# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] F. Jiang, J. 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. Google Scholar [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. Google Scholar [7] J. E. Littlewood, Some Problems in Real and Complex Analysis, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. Google Scholar [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. Google Scholar [9] L. F. Nie, Z. D. Teng, J. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] D. Qian, L. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] J. Sun, J. 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. Google Scholar [23] J. Sun, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] F. Jiang, J. 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. Google Scholar [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. Google Scholar [7] J. E. Littlewood, Some Problems in Real and Complex Analysis, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. Google Scholar [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. Google Scholar [9] L. F. Nie, Z. D. Teng, J. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] D. Qian, L. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] J. Sun, J. 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. Google Scholar [23] J. Sun, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar
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