scholarly journals Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 916
Author(s):  
Michal Fečkan ◽  
Július Pačuta
Keyword(s):  
Differential Equations ◽  
Inverted Pendulum ◽  
Averaging Method ◽  
Mechanical Systems ◽  
Second Order ◽  
Periodic Systems ◽  
Weakly Nonlinear ◽  
Averaging Methods ◽  
Second Order Differential Equations ◽  
Averaging Theorem

In this paper, we discuss the averaging method for periodic systems of second order and the behavior of solutions that intersect a hyperplane. We prove an averaging theorem for impact systems. This allows us to investigate the approximate dynamics of mechanical systems, such as the weakly nonlinear and weakly periodically forced Duffing’s equation of a hard spring with an impact wall, or a weakly nonlinear and weakly periodically forced inverted pendulum with double impacts.

scholarly journals Second-order differential equations with random perturbations and small parameters

2017 ◽  
Vol 147 (4) ◽  
pp. 763-779
Author(s):  
M. Kamenskii ◽  
S. Pergamenchtchikov ◽  
M. Quincampoix
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Second Order ◽  
Random Perturbations ◽  
Existence And Uniqueness Theorem ◽  
Second Order Differential Equations ◽  
Averaging Theorem ◽  
Small Parameters

We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero.

scholarly journals Path Following Control for a Class of Electro-Mechanical Systems and its Application

2011 ◽  
Vol 2-3 ◽  
pp. 501-506
Author(s):  
Mitsuru Taniguchi ◽  
Kenji Fujimoto
Keyword(s):  
Velocity Field ◽  
Differential Equations ◽  
Mechanical Systems ◽  
Path Following ◽  
Second Order ◽  
Hamiltonian Form ◽  
Second Order Differential Equations ◽  
Path Following Control ◽  
Field Control

This paper is devoted to path following control for electro-mechanical systems described by the port-Hamiltonian form. Path following control is investigated mainly for mechanical systems since the desired path is characterized by its ‘position’. Therefore, most of the existing results use the nature of second order differential equations since mechanical systems are described by them. The present paper proposes a new path following controller for 3rd order differential equations described by the port-Hamiltonian form. This is done by generalizing the authors’ former result on passive velocity field control for mechanical systems.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osama Moaaz ◽  
Choonkil Park ◽  
Elmetwally M. Elabbasy ◽  
Waed Muhsin
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
The Other ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

scholarly journals Fixed point theorems via Generalized WF-Contractions with applications

SeMA Journal ◽  
2021 ◽  
Author(s):  
Rosana Rodríguez-López ◽  
Rakesh Tiwari
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
New Class ◽  
Second Order Differential Equations ◽  
Fixed Point Result

AbstractThe aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations.

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

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