January  2019, 39(1): 431-445. doi: 10.3934/dcds.2019017

An application of Moser's twist theorem to superlinear impulsive differential equations

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: Xiong Li

Received  March 2018 Revised  March 2018 Published  October 2018

Fund Project: The second author is supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities.

In this paper, we consider a simple superlinear Duffing equation
$x''+2x^{3}+p(t) = 0\;\;\;\;\;\;\;\;(0.1)$
with impulses, where
$p(t+1) = p(t)$
is an integrable function in
$\mathbb{R}$
. In order to apply Moser's twist theorem, we need to ensure that the corresponding Poincaré map of (0.1) is quite close to a standard twist map but it is not usually achieved due to the existence of impulses. Two types of impulsive functions which overcome this problem with different effects in the Poincaré map are provided here. In both cases, there are large invariant curves diffeomorphism to circles surrounding the origin and going to the infinity, which confine the solutions in its interior and therefore lead to the boundedness of all solutions. Furthermore, it turns out that the solutions starting at
$t = 0$
on the invariant curves are quasiperiodic.
Citation: Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
References:
[1]

L. Bai, B. X. Dai and J. J. Nieto, Necessary and sufficient conditions for the existence of non-constant solutions generated by impulses of second order BVPs with convex potential, Electron. J. Qual. Theory Differ. Equ., 1 (2018), Paper No. 1, 13 pp. doi: 10.14232/ejqtde.2018.1.1.

[2]

D. Bainov and P. Simenov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, New York, 1993.

[3]

B. Dai and L. Bao, Positive periodic solutions generated by impulses for the delay Nicholson's blowflies model, Electron. J. Qual. Theory Differ. Equ., 4 (2016), 1-11.  doi: 10.14232/ejqtde.2016.1.4.

[4]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist-theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 79-95.  doi: ASNSP_1987_4_14_1_79_0.

[5]

F. JiangJ. Shen and Y. Zeng, Applications of the Poincaré-Birkhoff theorem to impulsive Duffing Equations at resonance, Nonlinear Anal. Real World Appl., 13 (2012), 1292-1305.  doi: 10.1016/j.nonrwa.2011.10.006.

[6]

V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. doi: 10.1142/0906.

[7]

J. E. Littlewood, Some Problems in Real and Complex Analysis, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968.

[8]

G. R. Morris, A case of boundedness of Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14 (1976), 71-93.  doi: 10.1017/S0004972700024862.

[9]

L. F. NieZ. D. TengJ. J. Nieto and I. H. Jung, State impulsive control strategies for a two-languages competitive model with bilingualism and interlinguistic similarity, Phys. A, 430 (2015), 136-147.  doi: 10.1016/j.physa.2015.02.064.

[10]

J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl., 205 (1997), 423-433.  doi: 10.1006/jmaa.1997.5207.

[11]

J. J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett., 15 (2002), 489-493.  doi: 10.1016/S0893-9659(01)00163-X.

[12]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680-690.  doi: 10.1016/j.nonrwa.2007.10.022.

[13]

J. J. Nieto and C. C. Tisdell, On exact controllability of first-Order impulsive differential equations, Adv. Difference Equ., 2010 (2010), Art. ID 136504, 9 pp. doi: 10.1155/2010/13650.

[14]

J. J. Nieto and J. M. Uzal, Positive periodic solutions for a first order singular ordinary differential equation generated by impulses, Qual. Theory Dyn. Syst., 17 (2018), 637-650.  doi: 10.1007/s12346-017-0266-8.

[15]

Y. M. Niu and X. Li, Periodic solutions of sublinear impulsive differential equations, Taiwanese J. Math., 22 (2018), 439-452.  doi: 10.11650/tjm/8190.

[16]

N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko and N. V. Skripnik, Differential Equations with Impulse Effects, De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110218176.

[17]

D. QianL. Chen and X. Sun, Periodic solutions of superlinear impulsive differential equations: A geometric approach, J. Differential Equations, 258 (2015), 3088-3106.  doi: 10.1016/j.jde.2015.01.003.

[18]

D. Qian and X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl., 303 (2005), 288-303.  doi: 10.1016/j.jmaa.2004.08.034.

[19]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science., World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.

[20]

I. M. Stamova and A. G. Stamov, Impulsive control on the asymptotic stability of the solutions of a Solow model with endogenous labor growth, J. Frankl. Inst., 349 (2012), 2704-2716.  doi: 10.1016/j.jfranklin.2012.07.001.

[21]

I. Stamova and G. Stamov, Applied Impulsive Mathematical Models, CMS Books in Mathematics Springer International Publishing, 2016. doi: 10.1007/978-3-319-28061-5.

[22]

J. SunJ. Chu and H. Chen, Periodic solution generated by impulses for singular differential equations, J. Math. Anal. Appl., 404 (2013), 562-569.  doi: 10.1016/j.jmaa.2013.03.036.

