scholarly journals Convergence to non-autonomous differential equations of second order

2015 ◽  
Vol 23 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Erdal Korkmaz ◽  
Cemil Tunç
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Autonomous Differential Equations

scholarly journals ON POLYNOMIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON A CIRCLE WITHOUT SINGULAR POINTS

2020 ◽  
Vol 12 (4) ◽  
pp. 33-40
Author(s):  
V.Sh. Roitenberg ◽  
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Vector Fields ◽  
Second Order ◽  
Singular Points ◽  
Free Term ◽  
Small Perturbations ◽  
Cylindrical Phase ◽  
Autonomous Differential Equations ◽  
The Right

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

scholarly journals On the boundedness of solutions of a kind of non-autonomous differential equations of second order with finitely many deviating arguments

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 529-538
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
Boundedness Of Solutions ◽  
Deviating Arguments ◽  
Deviating Argument ◽  
Autonomous Differential Equations

We study the boundedness of the solutions to a non-autonomous differential equation of second order with finitely many deviating arguments. We give two examples to illustrate the main results. By this work, we improve some boundedness results obtained for a differential equation with a deviating argument in the literature to the boundedness of the solutions of a differential equation with finitely many deviating arguments.

A second-order modified nonstandard theta method for one-dimensional autonomous differential equations

2021 ◽  
Vol 112 ◽  
pp. 106775
Author(s):  
Hristo V. Kojouharov ◽  
Souvik Roy ◽  
Madhu Gupta ◽  
Fawaz Alalhareth ◽  
John M. Slezak
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
One Dimensional ◽  
Autonomous Differential Equations ◽  
Theta Method
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