International Journal of Nonlinear Analysis and Applications
Periodic solutions for a class of perturbed fifth-order autonomous differential equations via averaging theory
Document Type : Research Paper
Authors
Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, Annaba, 23000, Algeria
Abstract
In this work, we use the averaging theory of first order to study the periodic solutions of the perturbed fifth-order autonomous differential equation
\begin{equation*}
x^{(5)}- \lambda \ddddot{x}+(p^{2}+1) \dddot{x}-\lambda(p^{2}+1)\ddot{x}+p^{2}\dot{x}-\lambda p^{2}x= \varepsilon F(x,\dot{x}, \ddot{x}, \dddot{x}, \ddddot{x}),
\end{equation*}
where $\lambda ,$ and $\varepsilon $ are real parameters, $p$ is rational number different from $-1,$ $0,$ $1$, $\varepsilon $ is a small enough and $F\in C^2 $ is a nonlinear autonomous function. we present some applications to illustrate our main results.
Keywords
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u +qu¨+pu =
εF(t, u, u,˙ u, ¨
...
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Gen. 45 (2012) , 55214.
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(2011), 1080—1083.
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[18] I.G. Malkin, Some Problems of the theory of nonlinear oscillations, Gosudarstv. Izdat. Tehn-Teor. Lit. Moscow,
(in Russian), 1956.
[19] M.S. Mechee and F.A. Fawzi, Generalized Runge-Kutta integrators for solving fifth-order ordinary differential
equations, Ital. J. Pure Appl. Math. 45 (2021), 600–610.
[20] G. Moatimid, F. Elsabaa, M. Zekry, Approximate solutions of coupled nonlinear oscillations: Stability analysis,
J. Appl. Comput. Mech. 7 (2021), no. 2, 382–395.
[21] S.N. Odda, Existence solution for 5th order differential equations under some conditions, Appl. Math. 1 (2010),
no. 4, 279–282.
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Springer, New York, 1985.[23] J.A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences,
Vol. 59, Springer, New York, 1985.
[24] N. Sellami and A. Makhlouf, Limit cycles for a class of fifth-order differential equations, Ann. Diff. Equ. 28
(2012), no. 2, 202–219.
[25] N. Sellami, R. Mellal, B.B. Cherif and S.A. Idris, On the limit cycles for a class of perturbed fifth-order autonomous
differential equations, Discrete Dyn. Nature Soc. 2021 (2021), Article ID 6996805, 18 pages.
[26] A. Tiryaki, On the periodic solutions of certain fourth and fifth order differential equations, Pure Appl. Math.
Sci. 32 (1990), no. 1-2, 11–14.
[27] C. Tun¸c, On the periodic solutions of certain fourth and fifth order vector differential equations, Math. Commun.
10 (2005), no. 2, 135–141.
[28] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, New York, 1996.
[29] X.-J. Wu, Y. Wang and W. Price, Multiple resonances, responses, and parametric instabilities in offshore structures, J. Ship Res. 32 (1988), 285.
Anal. Appl. 317 (2006), 613–619.
[2] A. Alomari, N.R. Anakira, A.S. Bataineh and I. Hashim, Approximate solution of nonlinear system of BVP
arising in fluid flow problem, Math. Prob. Engin. 2013 (2013).
[3] A. Buica, J.P. Fran¸coise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small
parameter, Commun. Pure Appl. Anal. 6 (2006), 103–111.
[4] Y. El-Dib and R. Matooq, The rank upgrading technique for a harmonic restoring force of nonlinear oscillators,
J. Appl. Comput. Mech. 7 (2006), no. 2, 782–789.
[5] M. El-Gamel, Sinc and the numerical solution of fifth-order boundary value problems, Appl. Math. Computation,
187 (2007), no. 2, 1417–1433.
[6] E. Esmailzadeh, M. Ghorashi and B. Mehri, Periodic behavior of a nonlinear dynamical system, Nonlinear Dyn.
7 (1995), 335-–344.
[7] J.O.C. Ezeilo, A further instability theorem for a certain fifth-order differential equation, Math. Proc. Cambridge
Phil. Soc. 86 (1979), 491–493.
