scholarly journals On the Limit Cycles for a Class of Perturbed Fifth-Order Autonomous Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Nabil Sellami ◽  
Romaissa Mellal ◽  
Bahri Belkacem Cherif ◽  
Sahar Ahmed Idris
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Limit Cycles ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Nonlinear Function ◽  
Averaging Theory ◽  
First Order ◽  
Autonomous Differential Equations ◽  
Fifth Order

We study the limit cycles of the fifth-order differential equation x ⋅ ⋅ ⋅ ⋅ ⋅ − e x ⃜ − d x ⃛ − c x ¨ − b x ˙ − a x = ε F x , x ˙ , x ¨ , x ⋯ , x ⃜ with a = λ μ δ , b = − λ μ + λ δ + μ δ , c = λ + μ + δ + λ μ δ , d = − 1 + λ μ + λ δ + μ δ , e = λ + μ + δ , where ε is a small enough real parameter, λ , μ , and δ are real parameters, and F ∈ C 2 is a nonlinear function. Using the averaging theory of first order, we provide sufficient conditions for the existence of limit cycles of this equation.

scholarly journals On the limit cycles for a class of eighth-order differential equations

2020 ◽  
Vol 6 (1) ◽  
pp. 53-61
Author(s):  
Chems Eddine Berrehail ◽  
Zineb Bouslah ◽  
Amar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Rational Numbers ◽  
Eighth Order ◽  
Existence Of Periodic Solutions

AbstractIn this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation {x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right), where A = p2λ2 + p2µ2 + λ2µ2 + p2 + λ2 + µ2, B = p2 λ2 + p2µ2 + λ2µ2 + p2λ2µ2, with λ, µ and p are rational numbers different from −1, 0, 1, and p ≠ ±λ, p ≠±µ, λ ≠±µ, ɛ is sufficiently small and F is a nonlinear non-autonomous periodic function. Moreover we provide some applications.

A further instability theorem for a certain fifth-order differential equation

1979 ◽  
Vol 86 (3) ◽  
pp. 491-493 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Trivial Solution ◽  
Constant Coefficient ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Non Linear ◽  
Image Position ◽  
Fifth Order

In our previous consideration in (1) of the constant-coefficient fifth-order differential equation:an attempt was made to identify (though not exhaustively) different sufficient conditions on a1,…,a5 for the instability of the trivial solution x = 0 of (1·1). It was our expectation that the conditions so identified could be generalized in some form or other to equations (1·1) in which a1,…,a5 were not necessarily constants, thereby giving rise to instability theorems for some non-linear fifth-order differential equations; and this turned out in fact to be so except only for the case:with R0 = R0(a1, a2, a3, a4) > 0 sufficiently large, about which we were unable at the time to derive any worthwhile generalization to any equation (1·1) in which a1, …,a5 are not all constants.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

DIFFERENTIAL EQUATIONS IN A HYPERCOMPLEX SYSTEM

2020 ◽  
Vol 69 (1) ◽  
pp. 7-11
Author(s):  
A.K. Abirov ◽  
◽  
N.K. Shazhdekeeva ◽  
T.N. Akhmurzina ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Inhomogeneous System ◽  
Homogeneous Solutions ◽  
First Order ◽  
Zero Divisors ◽  
Hypercomplex System ◽  
Independent Variable ◽  
The Right

The article considers the problem of solving an inhomogeneous first-order differential equation with a variable with a constant coefficient in a hypercomplex system. The structure of the solution in different cases of the right-hand side of the differential equation is determined. The structure of solving the equation in the case of the appearance of zero divisors is shown. It turns out that when the component of a hypercomplex function is a polynomial of an independent variable, the differential equation turns into an inhomogeneous system of real variables from n equations and its solution is determined by certain methods of the theory of differential equations. Thus, obtaining analytically homogeneous solutions of inhomogeneous differential equations in a hypercomplex system leads to an increase in the efficiency of modeling processes in various fields of science and technology.

Oscillation criteria for third-order nonlinear differential equations

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
B. Baculíková ◽  
E. Elabbasy ◽  
S. Saker ◽  
J. Džurina
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Third Order ◽  
Oscillation Criteria ◽  
The Third ◽  
Third Order Differential Equation ◽  
Oscillation Properties

AbstractIn this paper, we are concerned with the oscillation properties of the third order differential equation $$ \left( {b(t) \left( {[a(t)x'(t)'} \right)^\gamma } \right)^\prime + q(t)x^\gamma (t) = 0, \gamma > 0 $$. Some new sufficient conditions which insure that every solution oscillates or converges to zero are established. The obtained results extend the results known in the literature for γ = 1. Some examples are considered to illustrate our main results.

scholarly journals Weierstrass elliptic difference equations

1987 ◽  
Vol 35 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Elliptic Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Elliptic Difference ◽  
First Order ◽  
Second Order Differential Equation ◽  
Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

scholarly journals A note on solvability of three-point bound-ary value problems for third-order differential equations with p-Laplacian

2008 ◽  
Vol 39 (1) ◽  
pp. 95-103
Author(s):  
XingYuan Liu ◽  
Yuji Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Third Order ◽  
Third Order Differential Equation

Third-point boundary value problems for third-order differential equation$ \begin{cases} & [q(t)\phi(x''(t))]'+kx'(t)+g(t,x(t),x'(t))=p(t),\;\;t\in (0,1),\\ &x'(0)=x'(1)=x(\eta)=0. \end{cases} $is considered. Sufficient conditions for the existence of at least one solution of above problem are established. Some known results are improved.

scholarly journals Oscillation of even-order neutral differential equations via comparison principles

2014 ◽  
Vol 30 (3) ◽  
pp. 293-300
Author(s):  
J. DZURINA ◽  
◽  
B. BACULIKOVA ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Comparison Principles ◽  
Neutral Differential Equations ◽  
First Order ◽  
Even Order ◽  
Delay Differential

In the paper we offer oscillation criteria for even-order neutral differential equations, where z(t) = x(t) + p(t)x(τ(t)). Establishing a generalization of Philos and Staikos lemma, we introduce new comparison principles for reducing the examination of the properties of the higher order differential equation onto oscillation of the first order delay differential equations. The results obtained are easily verifiable.

scholarly journals New criteria for oscillation of nonlinear neutral differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Osama Moaaz
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Delay Equations ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
First Order ◽  
Riccati Substitution ◽  
Gamma S ◽  
New Criteria

AbstractThe aim of this work is to offer sufficient conditions for the oscillation of neutral differential equation second order $$ \bigl( r ( t ) \bigl[ \bigl( y ( t ) +p ( t ) y \bigl( \tau ( t ) \bigr) \bigr) ^{\prime } \bigr] ^{\gamma } \bigr) ^{\prime }+f \bigl( t,y \bigl( \sigma ( t ) \bigr) \bigr) =0, $$(r(t)[(y(t)+p(t)y(τ(t)))′]γ)′+f(t,y(σ(t)))=0, where $\int ^{\infty }r^{-1/\gamma } ( s ) \,\mathrm{d}s= \infty $∫∞r−1/γ(s)ds=∞. Based on the comparison with first order delay equations and by employ the Riccati substitution technique, we improve and complement a number of well-known results. Some examples are provided to show the importance of these results.

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