scholarly journals ON THE REFLECTING FUNCTION AND THE QUALITATIVE BEHAVIOR OF SOLUTIONS OF SOME NON-AUTONOMOUS DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 7 (2) ◽  
pp. 523-531
Author(s):  
Zhengxin Zhou ◽  
Keyword(s):  
Differential Equations ◽  
Qualitative Behavior ◽  
Reflecting Function ◽  
Autonomous Differential Equations

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

scholarly journals New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 934
Author(s):  
Shyam Sundar Santra ◽  
Khaled Mohamed Khedher ◽  
Kamsing Nonlaopon ◽  
Hijaz Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Discrete Equation ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Impulsive Differential Systems

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

Deterministic Stability Analysis of a Wind Loaded Structure

1978 ◽  
Vol 45 (1) ◽  
pp. 165-169 ◽  
Author(s):  
P. J. Holmes ◽  
Y. K. Lin
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Wind Loading ◽  
Qualitative Behavior ◽  
Stability Boundaries ◽  
Quantitative Estimates ◽  
Loading Problem ◽  
Loaded Structure ◽  
Excitation Conditions

We discuss the qualitative behavior of a pair of nonlinear differential equations arising in the study of a wind loading problem when turbulence terms are ignored. We obtain quantitative estimates of stability boundaries and are able to identify the most dangerous excitation conditions. This deterministic study provides the basis for further work on the full stochastic differential equations resulting when turbulence terms are included.

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.

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