[8] J.H. He, W.F. Hou, N. Qie, K.A. Gepreel, A.H. Shirazi and H. Mohammad-Sedighi, Hamiltonian-based frequencyamplitude formulation for nonlinear oscillators, Facta Univer. Ser. Mech. Engin. 19 (2021), no. 2, 199–208.
[9] S.N. Jator, Numerical integrators for fourth order initial and boundary value problems, Int. J. Pure Appl. Math.
47 (2008), 563.
[10] O. Kelesoglu, The solution of fourth order boundary value problem arising out of the beam-column theory using
Adomian decomposition method, Math. Prob. Engin. 2014 (2014).
[11] J. Llibre, J. Yu and X. Zhang, Limit cycles for a class of third order differential equations, Rocky Mount. J.
Math. 40 (2010), 581—594.
[12] J. Llibre and L. Roberto, On the periodic orbits of the third order differential equation ...
x−µ
..
x+
.
x−µx = εF(x,
.
x,
..
x),
Appl. Math. Lett. 26 (2013), no. 4, 425—430.
[13] J.Llibre and A. Makhlouf, Periodic orbits of the fourh-order non-autonomous differential equation ....
u +qu¨+pu =
εF(t, u, u,˙ u, ¨
...
u ), Appl. Math. Comput. 219 (2012) , 827–836.
[14] J.Llibre and A. Makhlouf, On the limit cycles for a class of fourth-order differential equations, J. Phys. Math.
Gen. 45 (2012) , 55214.
[15] J.Llibre and A. Makhlouf, Limit cycles for a class of fourth-order autonomous differential equations, Electronic
J. Differ. Equ. 2012 (2012) , 1–17.
[16] J. Llibre and E. Perez-Chavela, Limit cycles for a class of second order differential equations, Phys. Lett. A 375
(2011), 1080—1083.
[17] A. Malek and R.S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural
network-optimization method, Appl. Math. Comput. 183 (2006), 260.
[18] I.G. Malkin, Some Problems of the theory of nonlinear oscillations, Gosudarstv. Izdat. Tehn-Teor. Lit. Moscow,
(in Russian), 1956.
[19] M.S. Mechee and F.A. Fawzi, Generalized Runge-Kutta integrators for solving fifth-order ordinary differential
equations, Ital. J. Pure Appl. Math. 45 (2021), 600–610.
[20] G. Moatimid, F. Elsabaa, M. Zekry, Approximate solutions of coupled nonlinear oscillations: Stability analysis,
J. Appl. Comput. Mech. 7 (2021), no. 2, 382–395.
[21] S.N. Odda, Existence solution for 5th order differential equations under some conditions, Appl. Math. 1 (2010),
no. 4, 279–282.
[22] M. Roseau, Vibrations non lin´eaires et th´eorie de la stabilit´e, Springer Tracts in Natural Philosophy, Vol. 8,
Springer, New York, 1985.[23] J.A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences,
Vol. 59, Springer, New York, 1985.
[24] N. Sellami and A. Makhlouf, Limit cycles for a class of fifth-order differential equations, Ann. Diff. Equ. 28
(2012), no. 2, 202–219.
[25] N. Sellami, R. Mellal, B.B. Cherif and S.A. Idris, On the limit cycles for a class of perturbed fifth-order autonomous
differential equations, Discrete Dyn. Nature Soc. 2021 (2021), Article ID 6996805, 18 pages.
[26] A. Tiryaki, On the periodic solutions of certain fourth and fifth order differential equations, Pure Appl. Math.
Sci. 32 (1990), no. 1-2, 11–14.
[27] C. Tun¸c, On the periodic solutions of certain fourth and fifth order vector differential equations, Math. Commun.
10 (2005), no. 2, 135–141.
[28] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, New York, 1996.
[29] X.-J. Wu, Y. Wang and W. Price, Multiple resonances, responses, and parametric instabilities in offshore structures, J. Ship Res. 32 (1988), 285.
)
Volume 13, Issue 2
July 2022Pages 2479-2491
- Receive Date: 28 February 2022
- Revise Date: 21 June 2022
- Accept Date: 03 July 2022