March  2017, 22(2): 477-482. doi: 10.3934/dcdsb.2017022

Periodic solutions of some classes of continuous second-order differential equations

1. 

Departament de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

2. 

Department of Mathematics, Laboratory LMA, University of Annaba, Elhadjar, 23 Annaba, Algeria

We thank to Professor Rafael Ortega the information about the second--order differential equation $\ddot x + x^3= f(t)$, and to the reviewer his comments which help us to improve the presentation of this paper. The first author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014 SGR568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338

Received  March 2016 Revised  June 2016 Published  December 2016

We study the periodic solutions of the second-order differential equations of the form $ \ddot x ± x^{n} = μ f(t), $ or $ \ddot x ± |x|^{n} = μ f(t), $ where $n=4,5,...$, $f(t)$ is a continuous $T$-periodic function such that $\int_0^T {f\left( t \right)} dt\ne 0$, and $μ$ is a positive small parameter. Note that the differential equations $ \ddot x ± x^{n} = μ f(t)$ are only continuous in $t$ and smooth in $x$, and that the differential equations $ \ddot x ± |x|^{n} = μ f(t)$ are only continuous in $t$ and locally-Lipschitz in $x$.

Citation: Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022
References:
[1]

A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002.  Google Scholar

[2]

A. BuicăJ. Llibre and O. Yu. Makarenkov, On Yu.A.Mitropol'skii's Theorem on periodic solutions of systems of nonlinear differential equations with nondifferentiable right-hand sides, Doklady Math., 78 (2008), 525-527.  doi: 10.1134/S1064562408040157.  Google Scholar

[3]

A. Buică and R. Ortega, Persistence of equilibria as periodic solutions of forced systems, J. Differential Equations, 252 (2012), 2210-2221.  doi: 10.1016/j.jde.2011.06.006.  Google Scholar

[4]

A. CapiettoJ. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.2307/2154076.  Google Scholar

[5]

T. CarvalhoR. D. EuzébioJ. Llibre and D. J. Tonon, Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1-11.  doi: 10.3934/dcdsb.2016.21.1.  Google Scholar

[6]

T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

[7]

Y. A. Kuznetzov, Elements of Applied, Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. Google Scholar

[8]

N. G. Lloyd, Degree Theory Cambridge University Press, 1978. Google Scholar

[9]

G. R. Morris, An infinite class of periodic solutions of $\ddot x +2x^3 =p(t)$, Proc. Cambridge Philos. Soc., 61 (1965), 157-164.   Google Scholar

[10]

R. Ortega, The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom, Ergodic Theory Dynam. Systems, 18 (1998), 1007-1018.  doi: 10.1017/S0143385798108362.  Google Scholar

show all references

References:
[1]

A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002.  Google Scholar

[2]

A. BuicăJ. Llibre and O. Yu. Makarenkov, On Yu.A.Mitropol'skii's Theorem on periodic solutions of systems of nonlinear differential equations with nondifferentiable right-hand sides, Doklady Math., 78 (2008), 525-527.  doi: 10.1134/S1064562408040157.  Google Scholar

[3]

A. Buică and R. Ortega, Persistence of equilibria as periodic solutions of forced systems, J. Differential Equations, 252 (2012), 2210-2221.  doi: 10.1016/j.jde.2011.06.006.  Google Scholar

[4]

A. CapiettoJ. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.2307/2154076.  Google Scholar

[5]

T. CarvalhoR. D. EuzébioJ. Llibre and D. J. Tonon, Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1-11.  doi: 10.3934/dcdsb.2016.21.1.  Google Scholar

[6]

T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

[7]

Y. A. Kuznetzov, Elements of Applied, Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. Google Scholar

[8]

N. G. Lloyd, Degree Theory Cambridge University Press, 1978. Google Scholar

[9]

G. R. Morris, An infinite class of periodic solutions of $\ddot x +2x^3 =p(t)$, Proc. Cambridge Philos. Soc., 61 (1965), 157-164.   Google Scholar

[10]

R. Ortega, The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom, Ergodic Theory Dynam. Systems, 18 (1998), 1007-1018.  doi: 10.1017/S0143385798108362.  Google Scholar

[1]

Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451

[2]

Jaume Llibre, Amar Makhlouf, Sabrina Badi. $3$ - dimensional Hopf bifurcation via averaging theory of second order. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1287-1295. doi: 10.3934/dcds.2009.25.1287

[3]

Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529

[4]

Zaihong Wang. Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 751-770. doi: 10.3934/dcds.2003.9.751

[5]

Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373-381. doi: 10.3934/proc.2007.2007.373

[6]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[7]

Kunquan Lan. Eigenvalues of second order differential equations with singularities. Conference Publications, 2001, 2001 (Special) : 241-247. doi: 10.3934/proc.2001.2001.241

[8]

Juan Campos, Rafael Obaya, Massimo Tarallo. Favard theory and fredholm alternative for disconjugate recurrent second order equations. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1199-1232. doi: 10.3934/cpaa.2017059

[9]

Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207

[10]

C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 25-34. doi: 10.3934/dcds.1997.3.25

[11]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209

[12]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861

[13]

Yuan Guo, Xiaofei Gao, Desheng Li. Structure of the set of bounded solutions for a class of nonautonomous second order differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1607-1616. doi: 10.3934/cpaa.2010.9.1607

[14]

Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906

[15]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[16]

Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias, Mai Zurwatul Ahlam Mohd Jaffar. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 89-94. doi: 10.3934/naco.2017005

[17]

Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91

[18]

Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155

[19]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945

[20]

Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (19)
  • HTML views (86)
  • Cited by (1)

Other articles
by authors

[Back to Top]