulum-type system, for which, in the absence of equilibria, the existence of exactly three second-kind limit cycles on the entire phase cylinder is proved.

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