[23]

J. SunH. Chen and J. J. Nieto, Infinitely many solutions for second-order Hamiltonian system with impulsive effects, Math. Comput. Modelling, 54 (2011), 544-555.  doi: 10.1016/j.mcm.2011.02.044.

[24]

S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-015-8893-5.

[25]

H. Zhang and Z. Li, Periodic and homoclinic solutions generated by impulses, Nonlinear Anal. Real World Appl., 12 (2011), 39-51.  doi: 10.1016/j.nonrwa.2010.05.034.

show all references

References:
[1]

L. Bai, B. X. Dai and J. J. Nieto, Necessary and sufficient conditions for the existence of non-constant solutions generated by impulses of second order BVPs with convex potential, Electron. J. Qual. Theory Differ. Equ., 1 (2018), Paper No. 1, 13 pp. doi: 10.14232/ejqtde.2018.1.1.

[2]

D. Bainov and P. Simenov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, New York, 1993.

[3]

B. Dai and L. Bao, Positive periodic solutions generated by impulses for the delay Nicholson's blowflies model, Electron. J. Qual. Theory Differ. Equ., 4 (2016), 1-11.  doi: 10.14232/ejqtde.2016.1.4.

[4]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist-theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 79-95.  doi: ASNSP_1987_4_14_1_79_0.

[5]

F. JiangJ. Shen and Y. Zeng, Applications of the Poincaré-Birkhoff theorem to impulsive Duffing Equations at resonance, Nonlinear Anal. Real World Appl., 13 (2012), 1292-1305.  doi: 10.1016/j.nonrwa.2011.10.006.

[6]

V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. doi: 10.1142/0906.

[7]

J. E. Littlewood, Some Problems in Real and Complex Analysis, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968.

[8]

G. R. Morris, A case of boundedness of Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14 (1976), 71-93.  doi: 10.1017/S0004972700024862.

[9]

L. F. NieZ. D. TengJ. J. Nieto and I. H. Jung, State impulsive control strategies for a two-languages competitive model with bilingualism and interlinguistic similarity, Phys. A, 430 (2015), 136-147.  doi: 10.1016/j.physa.2015.02.064.

[10]

J. J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl., 205 (1997), 423-433.  doi: 10.1006/jmaa.1997.5207.

[11]

J. J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett., 15 (2002), 489-493.  doi: 10.1016/S0893-9659(01)00163-X.

[12]

J. J. Nieto and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680-690.  doi: 10.1016/j.nonrwa.2007.10.022.

[13]

J. J. Nieto and C. C. Tisdell, On exact controllability of first-Order impulsive differential equations, Adv. Difference Equ., 2010 (2010), Art. ID 136504, 9 pp. doi: 10.1155/2010/13650.

[14]

J. J. Nieto and J. M. Uzal, Positive periodic solutions for a first order singular ordinary differential equation generated by impulses, Qual. Theory Dyn. Syst., 17 (2018), 637-650.  doi: 10.1007/s12346-017-0266-8.

[15]

Y. M. Niu and X. Li, Periodic solutions of sublinear impulsive differential equations, Taiwanese J. Math., 22 (2018), 439-452.  doi: 10.11650/tjm/8190.

[16]

N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko and N. V. Skripnik, Differential Equations with Impulse Effects, De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110218176.

[17]

D. QianL. Chen and X. Sun, Periodic solutions of superlinear impulsive differential equations: A geometric approach, J. Differential Equations, 258 (2015), 3088-3106.  doi: 10.1016/j.jde.2015.01.003.

[18]

D. Qian and X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl., 303 (2005), 288-303.  doi: 10.1016/j.jmaa.2004.08.034.

[19]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science., World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.

[20]

I. M. Stamova and A. G. Stamov, Impulsive control on the asymptotic stability of the solutions of a Solow model with endogenous labor growth, J. Frankl. Inst., 349 (2012), 2704-2716.  doi: 10.1016/j.jfranklin.2012.07.001.

[21]

I. Stamova and G. Stamov, Applied Impulsive Mathematical Models, CMS Books in Mathematics Springer International Publishing, 2016. doi: 10.1007/978-3-319-28061-5.

[22]

J. SunJ. Chu and H. Chen, Periodic solution generated by impulses for singular differential equations, J. Math. Anal. Appl., 404 (2013), 562-569.  doi: 10.1016/j.jmaa.2013.03.036.

[23]

J. SunH. Chen and J. J. Nieto, Infinitely many solutions for second-order Hamiltonian system with impulsive effects, Math. Comput. Modelling, 54 (2011), 544-555.  doi: 10.1016/j.mcm.2011.02.044.

[24]

S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-015-8893-5.

[25]

H. Zhang and Z. Li, Periodic and homoclinic solutions generated by impulses, Nonlinear Anal. Real World Appl., 12 (2011), 39-51.  doi: 10.1016/j.nonrwa.2010.05.034.

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